Letters to the Editor
The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.
Can Two Periodic Functions with Incommensurable Periods Have a Periodic Sum? This note adds to the arsenal of coun terexamples in elementary analysis. It is well known and easily proved that the sum of two periodic functions on the real line is periodic if the periods of the
two functions are commensurable, i.e.,
their quotient is a rational number. It seems reasonable to assume that the sum of two periodic functions with in commensurable periods must be aperi odic, as has been done in at least one
textbook on differential equations. 1 But
Proof Choose y such that a,{3, yare lin early independent over IQ. Then, if l, m, and n are integers for which la + mf3 + ny = 0, we must have l m = n 0. On this foundation I construct!, g, and h. =
=
Define G G(a,{3,y) = { lt,m,n = la + m{3 + nyl {l,m,nj C Z}, and let!, g, and h be functions that vanish on the com plement of G, i.e., for all t ft. G, but are =
.f(t)
= mf3 + ny g(t) la- ny h(t) = la + mf3 =
this assumption is incorrect. My examples are everywhere dis continuous. This suggests the question whether less pathological examples exist. I will give a partial answer to this question at the end. To be precise, saying thatj:
IR � IR
is a periodic function of period U
.f(t) for all t E ( -x,oo) , and that U is the smallest pos means that f(t + U)
=
itive value for which this is true. Note that by this definition a constant func tion is not periodic, nor is the charac teristic function of the rationals.
lt,m,n E G
(1.1)
termines the values of m and n. Similarly,
a non-zero value of g(t) or h(t) uniquely
determines the values of l and n, or l and
m, respectively. Thus, for a fixed pair of integers m and n, not both of which are zero, j(t) = mf3 + ny only at the points t la + m{3 + ny, for arbitrary l E 1:'. Since j( t) = j( t + a) = 0 for all t ft. G, this shows that .f has period a. Similar remarks apply to g and h. In summary: =
1. the period off is a, 2. the period of g is {3,
there exist
But h
and
=
The number y was chosen to ensure
3. the period of
JIIR � IR
t
that a non-zero value ofj( t) uniquely de
THEOREM 1. Let a > 0 and {3 > 0 be in commensurable real numbers. Then functions
)
otherwise defined as follows:
=
h is y.
.f + g.The theorem is proved.
g IR � IR with periods a and {3, respec tively, such that h = f + g is periodic.
D
Comment. Since the set {qta + Q2f31{qt,Q2} C IQ} is countable, there ex ist uncountably many real numbers y such that the set {a,{3,y} is linearly in
The class of triplets f, g, and h con
dependent over IQ. We will need the ex
istence of such a y below. For a simple example, take
Vs.
VJ;;,
a =
1, {3
=
V2
and y
For a class of examples, let a
{3
=
VJ;;.,
y
=
�.
= =
for distinct
primes Pi· Or, let o E IR be transcen
Extending the class of examples
structed in the proof above can be en larged. Let
0, leads to a contradiction. Note that W must be incommensurable with both U and V. The function F attains a maximum F*, and G and H have finite least up per bounds, say G* and H*, respec tively. I will show that H* = F* + G*. Because W must be incommensurable with U, the latter equality places high demands on F, sufficient to prove that F must be constant. Given s > 0, choose x and y such that F(x) = F* and G(y) > G* - s. Since F is uniformly continuous, there is a 15 > 0 such that F I (t) - F(t' } < s for I t - t' l < 15. I need two facts concerning a pair of incommensurable numbers (,TJ E lhbo: (1) 7L( mod TJ is dense in the in terval [0, TJL and hence (2) for any s1 > 0, there exist positive integers m and n such that jx + m( - y - nTJI
0, the set 7L( mod 1 is dense in [0,1]. According to inequality (1.3), we may choose m and n so that
lx + mU -
y- n vj < 15.
(1.4)
By the choice of 15, IF(x + mU) F(y + n V) l < s, and therefore H*::::: H(y + nV) = F(y + nV) + G(y + nV) > F* + G* - 2s.
(1.5)
As clearly H* :s: F* + G*, it follows that H* = F* + G*. We will exploit the fact that at val ues of the argument for which H is near its maximal value, so must F and G be near their maximal values. As evident in (1.5), such a value of the argument is afforded by z = y + nV. Thus, be cause H is presumed to have period W, and referring again to (1.5), H(z + jW) = F(z + jW) + G(z + jW) = H(z) > F* + G* - 2s (Vj E 7L). It follows that F(z
+
jW) >F* + G* - G(z + jW)- 2s >F*- 2s (Vj E 7L). (1.6)
Finally, the period ofF is U, U and W are incommensurable, and so the set ltJ = (z + jW) modU I J E 7L} is dense in [O,U]. And, by periodicity of F, in equality (1.6) implies that F(t) >F*2s for all t in the dense set l tJl· This is possible only if F = F*. This contradiction proves that H cannot be periodic. D Michael R. Raugh Department of Mathematics Harvey Mudd College Claremont, CA 9 1 7 1 1 -0788 USA e-mail:
[email protected] The Pythagorean Theorem Extended- and Deflated
In my paper "N-Dimensional Variations on Themes of Pythagoras, Euclid, and Archimedes" (Mathematical Intelli gencer 26 (2004), no. 3, 43-53), I pro posed a generalisation of the usual Pythagorean theorem in the form THEOREM OF PYTHAGORAS ND. The square of the (N-1)-dimensional vol ume of the hypoteneusal face of an N-dimensional orthosimplex is equal to the sum of the squares of the vol umes of its N orthogonal faces. This was a rediscovery; I mentioned I had been anticipated by H. S. M. Cox eter & P. S. Donchian, Math. Gazette 19 (1935), 206. And by many others! Rajendra Bhatia traced out an In dian path-which is, after all, satisfy-
------- ·----·-----
ing, given that ancient Indian mathe maticians seem to have known what we call the Pythagorean Theorem well before the Greeks. K. R. Parthasarathy published a proof based on volume in tegrals calculated by using Gauss's for mula for the volume of convex poly topes: "An n-dimensional Pythagoras Theorem," Math. Scientist 3 (1978), 137-140. This impelled S. Ramanan to give a simpler (unpublished) proof us ing antisymmetric tensors. After an other elaborate proof was indepen dently published by S. Y. Lin and Y. F. Lin in Lin. Multilin. Algebra 26 (1990), 9-13, R. Bhatia sent a letter giving Ra manan's proof (Lin. Multilin. Algebra 30 (1991), 155), and included it as prob lem 1.6.6 in his book Matrix Analysis. More recently, French colleagues also stumbled on the results: J.-P. Quadrat, J. B. Lasserre, and J.-B. Hiriart Urruty, "Pythagoras' Theorem for Ar eas," American Mathematical Monthly 108 (2001), 549-551. They pointed out a French connection, at least for the 3dimensional case, which has been known for quite some time (though its analogy with the standard Pythagorean Theorem was apparently not stressed). The result was very likely known to R. Descartes himself, according to P. Costabel (see his edition of Descartes's Exercices pour la Geometrie des Solides (De Solidorum Elementis), Presses Universitaires de France, Paris, 1987). In any case, the (3-dimen sional) theorem is found in J.-P. Gua de Malves's memoirs of 17831, and L. N. M. Camot stated the result (re ferring to it as already known) in his Geometrie de Position, Crapelet, Paris, 1803. It also found its way into text books, such as P. Nillus, Ler;ons de cal cui vectoriel (t. I), Eyrolles, Paris, 1931. Now the publication of the paper by J.-P. Quadrat et al. brought new refer ences. The Editor, B. P. Palka, quotes but two comments (see "Editors' End notes" in the Monthly 109 (2002), 313-314). G. De Marco, from Padova, mentions an equivalent result involving N-dimensional parallelotopes, to be found in F. R. Gantmacher, Theone des Matrices (t. I), Dunod, Paris, 1966.
-----------
1The abbot Gua de Malves is a most interesting character. A typical polymath of the Enlightenment, he was in fact the first editor of the Encyc/opedie, before handing over the task to Diderot and D 'Aiembert.
© 2005 Springer Science+ Business Media, Inc., Volume 27,
Number 2 , 2005
5
J. Munkres recalls that a more general result is given in his book Analysis of Manifolds, Westview Press, 1991, pp. 184-187: THEOREM.
Let u be a k-simplex in W'. Then the square of the area of u equals the sum of the squares of the areas of the k-simplices obtained by projecting u orthogonally to the various coordi nate k-planes of W'. An elementary proof of a similar result for parallelotopes was published by G. J. Porter in the Monthly 103 (1996), 252-256. There have been many other publi cations of the Theorem. A very cursory Google search led me to a note by Eric W. Weisstein on MathWorld [http:// mathworld.wolfram.comldeGuasTheo rem.html]; the 3-dimensional case, re ferred to as "de Gua's Theorem," is said there to be a special case of a general theorem presented by Tinseau to the Paris Academy in 1774 (slightly before de Gua's own publication), quoted in the textbooks by W. F. Osgood and W. C. Graustein, Solid Analytic Geom etry, Macmillan, New York, 1930, Th. 2, p. 517, and N. Altshiller-Court, Modern Pure Solid Geometry, Chelsea, New York, 1979, pp. 92 and 300. As for the general case, I found a reference to a pa per by R. F. Talbot, "Generalizations of Pythagoras Theorem in n Dimensions," Math. Scientist 12 (1987), 117-121, probably following Parthasarathy's 1978 publication in the same journal. A charming sequel is the recent (2002) posting by Willie W. Wong, a Princeton University student, of his proof of "A generalized N-dimensional Pythagorean Theorem" on his site [sep.princeton. edu/papers/gp.pdf]. It may still be that Coxeter and Donchian have the first occurrence in print of the result for N > 3. We may well ponder the significance of the re curring rediscovery of this result-and of its remaining so little known; its aes thetic and didactic merits certainly earn it a high place in textbooks or in the oral tradition. The least we can say is that our recording and referencing system clearly shows here its lacunae. The irony of the situation is that this discussion amounts to much ado about little. Indeed, as pointed out by
6
THE MATHEMATICAL INTELLIGENCER
J. Munkres in the aforementioned book, the theorem holds not only for simplices and parallelotopes, but (surprisingly at first) also for arbitrary sets lying in a k plane of Rn (k < n)! This generalisation is all the more interesting in that it only takes a meaning for higher dimension alities than the k = 1, n = 2 case of the standard Pythagorean Theorem. How ever, far from being a deep theorem, it is almost trivial, at least in the case k = n - 1 considered up to now. LetS be an arbitrary set contained in an (n - 1 ) plane P of Rn, and call its volume A. Let vp be the unit vector orthogonal to P. Consider now the n projections Si (i = 1,2, . . . n) of S onto the (n - I)-dimen sional subspaces orthogonal to the unit vectors vi (i = 1,2, . . . n) of an orthog onal basis of Rn. Their respective vol umes Ai are obtained by projection and are given by Ai = (vp. vi)A. Since, by the usual n-dimensional Pythagorean Theo rem (or the so-called cosine law), one has I11Cvi, Vp)l2 = llvPII2 = 1, we obtain immediately the result announced, to wit I1 A� = A2• Hardly more than a Lemma!
The crux of the matter is that going from a !-dimensional segment to a k dimensional simplex is not the relevant generalisation here. In the present con text, a !-dimensional segment should be considered as an arbitrary con nected !-dimensional set. Here, as so often, a result proved in special cases through rather sophisti cated means finds an elementary proof showing its intrinsic nature once it is formulated in more general terms. This anticlimax only deepens the question of why the result has not been better understood by its many rediscover ers-including the present one. It is a pleasure to thank R. Bhatia, J. Holbrook, and J.-B. Hiriart-Urruty for a first introduction to the literature I had overlooked. Jean-Marc Levy-Leblond Physique Theorique Universite de Nice Sophie-Antipolis Pare Valrose 061 08 Nice Cedex France e-mail:
[email protected] HOLY GRAIL OF MATHEMATICS FOUND FERMAT'S PROOF TO HIS "LAST THEOREM" (A Restoration]
After some 370 years a 17th-Century proof to
the greatest enigma in mathematics is presented as the restoration of Fermat's letter to a dear friend divulging the origin and rationale of both the
mathematical AND geometrical proofs as examples of his descent infinite/indefinite
discussed in his note on the impossibility of the area of a rectangular triangle being an integer (newly translated) and his August 1659letter to Carcavi (the only translation).
Traces the proof from Euclid and Pythagoras.
A MUST FOR EVERY MATHEMATICIAN vii+ 22 pp. +illustrations $12.00+ $2.50
S&H + NJ 6% tax (U.S. $'s only)
Institutional checks or money orders only
Akerue Publications LLC Elizabeth, NJ 07202
•
PO Box 9547
c.J.ii.U.J.M
Knowledge and Community in Mathematics Jonathan Borwein and Terry Stanway
Mathematical Knowledge-As We Knew It
interrelationship
between
language,
meaning, and society that are com monly considered to fall under the um
Each society has its regime of truth, its "general politics" of truth: that is, the types of discourse which it accepts and makes function as true; the mech anisms and instances which enable one to distinguish true andfalse state ments, the means by which each is sanctioned; the techniques and proce dures accorded value in the acquisi tion of truth; the status of those who are charged with saying what counts as truth.1 (Michel Foucault)
brella of postmodernism. Stating that "attempts to make sense of this elu sive concept threaten to outnumber at tempts to square the circle," he focuses his attention on two relatively well developed aspects of postmodern the ory: "poststructuralism" and "decon
struction."3 He argues that the develop ment of these theories, in the works of Derrida and others, resonates with the debates surrounding foundation ism which preoccupied the philosophy of mathematics in the early stages of
Henri Lebesgue once remarked that
The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical
the last century and may even have
"a mathematician, in so far as he is a
been partly informed by those debates.
mathematician,
Our present purpose is not to revisit the
need not preoccupy
himself with philosophy." He went on to
connections between the foundationist
add that this was "an opinion, moreover,
debates and the advent of postmodern
which has been expressed by many
thought, but rather to describe and dis
philosophers."2 The idea that mathe
cuss some of the ways in which episte
community. Disagreement and
maticians can do mathematics without
mological relativism and other post
controversy are welcome. The views
a precise philosophical understanding
modern perspectives are manifest in the
of what they are doing is, by observa
changing ways in which mathematicians
tion, mercifully true. However, while a
do mathematics and express mathemat
and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-in-chief endorses or accepts responsibility for them. An Opinion
neglect of philosophical issues does not
ical knowledge. The analysis is not in
impede mathematical discussion, dis
tended to be a lament; but it does con
cussion about mathematics quickly be
tain an element of warning. It is central
comes embroiled in philosophy, and
to our purpose that the erosion of uni
should be submitted to the editor-in
perforce encompasses the question of
versally fixed perspectives of acceptable
the nature of mathematical knowledge.
practice in both mathematical activity
chief, Chandler Davis.
Within this discussion, some attention
and its publication be acknowledged as
has been paid to the resonance between
presenting significant challenges to the
the failure of twentieth-century efforts
mathematical community.
to enunciate a comprehensive, absolute foundation for mathematics and the postmodern deconstruction of meaning
Absolutism and Typographic Mathematics
and its corresponding banishment of en compassing philosophical perspectives from the centre fixe. Of note in this commentary is the contribution of Vladimir Tasic. In his book, Mathematics and the Roots of Postmodern Thought, he comments on the broad range of ideas about the
I believe that mathematical reality lies outside us, that ourfunction is to discover or observe it, and that the theorems which we prove, and which we describe grandiloq u en tl y as our "creations," are simply the notes of our observations. 4 (G. H. Hardy)
1Michel Foucault, "Truth and Power," Power/Knowledge: Selected Interviews and Other Writings 1972-1977, edited by Colin Gordon. 2Freeman Dyson, "Mathematics 1n the Physical Sciences," Scientific American 21 1 , no. 9 (1964):130. 3VIadimir Tasic.;, Mathematics and the Roots of Postmodern Thought (Oxford: Oxford University Press, 2001 ), 5.
© 2005 Springer Science+ Business Med1a,
Inc., Volume 27, Number 2 , 2005
7
We follow the example of Paul Ernest and others and cast under the banner of
absolutism descriptions of math
eprints which are read reprints which are cited is
junct between digitally "published" and typographically published
ematical knowledge that exclude any element of uncer
quite striking. Rather, it is a description of a time-honoured
tainty or subjectivity. 5 The quote from Hardy is frequently
and robust definition of
cited as capturing the essence of Mathematical Platonism,
ing environment. In the latter part of the twentieth century,
merit in a typographical publish
a philosophical perspective that accepts any reasonable
a critique of absolutist notions of mathematical knowledge
methodology and places a minimum amount of responsi
emerged in the form of the experimental mathematics
bility on the shoulders of the mathematician.
methodology and the social constructivist perspective.
An undigested
Platonism is commonly viewed to be the default perspec
In
the next section, we consider how evolving notions of
tive of the research mathematician, and, in locating math
mathematical knowledge and new media are combining to
ematical reality outside human thought, ultimately holds
change not only the way mathematicians do and publish math
the mathematician responsible only for discovery, obser
ematics, but also the nature of the mathematical community.
vations, and explanations, not creations. Absolutism also encompasses the logico-formalist schools as well as intuitionism and constructivism-in short, any perspective which strictly defines what constitutes mathe matical knowledge or how mathematical knowledge is cre ated or uncovered. Few would oppose the assertion that an absolutist perspective, predominately in the de facto Pla tonist sense, has been the dominant epistemology amongst working mathematicians since antiquity. Perhaps not as ev ident are the strong connections between epistemological perspective, community structure, and the technologies which support both mathematical activity and mathemati cal discourse. The media culture of typographic mathe matics is defined by centres of publication and a system of community elites which determines what, and by extension
wlw, is published. The abiding ethic calls upon mathemati cians to respect academic credentialism and the systems of publication which further refine community hierarchies. Community protocols exalt the published, peer-reviewed ar ticle as the highest form of mathematical discourse. The centralized nature of publication and distribution both sustains and is sustained by the community's hierar chies of knowledge management. Publishing houses, the peer review process, editorial boards, and the subscription based distribution system require a measure of central con trol. The centralized protocols of typographic discourse resonate strongly with absolutist notions of mathematical knowledge. The emphasis on an encompassing mathemat ical truth supports and is supported by a hierarchical com munity structure possessed of well-defined methods of knowledge validation and publication. These norms sup port a system of community elites to which ascension is granted through a successful history with community pub lication media, most importantly the refereed article. The interrelationships between community practice, structure, and epistemology are deep-rooted. Rigid episte mologies require centralized protocols of knowledge vali dation, and these protocols are only sustainable in media
Towards Mathematical Fallibilism
This new approach to mathematics-the utilization of ad vanced computing technology in mathematical research is often called experimental mathematics. The computer provides the mathematician with a laboratory in which he or she can perform experiments: analyzing examples, testing out new ideas, or searching for patterns. 6 (David Bailey and Jonathan Borwein) The experimental methodology embraces digital com putation as a means of discovery and verification. De scribed in detail in two recently published volumes, Math ematics by Experiment: Plausible Reasoning in the 21st Century and Experimentation in Mathematics: Compu tational Paths to Discovery, the methodology as outlined by the authors Uoined by Roland Girgensohn in the later work) accepts, as part of the experimental process, stan dards of certainty in mathematical knowledge which are more akin to the empirical sciences than they are to math ematics.
As an experimental tool, the computer can pro
vide strong, but typically not conclusive, evidence regard ing the validity of an assertion. While with appropriate validity checking, confidence levels can in many cases be made arbitrarily high, it is notable that the concept of a
"confidence level" has traditionally been a property of sta tistically oriented fields. It is important to note that the au thors are not calling for a new standard of certainty in mathematical knowledge but rather the appropriate use of a methodology which may produce, as a product of its methods, definably uncertain transitional knowledge. What the authors do advocate is closer attention to and acceptance of degrees of certainty in mathematical knowl edge. This recommendation is made on the basis of argued assertions such as: 1. Almost certain mathematical knowledge is valid if treated
appropriately;
As an aside, we emphasize that this
2. In some cases "almost certain" is as good as it gets; 3. In some cases an almost certain computationally derived
is not meant as an indictment of publishers as bestowers
assertion is at least as strong as a complex formal as
of possibly unmerited authority-though the present dis-
sertion.
environments which embrace centralized modes of publi cation and distribution.
4G. H. Hardy, A Mathematician's Apology (London: Cambridge University Press, 1 967), 21. 5Paul Ernest, Social Constructivism As a Philosophy of Mathematk:;s (Albany: State University of New York Press, 1998), 13. 6J. M. Borwein and D. H. Bailey, Mathematics by Expenment: Plausible Reason1ng 1n the 21st Century, A. K. Peters Ltd, 2003. ISBN: 1-56881 -21 1 -6, 2-3.
8
THE MATHEMATICAL INTELLIGENCER
The first assertion is addressed by the methodology it self, and in
Mathematics by Experiment, the authors dis
cuss in detail and by way of example the appropriate treat
Zucker.1° Computational confirmation to very high preci sion is, however, easy. Further experimental analysis involved writing
w3 as a
ment of "almost certain" knowledge. The second assertion
product of only r-values. This form of the answer is then
is a recognition of the limitations imposed by Gi:idel's In
susceptible to integer relation techniques. To high preci
completeness Theorem, not to mention human frailty. The
sion, an
third is more challenging, for it addresses the idea that cer
0= -1.* log[w3] + -1.* log[gamma[l/24]] + 4.*log[gamma[3/24ll
tainty is in part a function of the community's knowledge validation protocols. By way of example, the authors write,
. . . perhaps only 200 people alive can, given enough time, digest all of Andrew Wiles' extraordinarily sophisticated proof ofFermat's Last Theorem. If there is even a one per cent chance that each has overlooked the same subtle er ror (and they may be psychologically predisposed so to do, given the numerous earlier results that Wiles' result relies on), then we must conclude that computational re sults are in many cases actually more secure than the proof of Fermat's Last Theorem. 7
Our first and pithiest example answers a question set by Donald Knuth,8 who asked for a closed form evaluation of the expression below.
00
I
k�l
{
-
"
}
1 = -o.o840695o872765599646t ,v;c;-;2 �7Tk
It is currently easy to compute
.
.
0.0840695087276559964
=
c\13- 1)2/96.
Similar searches suggest there is no similar four-dimen
W4.
Fortunately, a one-variable inte
W4 := fo exp( -4t)Ig(t)dt, I0 is the Bessel integral of the first kind. The high
gral representation is at hand in where cost
of
four-dimensional
exp( -t) for
numeric
integration is
thus
100 places in under 6 seconds and to 500 in 40 sec 0
The second example originates with a multiple integral which arises in Gaussian and spherical models of ferromag netism and in the theory of random walks. This leads to an impressive closed form evaluation due to G.
N. Watson:
I0(t), using
Io(t) = exp( -t)
t2n n�o 22n (n!)2
I
t up to roughly 1.2 d, where d is the number of signif ·
icant digits needed, and exp( -t)
+
onds. Arguably we are done.
Io( t) =
1
--
�
L
(2k- 1)2 (8t) nn!
II��1
V2m n�o
for larger
t, where the limit N of the second summation n such that the summand is less than w-d. (This is an asymptotic expansion, so taking more terms than N may increase, not decrease the
is chosen to be the first index
error.) Bailey and Borwein found that a product of powers of
W4 is not expressible as
f(k/120) (for 0 < k < 120) with co efficients of less than 12 digits. This result does not, of
course, rule out the possibility of a larger relation, but it
Example 2:
I-TT I-TT I-TT 3 TT TT TT
representation of
� �.
We thus have a prediction which Maple 9.5 on a laptop con
�
Proving this discovery is achieved by comparing the out come with Watson's result and establishing the implicit[
.
20 or 200 digits of this Sym
bolic Calculator9 rapidly returns
%=
+ 18. *log[gamma[15 I 24]] + -2.*log[gamma[l7/24]]-7.*log[gamma[l9/24]]
careful computation of exp( - t)
sum. Using the "smart lookup" facility in the Inverse
firms to
+ -6.*log[gamma[ll/24ll + -9.*log[gamma[l3/24]]
avoided. A numerical search for identities then involves the
1 : Evaluate
kk --,-::k k. (::"
+ -8. *log[gamma[5/24ll + l.* log[gamma[7/24]] + l4.*log[gamma[9/24]]
sional closed form for
Three mathematical examples
Example
Integer Relation algorithm returns:
does cast experimental doubt that such a relation exists
1 -
cos(x) - cos(y) - cos(z)
=
cV3 96
1
r2
���
( ) ( )
_!__ r2 .!.!. " 24 24
more than enough to stop one from looking.
0
The third example emphasizes the growing role of visual discovery. Example
3: Recent continued fraction work by Borwein
The most self-contained derivation of this very subtle
and Crandall illustrates the methodology's embracing of
Green's function result is recent and is due to Joyce and
computer-aided visualization as a means of discovery. They
7Borwein and Bailey, p. 10. 8Posed as MAA Problem 1 0832, November 2002. Solution details are given on pages 1 5-1 7 of Borwein, Bailey, and Girgensohn. 9At www.cecm.sfu.ca/projects/ISC/ISCmain.html
10See pages 1 1 7-1 21 of J. M. Borwein, D. H. Bailey, and R. Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A.K. Peters Ltd, 2003. ISBN: 1 -56881 - 1 36·5.
© 2005 Springer Science+ Business Mecia,
Inc., Volume 27, Number 2, 2005
9
For our purpose, it is useful to note that the philosoph ical aspects of the experimental methodology combined
...
with the social constructivist perspective provide a prag matic alternative to Platonism-an alternative which fur thermore avoids the Platonist pitfalls. The apparent para dox in suggesting that the dominant community view of
.01
mathematics-Platonism-is at odds with a social con structivist accounting is at least partially countered by the observation that we and our critics have inhabited quite distinct communities. The impact of one on the other was
Fig. 1 . The starting point depends on the choice of unit vectors, a and
b.
investigated the and
tn where
wn
=
dynamical system defined by: to : = t 1
�
�
tn�l + Wn�l
(1- �)
:=
1
tn�2,
a2,b 2 are distinct unit vectors, for n even, odd,
respectively-that occur in the original continued fraction. Treated as a black box, all that can be verified numerically is that
1.
tn � 0 slowly. Pictorially one learns more, as illus
trated by Figure Figure
2 illustrates the fine structure that appears when
the system is scaled by coloured distinctly.
Vn
and odd and even iterates are
With a lot of work, everything in these pictures is now explained. Indeed from these four cases one is compelled to conjecture that the attractor is finite of cardinality N ex actly when the input,
a or b, is an Nth root of unity; other
wise it is a circle. Which conjecture one then repeatedly may test.
D
well described by Dewey a century ago:
Old ideas give way slowly; for they are more than ab stract logical forms and categories. They are habits, pre dispositions, deeply engrained attitudes of aversion and preference. . .. Old questions are solved by disappearing, evaporating, while new questions corresponding to the changed attitude of endeavor and preference take their place. Doubtless the greatest dissolvent in contemporary thought of old questions, the greatest precipitant of new methods, new intentions, new problems, is the one effected by the scientific revolution that found its climax in the "Origin of Species. "13 (John Dewey)
New mathematics, new media, and new community protocols
With a proclivity towards centralized modes of knowledge validation, absolutist epistemologies are supported by well defined community structures and publication protocols. In contrast, both the experimental methodology and social constructivist perspective resonate with a more fluid com munity structure in which communities, along with their implicit and explicit hierarchies, form and dissolve in re
The idea that what is accepted as mathematical knowl
sponse to the establishment of common purposes. The ex
to some degree, dependent upon a community's
perimental methodology, with its embracing of computa
methods of knowledge acceptance is an idea that is cen
tional methods, de-emphasizes individual accomplishment
edge is,
tral to the social
constructivist school of mathematical phi
losophy.
by encouraging collaboration not only between mathe maticians but between mathematicians and researchers from various branches of computer science.
The social constructivist thesis is that mathematics is a social construction, a cultural product, fallible like any other branch of knowledge. 11 (Paul Ernest)
munities, the social constructivist perspective is inherently
Associated most notably with the writing of Paul Ernest,
from easily identified elites and in the direction of those
Conceiving of mathematical knowledge as a function of the social structure and interactions of mathematical com accepting of a realignment of community authority away
an English mathematician and Professor in the Philosophy
who can most effectively harness the potential for collab
of Mathematics Education, social constructivism seeks to
oration and publication afforded by new media. The ca
define mathematical knowledge and epistemology through
pacity for mass publication no longer resides exclusively
the social structure and interactions of the mathematical
in
Social Construc tivism As a Philosophy of Mathematics, Ernest carefully
equipped with a LATEX compiler and the appropriate in terpreters is all that is needed. The changes that are oc
traces the intellectual pedigree for his thesis, a pedigree
curring in the ways we do mathematics, the ways we
that encompasses the writings of Wittgenstein, Lakatos,
publish mathematical research, and the nature of the math
Davis, and Hersh among others. 12
ematical community leave little opportunity for resistance
community and society as a whole. In
11 Ernest, p. 39ff. 12Ernest, p. 39ff. 13Quoted from The Influence of Darwin on Philosophy, 1 9 1 0.
10
THE MATHEMATICAL INTELLIGENCER
the
hands
of publishing
houses;
any workstation
Fig. 2. The attractors for various
ial lbl =
=
1.
or nostalgia. From a purely pragmatic perspective, the com munity has little choice but to accept a broader definition
In the epistemological universe, mathematics is con ceived as a large mass about which orbit many other bod
of valid mathematical knowledge and valid mathematical
ies of knowledge and whose gravity exerts influence
publication. In fact, in the transition between publishing protocols based upon mechanical typesetting to protocols
throughout. The medieval recognition of the centrality of mathematics was reflected in the quadrivium, which as
supported by digital media, we are already witnessing the
cribed to the sciences of number-arithmetic, geometry,
beginnings of a realignment of elites and hierarchies and a corresponding re-evaluation of the mathematical skill-set.
astronomy, and music-four out of the seven designated
curring in mathematics, we turn our attention to some per
liberal arts. Today, mathematics is viewed by many as an impenetrable, but essential, subject that is at the founda tion of much of the knowledge that informs our under
haps immutable aspects of mathematical knowledge.
standing of the scientific universe and human affairs. We
Before considering more carefully the changes that are oc
are somehow reassured by the idea of a Federal Reserve
Some Societal Aspects of Mathematical Knowledge
The question of the ultimate foundations and the ultimate meaning of mathematics remains open: we do not know in what direction it will find its final solution or even whether a final objective answer can be expected at all. "Mathematizing" may well be a creative activity of man, like language or music, of primary originality, whose historical decisions defy complete objective rationalisa tion.14 (Hermann Weyl)
Chairman who purportedly solves differential equations in his spare time. The high value that society places on an understanding of basic mathematics is reflected in UNESCO's specifica tion of numeracy, along with literacy and essential life skills, as a fundamental educational objective. This place of privilege bestows upon the mathematical research com munity some unique responsibilities. Among them, the ar ticulation of mathematical ideas to research, business, and public policy communities whose prime objective is not the
furthering of mathematical knowledge. As well, as con cerns are raised in many jurisdictions about poor perfor
Membership in a community implies mutual identification with other members which is manifest in an assumption of
mance in mathematics at the grade-school level, research communities are asked to participate in the general dis
some level of shared language, knowledge, attitudes, and
cussion about mathematical education.
practices. Deeply woven into the sensibilities of mathemati cal research communities, and to varying degrees the sensi
The mathematical canon
bilities of society as a whole, are some assumptions about the
role of mathematical knowledge in a society and what con stitutes essential mathematical knowledge. These assump tions are part of the mythology of mathematical communities and the larger society, and it is reasonable to assume that they will not be readily surrendered in the face of evolving ideas about the epistemology of mathematics or changes in the
I will be glad if I have succeeded in impressing the idea that it is not only pleasant to read at times the works of the old mathematical authors, but this may occasionally be of use for the actual advancement of science. 16 (Con
methods of practicing and publishing mathematics.
The mathematical community is the custodian of an ex tensive collection of core knowledge to a larger degree than
Mathematics as fundamental knowledge
stantin Caratheodory)
any other basic discipline with the arguable exception of the combined fields of rhetoric and literature. Preserved
Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field.15 (Paul Dirac)
largely by the high degree of harmonization of grade-school and undergraduate university curricula, this mathematical canon is at once a touchstone of shared experience of com-
14Cited in: Obituary: David Hilbert 1862-1943, RSBIOS, 4, 1 944, pp. 547-553. 15Dirac writing in the preface to The Prn i ciples of Quantum Mechanics (Oxford, 1 930). 16Speaking to an MAA meeting in 1 936.
© 2005 Springer Science+Business Media, Inc., Volume 27, Number 2, 2005
11
munity members and an imposing barrier to anyone who might seek to participate in the discourse of the commu nity without having some understanding of the various re
To see a World in a Grain of Sand; and a Heaven in a Wild Flower; Hold Infinity in the palm of your hand; And Eternity in an hour. (William Blake)
lationships between the topics of core knowledge. While the exact definition of the canon is far from precise, to vary ing degrees of mastery it certainly includes Euclidean
Freedom and Discipline
geometry, differential equations, elementary algebra, num
In this section, we make some observations about the ten
ber theory, combinatorics, and probability. It is worth not
sion between conformity and diversity which is present in
ing parenthetically that while mathematical notation can
the protocols of both typographically and digitally oriented
act as a barrier to mathematical discourse, its universality
communities.
helps promote the universality of the canon. At the level of individual works and specific problems, mathematicians display a high degree of respect for histor ical antecedent. Mathematics has advanced largely through the careful aggregation of a mathematical literature whose
The only avenue towards wisdom is by freedom in the presence of knowledge. But the only avenue towards knowledge is by discipline in the acquirement of ordered fact.19 (Alfred North Whitehead)
reliability has been established by a slow but thorough sciences, mathematical works and problems need not be re
The Rhythmic Claims ofFreedom and Discipline, Whitehead's comments
process of formal and informal scrutiny. Unlike the other
Included in the introduction to his essay
cent to be pertinent. Tom Hales's recent computer-assisted
about the importance of the give and take between free
solution of Kepler's problem makes this point and many oth
dom and discipline in education can be extended to more
ers. Kepler's col\iecture-that the densest way to stack
general domains. In the discourse of mathematical re
spheres is in a pyramid-is perhaps the oldest problem in
search, tendencies towards freedom and discipline, decen
discrete geometry. It is also the most interesting recent ex
tralization and centralization, the organic and the ordered,
ample of computer-assisted proof. The publication of
coexist in both typographic and digital environments. While
Hales's result in the
Annals of Mathematics, with an "only
ggo,.b checked" disclaimer, has triggered varied reactions.17
it may be true that typographic norms are characterized by centralized nodes of publication and authority and the com munity order that they impose, an examination of the math ematical landscape in the mid-twentieth century reveals
The mathematical aesthetic
strong tendencies towards decentralization occurring in
mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics. 18 (G. H. Hardy) The
dependently of the influence of digital media. Mutually re inforcing trends, including an increase in the number of PhD's, an increase in the number of journals and published articles, and the application of advanced mathematical methods to fields outside the domain of the traditional mathematical sciences combined to challenge the tendency
Another distinguishing preoccupation of the mathemat
to maintain centralized community structures. The result
ical community is the notion of a mathematical aesthetic.
was, and continues to be, a replication of a centralized com
It is commonly held that good mathematics reflects this
munity structure in increasingly specialized domains of in
aesthetic and that a developed sense of the mathematical
terest. In mathematics more than in any other field of re
aesthetic is an attribute of a good mathematician. The fol
search, the knowledge explosion has led to increased
"infinity in the palm of your hand"
specialization, with new fields giving birth to new journals
lowing exemplifies the
encapsulation of complexity which is one aspect of the aes thetic sense in mathematics.
1
1
1
+ 22 + 33 +
1 44
+
.
.
.
=
11 0
and the organizational structures which support them. While the structures and protocols which describe the digital mathematical community are still taking shape, it
1
would be inaccurate to suggest that the tendency of digital
:ff dx
media to promote freedom and decentralized norms of
Discovered in
1697 by Johannes Bernoulli, this formula has been dubbed the Sophomore's Dream in recognition of the
knowledge-sharing is unmatched by tendencies to impose
surprising similarities it reveals between a series and its in
typographic mathematics manifest decentralization
tegral equivalent. Its proof is not too simple and not too
knowledge fragmentation, we are presently observing ten
hard, and the formula offers the mix of surprise and sim
dencies emerging from digital mathematics communities to
plicity that seems central to the mathematical aesthetic. By
find order and control in the
contrast several of the recent very long proofs are neither
results from the codification of mathematical knowledge at
simple nor beautiful.
the level of micro-ontologies. The World Wide Web Con-
17See "In Math, Computers Don't Lie. Or Do They?", The New York Times, April 18G. H. Hardy, A Mathematician's Apology (London: Cambridge University Press, 19Aifred North Whitehead, The Aims of Education (New York: The Free Press,
12
THE MATHEMATICAL INTELLIGENCER
control and order. If the natively centralized norms of
6, 2004.
1 967), 21.
1957), 30.
as
knowledge atomization that
MathML initiative and the European Union's OpenMath project are complementary efforts to
thorship, such as are common in the Open Source pro
construct a comprehensive, fine-grained codification of
cal research. Michael Kohlhase and Romeo Anghelache
sortium (W3C)
gramming community, may find a place in mathemati
mathematical knowledge that binds semantics to notation
have proposed a version-based content management
and the context in which the notation is used. 2 The tongue
system for mathematical communities which would per
0
in-cheek indictment of typographic subject specialization as producing experts who learn more and more about less
mit multiple users to make joint contributions to a com
mon research effort. 23 The system facilitates collabora
and less until achieving complete knowledge of nothing-at
tion by attaching version control to electronic document
all becomes, under the digital norms, the increasingly de
management. Such systems, should they be adopted,
tailed description of increasingly restricted concepts until
challenge not only the notion of authorship but also the
one arrives at a complete description of nothing-at-all. On
idea of what constitutes a valid form of publication.
"non
2. The ascendancy of gray literature : Under typo
tologies. " If typographic modes of knowledge validation and
graphic norms, mathematical research has traditionally
tologies become micro-ontologies and risk becoming
publication are collapsing under the weight of subj ect spe
been conducted with reference to j ournals and through
cialization, the digital ideal of a comprehensive meta-math
informal consultation with colleagues. Digital media,
ematical descriptive and semantic framework which em
with its non-discriminating capacity for facilitating in
braces all mathematics may also prove to be overreaching.
stantaneous p ublication, has placed a wide range of
Some Implications
Ranging from Computer Algebra System routines to
sources at the disposal of the research mathematician. Home
Communication of mathematical research and scholar ship is undergoing profound change as new technology creates new ways to disseminate and access the litera ture. More than technology is changing, however, the cul ture and practices of those who create, disseminate, and archive the mathematical literature are changing as well. For the sake of present and future mathematicians, we should shape those changes to make them suit the needs of the discipline. (International Math Union Committee
21
on Electronic Information and Communication)
Pages
and
conference
programmes,
these
sources all provide information that may support math ematical research. In particular, it is possible that a p ublished paper may not be the most appropriate form of p ublication to emerge from a multi-user content management
such as proposed
by Kohlhase
and
Anghelache. It may be that the contributors deem it more appropriate to let the result of their efforts stand with its organic development exposed through a his tory of its versions. 3. Changing modes of knowledge authentication: The refereeing p rocess, already under overload-induced
. . . to suggest that the normal processes of scholarship work well on the whole and in the long run is in no way contradictory to the view that the processes of selection and sifting which are essential to the scholarly process are filled with error and sometimes prejudice. 22 (Kenneth Arrow)
stress, depends upon a highly controlled publication process. In the distributed p ublication environment af forded by digital media, new methods of knowledge au thentication will necessarily emerge. By necessity, the idea of authentication based on the ethics of referees will be replaced by authentication based on various types of valuation p arameters. Services that track ci
Our present idea of a mathematical research community is built on the foundation of the protocols and hierarchies which define the practices of typographic mathematics. At this p oint, how the combined effects of digital media will affect the nature of the community remains an open ques tion; however, some trends are emerging:
1. Changing modes of collaboration: With the facilita tion of collaboration afforded by digital networks, indi
tations are currently being used for this p urpose by the Web document servers
CiteSeer and citebase, among
others.24 Certainly the ability to compute informedly with formulae in a preprint can dramatically reduce the reader's or referee's concern about whether the result is reliable. More than we typically admit or teach our students, mathematicians work without proof if they feel
secure
in
the
correctness
of
their
thought
processes.
vidual authorship is increasingly ceding place to joint
4. Shifts in epistemology: The increasing acceptance of
authorship. It is possible that forms of community au-
the experimental methodology and social constructivist
2°For background on these projects, see: www.w3.org/Math/ and www.openmath.org, respectively. 21The IMU's Committee on Electronic Information and Communication (CEJC) reports to the IMU on matters concerning the digital publication of mathematics. See www.ceic.math.ca/Publications/Recommendations/3_best_practices.shtml 22 E. Roy Weintraub and Ted Gayer, "Equilibrium Proofmaking," Journal of the History of Economic Thought,
23 (Dec. 2001), 421--442. This provides a remarkably de
tailed analysis of the genesis and publication of the Arrow-Debreu theorem. 23Michael Kohlhase and Romeo Anghelache, "Towards Collaborative Content Management and Version Control for Structured Mathematical Knowledge," Lecture Notes in Computer Science no.
2594: Mathematical Knowledge Management: Proceedings of The Second International Conference, Andrea Asperti, Bruno Buchberger, and
James C. Davenport editors, (Berlin: Springer-Verlag,
2003) 45.
24citeseer.ist.psu.edu and citebase.eprints.org, respectively.
© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 2, 2005
13
', )
·�
' .,
:
,,
.·
,I
' ,
Fig. 3. What you draw is what you see. Roots of polynomials with coefficients 1 or -1 up to degree 1 8. The coloration is determined by a normalized sensitivity of the coefficients of the polynomials to slight variations around the values of the zeros, with red indicating low sen sitivity and violet indicating high sensitivity. The bands visible in the last picture are unexplained, but believed to be real-not an artifact.
perspective is leading to a broader definition of valid
ships. Collaborations, when they arise, are often career
knowledge and valid forms of knowledge representa
long, if not life-long, in their duration. The highly pro
tion. The rapidly expanding capacity of computers to fa
ductive friendship between
cilitate visualization and perform symbolic computa
wood provides a perhaps extreme example.
tions is placing increased emphasis on visual arguments
long-term collaborations are not excluded, the form of
G. H.
Hardy and J. E. Little While
and interactive interfaces, thereby making practicable
collaboration supported by digital media tends to admit
the call by Philip Davis and others a quarter-century ago
a much more fluid community dynamic. Collaborations
to admit visual proofs more fully into our canon.
and coalitions will form as needed and dissolve just as
The SIAM 100-digit Chal lenge: A Study In High-accuracy Numerical Comput ing25 never met while solving Nick Trefethen's 2002 ten
quickly. The four authors of
The price of metaphor is eternal vigilance
(Arturo
Rosenblueth & Norbert Wiener) For example, experimentation with various ways of rep resenting stability of computation led to the four images in Figure
3. They rely on perturbing some quantity and
recomputing the image, then coloring to reflect the change. Some features are ubiquitous while some, like
the bands, only show up in certain settings. Nonethe less, they are thought not to be an artifact of roundoff or other error but to be a real yet unexplained phe nomenon.
5. Re-evaluation of valued skills and knowledge: Com plementing a reassessment of assumptions about math
challenge problems which form the basis for their lovely book. At the extreme end of the scale, distributed com puting can facilitate virtually anonymous collaboration. In
2000, Colin Percival used the Bailey-Borwein-Plouffe 1,734 machines from 56 coun
algorithm and connected
tries to determine the quadrillionth bits of
7T.
Accessing
an equivalent of more than
250 cpu years, this calcula tion (along with Toy Story Two and other recent movies) ranks as one of the largest computations ever. The computation was based on the computer-discov ered identity
ematical knowledge, there will be a corresponding reassessment of core mathematical knowledge and methods. Mathematical creativity may evolve to depend less upon the type of virtuosity which characterized twentieth-century mathematicians and more upon an ability to use a variety of approaches and draw together and synthesize materials from a range of sources. This
which allows binary digits to be computed indepen dently.26
A Temporary Epilogue
is as much a transfer of attitudes as a transfer of skill sets; the experimental method presupposes an experi
The plural of "anecdote" is not "evidence." 27 (Alan L. Lesh
mental mind-set.
ner)
6. Increased community dynamism: Relative to com puter- and network-mediated research, the static social
These trends are presently combining to shape a new
entities which intermesh with the typographic research
community ethic. Under the dictates of typographic norms,
environment extend the timeline for research and pub
ethical behaviour in mathematical research involves ad
lication and support stability in inter-personal relation-
hering to well-established protocols of research and publi-
25Folkmar Bornemann, Dirk Laurie, Stan Wagon, Jbrg Waldvogel, SIAM 2004. 26See Borwein and Bailey, Chapter 3.
27The publisher of Science speaking at the Canadian Federal Science and Technology Forum, Oct 2, 2002.
14
THE MATHEMATICAL INTELLIGENCER
cation. While the balance of personal freedom against com
taneous weakening of community authority structures as
munity order which defines the ethic of digitally oriented
typographic elites are rendered increasingly irrelevant by
it seem as though
mathematical research communities may never be as firm
digital publishing protocols may make
or as enforceable by community protocols, some principles
the social imperatives that bind the mathematical commu
best current prac
nity have been weakened. Any sense of loss is the mathe
provides a snapshot of the de
matician's version of postmodern malaise; we hope and
are emerging. The CEIC's statement of
tices for mathematicians
"those who write, disseminate, and store mathematical litera ture should act in ways that serve the interests of math ematics, first and foremost, " the recommendations advo
veloping consensus on this question. Stating that
predict that, as the community incorporates these changes, the malaise will be short-lived. That incorporation is tak ing place, there can be no doubt. In higher education, we now assume that our students can access and share infor
cate that mathematicians take full advantage of digital
mation via the Web, and we require that they learn how to
media by publishing structured documents which are ap
use reliably vast mathematical software packages whose
propriately linked and marked-up with meta-data.28 Re
internal algorithms are not necessarily accessible to them
searchers are also advised to maintain personal homepages
even in principle.
with links to their articles and to submit their work to pre print and archive servers.
One reason that, in the mathematical case, the "unbear able lightness" may prove to be bearable after all is that
Acknowledging the complexity of the issue, the final CEIC recommendation concerns the question of copyright:
while
fundamental
assumptions
about
mathematical
knowledge may be reinterpreted, they will survive. In par
it makes no attempt to recommend a set course of action,
ticular, the idea of mathematical knowledge as being cen
but rather simply advises mathematicians to be aware of
tral to the advancement of science and human affairs, the
copyright law and custom and consider carefully the op
idea of a mathematical canon and its components, and the
The
idea of a mathematical aesthetic will each find expression
tions. Extending back to Britain's first copyright law,
Statute of Anne,
enacted in 1710, the idea of copyright is
in the context of the emerging epistemology and protocols
historically bound to typographic publication and the pro
of research and publication. In closing, we note that to the
tocols of typographic society. Digital copyright law is an
extent that there may be an opportunity to shape the epis
emerging field; it is presently unclear how copyright, and
temology, protocols, and fundamental assumptions that
the economic models of knowledge distribution that de
guide the mathematical research communities of the future,
pend upon it, will adapt to the emerging digital publishing
that opportunity is most effectively seized upon during
environment. The relatively liberal epistemology offered
these initial stages of digital mathematical research and
by the
experimental method and the social constructivist per-spective and the potential for distributed research and
publishing.
publication afforded by digital media will reshape the pro
Whether we scientists are inspired, bored, or infuriated by philosophy, all our theorizing and experimentation de pends on particular philosophical background assump tions. This hidden influence is an acute embarrassment to many researchers, and it is therefore not often ac knowledged. Such fundamental notions as reality, space, time, and causality-notions found at the core of the sci entific enterprise-all rely on particular metaphysical as sumptions about the world. 30 (Christof Koch)
tocols and hierarchies of mathematical research commu nities. Along with long-held beliefs about what constitutes mathematical knowledge and how it is validated and pub lished, at stake are our personal assumptions about the nature of mathematical communities and mathematical knowledge. 29 While the norms of typographic mathematics are not without faults and weaknesses, we are familiar with them to the point that they instill in us a form of faith; a faith that if we play along, on balance we will be granted fair ac
The assumptions that we have sought to address in this
As the centralized protocols of typo
article are those that define how mathematical reality is in
graphic mathematics give way to the weakly defined pro
vestigated, created, and shared by mathematicians work
cess to opportunity.
tocols of digital mathematics, it may seem that we are
ing within the social context of the mathematical commu
ceding a system that provided a way to agree upon math
nity and its many sub-communities. We have maintained
ematical truth for an environment undermined by rela
that those assumptions are strongly guided by technology
tivism that will mix verifiably true statements with state
and epistemology, and furthermore that technological and
ments that guarantee only the probability of truth and an
epistemological change are revealing the assumptions to be
environment which furthermore is bereft of reliable sys
more fragile than, until recently, we might have reasonably
tems for assessing the validity of publications. The simul-
assumed.
28CEIC Recommendations. See: http://www.ceic. math.ca 29As one of our referees has noted, "The law is clearly 25 years behind info·technology. " He continues, "What is at stake here is not only intellectual property but the whole system of priorities, fees, royalties, accolades, recognition of accomplishments, jobs." 301n "Thinking About the Conscious Mind," a review of John R. Searle's Mind. A Brief Introduction, Oxford University Press, 2004.
© 2005 Springer Science+ Bus,ness Media,
Inc., Volume 27, Number 2, 2005
15
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16
THE MATHEMATICAL INTELLIGENCER
•
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Computer Algebra
c.m;.u.u;
Evolution and Design Inside and Outside Mathematics Eric Grunwald
L
et me start with three dichotomies.
political views, they are a useful lan
Although the first one (at least) is
guage for describing real-life atti
very old, I will refer to them for conve
tudes and positions.
nience by the names of twentieth
Both Gowers and Atiyah recog
century mathematicians who have dis
nize this explicitly. Gowers points
cussed them.
out
that
most
mathematicians
would say that there is truth in both
Hardy's Dichotomy
points of view. The distinction is, as
. . . there is no sort of agreement about the nature of mathematical reality among either mathematicians or philosophers. Some hold that it is 'mental' and that in some sense we construct it ourselves, others that it is outside and independent of us [ 1}.
Gowers says, between the priori ties of mathematicians. Some math ematicians prefer to develop a gen eral understanding of mathematics and mathematical theories; others, to solve specific problems; but it's hard to imagine an intelligent math ematician thinking that theories are
Gowers's Dichotomy
bunk or that problems don't matter.
Loosely speaking . . . the distinction between mathematicians who regard their central aim as being to solve problems, and those who are more concerned with building and under standing theories [2}.
The extreme points of Atiyah's dichotomy also tend not to exist in isolation. One of the great glories of mathematics is the interconnected ness of algebra and geometry, to the extent that it's sometimes hard to tell whether a piece of mathematics
Atiyah's Dichotomy
Geometry and algebra are the two formal pillars of mathematics. . . . Geometry is, of course, about space. . . . Algebra, on the other hand . . . is concerned essentially with time. Whatever kind of algebra you are do ing, a sequence of operations is per formed one after the other and 'one af ter the other' means you have got to have time. In a static universe you cannot imagine algebra, but geometry is essentiaUy static [3}. My questions in this paper are these: are the three dichotomies related, and are they special cases of a wider di chotomy that operates outside as well as inside mathematics? My answer to both questions is yes.
is actually geometry or algebra. Nevertheless, some mathematicians certainly prefer geometry and think geometrically, whereas others are happier with algebra. It does seem possible, on the other hand, for an intelligent person to take an extreme position on Hardy's dichotomy. Indeed, Hardy himself goes on to say, " . . . I will state my own position dogmatically. I believe that mathematical reality lies outside us, that our fimction is to discover or observe it, and that the theorems
which
we
prove,
and
which we describe grandiloquently as our 'creations', are simply the notes of our observations." For an al ternative view, that mathematics is a social and cultural creation, see [4].
The Three Dichotomies Notice the following about these di chotomies:
2. The dichotomies operate on different levels. Hardy's dichotomy is at the top level. It is concerned with mathemat ics as a whole and the question, what
1. Cowers's and Atiyah's dichotomies describe extreme points on a spec trum. These extreme points can
acts at the level of mathematicians
probably not be attained, but like
tions. And Atiyah's dichotomy deals
"left-wing" and "right-wing," over
with the structure of mathematics.
simplifications of highly complex
When we do mathematics we move
is mathematics? Gowers's dichotomy and their preferences and motiva
© 2005 Spnnger Sc1ence+ Business Media, Inc., Volume 27, Number 2, 2005
17
around between these levels. Mathe maticians may only rarely think ex plicitly about the Hardy question, but at a deep level their attitude to it must affect everything they do. Atiyah, for example, likes to "move around in the mathematical waters" [5]: but in or der to do that, he must think of the waters as being there for him to move around in. When mathematicians do mathematics their
choices
about
what to do and the ways they go about doing it are shaped by their po sition on Gowers's spectrum, and
particularly relevant to the main aims of mathematics {2}.
dichotomy.
3. The three dichotomists are not neu tral
dichotomy-observers.
They
Evolution and Design For thousands of years, the stories through which we made sense of the
your sales and minimise your costs. The
world were founded on the idea of a solid design. Heroes did great deeds; vil lains tried to stop them. The gods, or the one God, duly rewarded or punished
speaks from the heart:
. . . algebra is to the geometer what you might call the 'Faustian Offer'. As you know, Faust in Goethe's story was offered whatever he wanted . . . by the devil in return for selling his soul. Algebra is the offer made by the devil to the math ematician. The devil says: 'I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine'. . . . the danger to our soul is there, be cause when you pass over into al gebraic calculation, essentially you stop thinking; you stop thinking geometrically, you stop thinking about the meaning {3}.
butcher needs to cut as much meat as he can of a type that people are prepared to pay a high price for, using the most efficient equipment he can afford, and to beat down as much as possible the price
wrong, nothing fundamentally changed, and we all knew where we stood.
farmers, the slaughterers, and the dis
Zeus
spoke, and nodded with his dark ish brows, and immortal locks feU for wardfrom the lord's deathless head, and he made great Olympus tremble {6}. When Zeus nodded you were well ad
vised to jump to it. As late as the sev
of the carcasses he buys. In this way, the
tributors will have to become more effi cient, and the butcher's customers will enjoy good nourishing food, and, so for
tified, go out and enrich themselves and the rest of us. And so on. There is no need at all for Zeus to tell us what to do-all he needs to do is to set up the rules of competition, make sure we all want to better ourselves, and let rip. We
enteenth century, Newton seemed to
are guided, not by Zeus, but by the in
have shown that the world operated me
visible hand detected by Adam Smith.
chanically, like a clock. There may have been superficial variations as planets moved and apples fell, but the funda mental design was unchanging, perma nent. Newton himself was never a fully
And this hand will make sure that the
world won't stay the same: it will get richer. Smith's story is based on the al gebra of Atiyah's dichotomy: a process
committed Newtonian, by the way. Like
that moves over time. In this algebraic story there is no end-point, no objective,
some other great scientists, he felt un
only rules of procedure.
comfortable with the consequences of his own discoveries. He wanted to look
In the nineteenth century, Charles Darwin did the same trick for biology. His story told how life on earth, with
through nature to see God. He believed that God, like Zeus, acted in the world and retained a perpetual involvement
its astonishing variety and beauty, was
can't predict. Newton's world was de signed, redesigned, and re-redesigned. Then, in the eighteenth century, Adam Smith introduced the idea of the
the result of random variations to gether with a rule of procedure called natural selection. This ensures that or ganisms that manage to survive and re produce will produce more organisms much like themselves, while the ones that can't feed themselves or find mates will simply die out. Despite ap pearances, we are, according to Dar win, not the result of a design at all, but
makes a plea on behalf of the prob
invisible hand. According to Smith,
the product of a battle for survival and
lem-solvers:
people's desire for self-betterment,
resources, rather like Adam Smith's
And Gowers calls his paper The
Two Cultures of Mathematics, and
. . . the subjects that appeal to the ory-builders are, at the moment, much more fashionable than the ones that appeal to problem solvers. Moreover, mathematicians in the theory-building areas often regard what they are doing as the central core (Atiyah uses this ex act phrase) of mathematics, with subjects such as combinatorics thought of as peripheral and not 18
der to get rich you need to maximise
them, and the world stayed much the same. Right was right, wrong was
make their own positions pretty clear. Hardy is dogmatic. Atiyah
less design. There was no geometry, fixed in space, in Smith's story. Rather, it was about rules of procedure. In or
when we analyse the mathematics they produce we are deep in Atiyah's
tremble as he placed us all in his death
THE MATHEMATICAL INTELLIGENCER
with and control over His creation [ 19]. Newton's God didn't merely set up the machine and press the "start" button: He was always fiddling with the world and adjusting it in ways that mortals
guided by their reason, together with
competitive game involving bakers,
the forces of competition that ensure
farmers, and bread-eaters. We are not,
ever greater efficiency, ensure the well
in Darwin's story, going anywhere, let
being of all:
alone trying to achieve perfection in or
It is not from the benevolence of the butcher, the brewer, or the baker, that we expect our dinner, but from their regard to their own interest [7}. The story Adam Smith was telling, unlike Homer's, had no Zeus making us
der to deflect the nods from a god's darkish brows. We are merely trying to compete: to be more attractive than the next man or woman, and to survive, if necessary at the expense of others. And in the twentieth century, Karl Popper told an evolutionary story about science itself. According to his
ideas, scientists do not use a technique called "induction" to analyse data and approach ever closer to the true design of nature. Rather, they create theories about the world that compete with each other according to certain crite ria, the winner emerging to face chal lenges from new rivals. The evolutionary stories, like alge bra, tell of processes through time, op erating according to rules of proce dure. The design stories are based on the assumption of a timeless geomet ric structure, often a goal, an ideal of perfection towards which we are en joined to strive. Just as algebra is a more recent dis covery (to use the language of Hardy's dogma) or invention than geometry, so the algebraic, evolutionary stories, such as Adam Smith's invisible hand and Charles Darwin's natural selection, are much more recent than the ancient design stories. Betty S. Flowers [8] tells the story of how American society has developed under the influence of a se quence of myths (where the word "myth" is not intended to imply either truth or falsity), from the hero myths of Greek warriors and gun-slinging white-hatted cowboys through the "re ligious myth," the "enlightenment myth" or "democratic myth," to what she believes to be the currently pre dominant "economic myth." In the lan guage of Atiyah's dichotomy, the old hero myths are geometric in nature, based on a fixed design of right and wrong, good and evil. The religious myth is still geometric. The enlighten ment/democratic myth also involves an element of geometry through the idea of a search for pre-Popper scientific "truth" and the "best" solutions, al though it also contains a considerable element of algebra: democracy is a process of votes, elections, changes of policy and rulers in the light of experi ence, more elections and so on, with no fixed geometric end in view. The economic myth, which is so influential nowadays, is pure algebra, pure evolu tionary process. There is no end-point: the imperatives are to get more effi cient, get bigger, get richer! The evolutionary stories have a pe culiar power to discomfort us. Many of us are still trying to come to terms with their real meaning. We feel uncomfort-
able with stories that don't give us the comfort of clear moral rules, elegant de signs, and God-given systems of rewards and punishments. The initial objections to Darwin's evolutionary theory were not principally against the notion that species vary or even that we are de scended from apes, ideas that had been current in the scientific community for some time. Darwin's predecessors had assumed that evolution was a goal-di rected process, that each stage of evo lutionary development was a more per fect realisation of a plan that had always been in existence: a plan, perhaps, pres ent in the mind of God. Darwin disposed of this comforting idea [9] . It was the lack of a fixed goal that shocked Dar win's contemporaries, and continues to shock today. The fundamental objection is that Darwin moved us from geometry to algebra. Protesters who disrupt interna tional meetings to agitate against something called "globalisation" are also shocked by a shift from geometry to algebra. They are protesting against the economic myth. Statesmen and ex ecutives of multi-national companies may explain and illustrate the benefits of joining the global economy, but they find their motives questioned. From a geometric point of view, it is necessary to describe one's overall design, which soon acquires the status of a moral code. To a geometer, the algebraist's simple answer "The process works!" cuts no ice.
in the southern states of the USA, where it can be spelled you aU, you-aU, ya 'U, yawl, or yo-aU [ 10, p.449]. Nevertheless, in the linguistic equivalent of Hardy's di chotomy, Crystal stands firmly on the side of evolution. There is no fixed de sign for the English language, he says, to which we should all aspire. The lan guage is constantly changing through complex social processes of which we have only a limited understanding, and it has no end to aim at. Crystal tells how English has gone through phases when it has split into different species, and other periods when dialects have been brought closer together, sometimes prescrip tively. In mediaeval times, for example, regional variation was rife in England, but the introduction of printing, to gether with the establishment of writ ten laws, made standardisation impor tant: for the rule of law to work, it was essential that people could understand what was written and interpret it as far as possible in exactly the same way. The most highly prescriptive period reached its zenith in the eighteenth and nineteenth centuries, when English dictionaries were first published, and people could write things like
Language
According to Crystal, English is un dergoing a period of rapid evolution, with many new species developing round the world out of British and Amer ican English, the first two to emerge. The proponents of design in English tend to conflate their own strict views with a moral code. The following letter to a newspaper, with its morally loaded words, is a typical example:
The linguistic scientist David Crystal has written a book called The Stories of English [ 10]. The plural in the title de liberately suggests his viewpoint: that there is no such thing as "standard" Eng lish, spoken in England, with somewhat defective versions in the USA, India, Nigeria, and so on. Rather, English is an evolutionary system with different vari ants spoken in different places and at different times. The rules of procedure that underlie this evolution are not clearly understood: as yet, we haven't found the linguistic equivalent of Smith's invisible hand or Darwin's natural se lection. We don't know why, for exam ple, the second person plural has be come youse in Liverpool, ye and yiz in parts of Ireland and Scotland, and y'aU
harsh as the sentence may seem, those at a considerable distance from the capital, do not only mispronounce many words taken separately, but they scarcely pronounce, with purity, a single word, syllable, or letter [ 1 1 j.
. . . Oh, and the phrase "dumbing down. " A wholesale and quite horrible corruption of the verb, filtered infrom the US, I accept, but no more forgiv able for that. Dumb is a perfectly ho nourable word, meaning an inability to speak, and its modern usage is very lazy, unacceptable and a sign of ver bal and literate degeneracy [ 12].
© 2005 Spnnger Sc1ence+Business Med1a,
Inc., Volume 27. Number 2, 2005
19
The novelist Kingsley Amis has identified the two extreme points on Gowers's spectrum as they apply to the use of the English language. He calls them berks and wankers ( 13] . Berks don't care about the language's design, they are content to let it evolve hap hazardly. They are "careless, coarse, crass, gross. . . . They speak in a slip shod way." Left to berks, a language would "die of impurity. " They are ex treme problem-solvers in Gowers's language, using whatever expressions come to mind: they certainly don't care about grammar. Wankers, on the other hand, are obsessed with the mainte nance of what they mistakenly believe to be timeless rules of English. They are "prissy, fussy, priggish, prim. . . . They speak in an over-precise way. . . . " They write letters to newspapers, such as the one above, complaining about new usages. Left to wankers, a lan guage would "die of purity." Wankers are on the theory-building extreme of Gowers's spectrum: mathematical wankers (if they existed) would take so much care of the structural unity of mathematics that it would become too inward-looking to have any relevance outside itself. What about Atiyah's dichotomy? Can we fmd linguistic equivalents of fixed geometric space and an algebraic process over time? There is some evi dence that our brains do operate in time-like and space-like ways when we use regular and irregular verbs. (Regu lar verbs in English are those, such as "continue" or "risk," that add -d or -ed to form the past tense and past partici ple, as opposed to irregular verbs such as go-went-gone and ring-rang-rung.) The experimental psychologist Steven Pinker reckons that the brain produces regular and irregular verbs in two quite different ways [ 14]. Accord ing to him, regular verbs are produced through a series of rules, which the brain knows. One such rule is: "add -ed to make the past tense and the past participle." There's no need to look up the past forms in your memory: merely follow the rules. Irregular verbs, on the other hand, are retrieved from memory. But, ac cording to Pinker, this memory isn't a mere string of words, it includes a net work of connections between them. 20
THE MATHEMATICAL INTELLIGENCER
That's why irregular verbs show so many patterns: blow-blew, grow-grew, throw-threw, and so on. If irregular verbs were stored as an unconnected list, we would expect them to be quite random. It also explains why many verbs have become irregular. Ring-rang used to be ring-ringed, but it became at tracted to ing-ang-ung by analogy with words like "sing." Quit-quit became pop ular only in the nineteenth century: Jane Austen used "quitted." Light-lit, creep crept, kneel-knelt, dive-dove, catch caught are other examples. And sneak snuck is in the process of becoming standard in America. We can also see this process at work in experiments with meaningless verbs. For instance, 800/o of people suggest that the past tense of "spling'' is "splang" or "splung." I like to think of the irregular verb forms as crystals. These crystals have formed, somehow or other, out of the surrounding flow of general rules. Once a crystal has become established, it can grow by attracting other similar forms. Bear-bore attracted wear-wore, for example. A crystal will form more easily, and an irregular verb enter the mental dictionary more easily, if its pattern is repeated more frequently. That's why the irregular verbs tend to be frequently used: of the most com monly used verbs in English, the first regular ones are "use" (fourteenth place) and "seem" (sixteenth place) [ 15]. Most of the crystals that spring into existence by the sporadic use of a new irregular form of a verb will die away. In order to grow and survive, these crystals need a lot of use. The crystals are examples of design. Of course the crystals originally sprang up as the result of some sort of evolu tionary process, but once they have been created, their designs are so powerful that they become fixed and draw other words towards them, making them more like themselves. Like design stories, such as religions or Zeus with his im mortal locks, these designs are hard to resist. People who say that the past tense of "spling" is "splung" are partici pants in an old and gripping design story. So our brains use two ways of work ing-at least, as far as producing past tenses of verbs is concerned. Either we take an evolutionary path, consult our rule book, follow its instructions, and
add -ed or -d to the end of the verb; or we look up our special dictionary of de signs, with its network of connections, for the irregular verbs. These two modes of operation can both be highly creative. As we have seen, the networked dictio nary creates new irregular verbs all the time. And, although the word "rule" may sound dull and constricting, the general rule for past tenses can, and often does, lead to the creation of completely new expressions. Pinker has noted "he out Clintoned Clinton," and made-up chil dren's words such as "spidered" and "lightninged." Living in Malaysia, I was delighted when people "onned" and "offed" the lights. There seems, therefore, to be a "di chotomorphism" between mathemat ics and language under which the six dichotomists are mapped onto each other (Fig. 1). Discovery and Synthesis
Leaders in business organisations can, and do, use two kinds of approach. They can set up a series of rules to determine, for example, the processes by which a business should run. They might specify the circumstances under which parts of the business should be divested or, al ternatively, given extra resources. These rules might be in terms of return on cap ital, or profit, for example. And then they can simply allow the different parts of the business to have their heads but sub ject them to the rigour of the rules. This is an evolutionary approach: there is no objective, merely a process. The overall business is broken into separate parts, each of which is free to go its own way within the rules. This approach is simi lar to the way we make regular verbs: we break the word up into its con stituent parts, and apply a series of rules to each part. An extreme kind of holding com pany would use an evolutionary ap proach, an algebraic process though time. The Chief Executive Officer (CEO) would, in principle, say to each of the business managers something like, "Make a business plan that brings in a return on capital of at least 12%. Then go away and implement the plan. Come and see me again next year with your 12% return and with the next plan. If during the course of the year you can see that you won't make your 12%, let
Language
Mathematics
Level
when great chunks of the British econ omy were moved from the geometric central control of government owner
Whole
Hardy
Gowers
People
B
Crystal
B
Arnis
B
Pinker
ship and made to ply their trades ac cording to the algebraic rules of the mar ket, with no destination in view. It was a
time
when
government
gradually
stopped trying to pick the economic sec tors in which Britain should be a big player-steel or car manufacture, for ex
Structure
Atiyah
ample-and started trying instead to set a framework, a process within which the players would determine their own fu ture. The Department of Trade and In
Fig. 1. The "dichotomorphism" between mathematics and language.
dustry was transformed (to a large ex tent) from the British economy's
me know. But, unless there's a very
the business, and at different times. The
into its
good reason for it, watch out!"
regulator.
owner
It used to do geome
skill is to know when to break up and al
try: to try, at least, to determine the con
Such a CEO would have little idea of
low to evolve and when to bring together
figuration of Britain's industrial econ
the overall shape of the business in
into a known crystalline pattern which
omy. But now it does algebra: it sets the
three or four years' time. He or she
will grow in a relatively controlled way.
rules and allows the economy to develop
might not even care: as long as we're
Many companies go through phases of
over time. In Amis's linguistic terms, one
making a good return, why worry about
evolution,
might say that where wankers once tried
of breaking apart existing
what we're selling? This is a highly Dar
structures, when they give individual
unsuccessfully
winian, evolutionary approach. Nature
business managers freedom to diversify
economy, now berks don't care what it
to
shape the
British
doesn't care where each species is go
into new, apparently profitable, busi
looks like as long as it works.
ing. And, like the rule of natural selec
nesses. They generally do this in good
Just as Atiyah sees algebra as the
tion and the rules which shape regular
times, when the risk of a few loss-mak
devil's Faustian offer, many people felt
verbs, the rules of the market bring us
ing operations can be borne in view of
that some sort of Faustian bargain was
extraordinary new species.
the possibility of some bonanzas. And
being struck in 1980s Britain: that the
they also go through phases of design, of
country was giving something up, per
On the other hand, business leaders approach. They can
bringing together. In harder times, when
haps social cohesion or even decency
view the overall business as a whole and
companies decide they can no longer af
and kindness, by its pursuit of wealth in the turbulent marketplace. It felt at
can use a
design
impose a structure on it, fixed in space
ford to wait for experimental businesses
like a geometrical object. This kind of
to become profitable, businesses are
the time like a turning point, and with
approach leads to concepts like the
sold off or shut down, and the company
the benefit of hindsight we can see that
"core business." CEOs say, in effect,
retreats to its "core business."
indeed it was. Britain's long relative
"This is what we do, trying out other
Gary Hamel, well-known in the world
economic decline stopped. In 2002, for
lines of business diverts too much man
of business as a strategy expert, has de
the first time, the UK rate of unem
agement effort, we should stick to what
scribed strategy as a process of discov enJ and synthesis [16]. Discovery is the
ployment was the lowest of all the G 7
we do best." The parts of such a busi ness come together. It is reminiscent of
algebraic process of moving through
the 1970s, the UK economy overtook
countries. And for the first time since
the way our brains deal with irregular
time, of following clear rules of engage
that of France. This may explain why
verbs. First, you spot a crystalline pat
ment that allow businesses freedom to
Britain has, ever since, voted for more
tern: widget factories in West Europe
evolve and determine, on the basis of
or less Thatcherite governments led by
are good business. Then you focus on
their performance, whether they are to
Margaret Thatcher herself, John Major,
that pattern: our core business is widget
be shut down or expanded.
Synthesis is
and Tony Blair. If the British people
manufacture in West Europe. Then you
the geometric redesign in space, the
have sold their soul, at least the devil
grow the crystal slowly: we will invest a
bringing together of current knowledge
has kept his or her side of the bargain.
little in splodget factories in West Eu
and insights into what appears to be the
It's worth pointing out that, although
rope and, maybe, widget factories in
best business configuration at a partic
the British economy moved towards an
East Europe.
ular point in time.
evolutionary story,
through privatisa
tions and increased labour-market flexi
The two approaches--evolution and design-are quite different. But, just as
Right and Left
bility, in another respect the country be
the brain needs both a set of rules for
Another complex system, the British
came more centralised, more geometric.
regular verbs and a dictionary with con
economy, made a significant move from
By 2002, government in Britain raised
nections for irregular verbs, corpora
design towards evolution in the 1980s,
only
tions need to use both evolutionary and
under Margaret Thatcher. This was the
ment, compared with 1ZO/o in the US and
design approaches in different parts of
beginning of the era of privatisation,
100/o in France. Mrs Thatcher didn't trust
4% of taxes through local govern
© 2005 Spnnger Sc1ence+Bus1ness Media, Inc., Volume 27, Number 2, 2005
21
local councils to push the evolutionary
tionary world. The politics of evolution
revolution
versus design is being replaced by the
deed, geometry is sometimes used as a
search for design in an evolutionary
metaphor in this very sense:
through-probably
rightly.
Neither did her successors [ 1 7]. a Liberal Democrat
world. And we can see in the aftermath
Member of Parliament in the UK, has
Vince
Cable,
of the twentieth century's great design
written about the distinction between
theories-communism, socialism, and
left and right in politics, how it has
fascism-the beginnings of some design
careful, degenerate into stagnation. In
The impression is of two men at the height of their abilities but exhausted and immobilised by the fixed geome try of their power [21].
blurre d and lost meaning and signifi
theories for the twenty-first century: re
cance in recent years [ 18]. Following the
ligious fundamentalism and a strong
end of the Cold War, the issues that ex
form of environmentalism known to its
cite and divide people don't fall within
detractors as "eco-fascism."
nies that concentrate on synthesis to the
wing. Such questions as regional inte
Back to Mathematics
tant to
gration and loss of sovereignty (in the
The dichotomy of evolution and design
picked
EU, for example), minority rights, and
provides different points of view on a
Crystal [10] observed that the vast ma
Large, overly bureaucratic compa exclusion of discovery become resis
the framework of left-wing and right
change,
and are eventually
off by nimbler competitors.
immigration are not left-right matters,
number of complex systems including
jority of comments he received from lis
and the existing political parties, having
mathematics, science, language, busi
teners to his radio series about English
grown up in the old left-right era, are of
ness
and
were complaints about linguistic evolu
ten thoroughly divided when it comes to
politics. The design point of view, like
tion: nobody seemed to be worried that the language might be too static.
organisations,
economics,
these issues. Many of the "new fissures"
geometry, centres on an image fixed in
in the world, as Cable terms them, are
space, often one towards which we
People move between the evolution
along questions of identity. What nation
strive. The evolutionary point of view,
ary and design points of view, just as
do I feel I belong to? In what minority
on the other hand, has no end-point,
mathematicians switch between algebra
group, linguistic or racial or sexual or
and deals with a process through time.
and geometry: we all use both regular
professional, do I feel most at home?
In the case of mathematics, the two
and irregular verbs and have moments of
points of view are reflected in the three
linguistic experimentation and pangs of
We can now think of the old left-right distinction as a question of evolution
dichotomies with which this paper was
irritation when other people push the
versus design. The Left had a solid, so
introduced: they describe the distinc
boundaries of language. A mathemati
cialist design in mind. They knew what
tion between evolution and design as
cian need not be on the same side of each
they wanted the world to look like, and
it impinges upon the overall philoso
of the three dichotomies. Complex busi
were less clear on how to get there. The
phy of mathematics, the motives and
ness organisations can operate in evolu
Right, on the other hand, wanted a cer
interests of mathematicians, and the
tionary, discovery mode in one part of
tain process through time, a world op
structure of mathematics itself.
the business while retreating to the core
erating largely through the evolutionary
The evolutionary point of view is
business, in synthesis mode, in another.
processes of free markets, but they were
more recent than the design idea. Al
The British economy moved towards
unclear about where the world should
gebra was invented/discovered long af
evolution when state industries were
be heading. The Left did geometry. The
ter geometry, and the apparently sim
privatised while simultaneously moving
Right did algebra. One might say that
ple evolutionary ideas of Smith and
towards design when power moved
while the Left sometimes thought the
Darwin emerged thousands of years
from local to central government. We
ends justified the means, the Right be
later than the religious and mythical
shouldn't think of evolution and design
lieves the means justify the ends. By the
theories and stories. Science managed
as mutually exclusive modes of opera
end of the twentieth century, in the post
things an order of magnitude more
tion, rather as two distinct points of view
Cold War world, this conflict essentially
quickly: the gap from Bacon to Popper
that are often relevant in understanding
ended in victory for algebra. (A big sur
was merely hundreds of years.
and describing what's going on.
prise for many people who, like me, en
Evolutionary stories have the power
tered university in 1968, the Year of Rev
to discomfort us. Darwin's ideas are
ical
olutions.)
still the subject of heated debate, and
points of view do not have equal sta
In some fields, the theory of biolog species for example,
the two
Following the victory of evolution,
"globalisation" invokes fierce protests.
tus: as evidence is accumulated one
people are yearning for the comfort of a
Design theories often become associ
point of view becomes preferable to
solid design. An evolutionary world is a
ated with moral codes, so that Dar
the other. But in many cases, such as
cold, soulless place. The unease people
win's theory is seen as "anti-religious,"
mathematics,
feel with evolutionary theories like Dar
capitalism can be thought to be im
points of view enriches our under
win's natural selection, and the repulsion
moral, algebra can be said to be a Faus
standing of the whole.
with which many view the evolution of
tian offer, and Gowers writes a paper
Atiyah's dichotomy describes the
free markets under Smith's invisible
imploring theory-builders not to look
most exquisite conjunction between
hand, leads them to search for a new de
down on problem-solvers.
evolution and design that has ever been
the
existence
of two
sign, an identity through nationhood or
Geometry/design, on the other hand,
achieved in any complex system: the
language or colour or sexual orientation,
while generally more comforting than
interplay between algebra and geome
in an algebraic, ever-changing, evolu-
algebra/evolution, can, if we're not
try. Although geometry existed for a
22
THE MATHEMATICAL INTELLIGENCER
long time before the evolutionary in terloper appeared, and although there are no doubt many local skirmishes, peace has essentially broken out be tween algebra and geometry to the benefit of everyone: many of the liveli est parts of mathematics have names like algebraic topology, differential geometry, and analytic number theory. Gower is talking about evolution and design as they affect the politics of math ematics. Who sneers at whom? Who gets the plum jobs? Who gets the money? The situation he discusses seems eerily fa miliar to observers of complex business organisations. The people running the core businesses are trying to perfect a design. One senior executive even told me once that he thought of himself as the steward of a stately home: his role was to hand his successor the business in inunaculate condition. The evolution ists in big companies are the people dri ving the diversifications, people in an en ergy company trying to develop a timber business, for example, or those in a med ical insurance company developing a care homes business. Like Gowers's problem-solvers, they don't want to per fect the current design, they want to strike out in new directions, not caring much about the connections with the present design: they merely want to solve the problem of making more money for the business. To adapt Amis's words, left to the stately home stewards, the business would die of purity; left to the diversifiers it would die of impurity. The power in large companies usu ally lies with the big battalions of the core business, because, inevitably, that's where most of the money and the people are. Most of the new ventures fail, so the safest thing to do when times are hard is to abandon them al together. (I might add that it's gener ally easier to have an intelligent con versation about these matters in a business context than in mathematics, because in business there are more or less agreed financial criteria against which to judge proposals.) I find it hard to see how the situation could be much different. The evolutionists, mutating on the fringes, are in an uncomfortable and dangerous region. Most of them will probably fail. The designists in the core business of mathematics may make life difficult for them, and per-
haps one should view this as part of the competitive, evolutionary environment that ensures that the unsuccessful mu tations die quickly. In [22], Atiyah recognises the need for a balance between ensuring the continuity and unity of mathematics (i.e., taking care of its design) and al lowing the evolutionary possibility of exciting new discoveries that might at first appear to be disjoint from the mathematical core. In practice, when, for example, specific funding decisions have to be made that might affect this balance, the outcome may well depend on whether the decision-makers are natural evolutionists or designists. On the two sides of Hardy's di chotomy are people who believe in a timeless design of mathematics that it is our duty to discover, and those who think of mathematics as a human creation evolving through complex processes over time. Of the three di chotomies, this is the least capable of discussion on the basis of evidence or facts. It is truly a matter of point of view: what inspires you to do mathe matics? To what extent is it to under stand and explain the design of the world outside us, including mathemat ics itself; and to what extent is it to cre ate something beautiful and remark able that evolves and grows over time? Hardy's dichotomy generates a good deal of heat because, when people ar gue about it, they are arguing not about mathematics or even prestige, power, or money; they are arguing about them selves. The issues are their own deeply held beliefs. The stakes are really much higher than in debates on the other two levels. Lose the argument and what's left of you? Thus Hardy, sensibly enough, is merely dogmatic, while Mar tin Gardner (referring to the mathe matics of elementary particles) ad monishes, "To imagine that these awesomely complicated and beautiful patterns are not 'out there,' indepen dent of you and me, but somehow cob bled by our minds in the way we write poetry and compose music, is surely the ultimate in hubris'' [20]. Some be lievers in the design theory of mathe matics may want to batter the evolu tionists into submission. Judging from the infiltration of algebra into geome try's domain, and the successes of the
evolutionists in economics, politics, bi ology, epistemology, and linguistics, perhaps they have a point. REFERENCES
[ 1 ] G. H. Hardy, A Mathematician's Apology, Cambridge University Press, 1 969, 1 23. [2] W T. Gowers, The Two Cultures of Math ematics , Mathematics: frontiers and per
spectives, 65-78, Amer. Math. Soc. Provi dence, R . I . , 2000. [3] M. F. Atiyah, Mathematics in the Twentieth Century, Am. Math. Monthly (200 1 ) , 1 08,
no. 7, 654-666. [4] R. Hersh, What is Mathematics, Really?, Jonathan Cape, 1 997. [5] M . F. Atiyah, An interview with Michael A tiyah , Math. l ntelligencer 6 (1 984), no.
9-1 9. [6] Homer, The Iliad. [7] A. Smith, Wealth of Nations. [8] B. S. Flowers, The Economic Myth , Center for International Business Education and Research, University of Texas at Austin. [9] T. Kuhn, The Structure o f Scientific Revo lutions, 1 962, 1 7 1 .
[1 0] D. Crystal, The Stories of English, Allen Lane, 2004. [1 1 ] J. Walker, Pronouncing Dictionary of Eng lish, 1 774, quoted in [1 0] , 408.
[1 2] D. Taylor, The Times, letter, 21 Septem ber 2004, 32. [1 3] K. Amis , The King's English, Harper Collins, 1 997. [1 4] S. Pinker, Words and Rules, Weidenfeld & Nicholson, 1 999. [1 5] N. Francis and H . Kucera, Frequency Analysis of English Usage; Lexicon and Grammar, 1 984.
[ 1 6] G. Hamel, Masterclass at CBI Conference, November 1 999. [1 7] The Economist, 4 May 2002, 29. [1 8] V. Cable, The World's New Fissures: Iden tities in Crisis , Demos, 1 994.
[1 9] B. J . T. Dobbs, The Janus Faces of Ge nius, Cambridge U niversity Press, 1 99 1 .
[20] M. Gardner, Math. lntelligencer 23 (2001 ) , no. 1 , 7 . [2 1 ] S. Jenkins, The Times, 2 9 September 2004. [22] M. Raussen and C. Skau, Interview with Michael Atiyah and Isadore Singer, EMS
September 2004. Perhelion Ltd. 1 87 Sheen Lane London SW1 4 8LE UK e-mail:
[email protected] © 2005 Springer Sc1ence+ Business Media. Inc., Volume 27, Number 2, 2005
23
M a thematic a l l y Bent
The proof is in the pudding.
Col i n Ada m s , Editor
rienced the floor show. So it could not
The Mathematical Ethicist
influence their survey responses, a complete waste of what must be a ma jor component of your budget. Com mon sense, fella? -Dr. Brad
() ()() Dear Dr. Brad, I was recently invited to give a talk at
Colin Adams
the prestigious Oberwolfach Tricen
Opening a copy of The Intelligencer you
tennial Conference on Number Theory. Mathematical
may ask yourself
uneasily, "lt?tat is this anyway-a mathematical journal, or what?" Or you may ask, "lt?tere am I?" Or even "lt?to am I?" This sense of disorienta tion is at its most acute when you
This was quite an honor for me, par
Dear Dr. Brad, I have gotten in the habit of throwing a lavish banquet for my students the evening before I hand out the student course evaluations each semester. My question is whether or not it is appro priate to have the students fill out the forms at the banquet. There is a break in the festivities after the dinner but
open to Colin Adams's column.
preceding the floor show, which could
Relax. Breathe regularly. It's mathematical, it's a humor column, and it may even be harmless.
serve for this purpose.
ticularly since I am not a number the orist. However, when it was time for me to give my talk, the conference or ganizer introduced me by saying, "And here is the man who proved Fermat's Last Theorem, Andrew Wiles. " Unfor tunately, I am not Andrew Wiles, and it was at this point that I realized that a mistake had been made. It was never their intent to invite me at all. I looked out at the sea of expectant faces, and did the only thing I could do. I pre
Best,
tended to be Andrew Wiles for the next
Waldo Wendt
hour, receiving a standing ovation at
University of Westport
the end. Did I do the right thing? Andre Wilson
Dear Waldo I am assuming that the course evalua tions play a substantial role in the tenure process at your institution, and
Prinsetown University Dear Andre,
further, that you are junior faculty. It is
Under the
a rare senior faculty member who
the right thing. You certainly wouldn't
circumstances,
you
did
throws a banquet with floor show for
want to disappoint the audience, many
his or her students.
of whom had come a long way to hear it be
Dr. Wiles. Luckily, mathematics is eso
hooves you to behave in a manner that
teric enough that you can make terms
cannot possibly be interpreted to sug
up on the fly, and the audience mem
gest, in the slightest way, that you are
bers will be too embarrassed to ac
Given these
assumptions,
attempting to influence the outcome of
knowledge that they have no idea what
the student course surveys. Even the
the heck you are talking about. So they
hint of such impropriety could be
applaud at the end, even though they
smirch your career permanently. In
haven't understood. Would you con
Column editor's address: Colin Adams,
other words, do not give the surveys
sider giving a talk at my institution? We
Department of Mathematics, Bronfman
out between dinner and the floor show.
can't afford Wiles.
Science Center, Williams College,
Wait until the next day. Besides, if they
Williamstown, MA 01 267 USA
did fill them out right after the banquet,
e-mail:
[email protected] the students would not yet have expe-
24
THE MATHEMATICAL INTELLIGENCER
© 2005
Springer Science+ Business Media, Inc.
-Dr. Brad ()()()
Dear Dr. Brad, A paper of mine was recently published in the
Journal of Algebra.
pensive presents to your new Bulgar
theorem, two theorems? Brush. And
ian friend.
what about a corollary? Spit. Some
translation of a paper by an obscure Bulgarian author that appeared in 1972
times I find myself lost in thought, froth
()()()
It was a
dribbling from my open mouth, the sound of banging on the bathroom door
Dear Dr. Brad,
the Journal of the
At a recent conference, I saw a very
Bulgarian Academy of Science. How
nice talk on laminated deck transfor
ever, I did make a few changes in the
mations. Afterward, I suggested to the
spacing and in the numbering of the
speaker that he might want to extend
theorems. Unfortunately, the Bulgarian
his results to polylaminated deck trans
author discovered my translation (turns
formations. I was chagrined, six weeks
in
Bioavtomatika,
out he speaks English, who knew?) and
later, when the editor of a prestigious
he seems to be upset. Should I have ref
journal asked me to referee a paper by
erenced his paper in the bibliography?
this same speaker, in which he ex
Hortense Galbloddy College of St. Geronimo
plained laminated deck transforma tions and the extension to polylami nated deck transformations, with no mention of me whatsoever. I see that I
Dear Hortense,
have three alternatives.
Do ideas transcend the language in
1. I could contact the author and let
from desperate members of my family echoing in the background. But in your case, the time it takes to floss should suffice. It boils down to a single word. Polylaminated. In fact, your contribution wasn't even the whole word. It was actually just the prefix. Does a four-letter pre fix justify inclusion as a co-author in a paper. It turns out that there is a prece dent. In its landmark ruling of 1967, the Ethics Committee of the Canadian Mathematical Society determined that a prefix of three or fewer letters does not suffice to presume co-authorship.
which they are stated? And is it the ideas
him know I am the referee, making
with which we credit a creator? These
it clear there is no way I will rec
are questions that I ask myself some
ommend the paper for publication
times, when I am in the shower, and I
without my own name on it as co
don't feel like going to work After sev
author.
Hence the prefixes sub-, dis-, ir-, bi-, in-, co-, and non- do not cut the mustard. However, in an intricate argument I
will not attempt to recreate here, they determined that a prefix of four or more
the shower, a complete prune, and think
recommend that the editor reject
as long as at least one letter is from the last ten letters of the alphabet, does suffice. Hence, any of para-, trans-,
about what to have for lunch. But
the paper, and in the meantime,
null-, pseudo-, semi-, ortho-, or quasi
enough about me and my day.
write up my own version and sub
will do nicely. But endo- doesn't quite
mit it elsewhere.
make the cut. Of course, if the prefix
eral hours of consideration, I step out of
The short answer is yes, ideas do
letters,
2. I could submit my referee's report,
transcend the language in which they
3. I could send out a blanket e-mail to
are created. Hence, if the ideas in your
everyone in the field explaining how
is no lower limit on the number of let
paper are exactly the same as the Bul
this cretin tried to steal my idea.
ters necessary to warrant co-authorship.
garian's ideas, you must credit him with their discovery. But what is an idea? How does one decide if two ideas are identical? These are questions I reserve for my time in the tub. My con clusion is that there is no idea yardstick which can be utilized to determine the size of an idea, and to compare it with the size of other ideas. So who is to say if your translation actually captures the same ideas that the Bulgarian was at tempting to express? How can you
But whatever happens, I want to make sure that my actions are completely ethical and above reproach. I look for ward to hearing from you as soon as possible.
a-,
there
For God's sake, it's a Greek letter. In your case, poly- does the trick. This means that any of the three alter natives you outlined above would be fully justified, and you can rest assured, if nowhere else, you will find support on the Ethics Committee of the Cana
With great respect,
dian Math Society.
Dr. Donald Dumpstead Ullalah U.
-Dr. Brad
()()()
Dear Don,
know what was in his head? You can't.
The question of what constitutes a suf
So sleep easy. You need not feel guilty
ficient contribution to a paper to jus
for neglecting to include his paper in
tify inclusion as a co-author is one of
your references.
the most difficult and slippery in all of
-Dr. Brad
contains a Greek letter, such as
mathematics. It is a question that oc cupies my thought processes when I
This concludes another column. But re member, when you find yourself tan gled in the morass of mathematical morality, you are only an e-mail away. I hope you don't have to write often. But a letter once in a while wouldn't hurt.
P. S. Of course, the Ethics Committee
brush my teeth every morning. I reserve
at your institution may see it differ
that time to consider it. Can a single
Conscientiously yours,
ently. You might want to send some ex-
lemma be enough? Brush, brush. One
Dr. Brad Dearborn, Ph.D.
© 2005 Springer Science+Business Media, Inc , Volume 27, Number 2 , 2005
25
GOVE EFFI NGER, KENNETH HICKS, AND GARY L. MULLEN
I ntegers and Po ynom i a s · Com pari ng the Close Cous i ns Z and Fq [x]
umber theory is an enigmatic discipline; its fundamental simplicity is tempered by a very rich complexity. Few areas of human inquiry give rise to questions which are so easy to ask but so dif ficult to answer. The simplicity is obvious in the straightforward, even intuitive, definition of the
positive integers. The complexity arises when, among other things, one attempts to isolate the fundamental building blocks (the prime numbers) and their distribution. How ever, the integers are not the only ring with this enigmatic property. In this paper, we contrast and compare the ring of integers and the ring of polynomials in a single variable over a finite field. We explore a number of questions which have analogous versions in both settings. While these ques tions are quite easy to state, they are at the same time not easily answered in either setting. Why Fq[x] Is Special
As is standard, we denote the integers as Z and the posi tive integers as z + . Viewed algebraically, Z is a commuta tive ring, and, in fact, Z is a very nice ring in the sense that it is a unique factorization domain, which means that any integer can be written uniquely (up to the ordering of the factors) as the product of a unit and powers of prime num bers (the units, i.e., elements possessing multiplicative in verses in the ring, of Z are simply 1 and - 1) This unique factorization in Z is precisely the content of the Funda mental Theorem of Arithmetic. The basic idea in any unique factorization domain is that there are certain (non unit) elements, called irreducible elements or, in the case .
26
THE MATHEMATICAL INTELLIGENCER © 2005 Springer Science+ Business Media, Inc
of Z, primes, which form the "multiplicative building blocks" for the ring. Notice, by the way, that z+ is exactly those elements of Z which have 1 as the unit in their mul tiplicative representation. Because they do form the foundation for the multi plicative structure of Z, prime numbers play a central role in many of the questions which arise in classical number theory. For example, we know the role of primes in multi plication, but what about in addition? Is it true, for exam ple, that there are enough primes so that every even posi tive integer can be written as sum of two of them? This is precisely the famous Goldbach Conjecture, first formulated in 1 742. It is an easy question to ask, but it remains un solved to this day. For another example, two primes are called twins if they are two apart (e.g., 3 and 5, 5 and 7, 1 1 and 13, etc). Easy question: Are there infinitely many such pairs? The answer to this "Twin Primes Question" is also unknown. Generalizing now, we note that while Z is one example of a unique factorization domain, we have many other ex amples which may be less fundamental than Z but nonethe less are interesting and important in their own right. So let us investigate one class of such examples: polynomials in one variable x with coefficients in a field K. We denote this
set by K{x] and observe that it is in fact, like Z, a unique factorization domain (see, for example, [14], pages 289 and 3 19). What are the units of this ring? Well, they are pre cisely the (non-zero) elements of the field, i.e., the non-zero constant polynomials. This tells us then that the correct analogue to z+ in this domain is precisely the set of monic polynomials, i.e., polynomials whose leading coefficient is 1. Now, we see that the analogues to prime numbers in Z are simply monic irreducible polynomials in K{x] . All right, can we now do "number theory" in this new domain? For example, what would be a reasonable ana logue to the Goldbach Conjecture in this setting? Maybe one should ask, "Can every even monic polynomial be writ ten as a sum of two monic irreducible polynomials?" Though the meaning of this question is not completely clear (for example, what is an "even" polynomial?), it does seem to make some sense. If, however, we try the Twin Primes Conjecture in this setting, it's not even clear what "twin irre ducible polynomials" might be. Hence we see that questions which are asked in classical number theory about the inte gers may or may not even make sense in our new setting. If we want to raise or, even better, answer questions like the two just presented about polynomials, we need to know something about the irreducible ones. The basic idea would be: If irreducibles are more dense among the polynomials than primes are among the integers, then the answers should be yes; if irreducibles are less dense than primes, the answers may be no; and if the densities are similar, then the questions may be comparably hard to answer. But the nature and density of irreducible polynomials depends completely on the coefficient field K, and so we need to consider some different, common fields of coefficients. Those which come to mind immediately are the complex numbers C, the real numbers R, and the rational numbers Q. Let's think a little about irreducibles in the domains of polynomials over these three fields. The Fundamental Theorem of Algebra quickly settles the issue of irreducible polynomials in C [x] . Since every poly nomial of degree n over C has n roots in C, we see that every polynomial over C factors completely into linear polynomials, that is, the only irreducibles over C are the linear polynomials. Turning now to the reals R, we note that since complex conjugation (i.e., the function which takes a complex number a + bi to a - bi) is a field homo morphism, its natural extension to C [x] will take a poly nomialf(x) in R[x] to itself. But now factoring j(x) over C and applying conjugation, we see that any non-real roots f(x) might possess must occur in conjugate pairs. If such a pair is a + bi and a - bi, then (x - (a + bi))(x - (a bi)) = x2 - 2ax + a2 + b 2 is irreducible over R. We see then that over R,j(x) factors into a product of linear and/or quadratic polynomials; that is, all irreducibles over R are either linear or quadratic. Thus we see that, in some sense, irreducibles in C[x] and R[x] are relatively scarce. In particular, it is obvious that no analogue to the Goldbach Conjecture could possi bly be true in these domains. More generally, the obviously wide gap between the multiplicative structures of Z on the
one hand and C [x] and R[x] on the other make any com parable "number theory" of those domains unlikely. The situation with Q[x] is not so clear. There exist few criteria for determining the irreducibility of polynomials over Q, the Eisenstein Criterion being one of those few. It states that ifj(x) has integer coefficients and if there ex ists a prime number p which does not divide the leading coefficient of f(x), which does divide all the other coeffi cients, but whose square does not divide the constant co efficient, then}t:x) is irreducible. For example, x" + p is ir reducible for all exponents k and any prime p. In particular then, the Eisenstein Criterion shows us that, unlike the cases of C [x] and R[x] , there are in Q[x] irreducibles of ar bitrarily high degree. But are they more or less dense in the set of all polynomials in Q[x] than the primes in Z? This is difficult to answer precisely, but a strong hint is the fol lowing result of D. R. Hayes [20], whose proof is quite short and uses primarily the Eisenstein Criterion: Theorem 1 (A Goldbach Theorem for Q[x]) Every poly nomial over Q can be written as a sum of two irreducible polynomials.
The ease with which this result comes forth, in contrast to the great difficulty of the classical Goldbach Conjecture, certainly leads us to suspect that irreducibles are quite dense in Q[x] and that any "number theory" of Q[x] would (as in the cases of C[x] and R[x] but, in a sense, for the op posite reason) be quite different from that of Z. Though perhaps not as "famous" as the fields C, R, and Q, another important collection of fields is thefinitefields. If q is any power of a prime number p, then there exists a (unique up to field isomorphism) finite field, denoted Fq, which has q elements. Once again, for any such q, the ring Fq[X] is a unique factorization domain. What about its ir reducibles? We shall discuss these at some length in this paper, but let us start by saying that although recognizing irreducible polynomials over Fq can be tricky, counting them is not hard. Restricting our attention to monic poly nomials (as previously justified), it is obvious that the num ber of such of degree r over Fq is exactly qT since there are r coefficient slots and q choices for each. Of these qr poly nomials, how many are irreducible? It turns out not to be difficult to see that there are precisely
Nq(r)
=
_!_ ) JL(d)qTid r d[';.
( 1 . 1)
monic irreducibles of degree r over Fq, where f.L is the so called Mobius function, whose values are 0, - 1 and + 1 . See, for example, Theorem 3.25 o f [23] . The order (i.e., largest term) of the sum is qr, and hence we see that the order of the number of monic irreducibles of degree r over Fq is simply qr/r. In other words, about one out of every r monic polynomials of degree r over Fq is irreducible. How does this density compare with the density of primes in z + ? According to the famous Prime Number The orem of classical number theory, the number of primes less than a positive integer n is of order nllog(n), where log in-
© 2005 Springer Sc1ence+Business
Media, Inc . Volume 27, Number 2, 2005
27
dicates the natural logarithmic function. In fact, a more ac curate estimate for the number of primes below a real num ber y is the so-called logarithmic integral
(Y . Ll(y) =
Jz
dt log(t) '
(1 . 2)
which makes it clear that "near" an integer n, the density of primes is approximately 1/log(n). For an excellent dis cussion of these ideas, see Section I of Chapter 4 of [29]. Now, back in Fq [x] , we would like to have a way to mea sure the "size" of a monic polynomial of degree r. A nat ural such measure would be the following: Definition 2 The absolute value of a polynomial A of de gree r in Fq[x], denoted !AI, is qr.
Note that whereas only one positive integer n has (in the standard sense) absolute value n, numerous (in fact qry monic polynomials of degree r over Fq have (in this new sense) absolute value qr. Using this notion and observing that r logq(q1, we see the following fascinating and im portant fact: In both Z and Fq[X], the density of irreducible elements "near" an element is approximately 1 over the log of the absolute value of that element. In the case of Z the log is natural; in the case of Fq[x] the log is base q. This close connection between these two quite different looking unique factorization domains Z and Fq[X] means that their number theories may very well be quite similar. In the remainder of this paper we shall investigate these similarities, together with the inevitable differences which arise. Can we state and prove a precise "Goldbach Con jecture for Fq[x]"? What about a precise "Twin Irreducibles Conjecture for Fq[x]"? In what follows we consider these and the analogues of various other questions, answered and unanswered, from classical number theory. =
Factorization Both Z and Fq[X] being unique factorization domains, every element in each can be factored in a unique way, apart from order of the factors, into "prime" elements. In the ring Z this factorization is into a product of prime numbers while in Fq[X] it is into a product of irreducible polynomials. Given any unique factorization domain, some basic ques tions arise quite naturally: How can we determine if a given element is irreducible, and are there efficient algo rithms to make this determination? More generally, we
can ask the harder questions: Given an element of our do main, how can we determine its unique factors, and are there efficient algorithms to make this determination? In this section we address these questions briefly in the cases Z and Fq[x], saying more about the polynomial case since it is less well known. A partial answer to all of our questions in the case of Fq[X] is as follows: There are at least two efficient linear algebraic methods which can be used to factor polynomi als over Fq (and hence, of course, determine irreducibility),
28
THE MATHEMATICAL INTELLIGENCER
at least in the case when the field is "small," i.e., when the cardinality q of the field is smaller than the degree r of the polynomial. We deal here primarily with polynomial fac torization over small fields since in most applications the field size is small (e.g., in algebraic coding theory for the error-free transmission of information and in cryptography for the secure transmission of information, q is usually 2). The first of these two methods is known as Berlekamp 's algorithm, and is described in detail in Section 4.1 of [23] . The second method, developed by Niederreiter in [24] , uses non-linear ordinary differential equations in characteristic p. In both cases, in order to factor a polynomial A(x) of de gree r, one sets up an r X r matrix MA over Fq. One then uses linear algebra to reduce the matrix MA - I, where I is the identity matrix. It turns out that the rank of MA - I is r - k, where k is the number of irreducible polynomials in the factorization of A(x). Note then that the polynomial A(x) is irreducible if and only if the rank of the matrix MA - I is r - 1. Moreover, with further work we can ob tain the actual irreducible factors using the vectors in a ba sis of the nullspace of MA - I. Again, these algorithms are efficient; i.e., their running times are bounded by a poly nomial function of r. To contrast now with the case of Z, we remark also that the Berlekamp and Niederreiter algorithms are not only ef ficient but are also deterministic (as opposed to proba bilistic) in the sense that they are guaranteed to give a def inite and correct output upon any appropriate input. Thus, as far as Fq[x] with small q goes, things are as good as they can be in terms of irreducibility testing and factorization. This however, is not the case for integers. First, for pri mality testing, up until very recently the existence of a comparable algorithm for integers had eluded mathe maticians, though there are various deterministic but not efficient algorithms as well as efficient but probabilistic al gorithms for proving primality (see for example [ 1]). How ever, a deterministic polynomial time algorithm for prov ing primality was announced in August 2002 and will be published in [2], providing a significant break-through in the area of computational number theory. Nonetheless, it
would appear that irreducibility testing in Fq[x] is a somewhat easierproblem than primality testing in Z. The phenomenon of problems seeming to be a bit easier in the polynomial setting than in the integer setting is a theme we shall see repeatedly in this paper. Turning to factorization, the problem for Z is a notori ously difficult one; there is no known efficient algorithm to factor integers. This difficulty lies at the heart of modem cryptography, for current cryptographic systems such as RSA depend on the fact that multiplication is easy but fac torization is hard. The fastest methods known are based upon number field sieves or on elliptic curves; see [7], [22], and [33] , for example. If one wishes to speculate about the use of quantum computers (not a current reality), then Shor has provided a polynomial time algorithm in his 1994 pa per [32]. However, there are not even any efficient proba bilistic algorithms for integer factoring as yet! On the poly-
nomial side, we have seen that for small q, all is well, but if we remove restrictions on q, then there are again no known efficient deterministic polynomial factorization al gorithms, even using the generalized Riemann Hypothesis (see for example [ 15]). However, there are efficient prob abilistic polynomial time algorithms for polynomial factor ing, some with near quadratic running times (see [ 16]). In other words, factorization appears to be, in general, a hard problem in both Z and Fq[x], but, again, probably "less hard" for polynomials than for integers, and definitely not hard if q is small, i.e., less than r. The Distribution of Primes and Irreducible& When studying the distribution of primes in Z or of irre ducible polynomials in Fq[x], one can be concerned with "large-scale" or "small-scale" distribution. As noted in the opening section, the large-scale distribution is essentially understood in both domains. The Prime Number Theorem tells us that the number of primes below an integer n is asymptotically 1T (n)
�
n
--
log n
,
and the number of monic irreducible polynomials at or be low a given degree r is asymptotically
Nq(r)
� qr!r.
We observed there the remarkable similarity of these esti mates. We should also emphasize, in support of one of our themes, that whereas the proof of the Prime Number The orem is extremely deep and complex, the estimate for Nq(r) is quite easy to establish using basic facts from the field theory. The small-scale distribution of primes and irreducibles seems to be a much more difficult problem in both domains. In this and the next section we shall look at two problems which illustrate this difficulty: the classical and polynomial versions of the Twin Primes Conjecture and the Goldbach Conjecture. In each of these cases, it is the small-scale dis tribution which is at issue. Here the matter of the Riemann Hypothesis, which ties the small-scale distribution of primes to the zeros of the Riemann Zeta-function, and the analogue of that hypothesis in the polynomial setting must come into play. So let us consider now twin primes and irreducibles. The Twin Prime Conjecture (see, e.g., [ 19]) states that there are an infinite number of consecutive primes with a difference of two, that is, Pm+ 1 Pm 2 for infinitely many m. More specifically, it is conjectured that the number 1r2 (n) of such twin prime pairs less than n is asymptotically -
==
(3.3) The product is known as the "twin prime constant," de noted C2, whose value is about 0.66016. In other words then, the density of twin primes near an integer n is ap proximately 1 .32/(log n)2. The Twin Primes Conjecture has
never been proved, but there is excellent numerical evi dence in support of its truth (see, e.g., [25]). It is thought-provoking to note that if the probability of finding a prime near n is of order 1/log(n), then the prob ability of finding a twin prime pair near n should be of or der 1/(log n)2• A wonderful discussion of heuristic reason ing in the theory of numbers and the apparent connection (perhaps midleading) between these probabilities is given by P6lya [28] . It may seem remarkable that while the reciprocal sum of the primes diverges, the reciprocal sum of the twin primes converges, whether or not there are an infinite num ber of twin primes. This was proved by Viggo Brun in 1919 the sum, known as Brun's constant and denoted Bz has value approximately 1.90216 and has been tabulated to con siderable precision [25] . This result expresses the scarcity of twin primes compared with the primes, much like the divergence of the harmonic series as compared to the con vergence of the reciprocal sum of squares. We tum now to an analogous question in Fq[x]. Can one define "twin" irreducible polynomials, and if so, does the dis tribution of twin irreducibles mimic the distribution of twin primes? The answers to these questions are "yes" and have been presented in some detail in our recent paper [ 13]. Again, we will consider only monic polynomials, for these are the correct polynomial analogues to positive integers. In anal ogy to the integer case, we define two irreducible polyno mials to be "twins" if they differ by as little as possible.
[4];
Definition 3 Two polynomials P1 and Pz, both of degree r over Fq, are said to be twin irreducible polynomials, or simply twin irreducibles, provided that IPz - P11 == 4 if q = 2 or IPz - P11 = 1 otherwise.
In fact, it is easy to show [ 13] that for any fixed degree
r, there are infinitely many twin irreducibles as the field or der q goes to infinity. However, the true polynomial ana logue of the Twin Primes Conjecture is when the order q is fixed and the degree r goes to infinity. Using a line of reasoning similar to that used by Hardy and Wright [ 19], it is possible to derive an analogue of the Twin Primes Con jecture for irreducible polynomials. From [ 13] , this con jecture is:
Nz,q(r)
(
� 0
q - 1 -2
)
qr
r2
I)
(
1
-
ciPI
1 -
)
1)2 '
where Nz,q(r) is the number of twin irreducible pairs of de gree r over F q[x], and the product extends over all irre ducibles P provided q > 2 but does not include linear irre ducibles if q = 2. Also, o = if q = 2 and o 1 otherwise.
4
=
The infinite product on the right converges to a number which depends on q. The reader should note the intriguing similarity between N2,q(r) and 1r2 (n) (Equation 3.3), giving another nice example of the closeness of Fq[x] and Z. Though well supported by numerical evidence (see [ 13] for the polynomial case), both of these results remain conjec tures for now.
© 2005 Springer Science+ Bus1ness Med1a, Inc., Volume 27,
Number 2, 2005
29
We remark that it is necessary to separate out the case for the field Fz because only for this field is the smallest difference between irreducibles not a simple constant term of the polynomial. For example, when q = 2 and r 3, the irreducibles :r3 + x + 1 and :r3 + x2 + 1 differ by x2 + x, the smallest difference between irreducibles for all r > 2. In contrast, for q = 3 and r 3, :r3 + 2x + 1 and :r3 + 2x + 2 are twins differing by the constant 1 . Finally, for any given q, one can estimate an appropri ate analogue B2 ,q of Brun's constant. For example, in [ 13] it is shown that B2, 2 1.059 1 . . . . Thus it is true that whereas both the sum over reciprocal primes and the sum over reciprocal absolute values of irreducibles over Fq[x] (which the reader should check is essentially the harmonic series) diverge, both the sum over reciprocal twin primes and the sum over reciprocal absolute values of twin irre ducibles converge. This is another indication of the close connection between the distribution of primes in Z and the distribution of irreducibles in Fq[X]. One wonders if the Twin Irreducibles Conjecture may be easier to settle than the notorious unsolved Twin Primes Conjecture. =
=
=
The Goldbach Conjecture and the Riemann Hypothesis In 1742 the German mathematician Christian Goldbach conjectured in a letter to Leonard Euler that every positive integer (greater than 5) is a sum of three prime numbers. Euler observed in reply that this is equivalent to every even positive integer (above 2) being a sum of two primes (the reader should check this equivalence), and it is this latter formulation which is now commonly referred to as the Goldbach Conjecture. There are many excellent discus sions of the complicated mathematical history of this con jecture (see, for example, [29]); we will focus here on a spe cific strand of that history which is then tied to the corresponding problem for polynomials. In 1912, 170 years after Goldbach's letter, the distin guished German number theorist Edmund Landau declared in a lecture in London that the Goldbach problem was
"beim gegenwtirtigen Stande der Wissenschaft unan greijbar"-"in the current state of knowledge intractable." Two young British mathematicians, G. H. Hardy and J. E. Littlewood, took up the challenge and, over the next 15 years, produced a seminal series of papers entitled "Some Problems of 'Partitio Numerorem.' " In the third of these, subtitled "On the Expression of a Number as a Sum of Primes," they attacked the Goldbach problem with a brand new method and, though still failing to solve it, were able to " . . . show, however, that the problem is not 'unan
greijbar', and bring it into contact with the recognized methods of the Analytic Theory of Numbers" ([18], page 2). This method, known today as the "Circle Method" or (ap propriately) the "Hardy-Littlewood Method" (first used by Hardy and Ramanujan, see page 698 of [31]), uses complex analysis (more specifically, complex line integrals around the unit circle in the complex plane) to obtain asymptotic formulas for the number of representations of integers in various forms, including as sums of primes. It turns out, as
30
THE MATHEMATICAL INTELLIGENCER
we shall see, that the method can be fruitfully applied to other domains by using appropriate analogues to the com plex plane, its compact unit circle, and line integrals. Saving some details for a bit later, here is what Hardy and Littlewood discovered. Let us denote by Mk(n) the number of representations of a positive integer n as a sum of k odd primes. Assuming what they called "Hypothesis R," a generalization of the Riemann Hypothesis to Dirich let L-functions, they obtain an asymptotic formula for Mk(n), meaning a formula which contains a main term, which measures Mk(n) with greater and greater relative ac curacy as n � oo, and an error term, whose magnitude is given using "Big-0" notation. That formula can be summa rized as follows:
Mk(n)
=
Ck(n)
nk- 1 + O (nk/2 + nk - 1 - 2 + 2E, 2k - 2 - 2E > k k - 1 - E > k/2, and so we obtain, for k 2: 3,
Mk(n)
=
Ck(n)
nk - 1 + O (nk- 1 - E). (log n)k
Now the asymptotics have "kicked in," and we can use Hardy and Littlewood's analysis to prove the existence of representations of positive integers as the sum of k primes provided k 2: 3, and provided, of course, that Hypothesis R is true. The case k = 3 is of particular interest. We shall call this the Odd Goldbach or 3-Primes Conjecture: Every odd num ber greater than 5 is a sum of three primes. Let us write down here exactly what Hardy and Littlewood proved for this important case: Theorem 4 (Asymptotic 3-Primes) If Hypothesis R
is true, then every sufficiently large odd number can be repre sented as a sum of three odd primes, and the number of such representations is given asymptotically by:
M (n) 3
; Jl ( 2
�
(lo
n)3
1
+
(p
� 1)3 ) ·
!] (
als, we very briefly summarize the current state of knowl edge about the Goldbach Conjecture. In 1937 Vinogradov, 1
-
�
p2 - p + 3
)
where p runs over prime numbers as specified.
using a modification of the Hardy-Littlewood method to gether with numerous ingenious estimates of trigonomet ric sums, proved without
hypothesis that every sufficiently
large odd number is a sum of three primes [35]. His analy sis has been refined to the point that "sufficiently large" to
The latter two products here are the explicit form of
day means greater than about 1043000 [6], which is of course
C (n ). The first product does not depend on n and has value 3 about 1 . 15. The second does depend on n's prime factor
still far beyond the possibility of checking the cases below
ization and has a value at or just below .67 if 3 1 n, and just
it using computation. In 1997 it was shown that if the full GRH holds, then every odd number above 5 is a sum of
below 1 otherwise, provided n is odd. Notice also that if n
three primes [8]; that is, assuming the GRH allows us to
is even, this factor is 0 and so M (n) = 0, as it must be. 3 One can use a simple heuristic to see in general that the basic order of the main term for Mk (n) should be nk - 1/(log n)k . Here is the argument for the case k = 3. To write an
eliminate the "sufficiently large" part of the Odd Goldbach
nal Goldbach Conjecture, the best current result, obtained
odd number n as a sum of three odd primes, by the Prime
in 1973 by Chen [5] using sieve methods, is that every suf
Conjecture-this will be significant when we discuss the polynomial analogue below. Finally, concerning the origi
Number Theorem we have about nllog n choices for primes
ficiently large even number is a sum of a prime and an "al
below n in each of the 3 slots. Each of these n3/(log n)3
most-prime" (a number which is either prime or the prod
combinations adds up to an odd number below 3n, so if
uct of two primes).
these sums are uniformly distributed, about 1 out of every
Let us tum now to the polynomial case. If we hope to
3n/2, or 2n2/3(log n)3 of them, will add up to n itself. The
write a monic polynomial of degree
2/3 part of this obviously needs closer analysis, but the
three irreducible monic polynomials, then, in general, one
n2/(log n)3 part is now clear.
r as a sum of two or
of the irreducibles must also be of degree r and the other(s)
We now tum to the connection between Hardy and Lit
must be of lesser degree. It is not clear that the concepts
tlewood's analysis and the Riemann Hypothesis, which is
of "even" and "odd" make any sense in the polynomial con
the source of the
text, but, surprisingly, they not only make sense but also
in the error terms above. If
E
x is
a "nu
merical character" mapping
come into play, much as in the integer case. We make the
page 418), then the
following definition:
x,
denoted
isfying
Z to C (see, for example, [3], Dirichlet L-function associated with
L(s,x), is ffi(s) > 1, by
defined, for a complex number s sat
L(s,x)
= I x(n�) . n�l
Like their close relative the Riemann Zeta-function, the L functions can be extended analytically to the whole com plex plane with a unique pole at ( 1 , 0) and with "non-triv ial" zeroes in the strip 0 ::S ffi(s) ::S 1 . The Generalized Riemann Hypothesis (GRH) is that these functions all have the property that all their non-trivial zeros z actually sat isfy ffi(z) = 112. Hardy and Littlewood's "Hypothesis R" (or a Weak Generalized Riemann Hypothesis) is that there ex ists a ®, 1/2 ::S ® < 3/4, such that every such zero z satis fies ffi(z) ::S ®. Finally, E above is simply 3/4 - ®. The connection between the locations of the zeros of the Dirichlet L-functions and the numbers Mk(n) is far from transparent and can only be hinted at here. The interested
Definition 5 A polynomial A over Fa is called even if it is divisible by an irreducible whose absolute value is 2. Otherwise A is called odd. It is evident from the definition of absolute value that even polynomials exist
only over the field F2 and are pre cisely those polynomials which are divisible by x or x + 1 , the two even irreducibles. Hence a polynomial over F
is 2 even if it lacks a constant term or if it has an even number
of non-zero terms (check). Unlike
Z, it is not true, for ex like Z, the
ample, that "even plus odd is odd. " However
reader should check that no even polynomial can be the sum of three odd irreducibles, and so oddness is a neces sary condition for our desired "3-irreducibles representa tion. " In a similar vein, polynomials which are too "small, " e.g., linear ones, won't have such a representation, and in fact it turns out that if q is even, then A
=
x2 + a (a E Fq)
reader can profitably consult the original work [ 18] or sec
also won't have such a representation (check), being also
ondary sources such as [34]. We say here only that in the
a little too "small." But those tum out to be the only ex
course of the difficult but beautiful analysis done by Hardy
ceptions:
and Littlewood, the L-functions and in particular their log arithmic derivatives
L'(s,x)IL(s,x)
arise in a natural way.
The locations of the zeros of L(s,x) bear directly on the es timates being made, and what is of crucial importance is that the real parts of those zeros stay away from 1, in fact stay strictly to the left of 3/4. Unfortunately, this was un proved in 1922, and it remains unproved today. Before moving to the analogous problem for polynomi-
6 (A Complete "3-Irreducibles" Theorem) Every odd monic polynomial of degree r 2: 2 over every finite field Fq (except for the case of x2 + a when q is even) is (t sum of three monic irreducible polynomials. Theorem
The proof of this result, which is somewhat lengthy, is contained mostly in [ 12] and completed in [9], [ 10], and [ 1 1 ] .
© 2005 Springer Science +Bus1ness Media, Inc., Volume 27, Number 2 , 2005
31
Actually, the first asymptotic result in this direction was David Hayes's 1966 paper [21], but a drawback there was that it dealt with representations of A (of degree r) of the form aP1 + f3P2 + yP3, where P1 , P2, and P3 are all of de gree r and a + f3 + y = 1. Nonetheless, it pointed the way to the analysis done in [12], which we now discuss briefly. To apply the Hardy-Littlewood method in the polyno mial setting, one first needs an appropriate analogue to the unit circle in C on which to carry out the analysis. In [12], that analogue is the compact adele class group Aklk, where k is Fq(x), the field of rational functions over the finite field Fq, and Ak is the adele ring. This latter object Ak is a re stricted direct product of the completions of k at all its places, including the infinite place. The reader should con sult Chapter 4 of [12] for the details. Now the complex path integral around the unit circle used by Hardy and Little wood is replaced by the Haar integral on A�, and the analysis can go forward in a similar fashion. The classical Dirichlet £-functions come over into this setting in a way which is natural but somewhat complicated (see Chapter 5 of [ 12]). But now a major advantage occurs because of the pioneering work of Andre Weil, who in 1948 proved an analogue of the Generalized Riemann Hypothesis in this function field setting [36]. We write now the implication of his remarkable result for our analysis:
We hope the reader, in comparing this result to Hardy and Littlewood's Theorem 4, cannot help but be impressed with their remarkable similarities. Z and Fq[X] are indeed close cousins! Some Concluding Thoughts on Z and Fq[x]
Despite the many similarities between Z and Fq[x], there are differences (1) in structure and (2) in the depth of analysis needed to uncover that structure. Two obvious examples of the former are that Z has characteristic 0 while F q[x] has characteristic p (where q = pk for some k), and that whereas each positive integer has its unique ab solute value, qr monic polynomials over Fq have absolute value qr. Two examples of the latter which we have seen are the Prime Number Theorem (very deep) versus for mula (1.1) for the number Nq(r) of monic irreducibles in Fq[x] of degree r (not deep), and the Riemann Hypothesis (unsolved) versus Weil's Riemann Hypothesis for function fields (deep but solved). We present now another exam ple where things are a little bit easier to understand in Fq[x] than in Z. For d 2: 1, let P(x) = PaXd + + Po E Z[x] be a poly nomial of degree d with Pa = 1. If Q = Z [x]I(P(x)) denotes the quotient or factor ring of Z [x] modulo the ideal gener ated by the polynomial P(x), then each a E Q has a repre sentation of the form a = ao + a 1x + + aa - 1xd- l with ai E Z, 0 :::; i :::; d - 1. The pair (P(x), M) with M = (0, 1, . . . , !Pol - 1}, is called a canonical number system if each a E Q admits a unique representation in the form a = h + ahx for some h 2: 0 with each ai E M, a0 + a 1x + 0 :::; i :::; h and ah =t- 0 for h =F 0. The problem of characterizing canonical number sys tems over Z is a very difficult one. As an indication of this difficulty, for quadratic polynomials over Z the problem has been solved, but only partial results are known for cubic polynomials; see [30]. On the other hand, the following re sult from [30] provides a complete characterization of anal ogous digit system polynomials in the setting of Fq[x]. For the finite field analogue, let x, y be transcendental + bo E over Fq and let P(x,y) = bn y n + bn - lyn - l + Fq[x,y] with bi E Fq[x], bn =F 0, deg bn = 0, and deg bo > 0. Let N = (P E Fq[X] : deg p < deg b0} and let R denote the quotient ring Fq[x,y]I(P(x,y)). Then each r E R can be rep resented as r = r0 + r1y + + rn l Y n - l with ri E Fq[x]. Further, we say that r E R has a finite y-adic representa tion if r admits a representation of the form r = r0 + r1 y + h + rh y for some h 2: 0 with ri E N for 0 :::; i :::; h and rh =F 0 for h =F 0. Finally the pair (P(x,y), N) is called a digit system in R if each r E R has a unique finite y-adic repre sentation. A complete characterization of finite field digit systems is given as follows in [30]: A polynomial P(x,y) as above is a digit system polynomial if and only if n n:tax{deg bil < deg bo. ·
·
·
·
Theorem 7
(GRH for Function Fields) Given suitable re strictions on the character x, the function field £-func tion L(s, x) is a complex polynomial Px in q-s, and when factored into
each
n
satisfies
l ril
=
1 q 12•
Though the details are many, this result allows the analy sis to proceed very smoothly and eventually yields not only the asymptotic result below but also very sharp numerical estimates for the error terms (see Chapters 6 and 7 and the Appendix of [ 12]). These estimates then lead to the "com plete" Theorem 6. The asymptotic result is (Asymptotic "3-Irreducibles") Ifeither thefield order q or the degree r is sufficiently large, then every odd polynomial A of degree r over Fq is a sum of three odd ir reducible polynomials, and the number M3(A) of such rep resentations is given asymptoticaUy by Theorem 8
q�
M3 CA) - �
J}2(
1
+
1
c�l - 1)3
TI (
1-
1
�12 - 3�1 + 3
where P runs over irreducible polynomials over Fq specified.
)
,
as
Note once again that if A were an even polynomial, then the second product, and hence M3(A) , would be 0, as ex pected.
32
THE MATHEMATICAL INTELLIGENCER
·
·
·
·
·
·
·
·
·
·
·
·
·
·
t�l
Despite examples like this one where the gap between Z and Fq[x] seems somewhat wide, we hope that by this time the reader is struck more by the similarities between
these two domains than by their differences. Though we have gone into some detail on certain points, we have not attempted to be comprehensive in our comparisons, and so the reader can certainly pursue further comparisons on his or her own-in fact such studies might make excellent un dergraduate research topics. For example, what about a "Waring Problem" for polynomials-can a given monic polynomial over
Fq be written as a sum of a fixed number
[1 4] J .A. Gallian, Contemporary Abstract Algebra , Houghton-Mifflin, 2002. [ 1 5] S. Gao, On the Deterministic Complexity of Polynomial Factoring, J. Symbolic Computation, 31 (2001 ), 1 9-36. [1 6] J. von zur Gathen and J. Gerhard, Modern Computer Algebra , Cambridge Univ. Press, 1 999. [1 7] H . Halberstan and H.E. Richert, Sieve Methods , Academic Press, New York, 1 974.
of k-th powers of polynomials? What about a polynomial
[1 8] GH Hardy and J.E. Littlewood, Some Problems of 'Partitio Nu
analogue to Dirichlet's Theorem on primes in arithmetic
rnerorurn'; Ill: On the Expression of a Number as a Sum of Primes,
progression? And so on. Here we have simply tried to shed a little light on the wonderful and mysterious relationship between these two domains.
Acta Mathernatica 44 (1 922), 1 -70. [1 9] G . H . Hardy and E.M. Wright, An Introduction to the Theory of Num bers, 4th Edition, Clarendon Press, Oxford, 1 959.
[20] D.R. Hayes, A Goldbach Theorem for Polynomials with Integer Co efficients, Arner. Math. Monthly, 72 (1 965), 45-46. REFERENCES
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the Number Field Sieve, Lect. Notes in Math., 1 554, Springer-Ver
Springer-Verlag , 1 992. [2] M . Agrawal , N . Kayal, and N. Saxena, PRIMES is in P, Annals of [3] Z. I. Borevich and I. R. Shafarevich, Number Theory, New York,
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1/1 1
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. . . ou les denominateurs
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[31 ) A Selberg, Reflections around Ramanujan Centenary, Paper 41 in Collected Papers/At/e Selberg, Vol. I , Springer-Verlag, Berlin, 1 989.
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Athens, Georgia 1 992. [1 2] G. Effinger and D.R. Hayes, Additive Number Theory of Polyno mials over Finite Fields, Oxford University Press, 1 991 .
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[34] R.C. Vaughan, The Hardy-Littlewood Method (2nd ed .), Cambridge University Press, 1 997. [35] I . M . Vinogradov, Representation of an odd number as a sum of three primes , Cornptes Rendus (Doklady) de I'Academia des Sci
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© 2005 Springer SC1ence +Bus1ness Media, Inc., Volume 27, Number 2, 2005
33
A U T H O R S
QOVE EFFINGER
KENNETH HICK
Depart
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rOilOf'llV
, OH 45701
USA
USA
IIXEIS out
spare t
l)iafiO �rod doing com·
Parallel Histories
34
THE MATHEMATICAL INTELLIGENCER
M?•ffiJ•i§::Gihfii@i§#fii.i,i§lid
M i chael Kleber and Ravi Vaki l , Editors
Cartographiana M Michael Kleber
aps maps maps, we love maps. Herein a collection of carto graphical miscellania, each of which appeals to us mathematically for its own reason. Not Looking Over a Four-Leaf Clover
This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.
Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil, Stanford University,
Department of Mathematics, Bldg. 380, Stanford, CA 94305-2 1 25, USA e-mail:
[email protected] .edu
Item one is courtesy of the "Knotted Objects" section of University of Toronto professor Dror Bar-Natan's Image Gallery [ 1 ] . American drivers can generally tum right easily: at a standard intersection of two-way roads, a right tum does not require crossing any lanes of traffic. Turning left is more difficult; the "cloverleaf' highway interchange, for example, makes up for this by replac ing 90° left turns with 270° right turns. Dror points out that "in England, on the other hand, it is easier to make left turns than it is to make right turns. The lovely interchange of I-95 and I-695, northeast of Baltimore (Fig. 1), com bines the advantages of the two sys tems-it has an outer layer of Ameri can (right) turns, followed by a braiding of the lanes, followed by an in ner layer of English (left) turns!" Tom Hicks, the Director of Traffic and Safety for the Maryland State High way Administration, says this delight ful interchange probably would not be built today, primarily because it takes up too much space. The reliance on the unusual left exit ramp is also a bit of a strike against it, according to the Hu man Factors people (and couldn't we do with a few more of those in mathe matics?). But the interchange does work well. Geometrically, Hicks tells me that the curves of the roadway and the ramps are gentle enough that cars have to slow down only slightly from full high way speeds. It's also a win topologi cally. In a cloverleaf, two lanes of traf fic must cross one another, as cars coming off one ramp need to switch places with ones entering another ramp. This interchange avoids that
problem: drivers entering from any di rection can tum either left or right without crossing any other lanes of traffic . (Unlike the cloverleaf, though, one cannot use multiple exit ramps to make a free U-tum.) But I like it purely aesthetically. So does Dror, who writes, "Notice that the lanes of I-95 and of 1-695 are braided in a non-trivial way, proving that the de signer of the interchange cared about el egance. " We're not the only ones: this same interchange put in an appearance as the "Michael G. Koerner Highway Feature of the Week" in January 1999 [4], and from Koerner I also learned that there is at least one more instance of this design, where 1-2011-59 crosses 1-65 just west of Birmingham, Alabama [5]. (In Birmingham, though, the lanes ofthe individual highways are unbraided.) Hicks tells me that you can drive through the Baltimore interchange without ever noticing there's some thing unusual going on: to the driver it just looks like a left exit and a few ex tra bridges, and only the sharp-eyed will notice that the first bridge's cross traffic is going the wrong way. Hooray for maps. Cartograms
Item two is courtesy of the 2004 U.S. election. In the past several years, the media have taken to referring to "blue states" and "red states," those which respec tively vote Democrat or Republican; televised election-night returns fea tured maps which converged (albeit slowly!) to one like Figure 2a. Robert Vanderbei, a professor of Operations Research and Financial Engineering at Princeton, has championed in re sponse a Purple America map, linearly interpolating the colors on a county-by county level to show that we're not so divided after all. (His map differs from the one reproduced in Figure 2b in that this one is higher contrast: a county here is colored pure red or blue if its vote split is 700h-300,6 or more.)
© 2005 Springer Science+ Business Media. Inc . , Volume 27,
Number 2, 2005
35
Fig. 1 . Cartographic and photographic views of an inter change northeast of Baltimore.
36
THE MATHEMATICAL INTELLIGENCER
Fig. 2a. The standard red/blue map of the election.
Fig. 2b. Purple America, with county colors interpolated.
© 2005 Springer Science+ Business Media, lnc.,
Volume 27, Number 2, 2005
37
j
t
Fig. 3. Name this city. (For a hint, see "Where in the World," below.)
This coloration is more informative than the winner-take-all version, but it still looks quite unbalanced, for an elec tion where only a few percentage points separated the popular vote for the two candidates. The problem, of course, is that the map is an equal-area projection, while we would need one with counties sized in proportion to population. Such a map is called a cartogram. Given a map and a population density function on it, a cartogram is a trans formation of the map whose Jacobian at each point is proportional to the den sity function-so after transformation, the population is uniformly distributed. Of course, this is not enough to deter mine the map, and the art of car togramy to date has been in trying to
develop a way to find such a transfor mation which also looks good. Figure 2c is the work of Michael Gastner, Cosma Shalizi, and Mark Newman, of the Center for the Study of Complex Systems and Department of Physics at the University of Michi gan. The technique is a new algorithm, by Gastner and Newman [3], which al lows the population density to equili brate by flowing according to basic linear diffusion, with the map bound aries being dragged along for the ride. The physics does a remarkably good job; I think their map is beautiful. Their Web page [2] contains more pictures. Dror (see item one) tells me he has heard, from both Bill Thurston and
Yael Karshon, that "given a smooth density function on S2 there is a canonical (up to rotations) diffeomor phism of 82. that takes it to the uniform area density." So there is, for example, a canonical way to smoothly redraw the globe with each country's area proportional to its population-well, once you figure out how to populate the oceans. I wonder how it compares to Gastner and Newman's diffusion technique. Where in the World?
And finally courtesy of University of Toronto professor Balint Virag: what city is shown in the map in Figure 3? The answer appears below, slightly ob scured.
© 2005 Springer Science +Business Media, Inc., Volume 27, Number 2, 2005
39
Credits, thanks, answer
Thanks to Greg Slater of the Maryland State Highway Administration for the aerial photograph in Figure lb and the data behind Figure la, and to Lisa Sweeney, Head of GIS Services at MIT, for help in handling it. See the Web sites listed in the references for more interchange maps and aerial photos. The maps in Figure 2 are by Michael Gastner, Cosma Shalizi, and Mark Newman, and are reproduced here with their permission. Moreover, the work is licensed under a Creative Com mons License, and the images, includ ing the ones which appear here, may
t
\
I t I I I ">
be freely distributed and used to make derivative works for any purpose, as long as the original authors are given proper credit and any redistribution passes on these terms. The city pictured in Figure 3 is of Lpmohdnrth (eoyj drbrm ntofurd), if you type the city's name and notable mathematical attribute with your fin gers shifted one key to the right. The map is copyright Ardis Media Group, and used with permission. REFERENCES
Both editions have sewn bindings
40
THE MATHEMATICAL INTELLIGENCER
http://www-personal. umich .edu/� mejn/ election/ [3] Michael T. Gastner and M. E. J. Newman, Diffusion-based method for producing den sity equalizing maps, Proc. Nat/. Acad. Sci. USA 1 01 (2004), 7499-7504.
[4] Michael Koerner, "The Michael G. Koerner Highway Feature of the Week, 9 January 1 999." http://www.gribblenation.corn/hfotw/ [5] Michael Koerner, "The Michael G. Koerner
[1 ] Dror Bar-Natan. "Dror Bar-Natan's Image Gallery."
http://www.math.toronto.edu/�
drorbn/Gallery/
Highway Feature of the Week, 21 Novem ber 1 998." http://www.gribblenation.com/ hfotw/exit_33.html
A CALCULUS BOOK WORTH READING •
•
537 pp $46 softcover (98 1 -02-4904-7) $82 hardcover (981 -02-4903-9)
Newman, "Maps and cartograms of the 2004 US presidential election results."
exit_40.html
•
Calculus: The Elements MICHAEL COMENETZ
[2] Michael Gastner, Cosma Shalizi, and Mark
Clear narrative style Thorough explanations and accurate proofs Physical i nterpretations and applications
"Unlike any other calculus book I have seen . . . Meticulously written for the i ntelligent person who wants to understand the subject. . . Not only more intuitive in its approach to calculus, but also more logically rigorous in its discussion of the theoretical side than is usual. . . This style of explanation is well chosen to guide the serious beginner . . . A course based on it would in my opinion definitely have a much greater chance of producing students who understand the structure, uses, and arguments of calculus, than is usually the case . . . Many recent and popular works on the topic will appear intellectually sterile after exposure to this one." -Roy Smith, Professor of Mathematics, University of Georgia (complete review at publisher's website) "One has the feeli ng that it is a work by a mathematician still in close touch with physics . . . The author succeeds well in giving an excellent intuitive introduction while ultimately maintaining a healthy respect for rigor." -Zentralblatt MA TH (online)
A selection of the Scientific American Book Club
World Scientific Publishing Company http://www. worldscientific. com 1 -800-227-7562
Leray in Edelbach Anna Maria Sigmund, Peter Michor, and Karl Sigmund
Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels?
If so, we invite you to submit to this
column a picture, a description of its
mathematical significance, and either a map or directions so that others may follow in your tracks.
Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42,
8400 Oostende, Belgium e-mail:
[email protected] T
his is a most unlikely place for the mathematical tourist to visit In fact, it is off-limits for tourists of any kind. Photographing, filming, even drawing, is prohibited by law, as signposts tell you sternly, and trespassers will be punished. If they survive at all, that is. Indeed, the signposts also warn you of LEBENSGEFAHR, meaning mortal danger. You are in a military zone, and had better watch out. Don't step on any mines, and avoid getting shot, says an urgent inner voice. But this is ridiculous. We are in Aus tria, after all, with almost sixty years of peace and prosperity behind us. No body wants any trouble. Let's not get caught, that's all. Welcome to Edelbach, or what is left of it. The place is not easily found on a map: it ceased to exist many years ago, during the darkest days of Aus trian history. Nobody lives here any longer. The main road between Vienna and Prague is a couple of miles to the north, but it can be neither seen nor heard. An eerie silence hangs over the place. All that remains of the former village are a few stone-heaps between thickets of fir trees, and a small, aban doned graveyard. To the north of it, a modem fence surrounds a vast ammu nition depot. It is very well guarded, and you can be sure, by now, that binoculars are fixed on you.
This place was once a camp for pris oners of war, mostly French officers. An "Offizierslager"-or Oflag for short: the bureaucrats of the Third Reich were fond of abbreviations. Oflag XVIIA was the birthplace of a substan tial part of algebraic topology. Spectral sequences and the theory of sheaves were fathered here by an artillery lieu tenant named Jean Leray, during an in ternment lasting from July 1940 to May 1945 ([Sch 1990] [Eke 1999] [Gaz 2000]). In the annals of science one finds several examples of first-rate mathe matical research conducted by prison ers of war. The Austrian Eduard Helly, for instance, wrote a seminal paper
©
on functional analysis in the Siberian camp of Nikolsk-Ussurisk, during World War I; and a century before, the Napo leonic officer Jean-Victor Poncelet de veloped projective geometry while in Russian captivity for five years. This may sound as if the monastic reclusion and monotonic regularity of confmed life provided ideal conditions for con centrating the mind. And indeed, An dre Weil wrote that "nothing is more favourable than prison for the abstract sciences" [Weil 199 1 ] . He wrote this while he was in prison, and managed, during his months of captivity, to find some of his major theorems. But he had a prison cell for himself, could re ceive visits from his family, and knew assuredly, to use his words, "captivity from its most benign side only." The physical and psychic deprivations of years in a POW camp, with its over crowding, sickness, hunger, and biting cold, on top of the boredom and un certainty, were something else: in these conditions, intense intellectual pursuit must have been a desperate means for keeping hold of sanity. The prisoners of Edelbach founded a "University in Captivity." Of the 5,000 inmates of the camp, of which a few hundred were Polish and the rest French, almost 500 got degrees, and their diplomas were all officially con firmed in France after the war. The fact that Jean Leray had been the director, or recteur, of this impromptu univer sity must have helped with the French authorities. His academic credentials were impressive: he had received his doctorate at the elite Ecole Normale Superieure in Paris, and had been pro fessor at the Universite de Nancy before being drafted into the war. His joint work with the Polish mathematician Juliusz Schauder Oater a victim of the Holocaust) developed a topological in variant to prove the existence of solu tions of partial-differential equations. This earned him in 1940 the Grand Prix in mathematics from the Acadbnie des Sciences de Paris.
2005 Springer Science+ Business Media, Inc., Volume 27, Number 2, 2005
41
M I LITX R I S C H E S S PE R R G E B I ET 1 L e b e n s gefah r ! Betreten und B efahren , Fotografi eren, Fi lmen u nd Zeichnen gesetzlich verboten und strafbar!
Fig. 1. Tourists are not exactly welcome in Edelbach nowadays, but what can you expect from an ammunition depot?
But Leray was not the only distin guished scientist in the Oflag. There was the embryologist Etienne Wolff, by all testimonies a driving force behind the university, but obliged, for racial reasons, to keep discreetly in the back ground. Etienne Wolff later became professor at the College de France, and member of the Academie des Sciences de Paris as well as of the Academie Franr;aise. Another luminary was Fran 25 1024 Martin Gardner was right in saying [28]: "If it exists, its ·
plies that if a
.
numbers would be huge." 4. "He [Michael Schweitzer] gives the following specimens in which only one diagonal fails. 1 272
462
582
22
1 1 32
942
742
822
972
(MS 1)
1 882
1 942
1 1 62
42
1 482
2542
2262
1 642
922
2822
29 1 2
1 742
62
2222
38 1 2
3392
2462
1 382
(MS2)
(MS3)
The magic totals are squares in each case: 1 472, Does this have to happen?"
2942, 44 1 2•
Lee Sallows also found in it is the
1996 [26] the first specimen (MSJ): (LSi) already presented in the introduction. It is the
smallest possible example in which only one diagonal fails. But I remark that it is simply a Lucas magic square! Apply with (p,
q, r, s)
=
(ELl) (1, 3, 4, 1 1), and you will get exactly the same
square. And it explains why the magic total is a square:
(p2
+
q2 + r 2
+
s2) 2
=
( 12
+
32
+
42
+
1 1 2) 2 = 1472.
The second specimen (MS2) also belongs to Lucas's fam ily! With
(p , q, r, s) = (2, 7, 4, 15), columns 2 and 3 being
inverted. It explains also why the magic total is a square:
(pz
+
q2
+
r z + s2) 2
The third specimen
=
(22
+
7z + 42 + 152) 2
=
2942.
(MS3) does not belong to Lucas's
family. The same magic sum of 441 2 is, however, possible, for example with
(1, 2, 6, 20), (1, 4, 10, 18), (1, 10, 12, 14), (2, 4, 14, 15), . . . but giving squares in which the two diag
onals fail, instead of only one with this third Schweitzer ex ample. The solution
(3, 12, 12, 12) gives a square in which q = r s, the integers
only one diagonal fails, but because
=
are of course not all distinct. Bremner's first specimen (AB2), found before the above squares and published by Guy and Nowakowski in
1995 [29],
in which the two diagonals fail, belongs to Lucas's family.
AB2. This semi-magic square of squares by Andrew Bremner in 1 995, is a 3 X 3 Lucas square with (p, q, r, s) columns 2 and 3. S2
54
THE MATHEMAnCAL INnELLIGENCER
=
4225 = 652•
=
(2, 6, 4, 3) and inverting
Michael Schweitzer [26] was the first to fmd an example (MS4) in which again one diagonal fails but having a non square magic sum:
• •
pr + qs 0, a I c [ - d(pq + rs) - b(ps + qr)] I [ b(pq + rs) + d(ps + qr)]. =
=
The work o f Euler is linked to the theory o f quaternions 3'9
34951
29582
36422
2 1251
1 785'
2n!¥Z
2()582
3()05?
[2) [ 15] [36] [37], developed later in 1843 by William Hamil ton. In his (LE3) square, Euler reuses an identity that he found and sent to Christian Goldbach in
MS4. Example with a non-square magic sum 52 = 20966014, in which one diagonal fails.
1748 [2 1 ) :
(a2 + bz + c2 + d2)(p2 + q2 + r 2 + s2) = (ap + bq + cr + ds)2 + (aq - bp - cs + dr)2 Z + (ar + bs cp - dq) 2 + (as - br + cq - dp) . -
This identity also follows from the fact that the norm of the Because its magic sum is not a square, this example is of course not a member of the Lucas family.
12
1 82
382
1 12
212
chet, and Fermat. Using as a basis these partial results of Euler's, Lagrange published in
222
472
1 72
(AB3, with S2
332
1 32
=
2823)
432 � 32 31� As for the 3 X 3 magic square of squares problem in part 1, there is a forgotten work of an old and prestigious au thor on this subject. The first 4 X 4 magic square of squares was constructed by Leonhard Euler, in a letter [56] written in French that he sent in 1770 to Joseph Lagrange, not giv
ing any method. This letter with the (LE2) square was known by Legendre
[39) in 1830 and by Lucas [40] in 1876,
but not the method.
41
fliP
2'¥1
1 7'
31
tiP
'Ill'
zt'
11
.,.
at
1754 [ 17) in a partial proof the sum ofat most four square
integers, an old co{\jecture announced by Diophantus, Ba
"He [Andrew Bremner] [1 1 ] also gave the 4 X 4 magic square of squares: 232
Euler first used this identity in that every positive integer is
5.
372
product of two quaternions is the product of the norms.
371 .. 81
LE2. The first known magic square of squares, sent in 1 770 by Leon hard Euler to Joseph Lagrange. The 4 rows, 4 columns, and 2 diag
1770 (55] the first complete
proof of this four square theorem, the same year as the let ter from Euler with the first
4 X 4 magic square of squares.
The magic square received by Lagrange is a member of
(a, b, c, d, p, q, r, s) = (5, 5, 9, 0, 6, 4, 2, - 3). In 1942, Gaston Benneton [ 1) [2) published another 4 X 4 square of squares using Euler's method. Its magic sum is 7150, a smaller sum than the Euler example. In 2005, we
this family of squares with
can now say that the smallest member of Euler's family producing distinct numbers is one not found by Euler, (GEl). And it is a member of a nice sub-family (CB2).
48"
zt'
at
18'
21
•
33'
322
1
3111
13'
422
222
�
44'
8'
CB1 . The smallest magic square of squares of Euler's family, gen erated by (2, 3, 5, 0, 1 , 2, 8, -4), giving 52 = 3230.
onals have the same sum 52 = 851 5.
Euler gave his method (LE3) at the St Petersburg Acad emy also in
1770 [ 18]:
CB2. Sub-family of Euler's magic squares of squares, 52 = 85(k2 + 29). k = 0, 1, 2 do not produce distinct numbers. k = 3 produces the (CB1) square. See [9] for another sub-family producing Euler's (LE2) and Benneton's squares. LE3. Euler's 4 X 4 magic square of squares family. Magic sum 52 = (a2 + b2 + c2 + d2)(p2 + q2
+
r2 + s2).
Its magic sum is smaller than in Euler's example (LE2), smaller than Benneton's square, but bigger than the Brem
He needs two supplemental conditions in order to get the
ner example (AB3), which is not a member of Euler's fam
two diagonals to sum to S2:
ily. I can confirm that the solution with the smallest sum
u
© 2005 Springer Science+ Business Media. Inc., Volume 27, N mber 2, 2005
55
Leonhard Euler (Basel 1 107-5t. Petersburg 1 783) sent in 1 770 a 4 X 4 magic square of squares to Joseph-Louis Lagrange (Euler's original letter: Bibliotheque de l'lnstitut de France, photo C. Boyer).
Euler's method published in Latin at St. Petersburg (Bibliotheque de I'Ecole Polytechnique, photos C. Boyer).
56
THE MATHEMATICAL INTELLIGENCER
is the Bremner sample (AB3) (and all its permutations). I can also add that it is impossible to construct another ex ample with a sum smaller than 3230: it means that the sec ond smallest solution of this problem is my square (CBI), coming from Euler's family. See the supplement [9] for a complete numerical study of Euler's family. Warning: both Lucas's and Euler's families often gen erate incorrect squares, because all their numbers are not always distinct. For example, the square (CB3) is smaller than the Bremner example, but it unfortunately contains the same number twice: a small game, will you quickly lo cate it?
CB3. A member of Euler's family, generated by {3, 2, 4 - 1 , 2, 8, -4), but unfortunately an incorrect magic square!
Note that with a = p, b q, c = -r and d -s, Euler's family of squares becomes (LE3cb ) It is a disguised and permuted version of Lucas's family (ELl) of 3 X 3 squares: =
=
.
CB5. The second smallest 5 X 5 magic square of squares. 82
=
1831.
Bimagic squares
Now, let us switch to bimagic squares: magic squares stay ing magic when their entries are squared-or, if you pre fer, magic squares of squares which stay magic when their entries are not squared. The previous examples in this ar ticle were not bimagic, because they do not stay magic when their entries are not squared. The first published bimagic square was an 8 X 8 square made by G. Pfeffer mann in 1890, and published in January 189 1 [4] [7] [45] [47] [62]. Amazed, Edouard Lucas immediately published in 1891 [42], just before his accidental death, a compliment to Pfeffermann for his achievement, together with a proof that a 3 X 3 magic square using distinct integers cannot be bimagic. John R. Hendricks published in 1998 [32] a different, long proof. I propose here a third-and far easier-proof of the impossibility, proving also the new result that even 3 X 3 semi bimagic squares are not pos sible. -
LE3cb. Transition from Euler's 4 X 4 to Lucas's 3 X 3 magic square of squares, generated by a
=
p, b
=
q, c
=
-r, d
=
-s.
And what about 5 X 5 magic squares of squares? They are also possible. Here are two examples, (CB4) (CB5). They seem to be the first published 5 X 5 magic squares of squares, and the smallest possible examples. Some related squares are given below (CB9) and in the supplement [9].
CB4. The smallest 5 X 5 magic square of squares. 82
=S2
=S1
CB6. A putative 3 X 3 semi-bimagic square.
=
1 375.
If such a square exists (CB6), then: Sl = a + b + c e=b+c-d
=
a+d+e
And: S2 z b + c2 b2 + c2 0 (d - b)(d - c)
= a2 + b2 + c2 = a2 + d2 + e2 = d2 + e2 d2 + (b + c - d? = 2d2 + 2bc - 2bd - 2cd = 0. =
This implies that d = b, or d = c. But a magic square has to use distinct numbers: without this requirement we would have bimagic, trimagic, . . . squares with, for ex ample, 1 in each cell! So a 3 X 3 semi-bimagic square can not exist, implying of course that a 3 X 3 bimagic square cannot exist.
© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 2 . 2005
57
Now increment the size, searching for a 4 X 4 bimagic square. Cauchy worked in 1812 on a related subject. After partial proofs of Gauss (triangular case in 1796) and La grange (square case in 1 770, as already mentioned above), Cauchy was the first to prove completely what Fermat pro posed: every positive integer is a sum of at most three tri angular numbers, four square numbers, five pentagonal numbers, and n n-gonal numbers. . . . In his proof [ 14], Cauchy used the same system of two equations that we have to use in the search for a bimagic square: • •
k = t2 + u 2 + v 2 + w 2 s=t+u+v+w
It is not too difficult to find samples of semi-bimagic squares (4 bimagic rows, 4 bimagic columns), the smallest being (CB7). See also an example of a semi-bimagic square using only prime numbers in [9].
1
35
46
61
= 1 43
12
352
462
61 2
=
7063
37
71
13
22
= 1 43
372
71 2
1 32
222
=
7063
43
26
67
7
= 1 43 .....
432
262
672
72
62
11
17
53
= 1 43
622
1 12
1 72
532
= 1 43
=
1 43 = 1 43 = 1 43
=
7063
=
7063
=
7063
=
7063
nal. 51 = 1 43, 52 = 7063.
55
1 05
36
69
1 00
21
15
28
49
19
109
99
1
60
45
CBS. The smallest 4 X 4 semi-bimagic square with one magic diag onal. 51 = 205, 52 = 1 5427.
But in a 4 X 4 square having 4 bimagic rows and 4 bimagic columns, it seems very difficult-or impossible to get two simply magic diagonals; or one bimagic diago nal. And it is proved impossible to get two bimagic diago nals: answering puzzle 287, posed by Carlos Rivera and me in October 2004 [50], Dr. Luke Pebody (Trinity College, Cambridge, England) and Jean-Claude Rosa (Cluny, France) proved independently that a 4 X 4 bimagic square is impossible. It is unknown if a 5 X 5 magic square using distinct in tegers can be bimagic. The following (CB9) magic square has 5 bimagic rows and 5 bimagic columns. The two diag onals are magic but not bimagic: not yet a bimagic square, but a better result than the (CBS) square.
58
THE MATHEMATICAL INTELLIGENCER
44
16
34
35
1
12
38
41
8
24
40
7
10
36
47
13
14
32
4
28
11
45
The 6 X 6 magic square (GPJ) was published by G. Pfefferman in 1894 [46]. It has 6 bimagic rows and 6 bimagic columns. The two diagonals are magic, but not bimagic, as in (CB9). 6 X 6 magic squares with the same characteris tics were published by Huber in 1891 [35], Planck in 1931 [48], Planck modified by Lieubray also in 1931 [48], Venkat achalam Iyer in 1961 (RVIJ) [61 ] , Collison in 1992, page 147 of [31 ] .
CB7. The smallest 4 X 4 semi-bimagic square without magic diago
9
20
onals. 51 = 1 20, 52 = 3970.
7063
It is also possible to get one magic diagonal. For example (CBS),
37
CB9. The smallest 5 X 5 semi-bimagic square with two magic diag
= 7063 =
3
6
42
29
3
40
30
8
44
47
21
20
10
33
31
41
37
1
7
19
17
13
9
43
49
36
2
5
35
34
38
48
14
15
45
12
16
GP1. A 6 X 6 semi-bimagic square with two magic diagonals, pub lished by G. Pfeffermann in 1 894. 51 = 1 50, 52 = 5150.
4x+y
5x+6y
2y
6x+2y
x+6y
2x+y
5x+2y
y
4x+6y
2x+6y
6x+y
x+2y
6y
4x+ 2y
5x+y
x+y
2x+2y
6x+6y
0
4x+4y
5x+5y
x+5y
2x+4y
6x
5x+4y
5y
4x
2x
6x+5y
x+4y
4x+5y
5x
4y
6x+4y
X
2x+5y
RVI 1 . A 6 X 6 semi-bimagic square family with two magic diagonals, published by R. Venkatachalam lyer in 1 961. 51 = 18x
+
18y, 52 =
82x2 + 1 08xy + 82y2•
Open problem 3. What is the smallest bimagic square us ing distinct integers? Its size is unknown: 5 x 5, 6 x 6, or 7 x 7? My feeling is that 5 x 5 bimagic squares do not exist. Bimagic squares of izes 8 X 8 and above are al ready known, see part 7. Open problem 4. Construct a bimagic square using dis tinct prime numbers. [9) [50].
6.
"It's implicit in the work of Carmichael that there can be no 3 X 3 magic squares with entries which are cubes or are fourth powers"
The work of Euler implies already that there can be no 3 X 3 magic square with entries which are cubes. If z 3 is the number in the centre cell, then any line going through the centre should have x 3 + y 3 = 2z 3. Euler and Legendre [39] demonstrated that x 3 + y 3 = kz 3 is impossible with dis tinct integers, for k = 1, 2, 3, 4, 5. Adrien-Marie Legendre mistakenly announced that k 6 is also impossible: Edouard Lucas published the general solution for k = 6 in the Amer ican Journal of Mathematics Pure and Applied of J. J. Sylvester [41 ] , and gave the example 173 + 373 = 6 · 2 13. The equation x 3 + y 3 7z 3 has been known to be possible since Fermat, one of his examples being 43 + 53 7 33 . Legendre showed also that x 4 + y 4 2z 2 is impossible if x i= y. Because z 4 = (z 2)2 , this implies that there can be no 3 X 3 magic square with entries which are fourth pow ers. It's also implicit in the later work of Carmichael [ 13] that there can be no 3 X 3 magic square with entries which are cubes, or are fourth powers or 4k-th powers. Noam Elkies [26] points out that with Andrew Wiles's proof it can be shown that an + bn = 2cn has no solution for n greater than 2, and thus that there can be no 3 X 3 magic square with entries which are powers greater than 2. And as said in the D2 problem, "The Fermat problem," page 219 of Guy's book [30], "It follows from the work of Ribet via Mazur & Kamienny and Darmon & Merel that the equation xn + yn 2zn has no solution for n > 2 apart from the trivial x = y z." So, 3 X 3 magic squares of cubes are impossible. I think that 4 X 4 are also impossible with distinct positive inte gers. The 12 X 12 (WTl) trimagic square of part 7 below, when its numbers are cubed, is a magic square of cubes. If we accept negative integers, and using the interesting but obvious remark that n3 and ( - n3) are not equal (the rule in a magic square is to use "distinct" integers, and the trick is that they are distinct!), (CBJO) and (CEl l) are magic squares of cubes having a nuU magic sum. They seem to be the first published 4 X 4 and 5 X 5 magic squares of cubes. If you do not like the terminological trick I used, then Open problem 5 is for you! And the (CB12) square is a first step. =
=
=
·
=
=
=
Open problem 5. Construct the smallest possible magic square of cubes: 5a) using integers having different ab solute values, 5b) using only positive integers. Open problem 6. Construct a magic square of cubes of prime numbers [9]. 1 g3
(-3)3
(- 1 0)3
(- 1 8)3
(-42)3
213
283
353
423
(-21 )3
(-28)3
(-35)3
( - 1 9)3
33
1 03
183
CB10. A 4 X 4 magic square of cubes. 53 = 0.
113
( -20)3
1 23
1 33
1 43
(- 1 5)3
213
33
(- 1 0)3
(- 1 7)3
43
sa
(-5)3 1 73 (- 1 4)3
03 (-4)3 --1-
1 03
( - 3)3
(-2 1 )3
1 53
(- 1 3)3
( - 1 2)3
203
(- 1 1)3
CB1 1 . A 5 X 5 magic square of cubes. 53 = 0. A special property: four consecutive integers are used in the first row.
g3
473
543
643
963
233
973
63
483
(723
1 03
1 43
673
1013
423
1 1 03
363
213
33
283
403
703
983
1 83
383
CB12. The smallest 5 X 5 semi-magic square of cubes using posi tive integers. 53 = 1 408896.
7. "The following example [30] [34] , due to David Collison, is 'trimagic' in the sense that it is magic and stays so when you either square or cube the entries"
Collison's 16 X 16 trimagic square uses distinct but non consecutive integers. Bimagic and trimagic squares using consecutive integers, so more restricted than squares us ing non-consecutive integers, were published before this Collison square.
Wanting. In parts 1 to 6, we spoke about magic squares using distinct integers, but generally non-consecutive in tegers. ow, in parts 7 to 9, we speak about magic squares and cubes using consecutive integers. The first published n X n bimagic square using consec utive integers from 1 to n2 is an 8 X 8 square made by G. Pfeffermann in 1890, and published in January 1891 [45]. Various other bimagic squares are currently known with various sizes n ::=:: 8. Walter Trump and I showed in 2002 that an n X n bimagic square using consecutive integers is impossible for n < 8. The first published trimagic square, staying magic when you either square or cube the entries, is a 128 X 128 magic square made by Gaston Tarry in 1905 [59]. Later, smaller trimagic squares were found: 64 X 64 by General E. Caza las in 1933 [16], 32 X 32 by William H. Benson in 1976 [3], and 12 X 12 (WTl ) by Walter Trump in 2002 [60]. Even using consecutive integers, this trimagic square is smaller than Collison's square. Walter Trump and I showed in 2002 that an n X n trimagic square is impossible for n < 12. Nobody will ever construct a trimagic square smaller than Trump's.
© 2005 Springer Science+Business Media, I n c . , Volume 27, Number 2 , 2005
59
1
22
33
9
119
45
75
141
35
74
8
1 06
41
62
66
79
83
1 04
112
123
1 44
115
1 07
93
52
38
30
1 00
26
136
48
57
14
131
88
97
1 10
4
70
43
102
133
96
39
137
71
1--
140
101
12 49 1--11 24 42 60
1 22
76
1 42
86
55
27
95
1 35
- --
37
1 08
85
103
21
44
5
67
1 26
19
78
59
3
69
23
130
89
56
15
10
50
118
90
77
28
13
1 43
81
72
-
132
117
68
91
11
73
64
2
121
1 09
58
98
84
1 16
138
80
34
1 05
6
92
51
63
31
20
25
99
46 i113 32
-
134
--
54
-
-
36
24
1 29
7
29
61
47
87
1 27
18
53
1 39
40
111
65
1 28
17
1 20
1 25
1 14
82
94
16 -
--
-
WT1 . The first 12 X 12 trimagic square, constructed by Walter Trump in 2002. This square using consecutive integers from 1 to 144 is magic, and it is again magic when its numbers are squared or cubed. 51 = 870, 52 = 83810, and 53 = 9082800. It is impossible to construct a trimagic square of a smaller size (and using consecutive integers).
In 2001, I published the first known tetramagic and pen tamagic squares, staying magic when you square, cube, or
lines in his cube, but 64 remains an excellent result com pared to the needed 76 theoretical magic rows.
raise to the fourth or fifth power the entries, in an article
More than three centuries after Fermat's 4 X 4 X 4 cube,
of Pour La Science (the French edition of Scientific Amer
Richard Schroeppel showed in 1972 [53] that a 4 X 4 X 4 per
[4] after a joint work with Andre Viricel: 5 1 2 x 512
fect magic cube is impossible, and in 1976 [54] that, if a 5 X
ican)
tetra and 1024 X 1024 penta. Later, I constructed a smaller tetramagic square (256 X 256), and Li Wen a smaller pen
if n
tamagic square (729 X 729). It is unknown
X
n tetra 6U1' v
...r /
/ -...r / .23 / ....;;;?!? / -7 / .;...r / 4/CY / -- / .::;t.r 7
1/ - / ..2 / ..? / «r 1/
PF1 . Pierre de Fermat (Beaumont de Lomagnes 1601-Castres 1665) sent in 1640 a 4 X 4 X 4 nearly perfect magic cube to Mersenne. Among its 76 lines, 64 lines have the magic sum 51
=
130, but 12 lines have a different sum.
/ .2.!7 / ...r6" / - 7 ...r-7 ..9.:7 7 /.L.r .:F / .94" / - / ...r / ..9.T / / -<JC2 /�/ � / .2 / .T.!r / / 6"6"7 .?.::? 7 .2.T/...ra?/ - / / 6'".T / ...rB' /...:z:.r..?/...rPI6"/ .!7 7 / ..?...r / .T.T / .?-r / / .T.t? / / � / 6"- /..z.r.T 7 - 17 .r..Y"7 / .?.:7 /'...:z:.r B'/ � /..z:.? ..Y'712.?7 / .26'" / .?.9 / .R.? / - /1 -/ 1/'...:z:.rd"/ ...r..T / ...r- / .?:Y / ..9.!7 I/ / -..T / 6"...r / � / �liP7 4"6'" 7 /...r.t?.T/ 4/CY / � / ..Y'J v --7 / 4".9 / 6"4" -- / .:?..T / / .?.2 / ..9.? / - / 4CY / ..z19 / v -- / .!7.:7 / «r / 6:? 7 ..T..9 [7 / .J?...r / � /�/...r. �/ ...r.t? / / ...z:.2 / - / _,_ / B'..T1/..,r,e?,e?/ /...r �/ ..Y' /...r�7 4" / ..96" / /.L.r .JI'"/ .!7..T / .9 / dP / /101"7 / ..5"6'" /...z:.2.:7/ � / 11N, Euler a Goldbach, Berlin 4 mai 1 748, Cor respondance mathematique et physique de quelques celebres geornetres du XV/1/eme siecle , St-Petersburg (1 843), 450-455
(reprint by Johnson Reprint Corporation, 1 968) (22] Martin Gardner, Mathematical Games: A Breakthrough in Magic Squares, and the First Perfect Magic Cube, Scientific American 234 (Jan. 1 976), 1 1 8-1 23 REFERENCES
[23] Martin Gardner, Mathematical Games: Some Elegant Brick-Pack
(1 ] Gaston Benneton, Sur un probleme d ' Euler, Comptes-Rendus Hebdomadaires des Seances de I'Academie des Sciences
ing Problems, and a New Order-7 Perfect Magic Cube, Scientific American 234 (Feb. 1 976), 1 22-1 29
[24] Martin Gardner, Magic squares and cubes, Time Travel and Other
214(1 942), 459-461 . [2] Gaston Benneton, Arithmetique des Ouaternions, Bulletin de Ia So
Mathematical Bewilderments , Freeman, New York, 1 988, 2 1 3-
225
ciete Mathematique de France 71 (1 943), 78-1 1 1 .
[3] William Benson and Oswald Jacoby, New recreations with magic
(25] Martin Gardner, The magic of 3
x
3, Quantum 6(1 996), n°3, 24-26
[26] Martin Gardner, The latest magic, Quantum 6(1 996), n°4, 60
squares , Dover, New York, 1 976, 84-92
(4] Christian Boyer, Les premiers carres tetra et pentamagiques, Pour
[27] Martin Gardner, An unusual magic square and a prize offer, CFF, no. 45, February 1 998, 8
La Science N°286, August 2001 , 98-1 02 .
[5] Christian Boyer, Les cubes magiques, Pour La Science N°31 1 ,
[28] Martin Gardner, A quarter-century of recreational mathematics, Scientific American 279 (August 1 998), 48-54
September 2003, 90-95. (6] Christian Boyer, Le plus petit cube magique parfait, La Recherche,
[29] Richard K. Guy & Richard J. Nowakowski, Monthly unsolved prob lems, American Math. Monthly 1 02(1 995), 921 -926; 1 04(1 997),
N°373, March 2004, 48-50. [7] Christian Boyer, Multimagic squares, cubes and hypercubes web
[30] Richard K. Guy, Problem D 1 5 - Numbers whose sums in pairs
site, www.multimagie.com/indexengl.htm [8] Christian Boyer, A search for 3
x
967-973; 1 05(1 998), 951 -954; 1 06(1 999), 959-962
3 magic squares having more
than six square integers among their nine distinct integers, preprint,
make squares, Unsolved Problems in Number Theory, Third edi tion, Springer, New-York, 2004, 268-271
September 2004
[31 ] John R. Hendricks, Towards the bimagic cube, The magic square
magic squares of squares problem" , downloadable from (7], 2005
(32] John R. Hendricks, Note on the bimagic square of order 3, J.
[9] Christian Boyer, Supplement to the article " Some notes on the [1 0] Andrew Bremner, On squares of squares, Acta Arithmetica,
course, self-published, 2nd edition, 1 992, 41 1 Recreational Mathematics 29(1 998), 265-267.
[33] John R. Hendricks, Bimagic cube of order 25, self-published, 2000
88(1 999), 289-297. (1 1 ] Andrew Bremner, On squares of squares II, Acta Arithmetica,
[34] John R. Hendricks, David M. Collison's trimagic square, Math. Teacher 95(2002), 406
99(2001 ), 289-308. (1 2] Duncan A. Buell, A search for a magic hourglass, preprint, 1 999 (1 3] Robert D. Carmichael, Impossibility of the equation x3 + y3 2mz3 , and On the equation ax4 + by4 cz2 , Diophantine Analysis , John =
=
Wiley and Sons, New-York, 1 91 5 , 67-72 and 77-79 (reprint by Dover Publications, New York, in 1 959 and 2004) [1 4] Augustin Cauchy, Demonstration complete du theoreme general de Fermat sur les nombres polygones, CEuvres completes, 11-6(1 887), 320-353 [1 5] Arthur Cayley, Recherche ulterieure sur les determinants gauches,
[35] A. Huber, Probleme 1 96 - Carre diabolique de 6 a deux degres avec diagonales a trois degres, Les Tablettes du Chercheur, Paris, April 1 st 1 892, 1 01 , and May 1 st, 1 892 , 1 39 [36] Adolf Hurwitz,
Ueber
die Zahlentheorie der Quaternionen,
Nachrichten von der Konig/. Gesellschaft der Wissenschaften zu Gottingen, 1 896, 31 3-340 [Reprint in Mathematische Werke von Adolf Hurwitz, Birkhauser, Basel,
2(1 963), 303-330.]
[37] Adolf Hurwitz, Vorlesungen uber die Zahlentheorie der Ouaternio nen, Verlag von Julius Springer, Berlin, 1 91 9, 6 1 -72
Journal fur die reine und angewandte Mathematik 50(1 855), 299-31 3
(38] Martin LaBar, Problem 270, College Mathematics J. 1 5(1 984), 69
(1 6] General Cazalas, Camas magiques au degre n, Hermann , Paris,
[39] Adrien-Marie Legendre, Theorie des Nombres, 3rd edition, Firmin
1 934 (1 7] Leonhard Euler, Demonstratio theorematis Fermatiani omnem nu merum sive integrum sive fractum esse summam quatuor pau ciorumve quadratorum, Novi comrnentarii acaderniae scientiarurn Petropolitanae 5(1 754/5) 1 760, 1 3-58 (reprint in Euler Opera Omnia , 1-2, 338-372)
(1 8] Leonhard Euler, Problema algebraicum ob affectiones prorsus sin
Didot, Paris, 2(1 830), 4-5, 9-1 1 , and 1 44-145 (reprint by Albert Blanchard, Paris, in 1 955) [40] Edouard Lucas, Sur un probleme d ' Euler relatif aux carres mag iques, Nouvelle Correspondance Mathematique 2(1 876), 97- 1 0 1 (41 ] Edouard Lucas, Sur !'analyse indeterminee du troisieme degre Demonstration de plusieurs theoremes de M. Sylvester, American Journal of Mathematics Pure and Applied 2(1 879), 1 78-1 85
gulares memorabile, Novi cornrnentarii academiae scientiarum
[42] Edouard Lucas, Sur le carre de 3 et sur les carres a deux degres,
Petropolitanae, 1 5(1 770) 1 771 , 75-106 (reprint in Euler Opera
Les Tablettes du Chercheur, March 1 st 1 891 , 7 (reprint in [44] and
Omnia , 1-6, 287-3 1 5)
[1 9] Leonhard Euler, De motu corporum circa punctum fixum mobil
in www. multimagie. corn/Francais/Lucas. htm) (43] Edouard Lucas, Theorie des Nornbres , Gauthier-Villars, Paris,
ium, Opera posturna 2(1 862), 43-62 (reprint in Euler Opera Om
1 (1 89 1 ) 1 29 (reprint by Albert Blanchard, Paris, in 1 958 and other
nia , 11-9, 431 -44 1 )
years) (reprint by Jacques Gabay, Paris, in 1 992)
© 2005 Springer Science +Business Med1a, Inc , Volume 27, Number 2 , 2005
63
[44] Edouard Lucas, Recreations Mathematiques, Gauthier-Villars,
A U T H O R
Paris, 4(1 894) 226 (reprint by Albert Blanchard, Paris, in 1 960 and other years) [45] G. Pfeffermann, Probleme 1 72 -Carre magique a deux degres, Les Tablettes du Chercheur, Paris, Jan 1 5th 1 89 1 , p. 6 and Feb
1 st 1 891 , 8 [46] G. Pfeffermann , Probleme 1 506- Carre magique de 6 a deux de gres (imparfait), Les Tablettes du Chercheur, Paris, March 1 5th 1 894, 76, and April 1 5th 1 894, 1 1 6 [4 7] Clifford A. Pickover, Updates and Breakthroughs, The Zen of Magic Squares, Circles, and Stars, second printing and first paperback
printing, Princeton University Press, Princeton, 2003, 395-401 CHRISTIAN BOYER
[48] Planck and E. Lieubray, Quelques carres magiques remarquables,
53, rue
Sphinx, Brussels, 1 (1 931 ), 42 and 1 35
De Mora
95880 Enghien les Bains
[49] Landon W. Rabem, Properties of magic squares of squares, Rose
France
Hulman Institute of Technology Undergraduate Math Journal
e-mail:
[email protected] 4(2003), N . 1 [50] Carlos Rivera, Puzzle 7 9 "The Chebrakov's Challenge", Puzzle 287
Christian Boyer was born near Bordeaux, and graduated from
"Multimagic prime squares, " and Puzzle 288 "Magic square of
two of the "grandes
(prime) squares", www.primepuzzles.net
eccles" of engi neering He has worked .
for Microsoft France, and more recently co-founded a suc
[51 ] John P. Robertson, Magic squares of squares, Mathematics Mag
cessful start-up in software. {He even offers advice to others
azine 69(1 996), n°4, 289-293
creating new companies.)
[52] Lee Sallows, The lost theorem, The Mathematical lntelligencer
Beside mathematical research and
popularizing of mathematics, he is a sports-car aficionado . He
19(1 997), n°4, 5 1 -54
is married, with three daughters.
[53] Richard Schroeppel, Item 50, HAKMEM Artificial intelligence Memo M.l. T. 239 (Feb. 29, 1 972)
[54] Richard Schroeppel, The center cell of a magic 53 is 63 (1 976), Villars, Paris, 2(1 894), pp. 1 86-1 94 (partial reprint of the letter at
www. rnultimagie. com/Eng/ish/Schroeppe/63. htm
[55] J . -A.
Serret,
Demonstration
d'un theoreme d 'arithmetique,
CEuvres de Lagrange, Gauthier-Villars, Paris, 3(1 869), 1 89-201
[56] J . A . Serre! and Gaston Darboux, Correspondance de Lagrange -
www.multimagie. com/Francais/Fermat.htm)
[59] Gaston Tarry, Le carre trirnagique de 1 28, Compte-Rendu de /'As sociation Fran9aise pour /'Avancement des Sciences, 34em e ses
avec Euler, Lettre 25, Euler a Lagrange, Saint-Petersbourg , 9/20 mars 1 770, CEuvres de Lagrange, Gauthier-Villars,
Paris,
sion Cherbourg (1 905), 34-45 [60] Walter Trump, Story of the smallest trimagic square, January 2003,
1 4(1 892), 2 1 9-224 (reprint in Euleri Opera Omnia, IV-A-5, 477-482)
www. multimagie. com/Eng/ish/Tri 12Story. htm
[61 ] R. Venkatachalam lyer, A six-cell bimagic square, The Mathemat ics Student 29(1 961 ), 29-31
[57] Neil Sloane, Multimagic sequences A052457, A052458, A090037, A090653, A09231 2, A TT Research 's Online Encyclopaedia of
[62] Eri c Weisstein, Magic figures, MathWorld, http://mathworld. wolfram. comltopics/MagicFigures. html
Integer Sequences, www.research.att.com/ �njaslsequences
[58] Paul Tannery and Charles Henry, Lettre XXXV IIIb bis, Fermat a
[63] Eric
Mersenne, Toulouse, 1 avril 1 640, CEuvres de Fermat, Gauthier-
Weisstein,
Perfect
magic
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64
THE MATHEMATICAL INTELLIGENCER
cube,
MathWorld,
math world. wolfram. com/PerfectMagicCube.html
http:!/
MARTIN J. MOHLENKAMP AND LUCAS MONZON
Tri gonometri c dentiti es and Su ms of Se parab e Fu nctions
sin(z + 'Y - a) . sm( 'Y - a)
sin(y + {3- a) . . sm(x + y + z) = sm(x) . sm ({3 - a) sin(x + a - {3)
+
sin(a - {3)
sin(z + 'Y - {3) . sm(y) , . sm( 'Y - {3)
sin(x + a - y) sin(y + {3 - y)
+
sin(a - y)
sin(f3 - y)
(1)
cute. It illustrates a richness of structure that invites future study. Other mechanisms that allow representations of the form (2) with low separation rank are described in [2] .
sin(z)
hold for all values of x, y, and z? Motivation Modem computers have made commonplace many calcu lations that were impossible to imagine a few years ago. Still, when you face a problem with a high physical di mension, you immediately encounter the Curse of Dimen sionality [ 1 , p.94]. This curse is that the amount of com puting power that you need grows exponentially with the dimension. The simplest manifestation appears when you try to represent a function by its sample values on a grid. If a function of one variable requires N samples, then an analogous function of n variables will need a grid of N" samples. Thus, even relatively small problems in high di mensions are still unreasonably expensive.
A method has been proposed in [2] to address this prob lem, based on approximating a function by a sum of sepa rable functions: r
f(x l ,
·
·
. , xn)
=
L ¢{(xi) ¢�(x2) j �I
·
·
·
�
representation would require only r n N samples, so if the approximation can be made sufficiently accurate while keeping the separation rank r small, we can bypass the curse. We describe here a particular test of (2), when the "straight forward" approximation is exact but has very large separation rank. Although it may not be directly useful in applications, the result of this test is surprising, positive, and, we believe, This
For what value(s) of a, {3, 'Y does the equality
¢ (xn).
(2)
A Test Function Our test function is the sine of the sum of n variables, sin(�.f� 1 Xj), which is a wave oriented in the "diagonal" di rection in n-dimensional space. One could use complex ex ponentials to express it as the sum of two separable func tions, sin
( In ) xj
j� l
1
= ---;
n
.
1
n .
fl� l eur:J - ---; fl e -ur:J, 2t j� l
2t j
but in our test we allowed only real functions. You can use ordinary trigonometric identities to find such a representation. When n 2 we have =
sin(x + y) = sin(x) cos(y) + cos(x) sin(y),
(3)
which expresses sin(x + y) as a sum of two separable func tions. When n = 3 we have sin(x + y + z) = sin(x) cos(y) cos(z) + cos(x) cos(y) sin(z) + cos(x) sin(y) cos(z) - sin(x) sin(y) sin(z), (4)
This material is based in part on work supported by the National Science Foundation under grant DMS-9902365, and by University of Virginia subcontract MDA-97200-1-0016.
© 2005 Springer Sc1ence + Business Media, I n c. . Volume 27, Number 2, 2005
65
Figure 1. left: Graphical separated representation of sin(K + y + z) using the usual trigonometric identity (4). Each of the four rows gives the factors of a separable function. For example, the first row corresponds to sin(K) cos(y) cos(z). The separable functions from each row are then added. right: Graphical separated representation of sin(K + y + z) using (1) with
a = 0, {3
=
Tr/3, and y
= 2Tr/3. The amplitude has been
equidistributed. (The original illustration uses three colors, blue, red, and green, for the three curves in each graph.)
which uses four terms. The drawback to this approach is that, for
n
variables, the number of terms is
2n- l . This exponen
tial growth in the number of terms negates the benefit of us ing the form
(2) . Indeed, if this really is the minimal number
of terms needed, then the entire approach is doomed.
We then asked what the minimal number of terms is, and our program replied shown in Figure
1
for all values ofA, B, s({3 - a) * 0). Proof
With the notation c(x)
metric identity
s(A)
(right). After some investigation, we de
s(y)
(3)
Any representation of a function of a sum of n variables will have
�
- 1 free parameters, because one can include n shifts Xj Xj + ai and one linear constraint �J= 1 ai = 0 and have �J=l (Xj + aj) = �J= l Xj. The identity that we pre sent below in Theorem 2 has n 1 additional independent n
-
+
n
=
3,
( 1) holds for arbitrary a, {3, and y, as - {3) * 0, sin(a - y) * 0, and sin({3 - y) * 0.
Since these three parameters occur only as differences, only two of them are independent. One can introduce two additional parameters as phase shifts to make versions of
(1)
with different symmetries.
The Identity Lemma 1 The function s(x)
s(A + B) =
66
=
recognize as
satisfies the equation
s(A + a - {3)s(B) s(A)s(B + {3 - a) + s({3 - a) s(a - {3)
THE MATHEMATICAL INTELLIGENCER
(5)
s(B) (s(A)c(- y) + c(A)s( - y)) s ( - y) + c(A)s(B),
which we
s(A + B).
D
Theorem 2 Any junction s(x) that satisfies (5) also sat
isfies
s
( � ) - �l ( ) L
j= l
Xj -
L
j=
s Xj
n n
s(xk + ak - aj) . s(ak - aj) k = l ,k ¥j
(6)
for aU choices of { llJ} such that s(ak - aj) * 0 for aU j * k. The proof is by induction and is given in the Appendix. We can generate a more general form by introducing
a/ s
sin(x)
par
--
all terms cancel except for s(A)c(B)
it provides an answer for our opening
teaser. The identity long as sin(a
{3 - a,
Multiplying out and using that s(x) is odd and c(x) is even,
parameters, which play a structural role in our represen tation. When
cos(x) and y =
(s(B)c(y) + c(B)s( y))
termined the trigonometric identity that our program had un numerical, not symbolic, and so uncovered a trigonometric
=
to obtain
covered. What is most remarkable is that the program was identity without even knowing it was doing trigonometry!
and {3 such that s(a - {3) * 0 (and
tially expand the right-hand side using the usual trigono
and produced graphs such as that
"n"
a,
Ct
Xj +
it
)
n
shifts
ai
=
n
n
I s(Xj + aj) =n k l,k¥j j=l
s(xk + ak + ak - aj) s(ak - aj)
By choosing different ways to satisfy the linear constraint
�J= l ai = 0, we can produce a variety of identities similar
to (6) without modifying the parameters a1, which are the
by setting y1
structural elements of our representation. Note that in the
identity, and then multiplying by exp(i�J= l x1)!2i.
set { ak - a1}Mj only n - 1 parameters are linearly inde pendent, say { a1 - ai1J=2·
Other Functions that Satisfy the Same Identity Because of Lemma 1 and Theorem 2, we know that sin(�J= l x1) is exactly separated with separation rank n. Moreover, this function is peculiar in that sin(·) is the only function used in the separated representation. We now consider the problem of finding other functions s(x) sat isfying (6). Since the general case (6) is equivalent to the
2 case, it is enough to describe all functions that sat isfy (5). n
=
Lemma
3 The function s(x)
=
x satisfies the identity (5).
This and the following lemma may be verified directly. Lemma 4
f(C D)
=
f(C)f(D ¢/8) + f(C8/¢)f(D) !( 8/¢) f( ¢/8)
is equivalent to (5) with the substitutions C exp(A), D exp(B), 8 = exp (a), ¢ exp(/3) , and s(x) = f(exp (x)). Sim ilarly, (6) is equivalent to =
=F 0, b,
and c
f
( Il
j= l
Yi
)
=
ate condition is that s(x) be meromorphic.
meromorphic function s(x) satisfies (5) if
and only if a exp(bx)x
or
s(x)
=
a exp(bx) sin(cx)
for some complex constants a =F 0, b, and c
=
I rl
j = l k= ! ,k #j
In analogy to Lemma 4, from the particular solutions f(x)
ln(x) and f(x)
=
=
1 - x to (8) we can generate other solu
Remarks and Conclusions It is easy to extend our results to find similar identities for f(�J= l Xj), where f(x) could be cos(x), cos2(x), or sin2(x), for example.
We also tested the function of six variables sin(u + v + w) sin(x + y + z). Using ( 1) on each factor and then mul tiplying out yields a representation of the form (2) with 9 terms, but our program found a representation with 8
A survey on the problem of exact separated represen
s (a + ak - a1) s(ak - aj)
When s(x) = sin(x), this result is presented in [4] . For a proof using Lagrangian interpolation, see [5, page 272]. The approach of [4] and [5], however, does not produce the gen eral results of Theorems 2 and 5. Conversely, our results can be used to derive only a few of the identities listed in
[5, Section 2.4.5.3].
out in Problem 4 on page 158, to find a minimal rank rep resentation for a separated representation is still an open problem. We believe that our Theorems 2 and 5 are an ex ample of such minimal representations. Lemma 1 can be proven geometrically, in a way similar to the geometric proof of the usual identity (3). We have not been able to find a geometric interpretation of (6).
Appendix: Proofs Proof of Theorem 2
The case n = 2 is Lemma 1 with A = x1, B = x2, a = a1, and {3 = a2. The proof will be by induction in n, so we as sume (6) has been proven for n 1. We will use (5) to sep -
The situation is different if we consider Milne's identity
[ 6, 3]
(8)
tations is the book [7] by Rassias and Sim8a. As they pointed
Under the same conditions as in Theorem 2,
s(a)
f(Yk8k181) , fC8k/8j)
to find the formula analogous to (6) for this case.
=F 0.
Extensions and Relationships with Other Identities If in Theorem 2 we set x1 = a for all j, we obtain the fol lowing corollary.
s (na)
k = l ,k #j
terms. After considerable effort, we have still not been able
The proof is given in the Appendix.
Corollary 6
I f(yj) Il
j= l
ization of Milne's identity.
=F 0.
(5), and then ask if we have missed any others. We only
Theorem 5 A
=
=
for constants a, b, and c. In this way we obtain a general
wish to consider reasonably nice functions. The appropri
arate out the variable Xn, then cancel like terms and reduce 1 case. the n case to the n -
1-
n
I1
j= l
n
Yi
=
I (1
j= l
-
yj)
n
I1
k = l,k #j
1 - Yk8k/81 1 - 8kJe1
First, expand the left-hand side of (6) using (5) with A
·
(7)
We can obtain another proof of this identity by setting s(x) = 1 - exp(x), ai In 8j, and Xj = In Yi in (6). Theorem 5 applies to this function because 1 - exp(x) = -2i exp( ) sin( ). Conversely, (6) for s(x) sin(x) can be obtained =
�
Simply note that the identity
axb f(XC)
Starting with our two basic functions sin(x) and x, we can use Lemma 4 to construct other functions that satisfy
=
A "multiplicative" version of the identities that we have discussed can be derived by generalizing this observation.
If s(x) satisfies (5), then so does
for all complex a
exp( - 2ia1) in Milne's
=
tions to (8), namely
a exp(bx)s(cx)
s(x)
exp( - 2ix1) and e1
=
=
�
r;: f Xj, B = Xn, (X = O'.n - 1 ,
and {3
=
O'.n to obtain
S(Xn + O'.n - O'.n- 1) s( O'.n - O'.n- 1) +S
(nfl J= l
Xj + O'.n - 1 - an
)
s(Xn) . a ns( l - an)
© 2005 Springer Sc1ence+Business Media, Inc., Volume 27, Number 2, 2005
=
(9)
67
On the right-hand side of (6), first separate off the j n term in the sum. When j i= n, we expand the k n term in the product using (5) with A = a, - a1, B = Xn, a = an - 1 , and {3 = an. Explicitly, the k = n term is =
=
s(Xn + a, - aj) s(an - aj) X
=
(
=
1 s(a, - aj)
S (a,, - aj)S(Xn + Cin - an- 1) s(an - Cin - 1)
s(xn + a, - Cin- 1) s(a, - Cin- 1)
+
(
+
S( Cin- 1 - aj)S(Xn) S ( Cin- 1 - a,)
s( Cin- 1 - aJ) s(an - aj)
)(
s(xn) s(an - 1 - an)
)
)
·
Note that the first term does not depend onj, and that when j = n - 1 the second term is absent. Combining these ex pansions, we can express the right-hand side of (6) as 1 s(xk + ak - aJ) s(xn + - Cin- 1) s(Xj) s(an Cin- 1) s(ak - aj) k=1,k"'J J=1
(I
(�
)
fY
�
n 1 S (Xk + ak - aj) S ( Cin - 1 - aj) n + L s(Xj) s(ak - aj) s(a n - aj) k= 1,k #J j= 1 2
)
Proof of Theorem 5, given Lemmas 7 and 8
We have already shown that these functions satisfy (5), so we need only show there are no more solutions. We now assume s(x) satisfies (5) and will deduce its properties. Using Lemma 7, we know that h(x) = exp( -bx)s(x) is an odd function. By Lemma 4, h(x) also satisfies (5). Then, by Lemma 8, h(x) is either ax or a sin(cx), so s(x) = a exp(bx)x or s(x) = a exp(bx) sin(x), which completes the proof. D The proofs of Lemmas 7 and 8 use the fact that s(O) = 0. By setting {3 - a = A in (5) and subtracting s(A + B) from both sides we obtain 0
=
s(O)s(B)
valid for all A such that s(-A) i= 0 and for all B. Choosing B such that s(B) i= 0 implies that s(O) 0. =
Proof of Lemma 7
We define the auxiliary meromorphic function
F(x) =
Now compare our expansions (9) and (10) of the two sides of (6). Using the induction hypothesis at n - 1, we can see that the first terms in (9) and (10) are equal, and so cancel. The remaining terms all have a factor of s(xn) in the numer ator and s(a,- 1 - a,) in the denominator, which we can also cancel. Thus we have reduced the proof to showing that n-1 Xj + Cin- 1 - an S J= l
(�
)
s(-A) '
s(x) s( -x) '
which cannot be identically zero, and show that it satisfies the functional equation
F(x + w) = F(x)F(w).
=
s(x)s(w) s(x + w)
+
s( - w)s( -x) s( - x - w)
)
_
.
(13)
Using (5) with A x, B = -x, a = -x, and {3 w, we con clude that the right-hand side of (13) is equal to s(O) = 0. D =
=
(
(12)
This functional equation is satisfied only by exponentials, so we can conclude that F(x) = exp(2bx) for some con stant b. Rewriting this condition in terms of s, we have exp( -bx)s(x) = -exp(bx)s( -x), which is what we are try ing to show. To show (12), we substitute in ( 1 1) and manipulate to form the equivalent equation 0
Now make the substitutions Xn - 1 = Xn - 1 + an- 1 - an and an - 1 = an and rearrange to obtain n-2 n -2 s(in- 1 + Un-1 - aj) X s I Xj + X- n- 1 = I s(xj) SC an- 1 aJ) j=l j=l
(11)
- -
Proof of Lemma 8
Taking a derivative with respect to A in (5), using the fact that s is odd, and setting A = - a, B = a and {3 = - a, we obtain
s ' (O)s(2a) = 2s(a)s ' ( a). We recognize this equation as the n - 1 case of (6), which is true by the induction hypothesis. D The proof of Theorem 5 depends on two lemmas. Lemma 7 If a meromorphic function s(x) satisfies (5), then there exists a complex constant b such that exp( - bx)s(x) is an odd function.
Thus, s' (0) i= 0, and because of the invariance with respect to multiplication by constants, we can assume s ' (O) = 1. We have the system
{
s'(O) s(2a)
Since s(O) 0, we know that s is analytic around zero. We can write s(z) = kk=O akzZk + 1 and use the previous con ditions to obtain a recurrence for the sequence an, =
Lemma 8 An odd meromorphicfunction s(x) satisfies (5)
if and only if s(x) = ax
or
for some complex constants a 68
THE MATHEMATICAL INTELLIGENCER
s(x) i=
=
a sin(cx)
0 and c
i=
0.
{
a0 = 1 22n + 1 an = (2n
+
2)
k�=O an-kak.
(14)
A U T H O R S
MARTIN MOHLENKAMP
LUCAS MONZ6N
Department of MathematiCs
Department of Applied Mathematics
Ohio University
University of Colorado Boulder, CO 80309-0526
Athens, OH 45701 USA
USA
e-mail:
[email protected] e-mail:
[email protected] received his Ph.D. in 1 997 from Yale Univer
Lucas Monz6n, following his undergraduate degree from the Uni
sity. He spent a semester at MSRI in Berkeley and several years
versity of Buenos Aires, obtained his Ph.D. from Yale University.
Martin Mohlenkamp
at the U niversity of Colorado
in Boulder before moving to Ohio Uni
Besides computational harmonic analysis, he enjoys theater, po etry, and the visual arts. His partner is Mariana lurcovich, an in
versity. He works mainly in numerical analysis.
ternational consultant in public health.
The value of an for n > 1 is uniquely determined by the value of a1 , which is arbitrary. Setting A = 6a 1 we claim An (15) an = (2n + 1)! · When A 0 we have s(x) = x and when A =!= 0 we have s(x) sin(Ax) and the Lemma follows. We prove the claim by gen eralized induction on the variable n. Thus we assume (15) for 0 ::::; n ::::; N - 1, and show it for n = N. Using (14) with n = N, =
22N+ l aN
=
=
(2N + 2)
N
L
k �O
aN-kak
and thus
+-
N- 1
(
2N + 2
)
k2:.l 2k + 1 ' and the result follows because 2f� o (::�) = 22N+ l . aN = (2N + 1)! 22N l
2(2N + 2)
[ 1 ] Richard Bellman. Adaptive Control Processes: A Guided Tour.
Princeton University Press, Princeton, New Jersey, 1 96 1 . [2] Gregory Beylkin and Martin J. Mohlenkamp. Numerical operator cal
culus in higher dimensions. Proc. Nat!. Acad. Sci. USA , 99(1 6): 1 0246-1 0251 , August 2002. University of Colorado, APPM preprint
August 2001 ;
http://www.pnas.org/cgi/content!abstract/
1 1 2329799v1 .
k�
1
REFERENCES
#476,
N- 1 AN-k = 2(2N + 2)aoaN + (2N + 2) (2(N - k) + 1)! (2k + 1)! ' AN
that the identity (6) is valid for s(x) = x, and so inspiring Theorem 5.
[3] Gaurav Bhatnagar. A short proof of an identity of Sylvester. Int. J. Math. Math. Sci. , 22(2):43 1 -435, 1 999. [4] F. Calogero. Remarkable matrices and trigonometric identities I I . Commun. Appl. Anal. , 3(2) :267-270, 1 999. [5] F. Calogero. Classical many-body problems amenable to exact treatments. Lecture Notes in Physics, monographs m66. Springer
Verlag, 2001 . D
Acknowledgments
Thanks to Gregory Beylkin for leading us to separate sine in the first place. Thanks to Richard Askey for pointing out
[6] S. C. Milne. A q-analog of the Gauss summation theorem for hy
pergeometric series in u (n). Adv. in Math. , 72( 1 ) : 59-- 1 3 1 , 1 988.
[7] Themistocles M . Rassias and Jaromir
S msa.
Finite sums decom
positions in mathematical analysis. Pure and Applied Mathematics.
John Wiley & Sons Ltd. , Chichester, 1 995.
© 2005 Springer Science+ Business Media, Inc., Volume 27. Number 2 , 2005
69
GERALD L. ALEXANDERSON AND LEONARD F. KLOSINS
Mathemati c i ans and O d Books hose interested in book collecting, mathematical books in particular, have observed remarkable phenomenon over the past few years: the prices of rare mathematical m terials have no t been driven downward by the recent recession and the bursting of t. dot-com bubble. While prices for fine art definitely slumped and prices in other are'
of book collecting have declined or held steady, reflecting the state of the economy, prices of books in mathematics and science, particularly at the high end, have risen, even after extraordinary price increases in the 1980s and 1990s. In 1982 we wrote a short article on collecting rare math ematics books [1). Recently we learned that the article was scheduled to go into a mathematical anthology and, when we reread it, we found that it was very out of date, not only the prices but other aspects as well. We decided more could be said on the subject. Here we'll be talking mainly about first editions of mathematical classics, unless otherwise noted. Probably the most extraordinary price increases in mod em times appeared in the famous auction of the Haskell F. Norman collection at Christie's (New York) in 1998. Nor man had been a psychiatrist in San Francisco and collected rare books of high quality in science and medicine. His son, Jeremy Norman, the prominent San Francisco rare book dealer, upon his father's death, put the collection up for auc tion, and Christie's, aware that the collection would attract lots of attention, took some of the best pieces in the sale out on the road to show to collectors in Milan, Paris, Lon don, New York, Chicago, San Jose (!), and Tokyo, among other cities. The sale included many items that had been ac quired some years ago and are now seldom seen on the mar ket. There were unique items like a first edition of Euler's VoUstdndige Anleitung zur Algebra (1770) that Euler had presented to Lagrange, with an inscription-very desirable to a knowledgeable collector. In the 1980s one could find a copy of this book for about $1000; this very special copy in
70
THE MATHEMATICAL INTELLIGENCER © 2005 Springer Science + Business Media. Inc.
the Norman sale went for $19,550. The whole Norman c lection (632 lots) sold for almost $8,000,000. After the Norman sale the conventional wisdom arne collectors and dealers seemed to be that the high pri
o n
6 t a
� r n. 2to n�orb � u l er.
•
·•
��·�����������·�
srbnufr �19
e5 t. '}> c t u6 & t1 r g. 'll:cob. tlf 'll:hff•nfcll• rm
Cn' .l'i•9f·
mo.
Fig. 2. The cover page of Euler's Algebra, with the inscription to Lagrange. The writing is almost certainly that of an assistant since, by the time this was published, Euler was almost totally blind.
could have been a local maximum and resulted from two
cachet, and the best copies, just like great paintings, come
factors: (1) Norman was a well-known collector, and
with a long list of the names of previous, and preferably
(2) Christie's created hype surrounding the sale by arrang
prominent, owners.
ing the world tour of the principal books. Provenance is
The surprise is that prices have held at the high level
important. Coming from a famous collection gives a book
and are actually increasing. If prices were driven up by the
© 2005 Springer Science+ Business Media, Inc., Volume 27. Number 2, 2005
71
Norman sale and by a lot of dot-com millionaires with an interest in mathematics and science, why have the prices held after many of those dot-com fortunes vanished? We don't have an answer. As we write this, the huge and important Macclesfield sale is going on at Sotheby's in London. This extensive li brary in the history of science was housed in Shirbum Cas tle near Oxford, and, according to the catalogue, encapsu lated "almost everything published on the astronomical and mathematical sciences up to about 1750: from Regiomon tanus and Peurbach to Copernicus and his contemporaries, and from there through Tycho Brahe, Giordano Bruno, Gilbert, Galileo and Hevelius to Newton, Leibnitz, Whiston and Euler. . . . There are many books famous in the annals of science, such as the copy of Copernicus's De revolu tionibus . . . which Owen Gingerich saw in situ and on a well-chosen day (as he relates in The Book Nobody Read, he went on a Thursday which was, according to Lady Mac clesfield, 'the only day we have a cook, and we can invite you to lunch') . . . . The genesis of this remarkable collec tion . . . lies in the biographies and interests of two men, . . . John Collins, the mathematician [and friend of Newton] whose collection was acquired by William Jones about twenty-five years after his death, and Jones himself, who bequeathed his library to the second Earl [of Macclesfield]."
The first Macclesfield sale of mathematics books took place June 10, 2004, and covered only the books by authors with names beginning with A-C. (A second sale in the se ries was to take place November 4, 2004.) The first 251 lots sold in June brought in almost $6.5 million, and that's for only three letters of the alphabet! Prices in the sale so far indicate that they are not only holding their own but in creasing. The Copernicus of 1543 went for $1.2 million. It was a nice copy, heavily annotated by the astronomer John Greaves. Still, for most people that's a lot of money to spend on a book (In 1964 Ernst Weil in London listed the 1543 Copernicus in his catalogue for $2,380. In the Honeyman sale of 1978, there were two copies-one sold for $35,000, the other for $ 19,000.) This book and the great works of Galileo, the first edition of Newton's Principia of 1687, the 1482 Ratdolt Euclid (the first printed Euclid, produced roughly 40 years after Gutenberg's invention of moveable type), and other such rarities now go for very high prices; few copies remain in private hands. As with any other items of interest to collectors, prices fluctuate over time, moving upward with overall inflation, and both upward and downward depending on shifting fashion. Fermat's 1670 edition of Bachet's Diophantus (this edition was supervised by Fermat's son and published shortly after Fermat's death) contains the printed version
O B S E R V A T I O D O M I N I P E T R I D E F E R M A T. J'/,.,.
••l<m ;, "••s ••h•s , •111 f"•Jr•t•fuJr•u• "' ""'' '!••Jr4111J11Urlfl•l s•••r•lltiT """•• "' ;,,,;,.,. .,,,. , •• J,.,,.,,. p•�
72
THE MATHEMATICAL INTELLIGENCER
of what Fermat supposedly wrote in the margin about hav ing a proof but not having enough room in the margin to write it down. Not long after Andrew Wiles's 1995 proof of "Fermat's Last Theorem," a copy of the Diophantus ap peared in a London dealer's catalogue for $9,075. Then in the Norman sale in 1998 it sold for $25,300 (after an esti mate of $2,000-$3,000 by the auction house!), and later a New York dealer had a copy listed at $45,000. One of the authors of this article bought a copy in San Francisco in 1965 for $484. At a June 2004 auction in Paris a copy sold for $31 ,020. The increase is far beyond the rate of inflation. But, of course, not every book enjoys the stimulus of a widely publicized proof after 350 years, one that made "Nova" and the front page of the New York Times. Perhaps another book belongs in the category above: Gauss's Disquisitiones Arithmeticae of 1801, a later book than the Euclid and the Newton but one of considerable mathematical interest and rarity. One of us bought a copy in 1967 from the dean of West Coast book dealers, Warren Howell in San Francisco, for $ 1,350. In 2003 a special copy that had been owned by C. J. Doppler was listed by the London dealer, Bernard Quaritch, for $60,800, but a plain, ordinary copy sold at a recent Christie's sale in Paris for $49,350. It is perhaps not surprising that this, probably the greatest of Gauss's works, should have this amazing in crease in selling price, but a copy of his Theoria motus cor porum coelestium, a less well-known book, was purchased from Howell for $190 in 1963 and a London dealer listed a copy in 2001 for $6,800. So even lesser works by great math ematicians have shown a significant increase. What can explain prices like this? After all, the text of many of these works is available in modem editions or on the Internet. The splendid 1847 edition of Oliver Byrne's Euclid in brilliant colors printed by woodblock and still available on the market for something like $7,500 (one of us owns a copy bought in a London bookshop in 1960 for less than $ 13) can now be seen on the Internet at http://www.sunsite.ubc.ca/DigitalMathArchive/Euclidlbym e.html. It's a beautiful book either way, but looking at pages on the screen is not quite the same experience as holding a copy in your hand and flipping through the pages. Byrne, incidentally, seems to have had time on his hands; he is identified on the title page as "Surveyor of Her Majesty's Settlements in the Falkland Islands." Here are a few examples of the increases over the years for some important books in mathematics: •
•
•
•
the editio princeps of the works of Archimedes of 1544: Honeyman sale of 1978, $ 1,280; W. P. Watson in London, $ 1 12,000 in 2002; Jacques Bernoulli's Ars Conjectandi: Honeyman sale, $900 in 1978; Norman sale, $ 12,650 in 1998; W. P. Wat son, $21,000 in 2004; Jean Bernoulli's Opera Omnia: Quaritch, $75 in 1962; the same dealer, $7,200 in 1997; Euler's Introductio in Analysin Injinitorum: Zeitlin & Ver Brugge in Los Angeles, $75 in 1961; Jonathan Hill in New York, $ 13,500 in 2000;
•
•
•
Galileo's Dialogo of 1632: E. Weil, $294 in 1958; Zeitlin & Ver Brugge, $5,500 in 1978; Norman sale, $27,600 in 1998; Christie's in Paris, $49,761 in 2004; Lagrange's Mechanique Analitique: Joseph Rubinstein in Berkeley, $250 in 1973; Norman sale, $ 13,800 in 1998; Watson, $20,000 in 2002; Newton's 1687 Principia: E. Weil, $490 in 1948; Zeitlin & Ver Brugge, $15,000 in 1978; Norman sale, $321,500 in 1998; W. P. Watson, $356,000 in 2004.
Even the third edition of the Newton ( 1 726), of interest because it was the last edition published in Newton's life time, was recently listed at $35,000 by a Los Angeles dealer. One of us paid $236 for a copy in 1969. Why does a person want to collect old and rare mathe matics books? Why indeed? They are often in Latin, a language not understood by many these days, and they are often filled with notation unfamiliar to modem mathemati cians. But there is a certain thrill for someone who loves mathematics to pull a volume off the shelf and see, for ex ample, the first appearance in print of Euler's formula for polyhedra in "Elementa doctrinre solidorum," in the Novi Commentarii Academiae Scientiarum Imperialis Petro politanae (1758), 109-60, or the first appearance of the dif ferential calculus as most of us know it, Leibniz's "Nova methodus pro maximis et minimis," in the Acta Eruditorum of Leipzig for 1684. Sometimes one gets lucky and comes across something totally unexpected, like a short manuscript we bought from a New York dealer that was a report signed by Laplace and Legendre and sent to the Academie des Sci ences, Paris, saying in French, but here translated: "We have examined at the request of the Academy the description of a vessel capable of traveling under water at a determined depth for whatever amount of time, and diving to any depth for a certain amount of time. The means that the author pro posed did not seem to us to be practicable or worthy of the attention of the Academy." So much for the idea of a sub marine. On another occasion, a letter turned up, signed by Gauss. There was nothing so unusual about that except that it was in English. At first glance that might appear suspi cious, but perhaps less so when one realizes that many of Gauss's descendants ended up in the United States, some re putedly because they couldn't get along with him. Potential collectors should be concerned about the pos sibility of buying a fake. Obviously this is a problem in the art world, where skilled and successful forgers often have fooled even the museum experts. Autographs and manu scripts are also susceptible to forgery. Books have been rel atively safe, at least until the prices went so high, because the amount of work involved in typesetting with antique typefaces and printing on paper of the correct period, would be prohibitively time-consuming and expensive. Still, a collector has to be careful. In 1976 one of us bought a copy of Gauss's doctoral dissertation, published in Helm stadt in 1799 and containing an important result, the first generally accepted proof of the Fundamental Theorem of Algebra. The book was purchased for $ 1,000 from a dealer of impeccable reputation in San Francisco, acquired from
© 2005 Springer Sc1ence+ Business Media, Inc., Volume 27, Number 2, 2005
73
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