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.
On the Classical-to-Quantum Correspondence
Although the article by Effros [1] is specifically targeted for young (naive?) mathematicians, this old mathemati cal/theoretical physicist* was surprised to see no mention or reference to Dirac's co-discovery and subsequent refined development [2] of the Poisson bracket-to-commutator-bracket corre spondence for quantization. Indeed Born, who is credited by Effros as the sole originator of the quantization cor respondence, states that [3] These commutation laws (Born and Jordan, 1925) take the place of the quantum conditions in Bohr's the ory. . . . It may be mentioned in con clusion that this fundamental idea underlying Heisenberg's work has been worked out by Dirac (1925) in a very original way.
dence, at least for those who share a predilection for algebraic aesthetics. [ 1 ] E. G. Effros, Matrix revolutions: an introduction to quantum variables for young mathematicians, Math ematical Intelligencer 26 (2004), 53-59. [2] P. A. M. Dirac, Quantum Mechan ics, Clarendon Press ( 1930), par ticularly chapter 4. [3] M. Born, Atomic Physics, Hafner Publishing Co. ( 1957), p. 130. [4] G. Rosen, Formulations of Quan tum and Classical Dynamical Theory, Academic Press (1964). Gerald Rosen 415 Charles Lane Wynnewood, PA 19096 U.S.A. Department of Physics Drexel University Philadelphia, PA 19104 e-mail: www.geraldrosen.com
Young mathematicians might also enjoy the fact exploited in the 1960s (see for example [4]) that the Poisson and commutator brackets are both Lie product binary operations with the properties: [A + B, C)
=
[A, C) + [B, C) (linearity)
[A, B] = - [B, A]
(antisymmetry)
[ [A, B], C] + [ [B, C], A] + [ [C, A] , B] = 0 (integrability) In addition, both the Poisson and com mutator brackets have the property [AB, C]
=
A[B, C]
+
[A, C]B
with the direct product of the algebraic elements defined appropriately in either case. Hence, classical mechanics fea tures the Lie product according to Pois son, while quantum mechanics has the Lie product represented by a commuta tor of linear operators. This takes some of the mystery out of the correspon-
'A
mathematical physicist
THE MATHEMATICAL INTELLIGENCER © 2005 Spnnger Sc1ence+Bus1ness Media. Inc
T
here is an interesting postscript to my paper in The Mathematical In telligencer 21 (1999), no. 2, 50, on the birthplace of Felix Klein (1849-1925). Readers may remember I said the "Heinrich Heine University" could have been appropriately named the "Felix Klein University," because I consider Klein at least as important to the progress of science as Heine to litera ture. I also pointed out that tourist guides in Dusseldorf never mention the slightest fact about Felix Klein. I continue to visit the Computing Center of the Heinrich Heine Univer sity on a yearly basis, to collaborate with Professor Dr Jan von Knop, di rector of the Centre (our joint work on computational biology and chemistry is now thirty years old). On one of my recent visits to the Heinrich Heine Uni-
is one who doesn't have the skills to do
one who doesn't have the skills to do
4
More on Felix Klein in Dusseldorf
real
experiments.
real
mathematics; a
theoretical physicist
is
versity, I noticed that the outer wall of the main lecture hall shows a huge commemorative inscription titled "Fe lix Klein Horsaal." It is located in the building that houses the Departments of Mathematics and Physics, as it be longs to the Mathematisch-Naturwis senschaftliche Fakultat. I can't claim that my article inspired the authorities of the Fakultat to dedi cate their main lecture hall to Felix Klein, but it may be significant that
many requests for reprints of my arti cle in The lntelligencer came from the Heinrich Heine University. Anyway, it is a pleasure to report that now at least one place in Dussel dorf bears the name of Felix Klein.
Read Something Different
Nenad Trinajstic Rudjer Boskovic Institute HR- 1 0002 Zagreb Croatia e-mail:
[email protected] Women in Mathematics Bettye Anne Case and Anne M. Leggett, editors
Marjorie Senechal is known to many of our readers as a
This eye-opening
mathematician specializing in aperiodic tilings, as a math
the stories of dozens of women
professor at Smith College, and as a contributor and col
who hove pursued careers in
umn editor for The
Mathematical Intelligencer.
Did you
know that she also was for years the head of Smith Col lege's Kahn Liberal Arts Institute? Liberal arts! What do the liberal arts have to do with us? Well-humanism fits into her intellectual universe perfectly comfortably-as it does, I think, into The
Intelligencer's.
Marjorie has been in charge of our multi-faceted Com
book. presents
mathematics, often with inspiring tenacity. The contributors offer
their own narratives, recount the experiences of women who come before them, and offer guidonce for those who
paths.
career
will follow in their
"This astounding book
munities column since vol. 19. She now becomes co-Editor in-Chief. Yes, send your manuscripts to either Marjorie or
provides a wealth of important information on
me. The Communities column will continue; the rest of our
women in mathematics
features will continue, only more so. Anything we could do
•..
exploring how they entered the
with Davis as Editor we can do at least as well in the new
field, what excited them about it
system. If Marjorie's participation spurs you to submit such
in their youth, what excites them
elegant manuscripts as to make my past editing awfully pale
now, and the many ways these
by comparison-so be it! Go right ahead and submit thos gems, my feelings won't be hurt!
women have advanced the frontiers of mathematics, or have used mathematics to the benefit of
Chandler Davis
society
..•.
How wonderful that this
is all gathered in one volume of easy reading."
-Mildred Dresselhous, MIT Clolh $35.95 £22.95
Celebrating 100 Years of "Excellence PRINCETON University Press (0800) 243407 800-777-4726
.K.
math.pupress.princeton.edu
© 2005 Spnnger Sc1ence 1-Bus1ness Med1a, Inc., Volume 27, Number 3, 2005
5
MARKUS BREDE
On the Convergence of the Sequence Defining Euler's Number
lthough the famous and well-known sequence ing Euler's number e
=
2. 71828
{ev}vEN :={(1
+ _!_YlvEN v
defin-
is perhaps the most important nontrivial se-
quence, less or even nothing seems to be known about the general structure of its Taylor expansion at infinity. Of course, using a modern computer algebra system like Maple or Mathematica, it is no problem to de termine the first, say, five terms of that expansion: Using the Taylor expansion of log(1
+ x) at x =0, valid for lxl < + l_) v
1, we immediately get the Taylor expansion of vlog(l at v
=oo. Taking the exponential gives ev
( + �r
To prove the above result, I will use a few lemmata. They
are in terms of the following Definition
:= 1
---
2447e 7e l le e = e - - + -- - -- + +··· 2v
S1 stands for the Stirling numbers of the first kind. In particular, this shows that the numbers en all are ratio nal multiples of Euler's number e.
where
24v2
16v3
5760v4
lfRex > -1, lxl i= 0, and ltl ::s 1, let EtCx) :=exp(log(1 + tx)lx), and let Et(O) : =e1• The branch of the logarithm is chosen by log(1) 0. =
Then we have, evidently: But can we determine what is hidden behind those three dots? i.e., what is" . . . "?
I will deduce a closed and finite expression for the coef
ficients
en in the above expansion, which,
at the same time,
implies the following sequence of asymptotic statements: (v�
oo,
n0 E N).
The derived result seems to be new; [Todorov], [Broth
ers, Knox], and [Knox, Brothers] examine related problems,
but they do not determine the general structure of en, which will tum out as finite sums of some multiples of Stirling numbers: I will prove the remarkably simple formula
� en - e L _
v �O
6
S1(n + v, v) �v L (n + V) ' m� O •
(-1)m m.' '
THE MATHEMATICAL INTELLIGENCER © 2005 Springer Scrence +Busrness Medra, Inc
Lemma 1
i. For each It I ::s 1, Et(X) is holomorphic inRex> -1. For real xi= 0 and real t E [ -1, 1] we have Et(x) = (1 + tx}i. Et(x) possesses a Taylor expansion for lxl < 1: Et(X) =: k��o en(t) xn. ii. The elementsev : = (1 + -lv_)vof the sequence(ev}vEN admit an asymptotic expansion at v = oo; the coefficients are the numbers en :=en(1) defined in (i): iiv=El
( _!_V )
I
=
n �o
en(1) Vn
=
i
n �O
en Vn .
By Lemma 1, (ii), the determination of the numbers
en is
reduced to the computation of the coefficients of the Tay lor expansion of
with the aid of
E1(x) at x =0. Their evaluation succeeds
The fact that Pnis a polynomial of degree n in t now yields two different consequences: the irrelevant but interesting result
Lemma 2:
i. For Re
x > -1 and l ti :::; 1 we
have
%
I
I (_1)"
t"Pn(t) x", Et(X)=e 1 n�O ·
,=o
where Pn denotes a polynomial of degree n in ii. For Re
x > -1 and It.
:::;
1
t.
The series appearing in parentheses converges. Here S1 denotes the Stirling numbers of the first kind, defined by their generating function (x)., :=x(x- 1) ... (x-(v- 1)) =: ���oS1(v, n)x"; see, for instance, [Abra mowitz, Stegun], Section 24. 1. 3. B, Formula 1.
,
Proof i. Under the given assumptions we have
e
(1 + tx) E,(x)=exp og x
) expl:H =
�
�
·)J
(-1)'"
(I
form > n,
Sl (n,+v, v) + v).(m- v).I
)
tm. m�o By this and by Lemma 2, (i), we are able to give an explicit expression for the above Taylor expansion of E1(x) at x=0; in particular, for t = 1 we have E1(x)= e �;,�oPn(1)x". Using Lemma 1, (ii), we finally see that the numbers e11 are the coefficients in this expansion: _
e, - e Pn0)-e =
C i + C ?:;: . .
tx-
and the explicit expression for Pn:
p, (t) = I
we moreover have
S1(n +v, v) =0 (n +v)!(m- v)!
r
,�o
� ( t:o
_
(-1)"
m 1)
v v)
eI ( _1)" S1(n + , (n +v)! FO
(n
�( �o
I
rn=v
.,
S1(n +v, v) - v)! (-1)"' (m - v)! .
_
1)
(n + v )!(m
This gives the assertion.
Remark:
The definition of the numbers S1 implies that the sign of S1(v, n) is (-1y-n. This together with the theorem yields that the sign of en is (-1)".
REFERENCES
=e'
[ - t2 (t33 8t4)' - (t44 6t" t(i)
ii.
2x +
1
·
+
:x-2
+
+ 48
[Abramowitz, Stegun] : M. Abramowitz, lA Stegun, Handbook of Math 8
.r +
.
. .l'
J
ematical Functions, Dover Publications, New York, 1964 . [Brothers, Knox]: H.J. Brothers, JA Knox, New closed-form approxi mations to the logarithmic constant e, The Mathematical lntelligencer
because of absolute convergence. This shows the as sertion.
[Hansen]: E.R . Hansen, A Table of Series and Products, Prentice-Hall,
On the other hand, with the aid of the binomial for mula we easily see that
[Knox, Brothers] : JA Knox, H.J. Brothers, Novel series-based approxi
1
tJ )
E1(x)=.
S1
t (�r ��>
� (tx)" = o .
tv
'X;
I
=
.,�o
��
n)
20(4), 25-29, 1998 . Englewood Cliffs, N.J., 1975. mations toe, The College Mathematics Journal 30(4), 269-275, 1999.
-
[fodorov]: P.G. Todorov, Taylor expansions of analytic functions related to (1 + z)X
JJ
1 IoS1(v, v- n) x" . v. n�
Using the absolute convergence of this series, we may
1, Journal of Mathematical Analysis and Applications
132, 264-280, 1988. AUTHOR
change the order of sununation and get the assertion.
Now I can state the main result: Theorem:
For all n let
e71
:= �ih�o (-1)"'/m!.
Then we have
Proof
�
_
en- e L v�o
First, Lemma
S1(n +v, v) Bn-p· (n +v)!
2 implies the simple conclusion
et .
t"Pn(t)=
I
tJ-:-=fl
Sl(v,
�- n)
v.
MARKUS BREDE
Fachbereich 17- Mathematikllnfonnatik
t",
UniversHat Kassel 34127 Kassel
that is, Pn(t)=I m�o
m (-1)
or, finally, Pn(t)=
I
m=O
(-1)m
m!
(I
v�O
Germany
S (n + v, v) v t , tm . I l (n + v)! p= () ( -1 )"
Sl(n,+v,v) v).(rn - v).,
(n +
Markus Brede studied mathematics and physics at Kassel, re
ceiving a doctorate in mathematics i n 2001 . He is now work
)
ing toward his habilitation. His interests are in analytic num
m t .
ber theory, function theory, and special functions.
© :?005 Spnnger SCience+- Bus1ness Media, Inc , Volume 27, Number 3, 2005
7
M?Ffiijl§:.;ih¥11=tfiJ§4£ii,'l,i§,i'l
The Rotor· Router Shape Is Spherical Lionel Levine and Yuval Peres
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.
Michael Kleber and Ravi Vakil, Editors
n the two-dimensional rotor-router walk (defined by Jim Propp and pre sented beautifully in [4]), the first time a particle leaves a site x it departs east; the next time this or another particle leaves x it departs south; the next de parture is west, then north, then east again, etc. More generally, in any di mension d :=:::: 1, for each site x E ll_d fix a cyclic ordering of its 2d neighbors, and require successive departures from x to follow this ordering. In rotor router aggregation, we start with n par ticles at the origin; each particle in tum performs rotor-router walk until it reaches an unoccupied site. Let An de note the shape obtained from rotor router aggregation of n particles in ll_d; for example, in 7L2 with the ordering of directions as above, the sequence will begin A1 = {0}, Az = {0,(1,0)}, A3 = (0,(1,0),(0,-1)}, etc. As noted in [4], simulations in two dimensions indi cated that An is close to a ball, but there was no theorem explaining this phe nomenon. Order the points in the lattice ll_d ac cording to increasing distance from the origin, and let En consist of the first n points in this ordering; we call En the lattice ball of cardinality n. In this note we outline a proof that for all d, the ro tor-router shape An in ll_d is indeed close to a ball, in the sense that
I
the number of points in the symmetric difference AnD.Bn is o(n) .
Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil, Stanford University,
Department of Mathematics, Bldg. 380, Stanford, CA 94305-2125, USA e-mail:
[email protected] (1)
See [6] for a complete proof, and error bounds. Let E C [Rd denote a ball of unit volume centered at the origin, and let A� C [Rd be the union of unit cubes centered at the points of An; then (1) means that the volume of the symmet ric difference n-lldA�D.B tends to zero as n � oo. A novel feature of our argu ment is the use of random walk and Brownian motion to analyze a deter ministic cellular automation. A stochastic analogue of the rotor router walk, called internal diffusion limited aggregation (IDLA), was in-
troduced earlier by Diaconis and Ful ton [3].In IDLA one also starts with n particles at the origin 0, and each par ticle in tum walks until it reaches an unoccupied site; however, the particles perform simple random walk instead of rotor-router walks. Lawler, Bram son, and Griffeath [5] showed that the asymptotic shape of IDLA is a ball.Our result does not rely on theirs, but we do use a modification of IDLA in our analysis. Since the lattice ball En minimizes the quadratic weight Q(A) = IxEAir:ll2 among all sets A c ll_d of cardinality n, the difference Q(An) - Q(En) can be seen as a measurement of how far the set An is from a ball. We claim that Q(An ) ;S Q(En), (where an
anlbn :S 1).
:S
bn
(2)
means that lim sup
It is easy to prove that this implies (1). To bound Q(An), we use a property of the function lr:ll2: its value at a point x is one less than its average value on the 2d neighbors of x.For a set A c ll_d and a point x E ll_d, let 1f;(x,A) be the expected time for random walk started at x to reach the complement of A. If x Et: A, then 1f;(x,A) = 0, whereas if x E A, then 1f;(x,A) is one more than the average value of1f;(y,A) over the 2d neighbors y of x. This implies that h(x) = llxll2 + 'g(x,A) is harmonic in A: its value at x E A equals its average on the neighbors of x. Consider rotor-router aggregation starting with n particles at 0, and recall that An is the set of sites occupied by the particles when they have all stopped. Given a configuration of n par ticles at (not necessarily distinct) loca tions X1, . . . Xn, define the harmonic weight of the configuration to be W = W(x1, . .., Xn) =
n
L Cllxkll2 + 1f;(xk,An)).
k�l
We track the evolution of W during ro tor-router aggregation. Initially, W = W(O, . . ,o) n1f;(O,An). Because .
=
© 2005 Spnnger Science+Bus1ness Med1a, Inc., Volume 27, Number 3, 2005
9
the location of p after j steps, then the expectation of IJSU + 1)112 given S(j) equals IJSU)II2 + 1.Therefore
18(0,Bk)
Figure 1. Rotor-router (left) and lOLA shapes of 10,000 particles. Each site is colored ac cording to the direction in which the last particle left it.
every 2d consecutive visits to a site x result in one particle stepping to each of the neighbors of x, by hannonicity, the net change in W resulting from these 2d steps is zero.Thus the final hannonic weight determined by the n particles, Q(An) �xEAn �(X,An), equals the ini tial weight n�(O,An), plus a small error that occurs because the number of vis its to any given site may not be an exact multiple of 2d.It is not hard to bound this error (see [6)) and deduce that
standard d-dimensional Brownian mo tion started at the origin. (Their proof uses the spherical symmetry of the Gaussian transition density and the powerful Brascamp-Lieb-Luttinger [2] rearrangement inequality. ) Since ran dom walk paths are well-approximated by Brownian paths, the Brownian mo tion result from [1] can be used to prove that for any k-point set A c ll_d, the expected exit time �(O,A) for ran dom walk is at most �(O, Bk) plus a Q(An) = n�(O, An) - �xEAn �(X,An), small error term; details may be found where a-, = bn means that lim an/bn = 1. in [6]. The number of steps taken by The key step in our argument in the particle Pk +1 in our modified IDLA volves the following modified IDLA: is at most the time for random walk Beginning with n particles {pk}�= 1 at started at 0 to exit the set occupied by the origin, let each particle Pk in tum the stopped particles p 1 , . . , Pk· It fol perform simple random walk until it ei lows that ther exits An or reaches a site different n from those occupied by P1, . .. , Pk-1· IE(Tn) !S I �(O, Bk)· (4) k=l At the random time Tn, when all the n The final step in our argument is to particles have stopped, the particles that did not exit An occupy distinct show that �r=l Cf,(O,Bk) is approxi sites in An.If we let these particles con mately equal to Q(Bn). Fix k :s n and tinue walking, the expected number let a single particle p perform random of steps needed for all of them to exit walk starting at 0 and stopping at the An is at most �x E An �(X,An). Thus first time tk that p exits Bk. If S(j) is n18(0,An) ::; IE(Tn) + �X E An �(X,An). So far, we have explained why
+
.
Q(An)
=
n18(0,An)
IE (tk)
=
1ECIJSC tk)l l2).
(5)
(Formally, this follows from the Op tional Stopping Theorem for Martin gales.) Let V1, Vz, ...be an ordering of ll_d in increasing distance from the origin, and recall that Bk = {v1, . .. , vk). Since all points on the boundary of Bk are about the same distance from the ori gin, IEC I IS C tk)l l2) = llvkll2. Summing this over k ::; n and using (5) gives
n
n
I �(O, Bk) = kI= llvkll2 Q(Bn). k =l
l
=
Together with (3) and (4), this yields Q(An) !S Q(Bn), as claimed. D Concluding Remark
discovered by Jim Propp, simula tions in two dimensions indicate that the shape generated by the rotor-router walk is significantly rounder than that of IDLA. One quantitative way of mea suring roundness is to compare inra dius and outradius. The inradius of a region A is the minimum distance from the origin to a point not in A; the out radius is the maximum distance from the origin to a point in A. In our simu lation up to a million particles, the dif ference between the inradius and out radius of the IDLA shape rose as high as 15.2.By contrast, the largest devia tion between inradius and outradius for the rotor-router shape up to a mil lion particles was just 1.74. Not only is this much rounder than the IDLA
As
- L 18(x,An) XEAn
:S
IE(Tn). (3 )
To estimate IE(Tn), we want to bound, for each k < n, the expected number of steps made by the particle Pk+1 in the random process above: for this, we use a general upper bound on expected exit times from k-point sets in ll_d_ In 1982,Aizenman and Simon [1] showed that among all regions in jRd of a fixed volume,a ball centered at the origin maximizes the expected exit time for
10
=
THE MATHEMATICAL INTELLIGENCEA
Figure 2. Segments of the boundaries of rotor-router (top) and lOLA shapes formed from one million particles. The rotor-router shape has a smoother boundary.
shape, it's about
as close to a perfect as a set of lattice points can get!
[2] H. J. Brascamp, E. H. Lieb, and J. M. Lut
Because of error terms incurred along
for multiple integrals, J. Functional Analysis
circle
tinger, A general rearrangement inequality
the way, our argument in this note only shows that the rotor-router shape roughly spherical.
is
17 (1994), 227-237.
http://www.arxiv.org/abs/math. PR/0503251. Department of Mathematics
[3] P Diaconis and W. Fulton, A growth model,
It remains a challenge
toties for the rotor-router model in :zd,
a game, an algebra, Lagrange inversion,
University of California Berkeley, CA 94720
to explain the almost perfectly spherical
and characteristic classes, Rend. Sem.
shapes encountered in simulations.
USA
Mat. Univ. Pol. Torino 49 (1991), 95-119 .
e-rnail:
[email protected] [4] M. Kleber, Goldbug variations, Math. lntel ligencer 27 (2005), no. 1, 55-63 .
REFERENCES
[1] M. Aizenman and B. Simon, Brownian mo
Department of Statistics
[5] G . F . Lawler, M . Bramson, and D. Griffeath,
tion and Harnack inequality for Schrodinger
Internal diffusion limited aggregation, Ann.
operators, Comm. Pure Appl. Math. 35
Probab. 20 (1992), 2117-2140.
Berkeley, CA 94720 USA
[6] L. Levine and Y. Peres, Spherical asyrnp-
(1982) , no. 2, 209-273.
University of California
e-mail:
[email protected] c/Qf.b.gp.Sci e nt ific WorkPlace· Mathematical Word Processing
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I
"-sUs -
A8 K, Df = f', f E H.
To calculate the Hilbert space adjoint D* : K --'> H of D, sup pose g E K. Then (DJ,g)K = ( f',g)K =
(if). (�) if) (�)
= (
where 1T
(:)
.P
if) (�) (�) .
)K2
)K2 = ( f,1TP
)H,
)K2 = (P
= r, (r, s E K). Hence, D*g = 1rP
(�}
f E H,
By contrast, if D is considered to be a closed densely defined linear transformation on K (with domain precisely those elements of K which are also in H), then
(3)
D1g
=
-
Dg, for all g E H with g(O) = 0
=
g(l ) ,
the conventional adjoint of the derivative operator (again having a dense, non-closed domain). It's (loosely speak ing) the same transformation: the derivative. Yet having two different norms on the domain makes it have two dif ferent adjoints. Being clear about such occurrences is helpful in dealing with Sobolev gradients (to be intro duced shortly). Let's do this more generally. Assume that each of L and K is a Hilbert space, and T is a closed, densely defined lin ear transformation from L to K. The "graph" of T is
Gr =
{( ;x) :
X
}C
E the domain of T
L X K.
" T is closed" means that Gr is a closed subset of L X K, the Hilbert space which is the Cartesian product of L and K. (Anyone who thinks functions are sets of ordered pairs will want to say that Gr is T, but the terminology "graph of T" is very common.) Anyway, the domain of T is made into a Hilbert space H by defining the norm
l lxl lk = CllxiiL + II Txiiid � , x E D(T), the domain of T.
( (I + rtn- 1 T(I + TtT) - 1
)
rtu + rrtr 1 I - (I + TTt) - 1 ·
u E H.
So long as, for u E H, F'(u) has range dense in K, it fol lows that any critical point of cp is a zero of F, for
cp'(u)h = \F'(u)h, F(u))x,
u, h E H.
(5) is called a least-squares formulation of the problem of finding u E H so that F(u) = 0. Thus, PDE solving is, in a very large sense, a matter of critical-point-finding. In this note, "critical-point-find ing" and "variational method" are two ways of indicating the same things. Here are some results on critical points. Assume cp is bounded from below and V cp is locally lipschitzian. If x E H, there is a unique function z : [O,oo) � H such that
Theorem 3.
z(O)
=
x, z'(t)
=
- (Vcp)(z(t)),
t ::::: 0.
Moreover,
r IICV ¢)(z)ll2 0
< 00•
If u is an w-limit point of a function z as in (6), then (Vcp)(u) 0 . Definition. The statement that a C1 function cp : H � R Theorem 4.
=
T1 denotes the adjoint of T as a closed, densely defined lin ear transformation on K into H. The reader may check that (4) is idempotent, symmetric, fixes each point of Gr and has range in Gr too. Thus (4) is convicted of being the claimed orthogonal projection. It is an exercise to use this formula to get a workable expression for the orthogonal projection P in the example in which K £2([0, 1]), H H1•2([0, 1]). To get more gen eral Sobolev spaces which are also Hilbert spaces, take K L2 (fl) for some region in a Euclidean space, H to be a lin ear subspace of members of K which have a certain num ber of appropriate partial derivatives; take Tfto be a list of these partial derivatives of f Then take llfi iH to be the "graph norm" off in the same manner as above. A gradient is called a Sobolev gradient if it is taken with respect to a Sobolev inner product. These orthogonal projections are fundamental to the construction of Sobolev gradients in both function space and corresponding finite-dimensional approximations. The self-dual nature of such spaces is systematically used de spite the emotional attachment of many to the idea that such self-duality is useless (or worse) in the study of dif ferential equations. A novella could be written on this topic. =
cp(u) = l iFCu)ll 2/2,
(5)
(6)
Thus D(T) with the norm II · IIH is isometric to Gr with the "graph norm" it has as a subspace of L X K. A formula of von Neumann, [19],[15] shows that the or thogonal projection of L X K onto Gr is (4)
Some Zero-Finding in Hilbert Space
The problem of solving a system of PDEs can often be re cast as the problem of finding a critical point of an appro priate real-valued function cp on some Hilbert space H. Many important systems arise naturally in this way. Con sider a system which doesn't arise this way. Express the system as the search for a zero of a function F : H � K, where K is a second Hilbert space, and assume F is C1 . Define
=
=
satisfies a gradient inequality (tojasiewicz inequality) on a region fl C H means that there are c > 0, (} E (0, 1) such that
(7) Theorem 5. Under the above conditions on ¢, if (7) holds and z satisfies
z'(t)
(8)
=
- (V cp)(z(t)), z(t) E fl,
t ::::: 0,
then u lim1 __, "' z(t) exists and (Vcp) (u) 0. In [7) it is shown that (7) holds in finite-dimensional cases in a neighborhood of a zero of cp provided cp is analytic. A =
=
direct generalization to infmite dimensions does not hold (take T E L(H, K) to be compact and self adjoint (H, K infi nite-dimensional Hilbert spaces), and define cp(x) = 11Txllkf2, x E H). In [16), this inequality was extended to some infi nite-dimensional cases to study asymptotic limits of solu tions to time-dependent PDE (see also [2),[5]). Theorem 8 gives a sufficient condition for a gradient inequality to hold. Theorem 6.
Assume that G is a C1 function on H and cp(x)
=
llx iiA-12 + G(x),
x E H.
© 2005 Spnnger Sc1ence+ Bus1ness Med1a, Inc , Volume 27, Number 3, 2005
49
Assume also that cf> is coercive (cf>(x) � oo as llx i!H � ooJ , and that V(G) is compact (if{xkll:= 1 is a bounded sequence in H, then { V G(xk))k= l has a convergent subsequence) and locally lipschitzian. If z : [O,oo) � H satisfies (6), then z has an w-limit point, and each such point is a zero of Vcf>. For the next theorem, suppose that K is a second Hilbert space, F : H � K is C1 ,
(9) and
x E H,
cf>(x) = HF(x) llk/2,
cf> has a locally lipschitzian derivative. Note that (Vcf>)(x)
( 1 0)
where for x E H, joint of F'(x). Theorem 7.
=
F'(x)* : K � H is the Hilbert-space ad
In addition to (9), assume that
I ICV c/>)(v) I !H 2: ciiF(v) iix,
( 1 1)
x E H,
F'(x)*F(x),
(i.e. , cf> satisfies (7) with
(}
=
t ) . If
V
E Br (x)
II F(x) llx s: rc, then there is u E Br(O) so that F(u) = 0.
(12)
In the case of Theorem 7, a sufficient condition for the gradient inequality (7) to hold on n is the following from [ 1 1 ]:
Assume that 0 c H, F : 0 � K is C1, and there are given M, b > 0 such that if g E K and x E !1, then there is h E H with llhi !H s: M and Theorem 8.
!
!
IICVcf>)(u) ll 2: ccf>(x) ' ,
x E O.
Thus a gradient inequality is implied by sufficiently good uniform approximation to members g of K by elements F'(u)h. The finding of solutions or approximate solutions h to linear equations of the form
F'(u)h = g, where u, g are given, is central to the main point of [9]. I am leading up to a recent result which captures much of the spirit of [9], and whose proof is very close to one for a slightly different result in [ 13]. But first some back ground. In addition to more conventional metrics, a gradient can be taken with respect to a Riemannian metric. Assume F is a C1 function from H to H, and cf>(u) = IIFCu)i ii£, u E H. F induces a Riemannian metric on H by means of, given
u E H, ( 13)
(x,y)u = (F'(u)x, F'(u)y)H,
X,
y E H.
Assuming F' (u) is bounded below for all u E H, a gradient can, given u E H, be defined as gu such that
50
THE MATHEMATICAL INTELLIGENCER
h E H.
cf>'(u)h = (F'(u)h, F(u) )H, Thus
(F'(u)h, F(u) )H = (F'(u)h, F'(u)gu)H,
h E H,
with gu as in (13), and so F' (u)gu = F(u ). This suggests that the Newton vector field for F belongs to the same family of gradients to which Sobolev gradients belong. With this motivation here are two more results to add to the above list. The first might be compared with Theo rem 7, and is a zero-finding result; the second is a version of the Nash-Moser inverse function theorem. See [ 13] for arguments. For these two results, assume that each of H, J, K is a Banach space. Assume also that H is compactly embedded in J, in the sense that if x 1 , x2 , . . . is a sequence in H whose terms are uniformly bounded in norm by M, then this sequence has a subsequence convergent in the J topology to an element of H which has H norm not ex ceeding M. For x E H and r > 0, Br,sCx) and br,H denote, respectively, the closed and open balls in H with center x and radius r. Theorem 9. Given x0 E H and r > 0, assume that F : Br,H(x0) � K is continuous in the J topology, and that if u E br,sCXo), then there is h E Br, sCO) such that
l. F(x)) tlim --->0 + t (F(x + th) -
=
-F(x0).
Then there is u E Br,sCxo) such that F(u) = 0. Closer to Moser's main result in [9] is the following in verse function theorem:
(F'(x)h,g)x 2: b llgllx.
Then for c = 2- 2 b!M,
But also then,
Suppose M > 0 and g E K. Assume also that G : Br,sCO) � K, with G(O) = 0, is continuous in the J topology, and that ify E br,sCO) there is h E BM,sCO) such that
Corollary 1.
l tlim --->0 + t . (G(x + th) - G(x)) = g. Then
ifO s: t s: riM there is u E Br,sCO) such that G(u) tg. G(x) - g, x E BM,sCO), and apply The =
(Just take F(x) = orem 9).
Equation (6) is an ordinary differential equation in in finite dimensions. Solutions to (6) can be tracked nu merically to obtain approximations to solutions u to F(u) 0. This gives the prospect of a unified numerical approach to a very large collection of problems in PDE. For problems in the form (9), existence of a solution u to F(u) = 0 can be established if a gradient inequality can be shown to hold on a region containing a trajectory z of (6). Establishing a gradient inequality is equivalent, according to Theorem 8, to establishing the uniform boundedness of solutions to a certain collection of lin ear problems. =
Differential Equations: More Concrete Developments
A very simple example illustrates how to deal with equa tions for which no natural variational principle is in hand. Experience has shown that someone who codes success fully this example is prepared to code much more compli cated problems, problems of scientific interest. Example.
Find
Let us call R11 + 1 under this inner product H11: a discrete ver 2 sion of H H1 above. A finite-dimensional version c/Jn of (14) is given by =
c/Jn(u)
u in the Sobolev space H = H1•2 ([0, 1]) so
IID 1u - Doull�n/2,
cp�(u)h
=
=
u' - u = 0. Define 1
cp(u) = 2
(14)
1 1 (u' - u?,
u E H.
0
F(u) = u' - u, cp(u) = liFCu) lll/2,
((:} (: --u�)!KxK
n{{)
() (
)
h u - u' (u)h - ( h' , P lK K, u' u x
and so
(
_
)
cp' (u)h = (h, 1rP u - u' lH , consequently
=
this is the ordinary gradient of cp11, that is to say, the list of relevant partial derivatives. Now for u E R" + I , cp� (u) is also a linear functional on H, that is, on R" + 1 under the norm ll · lls,n· Accordingly there is the function Vs,n c/Jn for which
c/J' n (u)h = (h, (Vs,n c/Jn)(u) /s,n = (Dh, D(Vs,n c/Jn) (u) )R2"
1rP
u' - u
(u u' ) -
u' - u
,
h, u E H;
u E H.
A finite-dimensional counterpart now follows. It is based on the same simple example, but the considerations gen eralize rather easily. Pick a positive integer n and divide the interval [0, 1] into n pieces of equal length. Take 8 = l_, and define D0, n D 1 : R''+ 1 � R" so that if u = (uo, u1, . . . , U11) E /?" + 1 then
( u1 +2 Uo , . . . ' Un +2Uu - 1 ) ' Un - Uu - 1 ) . (U D l u = 1 Uo
Dou =
8
' . . . '
8
'
take
Du =
(��:}
Define a new inner product between elements u, v E R'' + 1:
(u,v/s,n =
and so
,
Denote by P the orthogonal projection of K X K onto Q in (2), and define 1r : K X K � K by = .f Then from (16)
(V cp) (u)
(D 1h - Do h, D 1 u - DoulR" (h, CD 1 - Do) l(DI - Do)u)R"+� ,
Thus,
cp'(u)h = (h' - h, u' - u)K =
,
n E R + l.
u E H.
Note that for u, h E H
cp
U
so that
Something that occurs for many systems of differential equations happens in this case: a zero of the correspond ing Sobolev gradient V cp is also a zero of cp. Here is a rep resentation of V cp in the present case. Define F : H � K = £2([0,1]) by
(16)
=
Note that for u, h E /?" + 1,
that
(15)
•
This relationship between ordinary numerical gradient and its Sobolev gradient counterpart is a general phenom enom. A numerical analyst recognizes this Sobolev gradi ent to be a preconditioned version of the ordinary one. In [4], [ 1 1 ] this statement is reinforced in detail to show that finite-dimensional Sobolev gradient theory gives an orga nized approach to preconditioning. This is a place to point out that if in (14) a gradient g in £2([0, 1]) were sought in order to represent cp', so that
(cp' (u))h = (h,g(u))£2([0, 1 ]), u, h E £2([0, 1]), the gradient g would be only densely defined on £2([0, 1]) and discontinuous everywhere it is defined. Such an object is not a promising one to approximate numerically. How ever, the ordinary gradient of c/Jn is just what one would try to use for such an approximation. Now it is legendary that such ordinary gradients, defined for problems in differen tial equations, perform very poorly numerically, the per formance degenerating dramatically as the number of mesh points increases. By contrast, the Sobolev gradient of cp is defined everywhere and is a differentiable function. The Sobolev gradient of c/Jn performs very nicely numerically: typically the number of iterations required does not depend on the number of mesh points chosen. This is an instance of this writer's First Law of Numerical Analysis:
Numerical difficulties and analytical difficulties always come in pairs. An orthogonal projection related to the above example is given by
(Du, Dv)R2n.
© 2005 Springer Sc1ence+Bus1ness Media, Inc., Volume 27, Number 3, 2005
51
It is the orthogonal projection of R2n onto the range of D. It is an exercise to reconcile this expression with (4). The above is an of indication how a variational princi ple can be established for many systems of differential equations. If a given system corresponds to the Euler Lagrange functional on a Sobolev space, there is another, di rect method available. Examples include energy functionals for Hamiltonians, elasticity, transonic flow, Ginzburg-Landau functionals for superconductivity, oil-water separation prob lems, minimal surfaces, Yang-Mills functionals. Generally, if an energy functional is available, it is better to look directly for critical points of the functional rather than forming a new functional based on the corresponding Euler-Lagrange equations (for one thing, those are of degree twice the maximum order of derivative appearing in the energy functional). For an illustration, assume n is a positive integer and 0 is a domain in R". Defme cf> by (17) for u E H, H being an appropriately chosen Sobolev space, so that G E C1 and V cf> is locally lipschitzian. Assume also that cf> is bounded below. Then for u, h E H, (18)
cf>'(u)h = J ((Vh,Vu)Rn + G'(u)h) fl
( )(
)
h (VG)(u) =< )HxK, Vh ' Vu
K being an appropriate L 2 space. Denoting by P the or thogonal projection of H X K onto
one has
cf>'(u)h
=
(h, 1TP
(19)
(Vcf>)(u) = 1rP =
U
( (V��u) ) .n(u - (VG)(u) )
_
1Tr
O
,
p(C��(u)). Thus
u E H.
On bounded regions with a smooth boundary the transfor mation
u�
(
.n u - (VG)(u)
1T r
0
J.J)
)
(where = f) may be seen, by examining the relevant projection P, to be compact. Furthermore, u is a critical point of cf> if and only if (20)
52
(Vcf>)(u) = 0.
THE MATHEMATICAL INTELLIGENCER
cf>(uk) = inf cf>(u), klim uEH ->"'
then attempting to extract a convergent subsequence of this sequence whose limit is a minimum of cf> and consequently a solution to the corresponding Euler-Lagrange equations. Continuous steepest descent (6) provides a constructive al ternative, so it is not surprising that its finite-dimensional versions yield viable numerical methods. Numerical considerations for directly computing critical points of energy functionals such as (17) are quite similar to those for (9). Sometimes for an energy functional it is helpful to define a second function J: J(u) = IICV 4>)(u)ff7I,
uEH
and then use VJ in place of V cf> in (6). This is used when there are saddle points for cf> for which numerical compu tations are unstable. Critical points of cf> become local min ima of J. An alternative point of view is found in [ 10], which gives a way to constrain trajectories in seeking a critical point numerically. Boundary Conditions
( (V��u) ))H,
where 1T picks out the first component of for a Sobolev gradient of c/>,
The Sobolev gradient equation (20) is a substitute for the corresponding Euler-Lagrange equations together with any "natural" boundary conditions arising from the required in tegration by parts used in obtaining these equations from cf>. The functional (17) contains only first derivatives, but the corresponding Euler-Lagrange equations require sec ond derivatives. Use of equation (20) avoids the old prob lem, already noted by Hilbert, of introducing more deriva tives than the basic problem presupposes. The Sobolev gradient is a continuous function whose ze ros are critical points of cf> which may be sought by means of limits at infinity of a solution z to (6). Contrast this with the usual nonconstructive (cf. [3] , [ 18]) approach to dealing with (17): defming a minimizing sequence (uklk'= 1 , that is,
So far little has been said about how these ideas relate to boundary conditions. Following are two rather distinct de velopments on this topic, both of which contain a guide to numerics. Traditionally, the role of boundary conditions in con nection with partial differential equations is to give condi tions on boundary values under which a given system has a unique solution. It should be recognized that specifying conditions on a boundary of a region (on which solutions are to be found) is often not known to be adequate for mak ing the solution unique. Consider for example a Tricomi equation [8] , a version of which is the following: Find u on 0 = [0, 1] x [ - 1, 1 ] so that (2 1)
u1 , 1 (x,y) + yu2,2 (x,y) = 0,
(x,y) E 0.
It is my understanding that, after decades of consideration, it is still not known what boundary conditions might be placed on a prospective solution u to (21) in order that there be one and only one solution. A main problem with (21) is that it is elliptic above the real line and hyperbolic
below. For elliptic, parabolic, or hyperbolic problems, ideas about boundary conditions are quite well understood. But many systems defy such categorization. The first of my two comments on the "supplementary condition" problem deals with problems in the form (9) but applies equally to those in the form (17). Assume each of H, K, S is a Hilbert space, F is a C1 func tion from H to K, and B is a C1 function from H to S. Typical problem.
Find u E H so that
F(u)
(22)
=
0,
B(u) = 0.
Think of the first equation in (22) as specifying a system of differential equations and the second equation as supply ing supplementary conditions. Thus if we had the simple example above but with the condition u(O) = 1 imposed on a solution u, we would de fine
F(u) = u' - u, B(u) = u(O) - 1 ,
(23)
u E H1 •2 ([0, 1]).
In the general case, take
;"1 m� ,.01,) 'lin �7 '1M"1 1!)0 K1:1 "The Honors Class" .C'l11V l'lll 'l.l1nnl.l C'!Vllt '"ll :'lltl.,::l M"1Pl 1:::11, 7:>
,Wit '1:::1,_:::1, ,!lC ::ln:> In;"!
Philip J. Davis Brown University, Providence, RI -
"I predict your book will be a great success; the historical comments are fascinating". Peter Lax - CIMS, New York, NY
·KHMra HanMCaHa 0\leHb lKHBO M �MTaeTes� c yBI1e'feHMeM. 0Ha noMolKeT MHOfMM nOHATb, "f'TO T8K08 38HATMA M8T8M8TMKOM, K8K M8T8M8TMKM lKMBYT, M KaK H8 HMX BnMAeT MX npocpeccMA." Yurl Manin - Max-Planck lnstitut fur Mathematik, Bonn, Germany
,[Yandell] liefert im ausfiihrlichen intellektuellen wie politischeu und
personlichen Portriit des grolkn so\\jetischen Mathematikers Andreij Nikolajewitsch Kolmogorov ... ein dichtes, gelungeues Zeit· und Denkbild, das seinem Buch einen angenehm unpomposen, iiberzeugenden SchluBakkord beschcrt". Dietmar Dath
-
FAZ, Germany
tJ1� t:1 �;Yii�?tr!�c�=+i!t*c:'fJJ��r&�rp�MlR����. tffiJ%���$�1iWW�IJ�. �M�Jt.����tcr:�zJfH�tcr:�to/1 Yum-Tong Siu - Harvard University, Cambridge, MA
In
To P l a ce O r d e rs
All other countries
Europe
Transatlantic Publishers Group www.transatlanticpublishcrs.com
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www.ak.peters.com
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·•saunders :Mac Lan� is an immense presence in modern mathematics.
One of the last Americans to be educated in G1ittingen prior to the Nazi era, Mac Lane was ideally and propitiously situated to become a leader in modem algebra and modem algebraic topology. His influ
ence on our subject has been powerful and widespread. This chronicle
tells
of an influential and important life." Steven G. Krantz,
Washington University in St. Louts
In Memoriam
1909 . 2005
© 2005 Springer Science+Business Media, Inc., Volume 27, Number 3, 2005
59
theorem on equidecomposability can be extended to three-dimensional fig ures. It was solved shortly after the Paris Congress by Max Dehn. The problem was, perhaps, one of the "eas iest" posed by Hilbert, but that does not detract from Dehn's overall standing as a mathematician, as Yandell points out. Andre Weil, who was not given to overly generous appraisals of his con temporaries, said of Dehn, "I have met two men in my life who make me think of Socrates: Max Dehn and Brice Parain [a French philosopher] . . . . [Dehn] left behind a body of work of very high quality." The story of Dehn is a fascinating one. When it became nec essary for him and his wife to leave Germany in 1940 they took roughly the same route that Godel and his wife had taken, across Russia and Manchuria via the Trans-Siberian Railway, thence to Japan, and on to San Francisco. But where Godel proceeded on to Princeton, Dehn was able to find a job only at the University of Idaho in Pocatello-now Idaho State University. (Yandell does not make it clear that this was not even the principal campus, which is in Moscow, Idaho, but the Southern Branch of the University, located in a fairly remote and small town in south eastern Idaho.) Dehn, a distinguished German professor who had solved one of Hilbert's famous problems, must have appeared rather an exotic crea ture in Pocatello, known then as now mainly as a section point on the Union Pacific Railroad. Dehn went on to teach at the Illinois Institute of Technology, St. John's Col lege in Annapolis (where he had to teach Homer's Odyssey in one week, in English), and eventually to Black Mountain College in North Carolina, where there were few students, almost none interested in mathematics. Still, he lived almost all of the rest of his life at Black Mountain (it eventually went out of business), and there he and his wife spent happy years in part because of the distinction of others on the fac ulty: poets Robert Greeley, Robert Dun can, Denise Levertov; painters Willem de Kooning, Franz Kline, Robert Moth erwell, Josef Albers, Robert Rauschen berg; composer John Cage; dancer Merce Cunningham; and filmmaker
60
THE MATHEMATICAL INTELLIGENCER
Arthur Penn. One of Dehn's friends was Buckminster Fuller, who built an early geodesic dome on the campus, but "it collapsed and was renamed the 'supine dome.' " Dehn is buried in the woods on what was once the campus.
H i l bert ' s p roblems .
.
came 1 n vanous forms . For this reason it is not clear j ust how many have been "solved . " So, although the mathematical life there was not that of Gottingen or Berlin, for Dehn and his wife, Black Mountain turned out to be a good in troduction to some of the best of Amer ican culture in the postwar years. Before moving on to the next set of problems I should explain that Hilbert's problems came in various forms. For this reason it is not clear just how many have been "solved." Yandell says it best: Hilbert's problems have the charac teristics of any good founding doc ument. Each one is a short essay on its subject, not overly specific, and yet Hilbert makes his intent re markably clear. He leaves room for change and adjustment. Hilbert's goal was to foster the pursuit of mathematics. He helps us fmd the vital center of each problem by us ing italics-often in more than one place. So the question of how many of the problems have been solved becomes part of a more ambiguous question: How many have been posed? Sometimes Hilbert suggests a plan for investigation of an area. Or he may be pursuing an intuitive feeling, and the problem can be paraphrased: Why don't you look in that direction? However, most of his problems have been identified-in something approaching a consen sus-with a single, clearly stated
mathematical question. That con sensus, when it exists, is what I have taken to be the problem. The fourth problem, on alternative geometries resulting from weakened axioms, was rather vaguely presented, and it wasn't entirely clear what con stituted a solution. But you can read of the contributions of Georg Hamel, Her bert Busemann, and A. V. Pogorelov. The fifth problem, on Lie groups, is technical and difficult to describe, and it was solved in two stages, the first by Andrew M. Gleason, the second by Deane Montgomery and Leo Zippin. The biographical material on all three is rich in detail and tells a fine story. Yandell cannot refrain from looking at patterns. He points out how much good mathematics seems to be done while mathematicians are out on long walks: Hilbert, Hurwitz, and Minkowski, Ein stein and GOdel, Ulam and Erdos, and Montgomery and Zippin. Also in the context of the fifth problem, he ob serves that Hilary Putnam made major contributions to the tenth problem without having a degree in mathemat ics, and Gleason at Harvard became the Hollis Professor of Mathematicks and Natural Philosophy (a chair en dowed in 1 727) yet "never received a single graduate credit toward a Ph.D., much less the degree." Of course, Glea son shared the distinction of not hav ing a doctorate with another eminent Harvard mathematics professor, Gar rett Birkhoff. Being on the faculty with out a Ph.D. was possible for Junior Fel lows at Harvard in those days. We also learn that when Gleason took the Put nam Competition at Yale, he was un happy because he had been able to do only thirteen of the fifteen problems. "It was probably the first time he could n't solve all the problems on a mathe matical test." Nevertheless, he placed in the top five contestants that year and was awarded the Putnam Fellowship to Harvard. In his sixth problem Hilbert pro posed the axiomatization of physics, which has not been done, at least to everyone's satisfaction. Yandell points out that if one of the recent theories in physics is right, it should be axiomati zable, in which case Hilbert's problem
could have a "clean" solution. But then he observes that "the relationship be tween mathematics and physics clearly a case of opposites attracting continues to be tempestuous." He suggests that "maybe string theory will go to live with mathematics-at the present time this 'physical' theory has no connection to experimental data." The section on number theory is one of the richest, with six problems (though one of them went to the sec tion on foundations, as we saw), this is perhaps not surprising, since Weyl claimed that Hilbert's best work was in number theory. In the introductory re marks Yandell describes what he calls "near misses," and gives as examples both L. G. Shnirelman's theorem that every even integer is expressible as the sum of no more than 300,000 primes (not quite Goldbach's conjecture that every such number is the sum of 2 primes) and Alan Baker's work that implies that if the difference between one perfect cube and twice another is 1, then neither can be larger than 1.5 X 10 1 3 17. Hilbert's seventh problem asked for a proof that 2Y2 and some similarly formed numbers are transcendental. This was proved independently in 1934 by A. 0. Gelfand and Theodor Schnei der, but with a contribution along the way by George P6lya and many con tributions by Siegel. Yandell points out that in 1920 Hilbert said in a lecture that "he thought no one [present] would live to see 2Y2 proved to be tran scendental . . . [but he] said he thought he himself might live to see Riemann's hypothesis proven . . . and that the youngest members of the audience might see Fermat's last theorem proven. He had the order of solution the wrong way around." The author is always alert to the pos sibility of making something clearer with a clever choice of words. He quotes Gleason as saying, " [The positive solu tion of the fifth problem] shows that you cannot have a little bit of orderli ness without a lot of orderliness. " In talking about Zermelo's work on sets, Yandell says, "it forbids infinite nests of brackets with nothing in them, like parallel mirrors in a deserted barber shop."
Yandell moves quickly through the history of the eighth problem, the Rie mann hypothesis. With several recent popular books devoted exclusively to this problem, anything here becomes somewhat redundant. Still, it gives Yandell a chance to talk about Hardy and Ramanujan, in particular Hardy's receiving Ramanujan's letter from In dia explaining some of the work he had done, essentially in a vacuum. Yandell quotes Thomas Gray's "Elegy Written in a Country Churchyard": "Some mute inglorious Milton here may rest." He then quotes H. L. Mencken: "There are no mute, inglorious Miltons, save in the hallucinations of poets. The one sound test of a Milton is that he function as a Milton." Yandell goes on: "Hardy had just received a letter from a marginally educated Indian, whose English was poor, who was in a mathematical sense functioning as a Milton." Further along we read that "Siegel used mainly alge bra in proving the Thue-Siegel theorem and for some time couldn't convince ei ther Schur or Landau he had done it. Schneider solved . . . the seventh prob lem using fairly elementary methods, apparently without being aware he was working on it." How many others have been out there working, but with their accomplishments unrecognized?
Raman ujan was no m ute i ng lorious M i lton . Throughout the first half of the twentieth century the contrast be tween classical methods and the more modem trends that took hold later is obvious. Yandell points out that, in a rather provocative letter to Weil, one of the founding members of Bourbaki, Siegel wrote in 1959: "It is entirely clear to me what circumstances have led to the inexorable decline of mathematics from a very high level, within about 100 years, to its present nadir. The evil be gan with the ideas of Riemann, Dedekind and Cantor, through which the well-grounded spirit of Euler, La grange and Gauss was slowly eroded. Next the textbooks in the style of Hasse, Schreier, and van der Waerden,
had further a detrimental effect upon the next generation of scholars. And fi nally the work of Bourbaki here pro vided the last fatal shove." Weil's re sponse is not recorded. The modernists were clearly not deterred by Siegel's conservative views. Yandell raises the question of how Siegel would have re sponded to Wiles's proof of the Fermat problem, where he used modem meth ods and structures extensively. There are so many great stories in this book-How much can one pass along in a review? In the discussions of the ninth, eleventh, and twelfth prob lems on class field theory and related problems, we read of a party given by Teiji Takagi at the 1932 ICM in ZUrich, with guests Chevalley, Iyanaga, Hasse, Nagano, Noether, Taussky, and van der Waerden, among others. Yandell points out that Noether so admired Takagi's work that she had started learning Japanese in order to converse with him. At the party "Noether enjoyed her self, talking the most, . . . and then, as the party ended, requesting that the Japanese teach her how to bow . . . 'She asked, bending forward . . . "Like this, or more?" Her funny look remained long in the memories of the young Japanese mathematicians.' " She was the senior mathematician at the party. We can only wish we could have been there. Building on Takagi's development of class field theory, Emil Artin solved the ninth problem, on a law of reci procity; Helmut Hasse, the eleventh problem, on quadratic forms. These sections are filled with a warm account of Artin and his work, along with a guarded appraisal of Hasse, who was later criticized for staying in Germany during World War II and cooperating with the Nazi regime as director of the Institute in Gottingen. Yandell points out that Hasse's father was descended, on his mother's side, from the Jewish composer, Felix Mendelssohn. How did that get past the Nazis? Sanford Se gal in his excellent Mathematicians under the Nazis [6] explains that Hasse had a great-great grandmother who was Jewish but baptized, so he was 1/16 Jewish. To be an academic in Germany at that time it was only nec essary to have no Jewish grandparents.
© 2005 Spnnger Sc1ence+ Bus1ness Med1a, Inc , Volume 27, Number 3, 2005
61
But that condition was not sufficient for him to be a member of the Nazi party, which he was. For that "honor" one could have no Jewish ancestors who were alive after 1800. This should have excluded Hasse from the party, but a series of bureaucratic blunders made it possible for him to remain a member until the war was over. Hasse came to the United States af ter World War II and taught at several universities. Yandell says, "Hasse was an excellent mathematician. But com pare him to Hilbert in the same time and place, who saw clearly even from senility's downward slope. A political order that destroys mathematics and makes mathematicians flee for their lives is almost certain to be evil. To our lrnowledge Hasse never saw this." Se gal put it this way: "[Hasse held to) some what rigid principles, patriotism, and na tionalism-Manfred Knebuschdescribes Hasse in the postwar years as having 'all the strengths and wealrnesses of a Prussian officer'-[that] caused him to view Hitler as a national hero and to ap ply for membership in the Nazi party in 1937." Segal goes on to say that Hasse thought that "slavery in America had been a good institution for blacks." Hasse may have been a member of "The Honors Class," but that did not make him an admirable human being. In the discussion of the fourteenth and seventeenth problems in algebraic geometry we have an essay on Masayoshi Nagata. The cast of charac ters in the chapter on the fifteenth problem-Schubert's Variety Show includes Solomon Lefschetz, van der Waerden, Edward Witten, all the way up to Maxim Kontsevich. We learn that Hermann Schubert "calculated 666,841,048 quadric surfaces tangent to nine given quadric surfaces and 5,819,539,783,680 'twisted cubic space curves tangent to 12 given quadric sur faces.' " And he did it without comput ers! But computers have since con firmed Schubert's calculations. The eighteenth problem also in volves counting. Hilbert asked whether the number of crystallographic groups in four dimensions is finite, which was only part of the problem and a question fairly quickly answered in 191 1-12 by Ludwig Bieberbach. Yandell treats the
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THE MATHEMATICAL INTELLIGENCER
life and career of Bieberbach in some detail. A talented mathematician but a complex man, Bieberbach was a stu dent of Felix Klein and Paul Koebe at Gottingen. He was at Konigsberg and Basel before going to Berlin to replace Constantin Caratheodory in 192 1 (after the post had been turned down by Brouwer, Herglotz, Weyl, and Heeke!). Louis de Branges's proof of the famous Bieberbach conjecture on schlicht functions was one of the great achieve ments in mathematics in the 1980s. But able as he was as a mathematician, Bieberbach acted very badly during World War II. P6lya claimed that while some mathematicians in Germany used the system for personal advance ment-Wilhelm Blaschke, for exam ple-Bieberbach was a true believer in
Yandell g ives n i ne p roblems not on H i l bert ' s l ist that cou ld have been on a l i st of Poi n care ' s . the Nazi cause. Yandell is careful to give a balanced view of Bieberbach, telling of his many accomplishments as a mathematician, an author of text books, and a teacher, mainly in his early years. But Yandell does not de fend his actions after 1933. By that time Bieberbach had joined the S.A. (the storm troopers), and from that point on he became more combative, eventually establishing the infamous journal Deutsche Mathematik, "which mingled race politics with 'good German math ematics' . . . According to Bieberbach there were two types of mathemati cians, the S type, short for Strahltypus (Jewish, French, etc.-bad) and the J type (German-possessing Anschau ung-good). The J type had three sub divisions J1, J2, J3. So the man who had helped [to) clarify the classification of crystals into types [the eighteenth problem], making his way in the world, extended classification into human life, an obscene and sloppy tiling."
The last section is on the problems from analysis (with a more admirable cast of characters), not taken in the natural order but starting with a long section on the twenty-second problem, on automorphic functions, with a care ful exposition of the work of Paul Koebe and Henri Poincare, who is of ten cited, along with Hilbert, as one of the last mathematicians who had a grasp of the whole field of mathemat ics. Yandell gives us an additional list of nine problems not on Hilbert's list not surprisingly somewhat more topo logical-that could have been on a list of Poincare's, had he outlined an agenda for the century. The nineteenth, twentieth, twenty first, and twenty-third problems are discussed in two rather short sections. They are all concerned with differen tial equations and the calculus of vari ations. The twenty-third problem is not a problem at all but a program for fu ture research in the latter. Hilbert es sentially called on the mathematical community to look again at the calcu lus of variations-in Hilbert's words: "[it] does not receive the general ap preciation which in my opinion, it is due." Yandell describes briefly the ex tensive work that this program in spired. There is interesting biographi cal material on Josip Plemelj, Andrei Bolibruch, S. N. Bernstein, and I. G. Petrovsky. In this section we learn that it is not always a straight path to a so lution to one of these problems. The twenty-first problem, on the existence of linear differential equations having a prescribed monodromic group, con cerns those equations "that have points where the coefficients . . . have singu larities . . . [Solutions) to this kind of problem, as with many things in com plex analysis, are not single-valued. One circles a singularity (like a horse in a hippodrome), jumping from one lo cal value of the solution to another. The monodromy group captures this." Hilbert produced a partial solution in 1905, and Plemelj in 1908 gave a sim pler proof of a more general result. In 1989 Bolibruch found a counterexample to Plemelj's work, not just a gap in the proof. The result was false. Something similar happened in the solution of the sixteenth problem.
The most moving essay in this sec tion is about A. N. Kolmogorov, who solved, along with V. I. Arnol'd, the thir teenth problem on the general equation of seventh degree, and who also made contributions to the unfinished sixth problem on axiomatizing physics. This biographical essay is warm and touch ing, even lyrical. We learn that Kol mogorov's mother died in childbirth and that he was brought up by an aunt, first on an estate in the country and subsequently in Moscow. The descrip tion of his childhood calls to mind beau tiful and poignant scenes described by Tolstoy and Chekhov. V. M. Tikhomirov, editor of Kolmogorov's collected pa pers, wrote of Kolmogorov's childhood home . . . "the grand old house with a second floor, constructed, most prob ably, at the beginning of the 19th cen tury . . . with four massive columns at the main entrance. . . . Around the house . . . were courtyards-front and back, flower gardens, a kitchen garden, a garden (so big that it could be called a park), barns, a cattle yard, a stable, a bathhouse-in a word, all those things infinitely remote from our present-day lives, and of which we now know only from books." The Kolmogorov family followed Tolstoy's moral philosophy, so it is not surprising that Kolmogorov was named for Tolstoy's estimable character, Prince Andrei Nikolaevich Bolkonski. In 1929 Kolmogorov and P. S. Alek sandrov, later to become a world renowned topologist, along with an other friend, rented a boat for a trip down the Volga. They took with them one book, Homer's Odyssey. Kolmogorov's description of floating down this great river reads in part: "During the first days of our journey we often swam at night, in the white sum mer nights, gliding along past the over grown osier beds on the bank, the air filled with bird song, made a lasting im pression on us. We wished that this could go on forever." (Those interested in American literature will be reminded of Mark Twain's descriptions of Huck and Jim floating down the Mississippi.) Twenty-one days and 1300 kilometers later, the trip on the river ended. The three turned in their small boat and continued by steamer to Baku and
Yerevan before returning to Moscow. Much later Kolmogorov wrote, "Pavel Sergeevich Aleksandrov and I date our friendship . . . which lasted 53 years [from the 1929 trip] . . . . For me these fifty-three years of close and indissolu ble friendship were the basic reason why all my life turned out to be filled, on the whole, with happiness, and the basis of my good fortune was the un ceasing thoughtfulness of Aleksandrov." Kolmogorov, though his name is usu ally associated with probability, worked in other areas as well, producing impor tant results in geometry, algebraic topol ogy, approximation theory, ergodic the ory, and, during World War II, ballistics
Kol mogorov was named for Tolstoy ' s esti mable character, Pri nce And rei N i kolaevich Bol konski . and turbulence. It was in the 1950s that Kolmogorov did the work that resulted in the solution of the thirteenth problem, with Arnol'd, who also contributed to the solution of the sixteenth problem on the topology of algebraic curves and sur faces. Arnol'd, who always insisted on clarity in writing, shared some views with Siegel, mentioned earlier. He re ferred to "criminal bourbakizers." In the 1960s Kolmogorov wrote a se ries on the metrical structure of Rus sian poetry, not a frivolous enterprise. A. A. Markov earlier had claimed "that the string of vowels and consonants in [Pushkin's] Eugene Onegin was as pure an example of a Markov chain as he could find." At this point we should note that many of the pictures in this book are a treat for the reader, often fresh and un common, taken as they are from vari ous private collections. Arnol'd is shown, for example, standing on the Golden Gate Bridge! The story ex plaining it is worth checking out.
Yandell, in summary, quotes from Hermann Weyl in his mid-century re view of progress in mathematics [7]: "David Hilbert . . . formulated twenty three unsolved problems which he ex pected to play an important role in the development of mathematics during the next era. How much better he predicted the future of mathematics than any politician foresaw the gifts of war and terror that the new century was about to lavish upon mankind! We mathemati cians have often measured our progress by checking which of Hilbert's questions had been settled in the meantime." Of course, in the end, the natural question is how many of the twenty three problems have been solved. Yan dell summarizes: "Sixteen problems have been solved in a discrete way. The core of the question Hilbert asked re ceived an answer that is complete with respect to the original question (within reason), mathematically rigorous, and unlikely to be dramatically improved upon in the future. These problems are 1, 2, 3, 4, 5, 7, 9, 10, 1 1, 13, 14, 15, 17, 18, 21, and 22. Four problems-12, 19, 20, and 23-are what amount to pro grams for research and are somewhat vague in their statement. These have prompted a great deal of successful re search, and I think it is time to count them as 'solved,' as problems of Hilbert's. Their concerns can be embodied in new problems written in new language. Three problems-6, 8, and 1 6--have not been solved." Not bad. Here I have given only a quick overview of some of the fascinating, amusing, and evocative accounts in this book There are plenty of other sections that are equally interesting, mathematically stimulating, and his torically illuminating. This is a splen did piece of work and an essential book for anyone interested in the vast panorama of twentieth-century mathe matics. It should be in every mathe matician's library. Further, we should recommend it to our friends or our stu dents who will be inspired by this ele gant book about the accomplishments, not of the ancients, but of twentieth century people, many of whom lived during our lifetimes and contributed richly to our mathematical community and to society as a whole.
© 2005 Springer Sc1ence+Bus1ness Media, Inc., Volume 27, Number 3, 2005
63
[2] Felix Browder, ed. , Mathematical Develop
While Ben had interviewed many peo
25, 2004,
ments Arising from Hilbert Problems. Pro
ple, including the grandchildren of Max
Bel\iamin Yandell, only 53 years old,
ceedings of Symposia in Pure Mathematics,
Dehn, I was able to supply some more
died of a heart attack in Pasadena, Cal
Vol. 28. Providence, Rl: American Mathe
ifornia. In 1993 he had been diagnosed
matical Society, 1 976.
Epilogue
On the morning of August
contacts, including Peter Lax, who pro
vided valuable information about recent developments. Finally, after a labor of
as having multiple sclerosis, but this
[3] Anita Burdman Feferrnan, Politics, Logic,
disease did not appear to be related to
and Love: The Life of Jean van Heijenoort.
love of nine years
his death. He is survived by his wife,
Wellesley, MA: A K Peters, 1 993.
formed) the final manuscript emerged.
Janet Nippell, and a daughter, Kate Louise, born in 1988. In the years since the appearance of
The Honors
(if I am correctly in
[4] lvor Grattan-Guinness, A Sideways Look at
Ben's intellectual curiosity and his
Hilbert's Twenty-three Problems of 1 900.
ability to communicate his insights in
Notices of the American Mathematical So
words that were of interest to a broad
ciety 47:7 (2000), 752-757.
audience including experts and ama
Class he had been working on two writing projects: (1) Chasing a Wave, a story of the discovery and defi
[5] Jeremy J. Gray, The Hilbert Challenge. Ox
nition of solitons that was to be a his
[6] Sanford L. Segal, Mathematicians under the
torical-biographical account, similar to
Nazis. Princeton, NJ: Princeton University
is hard to accept. The following text,
Press, 2003.
found in his papers, explains better
that in The Honors
Class, and (2) a short
biography of John von Neumann. Yandell's undergraduate years were spent at Occidental College and Stan
ford: Oxford University Press, 2000.
courses. Instead of pursuing a doctoral
tainly his future readers with a loss that
[7] Hermann Weyl, A Half-Century of Mathe
than I could Ben's motivation, born out
matics, American Mathematical Monthly 58
of an awareness of his talents and a
(1 95 1 ) , 523-553.
modest intention to use them in the best way he knew. We all are benefit
ford University, to which he trans ferred so he could take some graduate
teurs is impressive. His unexpected death left many of his friends and cer
ing from his decision.
Department of Mathematics & Computer Science
degree in mathematics, he spent the
Santa Clara University
One of the jokes about "pure" math
years after graduating from Stanford
Santa Clara, CA 95053-0290 USA
ematicians is that they are always
writing poetry and working as a televi
e-mail:
[email protected] concerned with proving a given
later in Los Angeles. During the latter
Reminiscences
be solved and to what extent the so
part of his career in television repair he
The following are recollections of Yan
lutions are unique. Perhaps they will
was sitting in on physics classes at Cal
dell's life and his book from some of
supply a general outline of how such
tech. His first book was coauthored
the people who knew him:
a solution might be accomplished.
Mostly on Foot: A Year in L.A. (Floating Island, 1989).
From Klaus Peters, Publisher,
interested parties. I was beginning
AK Peters, Ltd.
to think that maybe I was such a
Hilbert (Springer, 1970), Yandell wrote
Several years ago I answered a call in
pure mathematician that I would be
in some autobiographical notes: "I real
the office and the caller ordered a num
satisfied with the determination that
ized how well-written it was. I didn't
ber of high-level research monographs
yes, it was possible that I could be
know, at the time, that Freeman Dyson
on various mathematical subjects. After
a mathematician. I began to lose in
had written, of the book, 'beyond this it
I secured the order, I asked the caller:
terest in working out the details of
"Who are you? I cannot imagine anyone
the actual problem. The attitude of
class of equations can in principle
sion repairman, first in Palo Alto and
Any actual solution is left to other
with his wife:
Inspired by reading Constance Reid's
is a poem in praise of mathematics.' . . .
Hilbert
in this era of specialization, who would
those around me was that I had
I . . . was looking up into the sky. It was
be interested and able to read these
something like a duty to become a
clear and cold and I could see the stars.
books profitably."-"! am writing a book
mathematician. But the graduate
There was a light dry wind, and I was
on the Hilbert Problems and the people
students I knew also worked to
aware that one could aspire to writing
who solved them."-"Who
is your pub
counteract that. I felt, whether I was
One night when I had just read
lisher?" I responded, bracing myself for
right or not, that I had a little more
I might bring something to trying to un
some disappointing news.-"You are,"
raw ability than they did. I couldn't
derstand the mathematical picture of
he answered; and when I declared that
really know. The only way to deter
something like . . .
Hilbert. . . . I thought
It was then that he
I knew nothing of the plan, he answered,
mine that was for all of us to spend
started working on this remarkable
in what I later found to be characteris
the next
tic matter-of-factness, "That's because I
cians. But I didn't feel there was any
Hilbert's problems." book, The
Honors Class, an inspiring
legacy of this all-too-short career. REFERENCES
[ 1 ] P. S. Aleksandrov, ed. , Hilbertschen Prob
years being mathemati
have not sent it to you yet, but your
gross discrepancy. My adviser and
name is associated with most of the
others thought I could get into Har
books that I have used in my research."
vard
or
Princeton
for
graduate
Ben then graciously agreed to send
school. I, too, felt I would have done
me his current draft. I read it enthusias
well there. However, when I was
Hannelore Bernhardt. Leipzig: Akademische
tically, and we both knew that there
still at Occidental, Benedict Freed
Verlag, 1 97 1 .
were many open areas to be treated.
man had arranged for me to meet
leme. Translated from Russian to German by
64
20
THE MATHEMATICAL INTELLIGENCER
his son, Michael. In the short term
very hard interviewing different experts
original result was conditional . . . they
this had helped excite me about
in order to be accurate. Despite his hand
had to assume that there existed arbi
mathematics, but in the long term it
icaps he was lively, hard-working, and
trarily long arithmetic sequences of
supplied me with the suspicion that
full of ideas for the future. He had iden
prime numbers . . . (and] even as of this
I wouldn't be missed if I didn't be
tified solitons as the topic of his next
writing [that] has not been proved." In
come a mathematician. Michael was
book This alone showed great discern
1959 it required an additional contribu
just a graduate student when I met
ment, since solitons were one of the
tion by Julia Robinson to obtain the re
him,
about him
great discoveries of the second part of
sult without this assumption. However,
seemed clear: He had a powerful
the twentieth century with wide appli
today, thanks to the very recent sur
mind and a powerful desire to do
cations and deep mathematical theory. I
prising discovery by Ben Green and Ter
mathematics. When allowed insight
strongly encouraged him and gave him
but two things
to the workings of a great mind, as
what advice I could. I am sure he would
ence Tao, we know that such arbitrarily long arithmetic progressions of primes
in the Cohen book, one comes away
have produced an excellent book which
do exist, so that the original proof is no
with a sense of ease, power, grace,
would have attracted a wide readership.
longer "conditional." Were Ben Yandell
is sad that fate prevented the realisa
still alive, he really would have ef\ioyed
and intensity. Michael seemed to
It
have the mathematical energy to
tion of this project. Mathematics needs
achieve something like that. At the
people like Ben who can convey the in
time, I had no way to tell that he was
tellectual excitement of our subject to
From Peter Lax, Professor Emeritus,
n't just a very good graduate student.
the general public. I hope others will fol
Courant Institute of Mathematical
low in his footsteps.
Sciences, New York University
From Martin D. Davis, Professor
tremely pleasant. We had a long tele
It later became clear that I had, in fact, met someone in that first rank. But in
1971, when I met him, Michael
this new footnote to the story.
My contacts with Ben Yandell were ex
merely supplied me subliminally, and
Emeritus, Courant Institute of
phone conversation when his book on
then after the fact, with a feeling of
Mathematical Sciences, New York
the solvers of the Hilbert problems was
assurance that I could do what I
University and Visiting Scholar,
in the planning stage. We discussed
University of California, Berkeley
bringing in Poincare, who had strong
wanted with a clean conscience. I began to focus on the romantic
Although I never met Ben Yandell in per
views of the shape of things to come in
prospect of becoming a poet. Such a
son, we had an extensive interaction by
mathematics, and perhaps goaded by
choice would validate my indiscrimi
e-mail and telephone over a five-year pe
Hilbert's example, described it in a re
nate curiosity. In the wake of the '60s
riod, and his untimely death leaves me
port to the
with a very real sense of loss. When he
of Mathematicians. We agreed that hav
such a choice seemed to have special
integrity. I was sure I could make a
contacted me in April
living somehow. Poetry and writing in
proposal to write about the Hilbert prob
1997 about his
1908 International Congress
ing a foil for Hilbert would enhance the book We met in person at Cal Tech; I was
general were what I thought I might
lems and their solvers, I was struck by
have an abiding interest in. This has
the sheer audacity of his project. With
surprised
proved to be the case. I am interested
only an undergraduate's knowledge of
bound, which in no way diminished his
in working out the details here. I had
mathematics, he intended to explain to
zest for life. I was shocked and sad
little, if any, concrete evidence of tal
a general audience the
full sweep of the
dened to learn that he is gone, in the
ent, but I felt I could learn. I felt I had
mathematical ideas needed to make
the energy necessary to concentrate
sense of the Hilbert problems. With my
on the essential problems, and re
own limited expertise, I could only be of
gardless of the quality of the results,
assistance with the first four chapters.
I have proven to have that energy.
As the project developed I became more
Writing has kept my attention. I am
and more impressed by his pleasant, easy-going style and by his ability to con
happy when I write.
verge to sufficient understanding of the mathematical ideas.
From Sir Michael Atiyah,
It was fun for me to read his account
University of Edinburgh
I met Ben Yandell on the occasion of the
of the lives and interactions of the four
Berkeley panel discussion of The
people involved with the negative so
ors Class,
Hon
his account of the Hilbert
lution of the tenth problem. I am par
problems. I had very much ef\ioyed read
ticularly amused by the realization that
ing the book, which was designed for a
in one small detail, his account would
general audience, and so I was keen to
be
meet the author. I was surprised to find
different today.
Concerning the
proof by Hilary Putnam and me, that
to
find
him
wheelchair
midst of his labors.
Science in the Looking Glass: What Do Scientists Really Know�
by E. Brian Davies
NEW YORK, OXFORD U NIVERSITY PRESS 2003. $US 29.95 (hardcover) Pp. x + 295
REVIEWED BY JAMES ROBERT BROWN
B
rian Davies takes a shotgun ap proach in
Glass.
Science in the Looking
There are rambling discussions
someone who was not a professional
every recursively enumerable set of
of a very wide range of topics: Psy
mathematician, but who had a deep feel
natural numbers has an exponential
chological work on perception is de
ing for the subject and who had worked
Diophantine definition, he wrote: "The
scribed, an argument against Platon-
© 2005 Springer Science+Business Media, Inc., Volume 27, Number 3, 2005
65
ism is presented, mind-body dualism is
be easily expressed as one about the
not descriptions of independently ex
rejected, there are speculations on
existence of (timeless) triangles ori
isting entities, as Platonism holds. We
thinking machines, intrinsically hard
ented at angle (} for every (} from
make them the way we make chairs
0° to
problems, observation in quantum me
360°. There is an interesting question
and cities and constitutions and games.
chanics; evolutionary biology is upheld
here about the relation between the
But once made, we can make discov
and reductionism is rejected. That's
idealized physical examples in mathe
eries about their consequences; for
just a sample. Needless to say, nothing
maticians' thought experiments and
instance, we discovered that three
is pursued in depth. I can't hope to dis
the mathematical entities themselves.
legged chairs don't wobble and that a
cuss it all, but I will take up a few of
How does the one give us knowledge
given sequence of moves leads to
Davies's points.
of the other? But the fact that we can
checkmate. The view has all sorts of at
and do reason this way does not un
tractions for an anti-Platonist such as
dermine Platonism.
Davies. But it has drawbacks. Mathe
The arguments against Platonism are quite unconvincing. Davies (Pro fessor of mathematics, King's College,
Davies very briefly mentions (but
matical practice is concerned with a
London) cites GOdel and Penrose as
neither endorses nor rejects) an argu
great deal more than merely establish
representative speak of
Platonists who
seeing mathematical
often
objects.
He then reminds us of what he reported
earlier in his book about sense percep
ment that many philosophers accept. Readers of
The Intelligencer may like
ing logical truths of the form: If axioms
then theorem. We worry about the truth
to know what it is. It begins with what
of the axioms themselves. In some
is known as the causal theory of knowl
fields this is more evident than in oth
tion, in particular that what we see is
edge:
To know about X one must
ers. Set theorists, for instance, are con
not the way things actually are. This is
causally interact with X. For instance,
stantly proposing new axioms con
hardly damaging to Platonism. GOdel
I know about the coffee cup on my
cerning large cardinals and arguing
repeatedly stressed the fallibility of
table because
coming
their respective merits. They seem to
mathematics in general and of mathe
from it to me. If I didn't interact with
be searching for axioms that are true.
photons
are
matical perceptions or intuitions in par
the cup in some way, then I simply
This flies in the face of Davies's claim,
ticular. "I don't see any reason why we
wouldn't know about it at all. The sec
since he would have it that axioms are
should have any less confidence in this
ond premise of the argument is a stan
not true-they are merely interesting
kind of perception, i.e., in mathemati
dard feature of Platonism: Mathemati
or useful or fun. Davies's view also
cal intuition, than in sense perception,
cal entities are perfectly real, but they
clashes with Godel's theorem. The fa
which induces us to build up physical
do not exist inside space and time.
mous incompleteness theorem says
theories and to expect that future sense
Third, is the assumption that there
that no set of axioms can capture all
perceptions will agree with them and,
can be no causal interaction between
the truths of arithmetic. It's hard to
moreover, to believe that a question not
things inside and outside of space and
avoid the conclusion that the unprov
decidable now has meaning and may be
time. (Religious people might claim
able sentence is actually true, which
decided in the future. The set-theoreti
this can happen, but, of course, they
means that truth and theoremhood
cal paradoxes are hardly more trouble
also say it is a miracle when it hap
(i.e., being derived from axioms) are
some for mathematics than deceptions
pens.)
It follows from this that we can
distinct things. If this is so, then the
of the senses are for physics." 1 The fact
not know mathematical entities, if they
Peano axioms do not create the truths
that we don't directly see things as they
are as Platonism claims. Since we do
of arithmetic, but rather partially de
are in the physical world is not a rea
seem to have mathematical knowl
scribe something that exists indepen
son to say the physical world does not
edge, it follows that mathematics must
dently. Davies mentions Godel's theo
exist or that it is unknowable. Exactly
be about some other kind of thing.
the same can be said in defence of the Platonic realm.
rem in connection with minds and
This is the best argument against
machines, but not in connection with
Platonism. Not all philosophers accept
Platonism, where it seems to refute the
In another argument he asks us to
it. I don't, but I won't take the time to
kind of postulationism Davies advo
consider a small triangle inside a larger
say why, except to say that I think the
cates. I readily admit that the argument
one. The problem to consider is this:
first premise is false, in spite of seem
for Platonism from Godel's theorem is
Can the smaller be freely rotated inside
ing quite plausible. Nevertheless, it's a
contentious,
the larger without hitting the edges? In
much better argument than the kinds
Davies doesn't mention it.
thinking about this, we imagine rotat ing the smaller to see
if
it will go
of considerations Davies brings against Platonism.
but I'm surprised that
There is another view he also holds, a kind of finitism. Not only does he re
around all the way. Davies points out
Davies's own view of mathematics
ject actual infinities, but he takes large
that this rotation involves time, but the
is a kind of postulationism. Axioms
fmite numbers to be a kind of fiction,
Platonic world is timeless. Well, this is
such as Peano's are things we create.
as well. Constructivists and Intuition
hardly a problem, since the puzzle can
They are not discovered by us; they are
ists are well known to draw a line be-
1 Kurt Godel. "Russell's Mathematical Logic," reprinted in Benacerraf and Putnam (eds.), sity Press,
66
1 97 4, 484.
THE MATHEMATICAL INTELLIGENCER
Readings in the Philosophy of Mathematics,
Cambridge, Cambridge Univer
tween the finite and the infinite. The
interesting example that occurs in a
that this is an admission that conser
common view is that the finite can be
Newtonian framework of five bodies
vation of energy is not an automatic
calculated "in principle," no matter
set up in such way that their gravita
consequence of Newton's laws. In any
how large. Davies dismisses this view
tional interaction leads to one of the
isolated system, it is, but in an infinite
and takes the divide to be between
bodies flying off to infmity in a fmite
universe, it must be a separate as
what can and what cannot be calcu
time. Davies rightly finds the example
sumption. In his subtitle, Davies asks,
lated in practice. If one is going to
impressive. But instead of milking it for
"What do scientists really know?", sug
make any distinction, then Davies, I
its considerable philosophical content,
gesting that they don't know a lot of
think, is right to draw the line at what's
he remarks that it is physically unreal
things they think they do.
practical. However, the price to be
istic and that when bodies start mov
pointing out that they don't know that
Perhaps
paid-which is much worse than the
ing fast, special relativity takes over
Newtonian dynamics is deterministic
price to be paid for constructivism-is
and prevents this sort of thing from
might have been a juicy example.
A very great deal of
happening. He is much too practical
When it comes to quantum me
mathematics would be under a cloud.
minded. I suspect readers would pre
chanics, Davies's views are difficult to
But why, after all, should we think that
fer the musings of a more philosophi
pin down. After a briefaccount of some
everything real must be directly acces
cally
of the physics, he discusses EPR and
sible to human beings? We cannot see
sneers at reality and wants to explore
SchrOdinger's cat. He suggests that the
electrons and we cannot count all the
the nature of Newtonian dynamics, re
cat doesn't go into a state of superpo
members of a transfinite set, but these
gardless of whether Newton's theory is
sition. But he seems unaware of the
entities are central to the best beliefs
true. What can one say about this ex
cost of saying this. It means that there
we have.
ample? There are two remarkable con
are two physical realms, a micro-realm
sequences.
that obeys the quantum laws and a
simply too great.
Science and mathematics
would be impoverished without them. And that's a good reason to think them
minded
mathematician
who
First, Newtonian dynamics, our par
macro-realm that does not. You don't
adigm example of a deterministic the
have to be a mad dog reductionist to
Turning to physics, Davies takes up
ory, is not, after all, deterministic. The
chaos, determinism, and related themes
argument is simple: Start with a world
rebel at this. It's the desire for one phys
perfectly real and objective.
ical world that leads people such as Eu
in a chapter on mechanics and astro
Science in the Look
in which, among other things, the five
gene Wigner to say that the cat does in
physics. Because
particles are arranged so that one will
deed go into a state of superposition and that human consciousness is what puts
ing Glass
has philosophical aspira
be sent off to infinity. Let it run, say, a
tions, it might be useful to contrast
second longer. Since Newtonian dy
Davies's view with how philosophers
namics is time-reversible, use the lo
think about these issues. First, con
cations of the remaining particles at
sider the definition of
this stage with all the motions reversed
mathematical
as the initial conditions for a world W 1 .
world, not in the world itself. But he fails
determinism.
The old idea was this: If we know exactly/approximately the laws and ini
For a world W2 use exactly the same
it into an eigen-state. Davies doesn't like
this solution either and goes so far as to
claim that superpositions are just in the representation
of
the
to note that without actual superposi
tial conditions, then we can know ex
initial conditions. The initial conditions
tions in the physical realm, we are at a
actly/approximately the final conditions.
in these two worlds, though identical,
loss to explain interference effects such
This is the idea captured in Laplace's
do not require or forbid a particle com
as the pattern that results in a split
famous demon and often called Lapla
ing in from infinity one second after the
screen experiment.
cian determinism. Chaos has revealed
start. Their histories could be different,
When discussing EPR he claims that
a problem with this account, since
in spite of their being governed by
the entangled state shows that the re
even
approximate
Newton's laws. That is, in one world a
mote particles cannot be considered as
knowledge of the initial conditions
highly
accurate
particle comes in from infinity after one
separate entities.
won't let us predict very far ahead.
second, but in the other world no such
what about non-local effects? There is
prima facie
Well,
maybe,
but
Among philosophers, the current
particle enters. These are both com
a
favourite definition of determinism has
patible with the initial conditions and
relativity, but Davies, unfortunately,
Newton's laws. The moral should make
doesn't mention it.
abandoned any connection to knowl edge or predictability.
A theory is
de
conflict with special
our heads spin: Newtonian dynamics is
Chapters at the end are on Darwin
not deterministic, because there are
ian evolution, which he endorses, and
the theory with identical initial condi
models with identical initial conditions
on reductionism, which he does not.
tions have identical final conditions. I
but different fmal conditions.
One of Davies's reasons for rejecting
terministic provided that all models of
mention this change in the concept of
If we impose a relativistic speed
reductionism is a poor one: our inabil
determinism only in passing. The ex
limit, then determinism can be recov
ity to predict. Knowledge of physics
ample I now want to discuss is equally
ered. But this would not be Newtonian.
won't allow us to predict someone's
bizarre and instructive on either defi
We could also recover determinism if
behaviour. Reductionism, however, is
nition.
we demanded conservation of energy
a doctrine about reality, not about our
in the initial specifications. But note
knowledge of reality. The fact that we
Davies describes the wonderfully
© 2005 Spnnger Sc1ence+Business Med1a, Inc , Volume 27, Number 3, 2005
67
least
cannot predict is irrelevant. (Note the
those which are the
mathemati
2. The notion itself of mathematics has
relation of this issue to the rival defi
cal-for example evolution, plate tec
to be redefined. Joseph defines it as
nitions
tonics, and the existence of atoms." (v)
of
determinism
mentioned
above.) He does, however, mention more interesting examples that do put
This is not only true, it is profoundly true.
to numbers or spatial configurations
the doctrine of reduction under a great deal of pressure: money or subjective
any activity that arises out of, or di rectly generates, concepts relating together with some form of logic.
Department of Philosophy
Hence he includes in his study
consciousness, for instance. It seems
University of Toronto
proto-mathematics.
almost impossible to reduce a social in
Toronto M5S 1 A 1
stitution such as money to the laws of
Canada
physics. I should point out that the
e-mail:
[email protected] Joseph justifies this decision by There is a close link between mathe
philosophical literature on this is far
matics and
from any consensus. The
last
chapter,
"Some
Final
Thoughts," contains brief musings on chaos, the anthropic principle, Hume and Popper on induction, realism vs anti-realism, the sociology of science, and technology. This chapter, like the whole book, is a series of observations, often off-hand, that are sometimes in sightful and sometimes not. It's hard to know for whom such a book is written. It is rambling, undisciplined, and un focussed throughout.
This in itself
needn't be a bad thing. Littlewood's
A Mathematician's Miscellany is wildly unfocussed and rambling, but it is one
of the most interesting books I've ever
read. What's the difference? Science in
the Looking Glass seems to be written
for a general audience. That, too, is
fine. But then explanations are needed of the relevant mathematics, physics, biology, and philosophy. Almost none are given. Scientists who have never given a moment's thought to philo
sophical implications of chaos or quan
tum mechanics or Godel's theorem
might profit from this book, because
they won't need to be filled in on the
background that Davies is assuming. But those who have thought about
these issues will find many of the mus
ings here obvious or misleading.
But there are also some notable ob servations that offset the book's disap pointment. One of these occurs at the outset. "My conclusion is surprising, particularly coming from a mathemati cian. In spite of the fact that highly mathematical theories often provide very accurate predictions, we should
not, on that account, think that such
theories are true or that Nature is gov
erned by mathematics. In fact the sci entific
theories
most
likely
to
be
around in a thousand years' time are
68
three aspects of proto-mathematics:
THE MATHEMATICAL INTELLIGENCER
The Crest of the Peacock: Non-European Roots of Mathematics, new edition
by George Gheverghese Joseph
PRINCETON AND OXFORD, PRINCETON UNIVERSITY PRESS, 2000 ISBN: 0-691 -00659-8, $US 22.95, 4 1 6 pp.
REVIEWED BY E. KNOBLOCH
T
his extraordinary book, first pub lished in 1992, lives on in this sec
ond edition, an enlarged and improved
version of the first. It is in search of our "hidden" mathematical heritage, which can be found all over the world and cer tainly not only in European countries. It is a plea against "the parochialism that lies behind the Eurocentric per ception of the development of mathe matical knowledge" (p. 348).
The title itself is taken from an In
dian source of the fifth pre-Christian century
(Vendanga Jyotisa) : "Like the
crest of a peacock, like the gem on the head of a snake, so is mathematics at the head of all knowledge." Joseph re
veals his driving passion behind the book: the global nature of mathematical pursuits and creations. Hence he con tinuously emphasizes that scientific cre ativity and technological achievements existed long before the incursion into these areas by Europe. His guiding prin ciple is to recognize that different cul tures in different periods of history have contributed to the world's stock of mathematical knowledge.
This principle implies some crucial
consequences: 1. The study of the history of mathe matics should not be confined to
written evidence.
astronomy.
Early man
kind's capacity to reason and to con ceptualise was not different from that of today's modem peoples. Conjec tures about the mathematical pursuits of early humankind have to be exam ined in the light of their plausibility. Joseph would like to attribute math ematical thinking, practice, knowledge to as many peoples, and as early peo ples, as possible, pleading against a re strictive view of what is proof. Conse quently, he begins by speaking about numerical recording devices on bones of Central Equatorial Africa (35000 B.c.),
South American knots, counting sys
tems of people in Nigeria, Mayan nu meration. The next eight chapters deal with Egyptian, Babylonian, Chinese, In dian, and Arab mathematics. The rich ness of this information cannot conceal the fact that most is well known to pro fessional historians of mathematics. No serious historian of science will contest either the importance of these non-Eu ropean cultures or Joseph's statement
that mathematics is a pancultural phe
nomenon which manifests itself in a number of different ways as counting,
locating, measuring, designing, playing, explaining, classifying, sorting. Joseph's
too
general
criticism
("most standard histories," "many text books," etc.) is outmoded to a large ex tent. Several decades later, M. Kline's evaluation of 1962 does not represent the state of the art any longer (p. 125). Yet Joseph does not stop there. He be lieves in the existence of East-West links he hopes would come to light if research were to be channelled in the direction of a westward transmission of mathematics. Fortunately he avows that much research needs to be done before we can be more certain about the nature and mode of the interchange
of mathematical ideas that took place between China and the other cultural centres (p. 2 12). Yes, we should distin guish among claims, beliefs, and his torical facts. Sometimes Joseph does not notice that he disproves his own argumenta tion. He complains about Eurocen trism because it cannot bring itself to face the idea of independent develop ments in early Indian mathematics, even as a remote possibility. But he does not concede this possibility to the Greeks with regard to the earlier cul tures of the Near East. By all means, it is a too condescending attitude to con cede only "that the Greek approach to mathematics produced some [!] re markable results" (p. 346). Thus the reader is left with mixed feelings. While Joseph rightly rejects the hegemony of a Western version of math ematics, he is inclined to replace it by another one, although he explicitly states that "since the first edition we are no closer to gathering further definitive evidence of transmission of mathemati cal knowledge to Europe" (p. 354). lnstitut fUr Philosophie, Wissenschaftstheorie Wissenschafts- und Technikgeschichte Technische Universitat Berlin 1 0587 Berlin Germany e-mail: ehkn01
[email protected] Mathematics and Music. A Diderot Mathematical Forum
edited by G. Assayag, H. -G. Feichtinger, and J. F. Rodrigues BERLIN, HEIDELBERG, SPRINGER-VERLAG, 2002, 288 PP., US $84.95. ISBN 3-540-43727-4
• • • •
• •
ity," also the source of the Music of Spheres that Pythagoreans referred to when required to swear (Fig. 1 ) . It was perhaps because he was im pressed by the mathematical consis tency of consonance that Pythagoras devised the idea that Number is the substance of the Universe. Be that as it may, on an instrument consisting of a single taut string vibrat ing on a sounding board and fitted with keys that make it possible to select suit able lengths of the string being vibrated, one obtains with the Tetraktys the in tervals known as octaves, fifths, and fourths. Figure 2 represents such a sin gle-stringed instrument (e.g., the Vosges spinet, still used today by certain folk groups in Eastern France) with the cor responding modem names of the notes. Musical instruments such as the tetrachord lyre may also be built with four strings having these same lengths (L, U2, U3, U4) that produce simulta neous sounds. The respective tensions are adjusted so that the sound pro duced by each string is that of the string having the same length on the monochord. The article describes the improve ments brought to this theory by Philo laus and others. The chief result of that period was obtained by Archytas, who demonstrated the need for unequal divisions in order to obtain all the con sonants comprised in an octave. He recognised the importance of arithmetic, geometric, and harmonic means. This
• • •
•
Fig. 1
and calculation in music," while in Paris the Forum dealt with "Mathe matical logic and musical logic in the twentieth century." These three topics are covered in a fairly balanced way in this book, five articles dealing with the first topic, seven with the second, and four with the last. All these articles are of signif icant interest, whether from a histori cal or theoretical point of view. Bringing them together in the same publication sheds magnificent light on the dialogue and mutual enrichment that Mathe matics and Music have developed over the centuries [ 1 ] [2] [3] [4] . The first article, by Manuel Pedro Ferreira, deals with the musical the ory constructed by Pythagoras. Two sounds from the same taut string are said to be consonant when they are pleasing to listen to simultaneously. In the Greek cultural arena of that period such sounds are produced by lengths of string that are inversely proportional to the numbers 1, 2, 3, and 4. These compose the famous Tetraktys (1 + 2+3+4 10), a diagram of figured numbers symbolising pure harmony, the "vertical hierarchy of relation be tween Unity and emerging multiplic=
... ..
................
Length L/4 : obtained sound Sol3,fourth ofRe3,and octave of Soh ......... ........
......
Length L/3 : obtained sound Re3,fifth of Sob .................................. ........................
......
REVIEWED BY SERGE PERRINE
Length L/2 : obtained sound Soh ,octave of Sol1 .................
.......
he book under review brings to gether sixteen contributions to the Diderot Mathematical Forum held un der the auspices of the European Math ematical Society, simultaneously in Lisbon, Paris, and Vienna, with tele conference exchanges, on 3 and 4 De cember 1999. The conference in Lisbon covered "Historical aspects," the topic in Vienna was "Mathematical methods
T
Total length L of the vibrating string: its vibration gives the sound designated by So11
�
1
I
I
I
.
1
Sounding box with keys
Fig. 2
© 2005 Springer Sc1ence+ Business Med1a, Inc , Volume 27, Number 3, 2005
69
G1
G2
D3
G3
B3
D4
L
U2
U3
U4
U5
U6
Octave
Perlect
Perlect
Major
Minor
112
fifth
fourth
third
third
2/3
3/4
4/5
5/6
Fig. 3
allowed him to enrich the range of sounds used and their associated in
tervals (Fig.
3).
The reference
made
to Aristox
as new ways of combining rhythms.
such a project. Yet, once rid of its ab
However,
surd objective, this statement aptly
this
culmination
of the
pythagorean musical base that had de
sums up the concept of music prevail
veloped over many centuries eventu
ing in Renaissance and Baroque times, founded on number and its symbolism,
enus-a pupil of Aristotle who totally
ally degenerated in the following cen
rejected pythagorean harmony in fav
tury because it proved to be inadequate
a source of beauty and harmony. It ac
our of a musical theory based on the
for responding to the new aesthetic
tually sets it in an oriental tradition
continuous sounds perceived by the
trends that were appearing as well as the
considerably older than the Greeks,
ear, as well as on the tensions of the
practical needs of musicians. Those who
that considered number as the handi
strings and their relaxation time
were concerned with the tuning of their
work of God who ordered all things in
shows the rich diversity of musical
keyboard instrument were led to con
measure, number, and weight (Wisdom
thinking in Ancient Greece.
sider the problem of temperament
However, the most interesting as
[5].
Having had one's mind brilliantly
1 1 . 17) and on which all the
of Solomon
work of Man rests. Kronecker's well
pect of this article, whose numerous
stimulated by such an article, one is led
known phrase "God created number,
references provide ample scope for
to wonder about the Byzantine evolu
all the rest is the work of Man," draws its inspiration from the same source.
digging deeper, is certainly the de
tion, geographically so close to the
scription of the rich musical evolution
Greek source; regrettably, however,
In fact, Kircher's book develops the
flowing from the Greek roots into the
this aspect is not touched upon in the
musical ideas of the minim monk Mar
Latin world and right up to the four
article. One wonders too what was
inus Mersennus (Marin Mersenne ) , in
teenth century of our era. In St. Au
the contribution of the ancient manu
particular the combinatory approach
gustine's De Musica, written at the end
scripts passed on by the "sons of the
of the fourth century, rhythms are also
Greeks," as the Arabs of the time called
Harmonia Univer
contained in his
salis,
written in
1636. The article un
classified according to their propor
themselves. Fortunately the article refers
fortunately does not speak of Mersenne's
tions (the proportional notation used
to the influence of Arabian and Persian
activity as the science correspondent
today came much later). Then in the
music in the
of the whole of Europe, nor of his
ninth century, Carolingian policy in
Peninsula, and the contribution of the
creation
educational and ecclesiastical matters
reading of the ancients, thanks to the
Parisiensis, the ancestor of the future
It encouraged
translation in the twelfth century of the
Academie des Sciences; nor does it
musical treatise by AI Farabi.
mention the measurement of the speed
vides a novel answer by introducing
his discovery of the higher harmonics
defmed new practices.
the use of neumes that indicate the in flexions of the voice, but not the pitch of the sounds. The names Do, Re, Mi, Fa,
Sol, etc., appeared with Guido
d'Arezzo in the eleventh century, de
Cantigas
of the Iberian
Eberhard Knobloch's article pro
the concepts of Athanasius Kircher, who in
in
1635 of the Academia
of sound that he obtained in
1636, nor
of a string. No mention is made either
Musurgia Univer
of his systematic use of the notion of
Kircher quotes Hermes Tris
frequency, introduced at the time by
1650 wrote
riving from the syllables at the begin
salis.
ning of the stanzas (voces) of a hymn
megistos, the mystical author who was
Galileo Galilei. Mersenne was a student
addressed to St John the Baptist, writ
so loved by the Medicis and Pico della
of the latter's work and was familiar with the law that gives the frequency
770 A.D. The notes (claves)
Mirandola: "Music is nothing else than
are also designated by letters, a prac
to know the order of all things." This
of the fundamental vibrationf of a vi
tice that is still in use today in English
very pythagorean concept postulates
brating string having a length L with a linear mass p and with tension F:
ten around
speaking countries (La = A, Ti = B,
that Music is a part of Mathematics
Do = C, . . . ) and in Germany (with
(and
some specificities). Finally, polyphony
Kircher, this is a relevant concept
created
harmonic
when seeking to help someone having
mastery, the response coming from
virtually no knowledge of the mastery
This formula is merely mentioned in
Philippe de Vitry in the fourteenth cen
of sounds to acquire an in-depth knowl
the
edge of musical composition. Pythago
Galileo, the son of the musician Vin
ras doubtless would have disowned
cenzo Galilei, and it was written in this
new
needs
for
tury with his Ars Nova: in this work he
defined new musical notations as well
70
THE MATHEMATICAL INTELLIGENCER
consequently
a science).
For
f=
Discorsi,
� If·
2
written
in
1638 by
modem form only in 1715, by Brook Taylor. The limited part of Mersen ne's work mentioned in the book is nonetheless of major interest and sets the record straight regarding a nwnber of misconceptions as to the history of science at that time. In his Harmonia Universalis, Mer senne sets out the table of all the values of the nwnber of permutations with n elements up to n 64. He discusses non-repetitive arrangements P(n, p) n(n - 1) . . . (n - p + 1) and combina tions C(n, p) = P(n, p) P(n, p)/P(p, p). He solves the problem of calculating the number of combinations presented by a given type of repetitions. This he does thirty years before Leibniz succeeds in obtaining, with a few errors, the same results in his "schoolboy's essay" De Arte Combinatoria, and well before the combinatorial work of Fermat and Pascal. If n is the maximum number possible of notes for a song composed with p different notes, of which r1 dis tinct notes appear once, r2 distinct notes appear twice, . . . , and using in fact r r1 + r2 + . . . + r, distinct notes in all, Mersenne gives the total number of possibilities for the corre sponding songs:
tury bell-ringers and the rules laid down by Fabian Stedman [7] is but a short step, but one which none of the articles in this book dares to take. On the other hand, the approach taken does shed light on musical analysis, as may be seen in the article by Laurent Fichet, and makes it possible to extend one's horizon, as in Marc Chemillier's article dedicated to ethnomusicology. The formula mentioned earlier re lating to the fundamental frequency of a string, in tum allows a better under standing of the problem of tempera ment. It consists of seeking to divide an octave into twelve equal intervals, and therefore to identify rational num bers that simultaneously come as close as possible to the irrational real num bers 2(1/1 2), 2(2/1 2) , 2 (3/12) , . . . , 2c11112) , being aware that a trained ear will per ceive any deviation that is too signifi cant. This leaves plenty of margin for numerous systems, and the remarkable article by Benedetto Scimeni pre sents the choice proposed by Gio seffo Zarlino in his work Le Istitutioni harmonicae [8], published in 1558:
n! r1 !r2 ! . . . r,!(n - r)!
Galileo's father quarrelled with Zarlino because he preferred 18/17 to 10/9. But of course, whatever choice one makes, the practical issue is the tuning of instruments, in particular harpsichords with several octaves and the largest possible number of tones. The article referred to here mentions the remarkable work undertaken on these questions by Giuseppe Tartini, Daniel Strahle, and Christoph Gottlieb Schroter. One of the most fascinating aspects is the connection with the so lution to Pell!Fermat's equation in Tar tini's Trattatto di Musica:
=
=
=
=
For 22 possible notes, of which 7 dis tinct ones are repeated according to the type 2, 2, 1, 1, 1, 1, 1, Mersenne shows that there are 3,581 ,424 possible songs. Is it therefore not understand able that it was the analogy with the combinatorics derived from gaming that led Mozart to devise a musical game allowing the players to produce waltzes by throwing dice [6]? This then poses the question of the link between musical creativity and chance. One may indeed wonder if certain of Haydn's compositions were not in spired by similar methods. This point is not mentioned, even though his 41st piano sonata is quoted in the article by Wilfrid Hodges and Robin J. Wilson dedicated to musical forms. Speaking of combinatorics raises the possibility of using the group of permutations of objects that one arranges and com bines. . . . From there to seeing Galois's theory in the practice of sixteenth cen-
10/9, 9/8, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 16/9, 9/5.
x2 - 2y2
=
1.
In fact, this becomes obvious when one realises that the above also leads to coming as close as possible to the irra tional 2C6112) = v2 with a rational nwn ber, a classic problem of diophantine analysis, which is much simpler than the previous problem of simultaneous approximation of the twelfth roots of 2. The book is incomplete if one con siders it from the point of view of the
history of acoustics, a word invented by Joseph Sauveur, who professed mathematics at the College de France from 1686. Dumb until the age of seven and deaf for the whole of his life, it is he who looked more closely at the ob servation made by Mersenne that there exist higher harmonics: a string may vibrate in several parts around nodes that remain fixed. The book makes scant mention of the work of Bernoulli or Euler. It remains almost completely silent concerning the discovery in 1 747 by d'Alembert of the partial differential equation of vibrating strings: a2y
-
at2
=
�
a2y
-
ax2
·
It is in fact the solution of this equation that makes Sauveur's discoveries un derstandable. However, research ac tivity on sound was so extensive at the time that to describe it would be an al most impossible task We would need to mention Wallis, Newton, La Hire, not forgetting Bach, Rousseau, and so many others; one would necessarily have to be selective. The selection made in the book is particularly rele vant, but makes one want a new Fo rum, to take the question deeper by re ferring to the activities of other authors who have been left out. The article by Jean Dhombres ex plores another major historical mile stone, by referring to the interest shown by Lagrange around 1760 in musical texts and the theory of instruments. In his Recherches sur la nature et la prop agation du son he gives a definition of the integral of a function as a limit. No more nwnber theory and geometry. He shows that the same differential equa tion appears in the vibrations of strings and those of air. He discovers the or thogonal relationship of sine and cosine. Yet Lagrange cannot be considered to be the inventor either of series or of the Fourier analysis. It was indeed Fourier who recognised the universality of the calculus discovered by Lagrange in his study of musical sounds. From a math ematical point of view, the next stages in this millennia! adventure, which are not covered in this book, are the march towards distributions [ 10] and the deeper understanding of spectral analy sis [11] and group representations [ 12].
© 2005 Springer Sc1ence+Bus1ness Media, Inc., Volume 2 7 , Number 3, 2005
71
Today the Music of corpuscles and solitons is taking the place of the Mu sic of spheres and mermaids. Consid erations of the multiple infinitely small (chaos?) are replacing those on the sin gle infinitely great (the cosmos?). The bifurcation took place at the end of the eighteenth century, at the very moment when musicians were being pushed into the category of artists, whose role was to provide pleasure for the pre sent, and mathematicians into the cat egory of scientists, building the society of the future. The remainder of the book presents four articles by Giovanni De Poli and Davide Rocchesso, by Erich Neuwirth, by Xavier Serra, and by Jean-Claude Risset on the application of modem digital sound technology. There are new acoustic domains being explored, such as the impact of non-linearity, the hearer's perception, the use of computerised toolboxes to produce sounds, texture compositions, and acoustic illusions. The proliferation is huge and shows how that mathemati cal machine par excellence-the com puter-is invading music. Far from slackening, the interaction between the two fields is continuing to develop and is as strong as ever. The major change seems to have been that math ematics now has its instruments computers--whereas classical musical instruments are left standing. Experi mental practice seems to have provi sionally changed sides, but the process of mutual enrichment is continuing [ 13] . It is consequently natural to ask about the logic and meaning behind this evolution of the two fields [ 14] . Logic has always been essential to mathematics, but in the recent period it would appear to be less natural in music. Of course, one may consider the computerised machine learning music, as do Shlomo Dubnov and Gerard As sayag. But does this have anything to do with logic? Another article by Marie Jose Durand Richard, which retraces the history of logic, shows that the is sue isn't clear. It refers us to the arti cle by Franc;ois Nicolas dealing with just that question: What is the logic in music? The answer given by Nicolas is as anti-pythagorean as could be, be cause it results in the impossibility of
72
THE MATHEMATICAL INTELLIGENCER
defining this concept today, and hence leads to falling back on the study of the practices involved in musical produc tion, free from mathematical, acoustic (physical), and psycho-physiological tutelage. Such a loss of meaning is in total contradiction of the tradition of a relationship between music and logic, as illustrated in the double organisa tion of ancient knowledge of liberal arts in the form of trivium (grammar, rhetoric, logic) and mathematical quadrivium (arithmetic or the number in itself, geometry or the number in space, music or the number in time, as tronomy or the number in space and time). Yet it is thoroughly contempo rary. It also sets itself completely apart from the theories that Marin Mersenne proclaimed in his Traite de l'harmonie universelle published in October 1627 under the pseudonym Franc;ois de Ser mes [15]. Theorem 1: Music is a part of mathematics and consequently a sci ence, capable of showing the causes, effects, and properties of sounds, songs, concerts, and anything related thereto. Theorem 4: Music is both a speculative and practical science, and an art, and consequently is a virtue of understanding, which it leads to the knowledge of the truth. In this under standing, which one may consider to be outdated (wrongly, for the joint evo lution of the two fields is continuing, as the present book shows), the logic of music finds profound meaning, which the author of the article submits to the meditation of its readers. To this end, the book contains one last article that I have not yet mentioned. It is by Guerino Mazzola and is titled The Topos Geometry ofMusical Logic. (See the review by Shlomo Dubrov, this is sue.) On the mathematical side, he relies on the theory of categories and Grothen dieck constructions; on the musical side, on Riemann's harmony (not G. F. Bern hard but K. W. J. Hugo, i.e., not the math ematician but the author of Mathema tische Logik published in 1873!). He develops a Galois theory of musical con cepts which locks Beauty and Truth into the same kingdom. So might there after all still be some pythagoreans in our day and age, lost among our contempo raries, Guerino being one of them? At any rate, his article is fascinating from
an intellectual point of view. He con firms that the new alliance between mu sic and mathematics announced by Pierre Lamothe in 2000 on his Web site is forging ahead, though using paths other than those he had envisaged [16]. This new alliance between pleasure and science cannot but enrich both parties. It might even constitute a rem edy for the desertion from mathemati cal studies observed in our times, when knowledge and work are parcelled out piecemeal. The pleasure derived from reading this remarkable work is very great. The reviewer is convinced that other Mathematics and Music initia tives need to be taken, and that there is no lack of topics to be covered. Acknowledgement
I am indebted to the non-mathemati cian Andrew Wiles for the English translation. BIBLIOGRAPHY
(1 ] http://www. medieval.og/emfaq/harmony/ pyth.html [2] J. G. Roederer, Introduction to the Physics and Psychophysics of Music, Springer Verlag, New York and Berlin, 1 975. [3] N. H. Fletcher, T. D. Rossing, The Physics of Musical Instruments, Springer-Verlag, New York and Berlin, 1 991 . (4] P. Bailhache, Une histoire de J'acousti que musicale, CNRS editions, Paris, 2001 http://bail hache. human a. u niv-nantes. frI thmusiques [5] E.
Neuwirth,
Musical
Temperaments ,
Springer-Verlag, New York and Berlin, 1 997. [6] http://sunsite. u nivie .a c. at/Mozart/dice/ mozart.cgi [7] W. T. Cook, Fabian Stedman, 1 64Q-1 730, Ringing World, 29 October 1 982, pp. 900-901 . (8] http://virga.org/zarlino/ (9] J . Cannon, S. Dostrovsky, The Evolution of Dynamics: Vibration Theory from 1 687 to 1 742,
Springer-Verlag,
New York and
Berlin, [1 98 1 ] . [ 1 0] J . LUtzen, The Prehistory of the Theory of Distributions, Springer-Verlag, New York and Berlin, 1 982. [1 1 ] C. Gordon, D. Webb, S. Wolpert, You cannot hear the shape of a drum, Bull. Amer. Math. Soc. (N.S.) 27, pp. 1 34-1 38, 1 992. [1 2] A. W. Knapp, Group representations and
harmonic analysis from Euler to Lang
into notebooks known as "common
truth-values by a "subobject classifer,"
lands, Notices Amer. Math. Soc. 43,
place books." These notebooks were
which is something more general than
1 996, pp. 4 1 0-41 5, pp. 537-549, http:�
commonly indexed and arranged for
the Boolean algebra of True and False.
www.ams.org/notices/1 99604/knapp.pdf
easier reference, and maybe it is not sur
This,
[1 3] http://cnam .fr/bibliotheque/tables/table
prising that the book by Mazzola opens
Grothendieck on algebraic geometry,
by placing music in a new "encyclo
allows Mazzola to define complex mu
0302.html
together
with
the
work
of
[1 4] R . Steinberg (ed.), Music and the Mind
space," a space where human knowl
sical structures of Global Music corn
Machine, Springer-Verlag, New York and
edge production is assumed to be cou
positions, with earlier categories being
Berlin, 1 995.
pled to navigation in a topologically
embedded as "patchworks" of local ob
arranged concept space.
jects in a global theory, leading to cat
[1 5] Marin Mersenne, Traits de l'harmonie uni verselle, Corpus des oeuvres de philoso phie en langue franr;:aise, Fayard, 2003. [1 6] http://www.aei.ca/�plamothe/
In mathematics, category theory is known as a study of abstract mathe matical structures and relationships.
egorization of music as constructions on geometric manifolds. The book opens with a very gen
Groups are often used to describe sym
eral,
metries of objects, and they were used
tion, which seems vague or somewhat
philosophical-historical motiva
Conseil scientifique de France Telecom R&D
by music theoreticians for describing
ambiguous, to provide an intuitive basis
38-40 rue du General Leclerc
musical
scales,
for dealing with the forthcoming for malisms. In Part II the author goes from
properties
such
as
92794 lssy les Moulineaux Cedex 9
pitch classes, or rhythms. Every ele
France
ment of the group creates a corre
e-mail: serge.
[email protected] spondence to some other set of ob
tions to very abstract concepts of forms
jects, and Cayley's theorem states that
and denotators, assuming prior knowl
every group
The Topos of Music: Geometric Logic of Concepts, Theory, and Performance by Guerino Mazzola, with Stefan Goller and Stefan Muller BIRKHAUSER VERLAG, BASEL. BOSTON. BERLIN. 2002 1 368 PP. Hardcover. rncl. CD-ROM. € 1 28 ISBN 3-7643-573 1 -2
REVIEWED BY SHLOMO DUBNOV
I
n the context of classical Greek phi losophy, a
topos
(literally "a place")
G
is isomorphic to the
gories,
Yoneda lemma in category theory is a
reader to "recall" these concepts from
generalization of Cayley's theorem that
appendix G would probably require also
allows the embedding of any category
"recalling" earlier concepts from ap
into a category of mappings (called
pendices C-F on set theory, rings, alge
functors) defined on that category. Us
bras, and algebraic geometry, and so on.
and logic.
Asking the
Part III of the book deals with the
jects, categories of musical composi
next level of describing musical con
tions are defined as elementary objects
structs, such as scales and chords,
of music. Then, describing the Yoneda
terming them
Perspective, Mazzola claims that in re
This brings up a discussion of musical
lation to the arts, "understanding paint
symmetries in the local composition
"local
compositions."
ing and music is a synthesis of per
objects, such as Messiaen modi and se
spective
rial techniques. But there appears to be
variations."
Therefore,
by
considering functors as the represen
a deeper aspect of local composition
tations, one is led to defining art and
related to the use of functors and their
music as a set of operations (symme
concatenations, needed in preparation
tries) that leave the object invariant.
for Part IV. This aspect (culminating in
Music composition becomes the "in variant" or the "essence" of a set of per
the art of oration or per
topoi,
ing denotators to describe musical ob
referred to a method of constructing
rhetoric,
edge of advanced concepts of cate
group of its symmetric operations. The
and presenting an argument, being part of
concrete examples of note representa
formances,
the Yoneda Perspective) employs the
fact that in the denotator representa
an idea related also to
tion one has the mathematical struc
suasion. Plato greatly opposed rhetoric,
Adorno's esthetic principle in music.
ture of a topos, which offers properties
claiming that it values style or manner
The emphasis is on the rhetoric func
such as unions, products,
of persuasion over the discussion of
tion "as a means to express under
(somewhat as in set theory), and al
or limits
substance. Then carne Aristotle's "rec
standing, and in this respect perfor
lows for enumeration or classification.
onciliation" of rhetoric and dialectic
mance is not only a perspective of
Musical or visual examples could help
saying that while dialectical methods
action but instantiation of understand
in clarifying these developments, but
are necessary to find truth, rhetorical
ing, of interpretation given structures."
the author offers rather general claims about the utility of the mathematical
methods are required to construct an
Mazzola further assumes that math
argument in order to communicate it.
ematical study in the context of art will
methods to analysis of an Escher draw
The concept of topos was extended
lead to objects which are "meant to de
ing or appreciation of the fractal Julia
later to literature by a German scholar,
scribe beauty and truth. " This brings
set shape, without much detail. Amer
Curtius, as a study of ways to compile
topos theory to being a way for com
ican Music Set Theory is called "thor
knowledge by selecting and indexing
bining logic and geometry. In topos
oughly out of date from the point of
important phrases, lines, and/or pas
theory one replaces the set by a cate
view
sages from texts and writing them down
gory, function by a morphism, and
conceptualization."
of
20th-century
mathematical
