Letters to the Editor
The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to either of the editors-in-chief, Chandler Davis or Marjorie Senechal.
Euler Diagrams and Venn Diagrams
In his review of my book Cogwheels of the Mind, Hamburger [1] incorrectly says, 'The famous three-circle Venn di agram, which is known to most peo ple, had already been used by Euler.' It does not occur in either Euler's orig inal Lettres a une Princesse d'Allemagne of 1768 or Hunter's English translation of 1 795, the two works cited by Ham burger. It should have done, but it didn't. Euler simply made a mistake in drawing his diagrams for three sets. When he added a third set C to a two set diagram with overlapping sets A and B so as to overlap A partially, he drew the cases C wholly in B and C wholly outside B, but muddled the case C par tially in B, which would have given him a Venn diagram for three sets. An 1823 reprint of the erroneous di agram can be seen at math.dartmouth. edu/� euler/, E343 Plate 3 figure 27, which is the same as figure 26. Venn owned a copy of this book; I have ex amined it and he did not mark the er-
ror. Recent editors of Euler's work have corrected it. The relationships between Euler di agrams and Venn diagrams were well understood by Venn himself and are, I believe, correctly described in my book. Euler did draw a Venn diagram for two sets, but then, so did writers in the eleventh century [2] . In other respects it is best to draw a veil over Hamburger's tirade against my "dabblings." REFFERENCES
[1] P. Hamburger, "Cogwheels ofthe Mind. The Story of Venn Diagrams
by A.W.F. Ed
wards" (review), The Mathematical lntelli gencer 27 (2005), 36-38.
[2] C. Nolan, Music theory and mathematics, in T. Christensen (ed.) The Cambridge History of Western Music Theory,
Cambridge Uni
veristy Press (2002), 272-304. A.W.F. Edwards Gonville and Caius College Cambridge CB2 1 TA, U.K. e-mail:
[email protected] ERRATUM Thanks to Gregory Kriegsmann of New Jersey Institute
of Technology for spotting an error in the article "e: The Master of All" by Brian J. McCartin (Math Intelligencer 28
(2006), no. 1, 10-21). In equation (5), the denominator reads (2k)!; it should be (2k+1)!.
The author acknowledges the typographical error, and reassures us that the erroneous formula was not used in arriving at Figure 2. Just so! If it had been, the figure would have made the claims of great accuracy for the formula (5) looked odd indeed.
© 2006 Springer Science+ Business Media, Inc., Volume 28, Number 3, 2006
3
An Elementary Proof of the Gregory Mengoli-Mercator Formula In this note we present a novel, elementary proof of "the remarkable formula
1
-
1
- +
2
-
1
1
- - +
3
4
-
. . .
=
log 2
(1)
'
one of the relations whose discovery made a deep impression on the earliest pi oneers of differential and integral calculus" (Richard Courant [C)). The formula goes back at least to Pietro Mengoli (1626-1686) [Men] , James Gregory (16381675) [G], and Nicolaus Mercator (1620-1687) [Mer] ; it provided an unexpected link between the antique world view, with its well-ordered Pythagorean natural numbers, and the emerging seventeenth-century culture of mathematics as a tool for exploring the material world, with its transcendental concepts of logarithms, 1 infinitesimals, and the number e. In modem times, the left-hand side of (1) is likely to have been first encountered by readers of this note in an undergraduate course on calculus or analysis, in the early chapters on sequences and series: it provides a classic example of a series which is convergent but not absolutely convergent. We are taught why the limit ex ists, usually in connection with Leibniz's more general criterion that an alternating series whose terms decrease monotonically to zero in absolute value is convergent. But what is the limit? The textbooks inform us that it will later in the course be shown to equal log 2 , using differential/integral calculus. This is somewhat discouraging: neither left-hand side nor right-hand side appears to require cal culus. So why can't one establish equality directly? The answer is that one can. Our proof below requires only the definition of the natural logarithm as the logarithm to base e, i.e., the inverse of the expo nential to base e, log X= , e X
(2)
and Euler's famous formula well known from compound-interest calculations that e=
(
lim 1 +
n�(l;
)
_!_ n n
(3)
(one of several standard definitions of e). We do not require tools from calculus, unlike the usual derivations. One of the latter starts from the power series representation of the logarithm, obtained for instance by termwise integration of a geometric series, log(l +
x -dt
x) = L
o
=
1 +t
x2 x3 x4 x- - + - - - +2
3
4
(4)
An additional argument such as Abel's theorem [A] is then employed to justify equality of left-hand side and right-hand side at the point x = 1 , at which the underlying geometric series fails to converge. Alternatively one uses Taylor's the orem (see, e.g., [R]) to obtain the series, and a careful remainder analysis to jus tify its validity at x = 1 . We thank R . Burckel for pointing out another calculus-based treatment: One shows that the quantity DN
=
I
n=I
1 /n -
JN(l! t)dt, which compares a Rie1
mann sum to an integral, converges (the limit being known as Euler's constant) . Using this, the first equation in (4), and the identity (6) below, we have
2
IN
n=l
(-l)n+ljn-
log(2N) + log(N) = DzN- D N �
0, which together
with the addition rule log(2N) - log N = log 2 proves ( 1 ) . See [K].
1Which was already known empirically at the time from the so-called quadrature problem for the hyperbola, as the base which makes the first equation in
4
THE MATHEMATICAL INTELLIGENCER
© 2006 Springer Science+ Business Media. Inc .
(4) below valid.
decreasing, one considers � and argues similarly. ) Con
We proceed to give our own derivation of the following
The alternating harmonic series
THEOREM 1 -+-
converges to log
4
PROOF
sums SN
1.
:=
Preliminaries.
2.
1 1 1 - - +- 2 3
We need to show that the partial
I;Y=1 converge to log 2. Existence of a limit follows, e.g., from Leibniz's criterion, as discussed above. (-1)"+1/n
En+!
sequently we can estimate the right-hand side of (7) from
above and below by
2N
IT
n=N+l
(1 +
) n 1
-
eS,v =
lim
2.
N-->oo
Simplifying the partial sums.
2.
(5)
We note that the alternat
ing harmonic series from 1 to 2N is equal to the non
alternating harmonic series from N+ 1 to 2N,
I
c -l)n+l
n
n=l
=
I
n=N+l
n.
(6)
fractions and negative even fractions, and hence equal to
the sum of all fractions minus twice the sum of the even
3.
2I;Y=1 l/2n.
-
Using the law of exponents.
It suffices to consider, in (5),
partial sums from 1 to even integers 2N, because SzN+l S2N =
1
2N+l
--
ponents
converges to zero. By (6) and the law of ex-
ea+h = eaeh,
the left-hand side of (5) becomes
2N
e52lv 4.
IT e�.
=
(7)
n=N+l
Using a different approximation for each factor. Now the
above factors, in such a way that the exponent 2_ always
e
( 1 +�) n for each n
=
2Nl, and hence, heuristically,
2N
n
ll
elln +l
=
2N
n
ll
+l
1
(1 + ) --;;
+
E {N
1, . . . ,
.
N+l N
inequality to
both converge to
1 +
e by
2N+ 1 2N
en
=
=
) n
(1 +l_
n and E n
:=
(
1 +
n�
J
n, which
( 3), are increasing and decreasing,
) n
(
(1 +l_ (1-
�) )
1 +
(n
I
1 +1
n
+ 1)2
by Bernoulli's inequality ( natural numbers
n.
N+2 N+l
1 "
.
2N 2N-l
--
=
.
2. This reduces the
2N
IT
As N approaches infinity, the term on the left converges to
2, and therefore so does the term in the middle. This es
tablishes the theorem.
As the reader may have guessed, our motivation for devis
ductory comments. We were teaching the contemporary
canon of convergence theory for sequences and series, to
second-month mathematics undergraduates, and wanted to share relation ( 1) with our students.
But since our main arguments are explicit manipulations
of finite sums and products, our proof should be elementary enough to be appreciated without a precise theory of limits.
In fact, it is the latter situation which resonates more closely
with the context of discovery of formulae such as ( 1) by the early pioneers: underlying finite calculations, such as evalu
problems, were based on firm mathematical concepts, and "passage to the limit" was performed intuitively.
2 1 x + m(m1·+1 2 lx +
REFERENCES
[A] Niels Henrik Abel, Recherches sur Ia serie 1 + '!'_
m(m�.�(;+2l x3
+
. . . , Crelles Journal 1, 3 1 1 -339, 1 827. Reprinted
in L. Sylow & S. Lie (eds), Oeuvres completes deN. H. Abel, Tome
1
2: (1 +1_)(1 - -
)
=
Erster Band. Springer 1 955
[G] James Gregory, Vera Circuli et Hyperbolae Ouadratura, in Propria Sua Proportion is Specie,
lnventa & Demonstrata, Padua, 1 667.
[K] Max Koecher, Klassiche elernentare Analysis , Birkhauser, 1 987. [Men] Pietro Mengoli, Novae quadraturae arithmeticae, seu de additione fractionum.
Bologna 1 650.
[Mer] Nicolaus Mercator, Logarithmotechnia, 1 668. [R] Walter Rudin, Principles of Mathematical Analysis, 2nd edition, McGraw-Hill, 1 964 Gero Friesecke
n+ l
n+ l
n-
)!1
[H] Stefan Hildebrandt, Analysis 1 , Springer, 2002
respectively. ( Proof that the first sequence is increasing:
en+l
1
1 1 2- ---� en� 2. N+1 n=N+l
nung.
( �)
n=N+l
1 +
[C] Richard Courant, Vorlesungen uber Differential- und lntegralrech
N+ 2 N+ 3 =---·---· N+l N+2
:=
n=N+l
(
1 , 2 1 9 -250, Grondell & Son, 1 881 .
nlu
But the numerator of each factor cancels the denominator 2N+1 of the next, so the right-hand side equals--, which apN+1 proaches 2 as Nbecomes large. To make this rigorous, we use the well-known fact ( e.g., [ H, Chapter 1. 10)) that the sequences en
IT en � IT
ation of what we nowadays call Riemann sums in quadrature
key idea is to use a different approximation for each of the cancels. If N is large,
2N
1
ing this proof arose in the context described in our intro
l_
This is because the left-hand side is a sum of positive odd
fractions, I�0\ lin
2N
product gives--·--· . . .·
arithm, ( 2), and by continuity of the exponential function x� it suffices to show that
n�
The left product was already evaluated above, and the right
We denote the limit by S. By the definition of the natural log
ex, 2
n
1,
n n +1 1 +x)11 2: 1 +nx for x > -1 and
To show that the second sequence is
Jan Christoph Wehrstedt
Zentrum Mathematik
Zentrum Mathematik
Technische Universitat Munchen
Technische Universitat M unchen
D-85747 Garching b. M unchen
D-85747 Garching b. Munchen
Germany
Germany
Mathematics Institute
e-mail:
[email protected] University of Warwick Coventry CV4 7AL, U.K. e-mail
[email protected] 2or using the more elementary argument that the exponential function is increasing, and that SN < S < SN+J for N even, whence e5N:,; e5:,; e5N+1 © 2005 Spnnger Science+ Business Media, Inc., Volume 28, Number 3, 2005
5
A Heuristic for the Prime Number Theorem
l\ \ ·. \
/
l · ·
HUGH L. MONTGOMERY AND STAN WAGON hy does
e play such
a central role in the distri
bution of prime numbers? Simply citing the Prime
Number Theorem
( PNT),
which asserts
that
A complete proof of Chebyshev's First Theorem ( with
slightly weaker constants) is not difficult, and the reader is
encouraged to read the beautiful article by Don Zagier [9],
T 1 (x) �x/ln x, is not very illuminating. Here"�" means"is
the very first article published in this magazine ( see also
or equal to x. So why do natural logs appear, as opposed
sonable rough approximation to the growth of T 1 (x), but it
asymptotic to" and 1T(x) is the number of primes less than to another flavor of logarithm?
The problem with an attempt at a heuristic explanation is
that the sieve of Eratosthenes does not behave as one might
guess it would from pure probabilistic considerations. One
might think that sieving out the composites under x using
primes up to
Vx
would lead to x IIp 2 n 2:
II
n<po52n
p>
II
n<po5 2n
n =
n1T(2n)-1T(n)
This means that 1T(2 n) - 1r(n):::; (In 4) n/ln n. Suppose n is a power of 2 , say 2k; then summing over 2 :::; k:::; K, where K is chosen so that 2K:::; n < 2K+ l , gives 1r( n):::; 2 +
IK
k� 2
2k -. k In 4
Here each term in the sum is at most 3/4 of the next term, so the entire sum is at most 4 times the last term. That is, 1r( n):::; c n/ln n, which implies 1T (n)/n� 0. For any pos itive x > 0, we take n to be the first power of 2 past x, and then 1r(x)/x:::; 2 1T( n)/n, concluding the proof. D By keeping careful track of the constants, the preceding proof can be used to show that 1r(x):::; 8 . 2 x!ln x, yield ing one half of Chebyshev's first theorem, albeit with a weaker constant. The second lemma is a type of Tauberian result, and the proof goes just slightly beyond elementary calculus. This lemma is where natural logs come up, well, naturally. For con sider the hypothesis witii loge in place of ln. Then the con stant In c will cancel, and so the conclusion will be unchanged!
LEMMA 2 If W(x) In x,
then
W(x)
�
is decreasing and f3 W(t) in(t)/t dt
�
1 /ln x.
······························································································································································································································································
HUGH L. MONTGOMERY studied at the Uni
versity of Illinois and the Universrl:y of Cambridge.
He worl<s in analytic number theory, particularly on distribution of prime numbers. Department of Mathematics University of Michigan Ann Arbor, Ml 481 09 USA e-mail:
[email protected] STAN WAGON studied at McGill in Montreal and
at Dartmouth. Much of his work currently is on
using Mathematica to illustrate various concepts
of mathematics, from the Banach-Tarski Paradox
to dynamical systems; see, for instance, his recent
lnte/ligencer cover (vol. 27, no. 4). Another enthu
siasm of his which has been reported in this mag
azine is snow sculpture (see vol. 22, no. 4). Department of Mathematics Macalester College, St. Paul, MN 55 I 04 USA e-mail:
[email protected] © 2006 Spnnger Science+ Business Media, Inc., Volume 28, Number 3, 2006
1
PROOF. Let
E be small and positive; let
hypothesis implies g'
E In x-
r :
2 tox andx tox 1+€).
r X
+ l € In
f
t
dt = E
( + �)2 1
g�-, W(t) j(t) dt -2 E
1
'\' L x �
In x shows that
.
THEOREM If x/7T(X) is asymptotic to an increasing function, then 7T(x )-x/ln x.
PROOF Let L(x) be the hypothesized increasing function and
to x, and use the fact
r 7T(t) j'(t) dt 2
=
following sequence, which reaches the desired conclusion by
=
2
holds, for then
L(x)- In x.
x � 0, then Inx - g(x); we will use this several times in the
pk II n in the third ex largest power of p that divides
a chain of 11 relations. The notation
n;
the
the equality that follows the
ing each pm for 1 In x =
_!_ X
-1
_!_ I
I I
n �x pklln
sum comes from consider
n
(can be done by machine;
note 1)
k In p
_!_
=
I
I
1
'\'
X n�x pmln,m""i
X
'\'
-I f(p) + p:5x
_
II
I
In
pm:sx, 2:Sm
p!
-
I f(p)
p:sx
7TclxJ) j(x) - r 7T(t) j'(t) dt 2
_
r2 7T(t) j'(t) dt
x - - { t W(t)
)2
- {
x
)2
In
W(t)
In
t
( __!_t2 - t2 t ) dt In
t
dt
X
(error is small;
note 2)
(geometric series estimation; note 3)
(partial summation; note 4) because 7T(lxJ) j(x)/ln x :5 ?T(x)/x-> 0 by Lemma 1)
fn+l 7T(t) j'(t) dt n x
7T( lxJ) (j(x) - JClxJ) ) + I �� � 1 7T(n)(j(n + 1) - j(n))
7TclxJ) j(x) - Ip�x j(p)
L t Wt( ) �t dt yields W(1/xx)/x In x 2 1
5. L'Hopital's rule on --
x
W(x) = 1/L(x), which approaches 0 by Lemma 1.
No line in the proof uses anything beyond elementary cal
culus except the call to Lemma 2 . The result shows that if then that function must be asymptotic to x/ln x. Of
course, the PNT shows that this function does indeed do
the job.
This proof works with no change if base-c logarithms are used throughout. But as noted, Lemma 2 will force the natural log to appear! The reason for this lies in the indef
inite integration that takes places in the lemma's proof.
Conclusion
Might there be a chance of proving in a simple way that
x/7T(x ) is asymptotic to an increasing function, thus getting another proof of PNT? This is probably wishful thinking.
of x/7T(x) ( precise definition to follow ). The piecewise linear function L(x) is increasing because x/7T(X) � oo as x � oo. Moreover, using PNT, we can give a proof that L(x) is indeed asymptotic to x/7T(x). But the point of our work
But it is intriguing to see that
If
Mathe
this, using symbolic algebra. The sum
Mathematica
quickly returns 1 when asked for the limit of x In x/ln(x!) as x � oo.
8
1
function. Let L(x) be the upper convex hull of the full graph
D
(l'H6pital, note 5)
test ideas: start with the fact that the sum lies between
is just lnClxJ!) and
llxJ
lx J
7T(t) J' (t) dt + In-2 _-
However, there is a natural candidate for the increasing
(because 7T(t) - t W(t))
1. It is easy to verify this relation using standard integral
t dt and x In x. matica can resolve
x
7T(x),
p
NOTES
In
to x as a
there is any nice function that characterizes the growth of
In p [----;]In p;;L pm X pm-:::;L ; , J=::;;m x, l:Sm pm:::;x P
X
=
In
X n-:::;x
::s m ::s k.
2
n + 1] together with one from lxJ that 7T(t) is constant on such inter
vals and jumps by1 exactly at the primes. More precisely:
ln(t)lt. Note that if lnx- g(x) + h(x) where h(x)/ln
that the hypothesis of Lemma
k is
Lemma 1.
'\'oo In p In n p(p-1 ) :::; Ln�2 n(n-1 ) oo (n -1 ) 1/2 x1 s o that 1T(x1) < (1 + E) In x0 . It follows that A(.x{)) contains B: be yond .x{) this is because x0 > x1 ; below x1 the straight part is high enough at x = 0 and only increases; and between x1 and .x{) this is because the curved part is convex down,
In
•
by the choice of
�
REFERENCES
1.
D. Bressoud and S. Wagon, A Course in Computational Number Theory,
Key College, San Francisco, 2000.
2. R. Courant and H. Robbins, What is Mathematics?, Oxford Univer sity Press, London, 1941. 3.
J. Friedlander, A. Granville, A. Hildebrand, and H. Maier, Oscillation tions, J. Amer. Math. Soc. 4 (1 991) 25-86.
4. X.
Gourdon and P. Sebah. The 7T(X) project, http://numbers.
computation. free.fr/Constants/constants. html.
5. G. H. Hardy and E. M. Wright,An Introduction to the Theory ofNum bers,
4th ed. , Oxford University Press, London, 1 965.
6. I. Niven, H. S. Zuckerman, and H. L. Montgomery, An Introduction to the Theory of Numbers , 7.
lf-TEX Typesetting
•
2nd ed. , Wiley, New York, 1991.
G. Tenenbaum and M. Mendes France, The Prime Numbers and their Distrib ution ,
Amer. Math. Soc., Providence, R . I . , 2000.
8. S. Wagon, It's only natural, Math Horz i ons 13:1 (2005) 26-28. 9.
D. Zagier, The first 50,000,000 prime numbers, The Mathematical lntelligencer
0 (1977) 7-19.
Animate, Rotate, Zoom, and Fly
ScientificWorkPiace· Mathematical Word Processing
7T(x)
This means that the convex hull of the graph of
The slopes
e ,X{), so this is indeed convex . The PNT implies that for any E > 0 there is an x1 such that, beyond x1, (1 - E )
match at
x1 . x/1T(x) is contained in A(.x{)), because A(.x{)) is convex. That is, L(x) ::5 (1 + E) In x for x 2: x0. Indeed, (1 - E) In x ::5 x/7T(x) ::5 L(x) ::5 (1 + E) In x for all sufficiently large x . Hence D x/1r(x) L(x) .
would be, and that dominates
theorems for primes in arithmetic progressions and for sifting func
A(.x{)) whose boundary consists of the positive x-axis, the
(1 + E) In
and so the straight part is above where the curved part
New in Version 5.5
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Select and right click a word for links to internet
• import 1 To IMk• an -*"'t•CS tlbt plot 1 Type an upras.sion in one cr two wriables 2. With the W.sertlon poW WI the opression,. 3 Open the Plot Properties di*>g ID1,. on 1ha slm8variablli!ls The not �on shows a knot being�. � Plot 3D Animated +Tube
r
-10cotl:l'-2�.5tt)+I3N{Uf) -Uooo("')+ 10*'"-2.m(S:.)
l
Third Edition
The Gold Standard for Mathematical Publishing Scientific WorkPlace
......., -
�-.......,..
..
D""
'
•n�
5.5 makes writing,
baring, and doing mathematic easier. A click
of a button allows you to rype et your
documents in 1 f3n+1 2:: bn+ 1
(*)
and
K
=
f3n+1 · A n + A n- 1 f3n+1 · En + En- 1
A n · En- 1 - En · A n-1 '. Cf3n+1 · En + En- 1 ) - En
'
thus
i.e.,
(1) Since En- 1 > 0, there is also the slightly weaker upper bound (2) Furthermore, since bn+ 1 ::::: 1 ,
I
(3)
An
En
-
Kl
1 ((b n+ l + 1) · A n + A n- l ) ' B n
------
1
1
i.e.,
(A n+ l + A J · En
I ( �: ) I
(4' )
-
1 _____ K /K > -___(A n + l + A J · E n
These bounds (4'), ( 1 ') , (2'), and even (3') are indeed sometimes quite good, as Table 1 shows. Table 1. Error bounds of Wallis's continued fraction for the calculation of "'
(4')
n
+
(An+1
( 1 ')
(2')
An+1 · Bn
bn+1 · An · Bn
relative error
1(:: -+KI
A,) · Bn
(3')
1
1
4 . 00000 . 1 0-2
4.50703 . 1 o-2
4 . 54545
4.024 1 4 . 1 0-4
4.02499 . 1 o-4
4 . 29000 . 1 0-4
4.32900
2
1 .371 22 . 1 0-5
2 . 64896 . 1 0-5
2.65746 . 1 0-5
2 .83302
3
8.48081
1 o-8
8.49137 . 1 0-8
8.50976 . 1 o-8
4
1 .45001 . 1 0- 1 0
1 .83948 . 1 0- 1 0
2.89509 . 1 0-1 0
2 . 90497 . 1 0- 1 0
2 .90497 . 1 0 - 1 0
0.96284 . 1 0- 1 0
1 . 05560 . 1 0- 1 0
1 .44508 . 1 0- 1 0
2.88529 . 1 0- 1 0
2.88529 . 1 0- 1 0
0
5
·
In our example, we have for the fourth approximation
355 113
1 o-2
1
An · Bn
·
4 . 76190 . 1 0-2
3. 33333 . 1 0- 1
·
1o-4
6.49351 . 1 o-3
·
1 o-5
2 .83302 . 1 o-5
8.5371 0 . 1 0-8
2.49283 . 1 o-5
,
the true relative error 8 . 49137 · w - 8 , and the upper bounds (1') (2') (3')
103993 . 1 1 3
-1292
•
1-
-
355 . 1 13
1 355 . 1 1 3
= =
8. 50976 · w-s, 8.53710
.
w-s.
= 2.49283 · w- 5 .
Clearly, the upper bounds (2') are better, the bigger the partial denominator b n+l is. Thus, two rather near-by situated bounds ( 1 ') , (2') are obtained for n = 3 ( b4 = 292), and for n = 1 (bz = 1 5) . On the other hand, the bound (3') is weaker, the bigger b n+ l is. Nevertheless, this bound is useful, if one does not have at hand more than the denominator and numerator of the approximating quotient.
©
2006 Springer Science+ Business Media, Inc . . Volume 28, Number 3, 2006
41
Since An+ 1 > An holds, another lower bound, often much weaker, is obtained when in the lower bound
(4') An is replaced by An+ 1 . Thus
�
•
�
An+ · Bn
ume 2 8 , Number 3, 2006
43
G ode l 's V i e n na JOHN W. DAWSON J R . , AND KARL SIGMUND
Does your hometown have any mathematical tourist attractions such as
statues, plaques, graves, the cap?
where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical
y August 17, 1939, a European war was imminent. Two weeks before Hitler invaded Poland, Dr. Kurt Godel received a letter from his tailor: 'Sending repaired trousers. As I heard, you will journey to America again. You will certainly need a suit. . . . With German greetings, Decker.' Godel ordered the suit. His journey back to Princeton seemed to offer no problems. On August 30, 1939, a few days after the Stalin-Hitler pact, Godel blissfully announced to his friend Karl Menger his intention of returning to Princeton forthwith, in a letter which, in Menger's eyes, 'may well represent a record for unconcern on the threshold of world-shaking events' (Menger 1994). Two days later, Hitler informed a wildly cheering German Reichstag that 'since 5:45, the fire has been returned . ' Godel's outlook changed drastically. He had to write Oswald Veblen in No vember: 'It now seems likely that I will not be able to come to Princeton this academic year, because it will probably be impossible to obtain a German visa during the war-time. ' Godel was trapped in Vienna. He would spend the next few months in desperate attempts to leave for the US. Against all odds, he finally succeeded. But after the war, Godel would never return to Vienna again. He was through with it.
museums, and are shown the many dwellings of Beethoven and Mozart, or the churches where Haydn and Schu bert performed. Increasingly, tours in clude aspects of Vienna between the two world wars, most notably the re cent Leopold museum, with its paint ings by Klimt, Schiele, and Kokoschka, or the architectural monuments of Red Vienna, or the art deco villas built by Hoffmann and Loos. If, as a tourist, you relax in a coffee-house between visits to these sights, you will already be very close to Godel. Let us pick up his trail, a map of which appears at the end of this article. (For more details, see Daw son 1997.) Kurt Godel's parents were well off his father was manager, and part owner, of a textile firm in Brunn, a charming little town which used to be called 'the Czech Manchester', less than two hours by train north from Vienna. In 1919, the treaty of St. Germain had established a border between Austria and Czechoslo vakia, but for the large German-speak ing segment of what was by then Brno, Vienna as the former capital was still the focus, and obviously the place to go to study. At that time, young Godel could probably not have chosen a site more tailor-made to his talents any where in the world.
Vienna: A Logical Choice
Kurt Godel moved in with his brother Rudolf, four years his elder, who stud ied medicine under the illustrious fac ulty to which Freud had often dreamt of belonging. Kurt first enrolled for physics, but switched to mathematics under the spell of superb introductory lectures on calculus by Furtwangler and a 'survey of the major problems in phi losophy' by Gomperz (Sigmund 2006). In his fifteen years in Vienna, Godel lived in seven different apartments. Tourists will be reminded of Beethoven or Mozart, who also moved a lot. The Godel brothers obviously had a well-
significance, and either a map or directions so that others may follow in your tracks.
Please send all submissions to Mathematical Tourist Editor,
Dirk Huylebrouck, Aartshertogstraat 42,
8400 Oostende, Belgium
e-mail:
[email protected] 44
In 1924, when he arrived in Vienna as an eighteen-year-old from provincial Brno to study at the university, things had looked very different. Vienna had overcome years of hunger and misery, and the economy was picking up. The intellectual and cultural life underwent an amazing flowering. Very soon, Kurt Godel would contribute to it. His work may one day well be viewed as the most lasting achievement of that epoch. Today's tourists to Vienna follow the traces of Habsburg, visit the imperial
THE MATHEMATICAL INTELLIGENCER © 2006 Springer Science+ Business Media, Inc.
Settling Down
demics and students (but brutally dis figured today). Godel was a very quiet young man,
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but not always the hermit he later be came. He studied diligently, and was soon invited to join the Vienna Circle, a brilliant group of positivists gathered around the mathematician Hans Hahn (famed for his work on functional analy sis) and the philosopher Moritz Schlick. Both were professors at the university. The young German philosopher Rudolf Carnap had also just moved to Vienna and joined the Circle. Gbdel's closest stu dent friends were Marcel Natkin and Her bert Feigl, who studied mathematics and philosophy and were disciples of Schlick. They all met, every second Thursday, in a small lecture room of the Mathematis che Seminar (Stadler 2002). Informally, most of them also met at the Josephinum, or the Cafe Reichsrat, Cafe Central, and Cafe Arkadenhof, among other places filled with journals and tobacco smoke. These cafes were crowded with intellec tuals and world-reformers nurturing delusions of grandeur and talking phi losophy, literature, psychoanalysis, eco nomics, or politics late into the night.
A Nervous Splendor
Letter from Godel's tailor, August
Austrian politics was dominated by a fierce struggle between the Social Dem ocrats and the Conservatives. The for mer had the majority in Red Vienna, and were engaged in a sweeping pro gram of social reforms. The Conserva-
1939.
defined image in mind in their apart ment hunts: the seven houses look remarkably alike. All are massive four story buildings erected at the turn of the century, staid and stately. If you have seen one, you have seen them all. Only one of the houses has a plaque com memorating Kurt Godel, but this may change with the 2006 centenary-Gc)del was born on April 28, 1 906, and Vienna is set to celebrate. Most of the houses are close to the university, and especially close to the building on Boltzmanngasse where the institutes of physics and (in Godel's time) the Mathematische Seminar were located. On one occasion Godel lived just across the street, two floors above the Josephinum, one of the most dec orative cafes favoured by Viennese aca-
Cafe Josephinum. © 2006 Springer Science+Business Media, Inc., Volume 28, Number 3, 2006
45
9Jlaffage
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The Godel trail. Godel heard his philosophy lectures at the main building of the university (7), and his lectures on mathemat ics and physics in the complex building (8) between Boltzmanngasse, Strudlhofgasse, and Wahringerstrasse, which had opened in 1 9 1 5 . The Strudlhofstiege is close by.
could not offer him a job, he was forced to remain in Princeton. Moreover, as Rudolf could honestly point out, the German consul in New York had ex plicitly advised against an Atlantic cross ing; and the option of a return journey
sumed their contacts with Austria after the war, hut not Godel. He adamantly refused all honors conferred upon him by the University of Vienna and the Aus trian Academy of Sciences; and when he wrote, 'It seems that Vienna is chang
via Siberia was not open for long.
ing only very slowly, ' he did not mean
After the war, every other Sunday Godel wrote ro his mother ( who had moved back t o Vienna in 1 944, and thu s avoided the expulsion of the German-speaking population from Czechoslovakia) . Although Adele re turned to Vienna a number of times, GC'>del never joined her. In one letter, he confessed to his mother that he had had nightmares about being trapped in Vienna, unable to leave. Fi nally, his mother, by then almost eighty, flew ro New York to visit her famous son in Princeton. Her visit was such a success that it was repeated every second year until her death in 1 966. Both Morgenstern and Menger re-
it as a compliment. And, you may ask, what about the suit Godel had ordered from his tailor Decker� In 1 952 GC'>clel wrote to his mother: 'Decker's suit looks as good as new . · Made of pre-war quality cloth, that was to be expected.
Menger, Karl (1994) Reminiscences of the Vi enna Circle and the Mathematical Collo quium,
Kluwer, Dordrecht
Stadler, Friedrich (2001 ) The Vienna Circle, Springer-Verlag, Wien, New York Dawson John (2002), Max Dehn, Kurt Godel, and the Trans-Siberian Escape Route,
tices AMS 49, 1068-1 0 75
No
Sigmund, Karl (2006) Pictures at an exhibition, Notc i es AMS 53, 426--430
Taussky-Todd, Olga (1987) Remembrances of Kurt G6del, in Gddel remembered: Salzburg, 10-12
July 1983 (ed. P. Weingartner and L.
Schmetterer), Bibliopolis, Naples, 2 9-41
Galland, Louise and Sigmund Karl (2000) Exact Thought in a Demented Time- Karl REFERENCES
Alt. Franz (1998) Afterword to Karl Menger, Ergebnisse eines mathematischen Kol/oqui ums,
ed . by E. Dierker and K. Sigmund,
Springer-Verlag, Wien
Wellesley, MA
Sigmund, Karl A (1995) Philosopher's Mathe matician - Hans Hahn and the Vienna Circle, Math lntelligencer, 1 7(4), 16-2 9
Dawson, John (1997) Logical Dilemmas: The Life and Work of Kurt G6del, A
Menger and his Viennese Mathematical Col loqium, Math lntelligencer 22(3) , 34-45
K. Peters,
Kohler, Eckehard, et a/. (eds) (2002) Wahrheit und Beweisbarkeit,
Hi:ilder-Pichler-Tempsky,
Vienna
© 2006 Springer Science+ Business Media, Inc., Volume 28, Number 3, 2006
55
i;i§lh§i.lfj
Osmo Pekonen , Editor
Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book ofyour choice; or, ifyou would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.
I
� �
ir Roger Penrose, a leading mathematical physicist and a renowned � expositor of science, has taken on an enormous task in writing The Road to Reality, a vast survey of contempo rary mathematical physics. The "the" in the title is perhaps misleading, though, as a single path does not emerge from this book. Rather, the author leads the reader along many avenues in territory that is yet to be mapped. Starting from the birth of Greek science and funda mental definitions of numbers, symme tries, and manifolds, he proceeds to ex plain Relativity, Quantum Mechanics, Gauge Theory, the Standard Model, Quantum Field Theory, the Big Bang, Black Holes, Supersymmetry, String Theory, M-Theory, Loop Quantum Gravity, Twistor Theory, and you name it. The final chapter is tantalizingly en-
The Road to Real ity: A Complete G u ide to the Laws of the U niverse
by Roger Penrose PP.
$30.00,
ISBN:
REVIEWED BY PALLE E.
I
Column Editor: Osmo Pekonen, Agora Centre, 40014
University of Jyvaskylii, Finland
e-mail:
[email protected] -Osmo Pekonen, Reviews Editor
through more than 1 000 pages. I found the book inspiring, informative, and ex citing. Penrose's writing is calm and composed. It is also honest about what mathematics and physics can accom plish. I count Roger Penrose among the most outstanding scientists and exposi tors of his generation.
Aim and Scope
NEW YORK. ALFRED A. KNOPF, I N C . , 2005, xxviii +
1099
titled "Where lies the road to reality?" In a poetic epilogue, an imaginary Ital ian (female) scientist is brought to the scene to settle it all, some day, in per haps a not-so-distant future. In other magazine reviews, and in the blogosphere, the book has gener ated divided opinion. While everyone praises the insightful presentation of some fundamental issues and concepts of mathematical physics, many readers feel unhappy with a certain editorial sloppiness which has marred their reading experience. We exceptionally publish two re views of this exceptional book, reflect ing the extremes of the spectrum of opinion.
0-679-45443-8
T.
JORGENSEN
t's always a delicate balance for a sci ence book: encyclopedic versus well focused on a unifying theme. Pen rose succeeds admirably in striking this balance: the book offers a panorama of science, and its goal is quite ambitious. Yet the presentation is for the lay reader; it is engaging, and certainly not boring. Indeed, there are preciously few authors who manage to guide begin ning students into serious scientific top ics, especially when the aim is the panoramic picture of science. The nar rative flows well; Penrose captured my imagination and held my attention
I admire authors who succeed in com municating math to the man and woman "on the street." You can't argue with success: this book managed to hit a top spot on the Amazon.com best seller ranking, so Penrose must surely be doing something right. Achieving popular appeal with a serious science book is impressive. What author of math or of physics books would not envy this degree of circulation, or even a small fraction of it? While this is a popular book, its goal is not modest: a search for the under
lying principles which govern the be havior of our universe. It raises the ex
pectation level, yet the presentation is modest when modesty is called for. It shouldn't surprise us that the principles Penrose has in mind take the shape of
© 2006 Springer Science+ Business Media, Inc., Volume 28, Number 3, 2006
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mathematics (roughly the first half of the book) and physics (the good sec ond half). Despite the grand and ambitious goal, I didn' t feel cheated. But I was first skeptical as, browsing in my book store, I took the book from the shelf. As I read, I was pleased to find all the equations (not just hand waving). In deed, the reader is first gently prepared with explanations for the technical sec tions. And when the formulas come, the reader is ready and will then want the mathematical equations. They aren't just dumped on you! Penrose's book is likely to help high school students get ting started in science, and to inspire and inform us all. There is something for everyone: for the beginning student in math or in physics, for the educated layman/woman (perhaps the students' parents), for graduate students, for teachers, for scientists, for researchers; the list goes on. I believe Penrose proves that it is possible for one group of readers to be respectful of the needs of another.
What Is the Book all About? It is both a big idea and, especially, a unifying vision. What laws govern our universe? How may we know them? How will this help us understand? Yet despite its vastness, the subject is well organized, and it is fleshed out in the language of science. A small sample of topics from the contents will give you a taste. This is only a sample, as there are 34 substan tial and wide-ranging chapters in all. The roots of science. An ancient theo rem and a modern question. Geometry of logarithms, powers, and roots. Real and complex numbers. Calculus (a re freshing approach, I might add). Func tions and Fourier's vision. Surfaces and manifolds (plus calculus revisited). Sym metry groups. Fiber bundles. Tensor bundles and tensor calculus. Cantor's infinity, Turing machines, and Godel's theorem. The physics topics range from classical (Minkowski, Maxwell, La grange, and Hamilton) to modern, start ing with Einstein's theory of relativity and the pioneers of quantum mechan ics, Bohr, Heisenberg, Dirac, and Bohm. The modern topics further span quan tum field theory, the Big Bang, cos mology, the early universe, gravity, su persymmetry, and they all merge into
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THE MATHEMATICAL INTELLIGENCER
the final chapter, "Where lies the road to reality?" Save the Epilog for art! I believe that this book does a great job in (apparently effortlessly) moving the presentation from high school math to advanced topics (like Riemann sur faces, manifolds, and Hilbert space), and in physics (quantum theory, rela tivity, and cosmology) . In fact, I am hard pressed to come up with a book that is even a close second in this way. It is one of the very few science books of ambitious scope that is not viewed by students as intimidating. Penrose's clever use of Prologue and Epilogue en gaged me as an uninitiated reader. As the book has now become a best seller, I expect that it worked well for other readers too. In fact, Penrose adds an element of suspense, and he man ages to give the book the flavor of a novel. I can't begin to do justice to this book. Get it, and judge for yourself. I will also not give away the ending, other than by saying that the title of the book offers a hint. And you will be able to form your own opinion-your own take-and to shape your own ideas and draw your own conclusions. (You won't be spoon fed!) As with all good and subtle endings to novels, this one can be understood and appreciated on sev eral levels. It is no surprise that one of Penrose's unifying themes is attractive and pleas ing geometric images. They underlie both the mathematics (roughly one third of the book: modern geometry, Riemann surfaces, complex functions, Fourier analysis, visions of infinity), and the physics: cosmology (the big bang, black holes), gravity, thermodynamics, relativity (classical and modern: loop groups, quantum gravity, twisters), and quantum theory (wave-particle duality, atomic spectra, coherence, measure ments). In the case of this book, a line-by line overview from the table of contents is misleading. A compelling feature of the presentation of topics from mathe matics is that it is sprinkled with ex amples from physics. I wish this were done more in standard mathematics texts. Not only does this motivate and illuminate the concepts from mathe matics, it also serves to introduce ideas from physics. For Penrose's grand am bition, this is essential. And as a peda gogical principle it works: the student
will already have seen the areas of physics and cosmology that will be re visited in the second part of the book. Of the author's earlier research pa pers likely to have influenced the theme of the book, I would mention [Pe65] in which Penrose proved that, under con ditions which he called the existence of a trapped surface, a singularity in global space-time must necessarily occur at a gravitational collapse. Roughly, this is when space-time cannot be continued and classical general relativity breaks down. The present book leads the reader on a search for a unified theory combining relativity and quantum me chanics, since quantum effects become dominant at singularities. A second influential paper is [PeMa73l, in which Penrose introduced his twistor theory; again an attempt at uniting relativity and quantum theory. Not surprisingly, this grand mathemati cal scheme is directed at unification, combining powerful algebraic and geo metric tools! While it is true that the book is about the laws of the universe, the reader fa miliar with other Penrose books will probably detect the contours of the au thor's prolific scientific activities span ning several decades, including what is often called "recreational" mathematics. Reflecting on the versatility of Penrose's activities, it is worth remembering some of them: Roger Penrose, a professor of mathematics at the University of Ox ford, is known for his outstanding con tributions to mathematics, to physics (relativity theory and quantum me chanics), and to cosmology. In addition, he has for decades pursued his inter ests in writing and in recreational math, for example geometric tessellations. Penrose tiles-tile systems covering a surface with prescribed shapes, say kites and darts-at first glance seem to repeat regularly, but on closer exami nation do not: they are quasi-periodic. A "hobby" of apparently frivolous geo metrical puzzles, Penrose tiles are actu ally studied by solid-state physicists: some chemical substances are now known to form crystals in a quasi-peri odic manner. Readers interested in math illustra tion might find it intriguing that the au thor and his father are the creators of the so-called Penrose staircase and the impossible triangle known as the
"tribar." Both of these "impossible" fig
would be a commercial hit. I hope that
Department of Mathematics
ures have been used in the work of Maurits Cornelis Escher in his creation of structures such as the waterfall, where the water appears to flow uphill, and the building with impossible stair cases that rise or fall endlessly, yet re turn to the same level.
Penrose's book will now encourage book publishers to give our subject the attention it so richly deserves.
The University of Iowa
The Pictures In fact, I taught a geometry course this semester, and had a hard time present ing Riemann surfaces in an attractive way. It's a subject that typically comes across as intimidating in many of the classical books: take Herman Weyl's, for example. I found the graphics in Pen rose refreshing: his many illustrations are full of his own artistic touch. They are done with flair and are an antidote to the flashy computer-generated color graphics and special effects that are typ ical in textbooks. Readers will probably relate better to illustrations with a per sonal touch: his clever use of shading presents core ideas much better and appeals to imagination. And they are less intimidating: We sense that we our selves would have been able to make similar pencil sketches. Or at least we are encouraged to try! A key to books for the classroom is student involvement. The choice of exercises is essential: they help stu dents-and other readers-become part of the discovery. The pictures and the projects serve to bring to life the underlying ideas. Beginners might oth erwise get lost in the math and the equations, or in the encyclopedic panorama of topics.
Yes, the copy-editing of the book is sloppy. But by now there are websites with endless lists of tiny errors and omissions, misspelled names, etc. This will probably bother a few mathemati cians and other specialists. Given the length of the book, and the realities of science publishing, it didn't bother me. I don't really think that the various cor rection lists are alarmingly long. But it is a sad fact that modern-day book pub lishers tend to skimp on copy editing. The publisher was probably reluctant to spend big bucks on a book with for mulas, perhaps not expecting that it
USA e-mail:
[email protected] Postscript
I discovered The Road to Reality in my bookstore by accident, and I was at first apprehensive: the more than 1 000 pages and the 3.3 pounds are enough to intimidate anyone. But when I started to read, I found myself unable to put it down. And I didn't. I bought it and had several days of enjoyable reading. I am not likely to put it away to collect dust, either. It is the kind of book you will want to keep using, and to return to. Books like this are few and far between. I expect that readers will react dif ferently to the title, to the Prologue, to the math, and to the very ambitious scope. The choice of title gave me as sociations (perhaps intended), bringing to mind Douglas Adams's amusing lit tle book series, Hitchhiker's Guide to the Galaxy. Here's a sample of sub-ti tles in those hilarious books: Life, the universe, and everything and Mostly harmless. Another association was a fa vorite series of mine of popular science books by George Gamow dating back to my childhood. Several of Gamow's books were recently reprinted, for ex ample, Mr. Tompkins in Paperback. I only mention my associations with these more lighthearted books to en courage readers to be realistic in their expectations. Judging by its top rank ing at Amazon.com on release, Pen rose's book gained a rare entrance (for a science book) to the short list of pop ular best-sellers. This is impressive for
Is There Anything for a Reviewer to Complain About?
Iowa City, lA 52242- 1 4 1 9
a math book which is prolix, deep, thor ough-going, and which at the same time tackles serious philosophical questions. While Penrose's book is indeed serious, I didn't mind a personal flair and the warm sense of humor he brings to the subject. Only rarely do science books make me smile. REFERENCES
[I ] Penrose, Roger; Gravitational collapse and space-time singularities. Phys. Rev. Lett. 1 4 (1 965) 57-59.
[2] Penrose, R.; MacCallum, M . A. H . ; Twistor theory: an approach to the quantisation of
fields and space-time. Phys. Rep. 6C (1 973), no. 4, 241 -3 1 5 .
The Road to Real ity: A Complete G u ide to the Laws of the U niverse
by Roger Penrose NEW YORK, ALFRED A. KNOPF, INC., 2005,
1136 PP. $40.00, ISBN 0-679-45443-8 REVIEWED BY ROY LISKER
more accurate title might be "Many Roads to Reality . " Only a hidebound physicist (a class to which Roger Penrose assuredly does not belong) or religious fundamentalist would maintain that there is a unique road to reality. Roger Penrose has gained interna tional recognition for his research into cosmology, general relativity, geomet ric combinatorics, quantum field the ory, differential geometry, and related disciplines. He is passionately commit ted to the advancement of knowledge, with particular emphasis on the pro motion and defense of his own ideas. By turns provocative and profound, a number of them are re-introduced in this book. Penrose's ambitious overview con firms my assessment that the "road to reality" constructed by the theoretical physicists of the last century is a riddle within a quagmire inside the realms of Chaos and Old Night. This is particu larly notable in the final chapters of the book, which deal with quantum field theory, supersymmetry, the elec troweak standard model, loop gravity, string theory, twistors, and related top ics. However, the many subjects cov ered do not gain in transparency through the reading of this mastodonic treatise. In addition, the book suffers from a lackadaisical approach toward disciplines outside his fundamental re search interests. The reader wanders through a wilderness of confusion, pop
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science, tortuous prose, yet also (for those with a command of the basics treated in the first 16 chapters) a host of brilliant insights worthy of a Pen rose. The writing is mediocre at best. It is painfully obvious that lbe Road to Re ality was scantily edited, by both Pen rose and the publisher, Alfred A. Knopf. Promoting it to a scientifically unedu cated and ill-informed general public is at best disingenuous, at worst less than honest. The teacher mode is vintage "blackboard": breezy, redundant, with important and inessential observations mixed together with little attention to degrees of relevance. One finds fun damental ideas (such as "orthogonality" or "open set") hastily inserted as after thoughts following exposltlons in which knowledge of them has been as sumed. Penrose employs Feynman di agrams for two chapters before ex plaining what they are. My negative verdict on the haste, confusion, and sloppiness that charac terizes lbe Road to Reality is reinforced by the fact that Roger Penrose is not confused in the least with respect to its subject matter. Any knowledgeable reader will recognize the powerful command he brings to bear on com plex issues. Very few expositions of the ideas of modern theoretical physics present such a comprehensive vision of the whole picture. A summer's study of it is a valuable experience for someone like myself, that is to say someone with a solid grounding in the basics who, at
one time or another and in a sporadic manner, has had some exposure to the rest of its ingredients. For the select audience with the background for understanding lbe Road to Reality there is much to be learned from it. We therefore examine its positive qualities first. Among them are: (i) Shrewd observations only a Pen rose is likely to make; (ii) unique insights into subjects pre viously mastered by the reader; (iii) searching critiques of the deficien cies of most of the "unifying" the ories, or "theories of everything" of modern physics. (i) For the informed reader Penrose will, from time to time, nonchalantly drop one of his amazingly cogent in sights. For example, on page 153, after
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THE MATHEMATICAL INTELLIGENCER
discussing the unique role of analytic functions in modeling causation, he ob serves that this implies that "informa tion" must be transmitted in discrete packages. An insight from causal de terminism points directly toward quan tum theory. (ii) He wonders, somewhat rhetori cally, if quaternions might be useful to physics, thereby leading us to a num ber of observations that show why they aren't. His treatment of parallel trans port is one of the best I've come across anywhere; his discussions of octonions, Clifford algebras, Grassmann algebras, and Clifford bundles are first-rate. (How useful they can be to anyone who doesn't even know the technical definition of an algebr�not given anywhere in the book-is open to question.) His discussions of gauge connections, covariant derivatives, bun dles, and curvature unequivocally re veal his understanding of these notions. Whether he can convey this under standing to others who don't is another matter. In Chapter 2 Penrose does a great job of presenting a diversity of geometric models (Beltrami, Poincare, Klein, Minding) for hyperbolic geome try. Their relationship to M . C. Escher's woodcuts is cleverly indicated. Com puter transformations of the Escher pic tures allow one to see the special ad vantages of differing representations. The lack of a similar treatment of el liptic geometry weakens the effective ness of presentations, further on, of the properties of spinors, rotations in 3-
space, and projective geometry. Chap ter 17 on space-time is filled with orig inal ideas. Observing that the natural setting for Galilean relativity is a fiber bundle, and that the space-time of Newtonian gravitation is best under stood as an affine space, Penrose shifts the usual emphasis on the metric (which is analytic) to the null-cones (essentially topological) as the funda mental determinant of space-time struc ture. The treatment of Einsteinian space-time that follows is vintage Pen rose at its best. Clear presentations of the Mach-Sehnder experiment (p. 5 13) and the Elitzur-Vaidman experiment (p. 545) provide convincing evidence for two mutually incompatible proce dures at work in our quantum universe: the deterministic "U" process, and the statistical state reduction "R" process.
Penrose's perspective is particularly clear-sighted when he portrays the state of a highly confused science! His cele brated arguments for a "low entropy" Big Bang appear on pages 690 to 7 1 2 . (iii) A series o f critiques o f the "uni fying" theories of modern physics, those which attempt to bring together relativity, quantum theory, and ele mentary particle theory, begins in Chapter 28. They reflect the views and expertise of a mathematician who has thought long and hard about such mat ters. Briefly: On page 755 Penrose di rects the reader's attention to inconsis tencies in inflationary models. On page 758 he shows up the circularity of ar guments based on the Anthropic Prin ciple. The pitfalls in the so-called Eu clidization technique are noted on page 770. A truly impressive analysis of the six basic viewpoints on "quantum on tology" begins on page 786. Inadequa cies in ]. A. Wheeler's quantum foam hypothesis are pointed out on page 861 ; Penrose cogently observes that "quantum fluctuations" are insufficient to account for the formation of the galaxies. Throughout the book, Pen rose expresses his dissatisfaction with theories that require the introduction of new dimensions, such as supersymme try theories and string theory. So the book has many strengths. What's wrong with it? A great deal, un fortunately. In my opinion the negative features of lbe Road to Reality out weigh its merits. These are: (i) The fiction that it can be under stood by someone not familiar with the disciplines it deals with; (ii) the sub-standard writing; (iii) the incongruous manner in which popular cliches about science and scientists are mixed in together with insights at the forefront of the oretical physics; (iv) The bad pedagogy in expositions of unfamiliar mathematics. (i) Penrose claims that lbe Road to Reality can be profitably read by four audiences: The first class consists of people with so little aptitude for math ematics that they have "difficulty in coming to terms with fractions." Yet Penrose assumes quite a lot of prereq uisite knowledge, even erudition, from his readership. This may turn out, how ever, to not be much of an impediment to sales. It has ever been the case that
owning a Bible will be deemed more important than taking the trouble to
How could they have missed so much? Evidently the staff editors at Alfred A.
read it. Next come those who are willing to peruse mathematical formulae but who lack the inclination to verify them. Pen rose claims that the problems presented in the footnotes will help them do this. I did not find them helpful. Many are absurdly easy; others, forbiddingly dif ficult. Third are those readers who do have a mathematics background and wish to use Tbe Road to Reality as a textbook on the applications of mathematics to modern theoretical physics. Unfortu nately all too many of the mathemati cal expositions are flawed. Ideas and demonstrations are presented in the wrong order. The author has a ten dency to correct himself as he goes along. Many treatments are truncated, while a confusing rhetoric contributes to obscure even the best of them. The fourth group of readers consists of professionals working in areas of modern theoretical physics. Penrose is quite correct when he says: "You may find that there is something to be gained from my own perspective on a number of topics." Yet why should an expert spend $40 on a popularizer of a subject he already knows? Either he's already encountered his perspective through Penrose's many lectures, arti cles, and books, or he would probably prefer to visit the library, where he can devote a pleasant afternoon browsing the pages between chapters 26 and 34. (ii) Clear writing is essential to cor rect thinking. It really does matter when a scientific idea is poorly expressed. Readers come away with the sense that they understand certain ideas, only to discover, once they try to use them, the extent to which they have been short changed. This sentence on page 48 sets the tone: "For Hamilton found that ij = kj, ki ji, jk ik, which is in gross violation of the standard com mutative law . " What is a "gross viola tion" of the commutative law? Is it worse than a "genteel violation" of the same law? What is the "non-standard" commutative law? Sentences like this sug gest that little editing was done on a man uscript that Penrose claims took him 8 years to write. In the Acknowledgments, he generously credits Eddie Mizzi and Richard Lawrence for help with editing.
Knopf did virtually no editing, and a half hearted job at proofreading. Publishers of books and magazines tend to choose one of two approaches to scientific texts. The first, of which Scientific American is a notorious practitioner, is to systematically rewrite every manuscript accepted for publica tion in a dull-as-dust, predictable house style. Everything looks and feels as if it came from the same assembly line, and only people interested in a partic ular subject will take the time to read articles about it. The other is to assume that the ideas embedded in the turgid prose are so arcane, that to change even a single word might risk con demnation by the entire scientific es tablishment. This appears to have been the policy of the editors at Knopf. From the many examples of substandard writing I select two:
-
=
-
=
-
Yet, remarkably, according to the highly successful physical theories of the 20th century, all physical in teractions (including gravity) act in accordance with an idea which, strictly speaking, depends crucially upon certain physical structures pos sessing a symmetry that, at a funda mental level of description, is indeed necessarily exact! (page 247) I shall certainly not be able to go into great detail in my description of this magnificent profound difficult sometimes phenomenologically ac curate yet often tantalizingly incon sistent scheme of things. (page 657) An expositor of current science need not be a prose master like Bertrand Russell or D'Arcy Thompson. Prose like this, however, is beneath any accept able standard. (iii) Tbe Road to Reality combines an overly scrupulous concern for pri ority recognition with narratives of the life and thought of major figures, yet it shows little respect for historical schol arship. On page 81 Penrose feels the need to remind us that the properties of what is known variously as the "Ar gand diagram" or the "Gaussian plane" were first discovered by Caspar Wessel (unknown for anything else) in 1 797. To learn more about Wessel, I con-
suited Florian Cajori's History of Math ematics (1980). Paraphrasing page 295: "Caspar Wessel. Employed as surveyor by the Danish Academy of Sciences. Es say on the Analytic Representation of Direction. Published in Vol. V of the memoirs of the Danish Academy . . . article buried for a century. French translation published in 1897 . . . " Hav ing thus restored an unjustly neglected reputation, one might assume that Pen rose would behave with equal concern for such distinguished figures as Pythagoras, Plato, Leibniz, and Aristo tle. Such is not the case. Granted, only the most fastidious math historian would be outraged rather than amused by Roger Penrose's popular science ex humation of the life and works of Pythagoras. From page 5 of the Pro logue one learns that the "sage" Pythagoras maintained a "brotherhood" of 571 wise men and 28 wise women at Croton in southern Italy. On page 1 0 h e gives "dates" for "Pythagoras of Samos" as 572-497 B.C.E. Yet in a foot note he admits that "almost nothing re liable is known about Pythagoras, his life, his followers or their work . " Ig noring his own cautionary note, he then attributes to Pythagoras the dis covery of the idea of mathematical proof! The rest of page 10 is filled with a long list of accomplishments and dis coveries by these anonymous figures. Coming to Plato, the astounding reach of one of the greatest minds in history is encapsulated in a simple thumbnail phrase: Page 1 1 : " . . . Plato made it clear that the mathematical propositions [ . . ] inhabited a differ ent world distinct from the physical world. Today, we might refer to this world as the Platonic world of mathe matical forms. " My objections to Pen rose's one-line description of Platonism are not so much directed at his super ficial version of the "story of philoso phy" as to the obvious disdain that he and many research scientists manifest to ward the kindred disciplines of philos ophy and the history of science. Some how it never seems worth their time and trouble to "get it right. " Despite this, Penrose then devotes all of section 1 . 4 to a travesty o f serious philosophy i n a series of meditations on "the three deep mysteries": the connections between the physical, the mental, and the Platonic mathematical. The shift from arid Pia.
.
© 2006 Springer Science+ Business Media. Inc. • Volume 28, Number 3, 2006
63
tonism to penny-ante neo-Platonism has been made without missing a beat. It's threadbare pop philosophy from the pen of a brilliant mathematician and ma jor scientific figure. Even worse is his portrayal of "Aris totle" and "Aristotelian thought" in Chapter 17. On page 382 he writes: "In Aristotelian physics, there is a notion of Euclidean 3-space £3 to represent phys ical space." Aristotle's dates are 384-322 B.C.E. He directed the Lyceum in Athens, from 335 to 323. Euclid's Ele ments were published in Alexandria around 320, where he had passed 10 years working on them. In fact, Aristo tle's physical universe has very little to do with Euclidean 3-space, and less with any notion of a geometrized "space-time. " Penrose's perspectives o n Galilean, Newtonian, and Einsteinian space-time are less problematic. Not only does he know what he's talking about, he be longs among the handful of modern thinkers with the best understanding of them. I would have liked him to men tion the equally important universe models of Descartes and Leibniz. Des cartes's "horror of the vacuum" has resurfaced in our own day in Dirac's "electron sea," and Leibniz's brilliant critique of Newton's absolute space and time has delivered its delayed fruits in General Relativity. (iv) Many of Penrose's explanations are mystifying to all four classes of read ers, including the experts. Here is a par ticularly glaring example (p. 642): "A general U(2) transformation of the Her mitian matrix (which we must bear in mind involves both pre-multiplication by the U(2) matrix and post-multiplica tion by the inverse of that matrix) does 'churn around' the elements of this Her mitian matrix, in very specific ways, but its Hermitian character is always pre served. In fact this analogy is very close to the way in which U(l) indeed acts in electroweak theory (the only com plication being that we must allow for a linear combination of the diagonal el ements with the trace, in this identifi cation, related to the 'Weinberg angle' that we shall be coming to in §25 .7) . " This paragraph i s incomprehensible to someone who doesn't already know the subject. ( What "analogy"?) There are also notational inconsis tencies, but they are quite minor in
64
THE MATHEMATICAL INTELLIGENCER
comparison with his masterstroke of obfuscation: a frequent appeal to ten sor diagrams, starting on page 241. These are tiny drawings that must look peculiar to anyone not working in knot theory, topological quantum field the ory or the mathematics of " q-deformed matrices. " Totally mystifying to out siders, they reduce the readership to the tiny elite for whom they sometimes simplify the computational tedium as sociated with tensor analysis. Yet on page 260 Penrose allows that his read ers may not even know what a "deter minant" is! After a 4-page crash course on determinants, he proceeds to "prove" the fundamental theorem of determinants (det(AB) = detAdetB, A , B matrices) b y pointing t o a particu larly incomprehensible tensor diagram on page 264! The worst failing in this domain is, to my mind, the absence of any introductory material on topology. In summary, Tbe Road to Reality is really only written for persons who've previously studied most of its contents. It is virtually unedited. It is very com plete; indeed it tries to do too much. It can be used as a reference book at a rudimentary level for professional theo retical/mathematical physicists. For all other classes of readers, unfortunately, it is likely to be a disappointment. 8 Liberty Street #306 Middletown, CT 06457 USA e-mail:
[email protected] 1648), a seventeenth-century monk trained by the Jesuits at the famous col lege in La Fleche. The author, Jean-Pierre Maury, pre sents Father Mersenne as a founding fa ther of scientific research. But what does he understand by "scientific re search", given that no famous concepts or crucial inventions (except the Mersenne numbers) are named for his biographee? The phrase instead stands for scientific communication: Mersenne was one of the most prolific letter-writ ers of his period. Except for some trav els in France and abroad, he lived in his cell at the Parisian convent of his Order, the Minims, receiving visitors, or ganizing meetings between the most fa mous scientists of his time, experi menting, and writing. His role in the history of science can be best described as that of an intermediary, questioning through letters the savants from Leiden to Gdansk, and thus establishing con tacts among the members of a vast com munity of scientists. It has even been said that he originated the seventeenth century scientific community, whose boundaries coincide with those of his network of correspondents. His pub lished correspondence fills seventeen (not eleven, as the readers of the book are wrongly told) volumes, edited in the twentieth century by several genera tions of researchers. Thus Maury offers a highly modern understanding of sci entific research, putting emphasis less on the actors and their results than on their practices: traveling, visiting each
A I'Origine de Ia recherche scientifique: M ersenne
by Jean-Pierre Maury edited by Sylvie Taussig PARIS, VUIBERT, 2003, 311 PP., ISBN: 2-711-75291-7, €33.00
REVIEWED BY JEANNE PEIFFER
.,
•• Li he title of this book clearly reveals .
·
. · ·. .. .
. · · · .
;
its author's aim: to write a biography of Marin Mersenne ( 1588-
other on recommendation of interme diaries, meeting in private academies, experimenting, exchanging informa tion, debating, writing journals, letters, and books, some of them published. Research is pictured as a collective en terprise, a common effort toward the advancement of science. Mersenne's role was pivotal: he can be considered as one of the first mathematical intelli gencers in history. Maury has given his book an origi nal structure. The first part is devoted to three of Mersenne's friends: Pierre Gassendi, Nicolas Claude Fabri de Peiresc, and Rene Descartes, with each of whom he initiated his role as a great communicator in the 1620s. The second part describes Mersenne's activities dur ing the crucial year of 1634. Finally, in the last part, Maury discusses a prob-
!em that played an important role in the creation of modern science: the void. The first part presents young Mersenne's efforts to build his network of acquaintances and correspondents. Each of his friends, Peiresc, Gassendi, and Descartes, enables him to make contact with further savants. Through Peiresc, he is introduced to musicians . like Jacques Mauduit, an actor of Ba ifs academy, to Jean Titelouze, an organ player from Rouen, and to Italian con noisseurs of ancient music, like Gio vanni Battista Doni in Florence or Jacques Gaffarel in Venice. It is espe cially with Descartes, after he left for the Netherlands, that Mersenne achieved his apprenticeship as an "in telligencer". He aimed to be the only link between the philosopher and his homeland. This part of the book also gives the reader a flavor of the scien tific atmosphere in Paris when Mersenne arrived there in 1619. The monk took part in the heated debates on Paracelsian alchemy, and on the phi losophy of the Rosecrucians. In a com mentary on Genesis (1623), he con demns all kinds of heresy in the strongest terms. Maury interprets Mersenne's attacks against the English Paracelsian Robert Fludd as a first sign of Mersenne's future orientation toward science put in the service of religion. The second part, focusing on the year 1634, shows Mersenne's network devel oping, especially in southwest France, where he discussed the problems of the fall of a body or of the pendulum with correspondents like the lawyer Jean Trichet and the physicians Jean Rey and Christophe de Villiers. Maury's book gives a lively account of the outburst of scientific research during the first part of the seventeenth century, and we en counter a number of lesser-known ac tors whose biographies are fortunately provided by Sylvie Taussig in an ap pendix. In two of the five treatises he wrote in 1634, Questions inouyes and
Questions theologiques, physiques, etc.,
Mersenne extended the method of ques tioning he had developed with corre spondents like Descartes, Van Belmont, and Beeckman. He was looking for an swers to unsolved problems from his readers. One year after Galileo's con demnation, Mersenne published the Me chanicques de Galilee, Florentin and sig nificantly contributed to the circu-
lation of Galileo's ideas, especially in France. Also in 1634, Tommaso Cam
enterprise, has not extended this con cept to modern history of science. He
panella escaped from the Roman Inqui sition's prison and came to Paris. Maury
hardly makes any use of the important secondary bibliography; he does not even make the slightest reference to Ar
is excellent in characterizing the relations between the two men, and especially Mersenne's disappointment when he fi nally met Campanella. In his prison, Campanella had hardly been able to come to grips with the quick advance ment of science. Mersenne saw him as a man of the past, while he himself was gathering in his circle the most modern representatives of science and especially of mathematics: Etienne Pascal, Gilles Personne de Roberval, Claude Hardy, Claude Mydorge, etc. He was thus a fore runner of the modern academy. The third part is a study of Mersenne's contributions to a well-known problem in the history of science, the problem of the void: is it possible for any part of the universe to be absolutely empty? The ac cepted view at that time was that there could be no such thing, because of "na ture's abhorrence of a vacuum". Blaise Pascal confirmed the existence of a vac uum in 1648, thus bringing, in the au thor's eyes, the new science born with Galileo to maturity. Mersenne, traveling in Italy in 1644, brought news of Torri celli's barometric experiment to France. He served as an intermediary between Italian experimenters, like Gasparo Berti and Emmanuel Maignan in Rome, for in stance, and his French correspondents, especially the engineer Pierre Petit, who was able to repeat Torricelli's experiment with Etienne and Blaise Pascal in Rouen. It was left to the young Pascal to give an interpretation of the experiment. In 1647, he set a glass tube sealed at one end in a bowl of mercury and showed that the space not occupied by the ris ing liquid was empty. He then designed his famous experiment at Puy-de-Dome, in which he showed that, on a moun taintop, the height of the column of mer cury decreases due to the pressure of the air outside. This story is well known. Maury's narrative takes in a number of lesser known actors, for instance Valeri ano Magni experimenting in Poland, and describes in detail the contacts between the different actors, their travels, en counters, academies, written reports, and letters exchanged. It is a pity that Maury, who excels in giving an accurate picture of seven teenth-century science as a collective
mand Beaulieu, the editor of the last volumes of Mersenne's correspondence and also the author of a biography of Mersenne. Professional historians will be upset by the lack of references, some unfortunate references to "the dark mid dle ages", and the ignoring of their own work. But this book will be read with profit by all those whose interest in sev enteenth-century science is new and who want to know how modern sci ence took its present form. Jean-Pierre Maury, who died in 200 1 , was not able t o finish his book; it has been edited by Sylvie Taussig, a philoso pher and Gassendi scholar. She added footnotes and valuable appendices to the core of the work: a chronology, short bi ographies of all the actors of Maury's story, a bibliography of Mersenne's writ ings, and a substantial afterword putting emphasis on the links between science and religion in Mersenne's work and in the seventeenth century. Centre Alexandre Koyre (CNRS) 27 rue Damesme F-7501 3 Paris France e-mail:
[email protected] Irresistible Integrals: Symbol ics, Analysis and Experiments in the Eval uation of Integrals
by George Boros and Victor H. Moll NEW YORK, CAMBRIDGE UNIVERSITY PRESS,
xiv+306 PP., US $29.20 ISBN 0-521-79636-9 (pbk) REVIEWED BY J. J. FONCANNON
he physicist Richard Feynman once claimed that he acquired his initial professional reputation not as a physicist, but as a redoubtable eval uator of integrals. A colleague would
© 2006 Springer Science+Business Media, Inc., Volume 28, Number 3, 2006
65
approach him with a knotty integral dis covered during the investigation of some physical problem. A couple of hours later Feynman would return the integral, fully evaluated. The donor, typ ically, would react with bemused rev erence. Feynman, who had a mischie vous nature, wa� secretive about how he accomplished what he did, but he later revealed that he had developed an armamentarium of techniques. Promi nent among these was differentiation of the integral with respect to a parame ter to produce a differential equation, which he subsequently solved. Further more, Feynman had acquired an intu itive feel for how to introduce a pa rameter artificially when the original integral lacked one, and then he would work with the modified integral. This approach, of course, constitutes a sub case of a widely recognized mathemat ical ploy: if you cannot solve the orig inal problem, solve a more general problem. Differentiation with respect to a parameter is a simple technique the authors of the present book use often. Other techniques they use are to gen erate a recurrence for the integral and then solve the recurrence, or to employ partial fractions. Many current victories in the evaluation of challenging inte grals have been obtained through the use of modern mathematical develop ments, like group representation theory or arguments based on combinatorial reasoning. However, these accomplish
inition of the integral or in the evalua tion of the integral: Bessel functions, hy pergeometric functions, the error func tion, the exponential integral, sine and cosine integrals, etc . . Hypergeometric functions are mentioned in the present book, but not presented in enough de tail to give the reader a feel for them. All mathematical software has the ca pability of handling such functions, and I find it strange that the authors, who rely heavily on Mathematica in their ex position, have declined to define and provide the properties of such func tions. They utilize only the gamma and beta functions and some related func tions. I fully understand the authors' rea son for not including function-theoretic arguments or more material from spe cial functions, namely, the lack of ex perience among the undergraduates for whom the book was primarily written. But the authors have ended up with a book that will be of only provisional appeal to its chosen audience, and it certainly will not provide enough math ematical substance to interest those who will go on to become practicing physi cists or mathematicians. As a conse quence the book has a fussy, ad hoc, chase-your-own-tail quality, and too rapidly degenerates into a bricolage of formulas. I suspect that some of the ma terial was included not because the au thors considered it truly germane, but because they couldn't stop. (How well
ments lie far beyond the scope of the present book. Like all good things, skill at the eval uation of integrals increases with expe rience, and expertise does not depend, necessarily, on one's being a practicing mathematician. One of the most skillful practitioners I have ever known was an aeronautical engineer, who was a wiz ard at evaluating integrals around branch points in the complex plane. The present book is replete with riches. However, in my view, it suffers from two major shortcomings. One is in the paucity of material from the theory of special functions. The other is the lack of function-theoretical methods. Most integrals arising in mathematical physics call for the special functions of mathematical physics, either in the def-
I understand this affliction.) All the integrals the authors treat that are of the form
1 Erdelyi,
A.,
r 0
X'
(a + x)f3 (b + x)'Y
dx,
and the investigation of these com prises a substantial part of the text, are actually hypergeometric functions 1 , 2 F1 ' s. To speak a little more of these functions would have saved countless pages of busywork. Much of the rig marole in the book could have been avoided, leaving more room for so phisticated matters. As a number of current texts have shown, the tyro can rapidly be given the knowledge necessary to apply the residue theorem-precisely what one needs to evaluate integrals. The book
et a/., Higher Transcendental Functions, v. 1 , p 1 1 5, (5), New York, McGraw-Hill, 1 953.
2Seaborn, James B., Hypergeometric Functions and Their Applications, New York, Springer-Verlag, 1 991 .
66
THE MATHEMATICAL INTELLIGENCER
2 of Seaborn does this brilliantly, in a scant 30 pages. A student who can con ceptualize power series (and the au thors rely on these heavily) can deal with the rudiments of complex analy sis. The authors talk at great length about the evaluation of
Lm(a)
=
Ioo CXZ o
+
dx 2ax + l) m+l m = 0, 1 , 2,
A s previously pointed out, these are hypergeometric functions, but a more elegant way of proceeding is to evalu ate by residues the integral
J
c
x' dx
CXZ + 2ax +
l ) m+l
along a Mellin contour C-a circle with radius R centered at (0) cut along the positive real axis, R -7 eo--and then to let s -7 = 0. In Chapter 2, the authors show that satisfies the recursion
Lm(a)
Lm(a)
=
2m - 1 2 m( 1 - a2)
Lm-l (a)
a
but they state that obtaining a precise form for Lm(a) from this relationship "seems to be difficult. " No-it is easy. The equation is a linear first-order non homogeneous difference equation and can be solved by the standard tech nique: obtain the solution of the related homogeneous
equation,
and then a
particular solution by variation of pa rameters. The authors should have in cluded this procedure in their intro ductory material. A minor misgiving I have is the em phasis on "conjecturing closed formu las . " The authors are devoted to this arithmancy, and they invoke it re peatedly. Typically, they will compute several selected values of a number theoretic sequence Xn and, providing hints, ask the reader to conjecture a general formula for Xn . This approach reminds me of the hoary query I was posed on moving to what is now my hometown: What is the next entry in
the sequence
30, 34, 40, 46,
52,
60, 63,
0
0
0
?
The answer is Millboume, the next stop on the westbound Market Street El. Al though the answer will resonate only with a Philadelphian, the vaunted tech nique of conjecturing an answer, along with $2, will certainly entitle you to a ride on the El. Before I move to more positive things, I must remark on the numerous typos in the book-"cosideres, " "disc tinct, " etc. And then there are exposi tory blunders. Wallis's formula is de clared to be proved, but Wallis's formula is not clearly limned. However, who am I to complain? I sometimes think that my first book, Sequence Traniformations, could have been des ignated more accurately, Mistakes in Se quence Traniformations. At least my culpability is less; at that time, we did n't have spell checkers. The first chapter of this book dis cusses some basics: prime numbers, the binomial theorem, the ascending facto rial symbol:
(a) n = a(a + 1)(a + 2) . . . (a + n - 1), n = 1 , 2, 3, . . . , (a)o = 1 .
Chapter 2 discusses the integration of rational functions. Chapter 3 is entirely allocated to the integral
I(m, n) = Loo ( x +X' ) m+ 1 , m > n, qo o q1
m, n,
integers. (More hypergeometric functions!) The authors first show that
I(m,n) =
1 n m q1 n+ 1 qo
I
J�o
(- l)n�j m- 1
(!'?), 1
and then proceed, after many pages, to find a formula for the sum. If they didn't want to use 2F1 's, I don't under stand why they didn't at least make use of the easily proved formula
(*) The value of the integral I(m, 0) is im mediate. Using (*) , one differentiates this formula n times with respect to q1 , then replaces m by m-n to give the au thors' final formula. Of course, there is no reason to do any of this if one uses residues. The integral can be evaluated
for general n by the Mellin contour technique mentioned previously. Chapter 4 talks about power series. For a power series with coefficients Cn the authors give for the radius of con vergence R of the series the formulas
and
These formulas are both wrong, the first more so, since it isn't rescued by re placing the lim by a limsup, although the second formula is. A power series that illustrates the perils is furnished by the one with coefficients {1,1,1,4,1,42,1,43, 1,44, . . . }, having radius of convergence 1/2, though neither of the above limits exists. The authors next define the diloga rithm function and display one of my favorite-and one of the most arcane formulas in analysis, taken from the book by Lewin3:
c\15-n;2
Jo
ln(l
- x)
dx =
X
]n
2
( Vs- 1 ) 2
�
10 .
A treatment of Eulerian polynomials
(
)
d n 1 A n(X) = (1- x) n+ 1 X - -- , 1-x dx which have applications in the sum mation of series, closes the chapter. Chapter 5, on the exponential and log arithmic functions, is rudimentary, but there are interesting nuggets sprinkled throughout: the irrationality of e, a su perb section on Stirling's formula, and an extended treatment of Bernoulli num bers. Chapter 6 discusses the basic trig functions, 'TT, Wallis's product, more power series expansions for elementary functions, and the Riemann zeta function. In Chapter 7, the authors treat the quartic integral
r (x4 + 2 a�� + n m+ 1
suit from the known properties of Gauss's function, including the recur rence relations it satisfies. In addition, because of the form of its parameters, the function admits a large number of quadratic transformations. One of these (Erdelyi, p . 1 13, (30)) reveals that the integral is, essentially, a polynomial 2 F1 of degree m in a. Chapter 8 looks at some classical ma terial with fresh eyes. The topic is the normal integral,
v;
I = Loo e-xz dx = -- . 2 o The authors have, apparently, made a hobby of collecting information on the evaluation of l The first method they give is the one known to all of us: ex pressing I2 as an integral over the first quadrant in the x,y plane and switch ing to polar coordinates. But they give an even simpler method:
0 0
f"'
C x2
0
f"' e-xzyz X dy dx
= J'"'r· xe-xZ(l +yZ)dy dx 0
1 = 2
0
f"' 0
dy
1 +
y
=
7T
4•
They next advance a profoundly clever number-theoretic argument that relates I to the number of representations of an integer n as the sum of two squares, rin). The derivation utilizes the identity
(
00
n�oo
) � r2(m)xm. 0
2 xn2 =
00
I wish I could reproduce this soul-sat isfying derivation in its entirety. The student is asked to prove the above identity as an exercise-no hints. This should separate the wheat from the chaff. The next chapter provides a nice in troduction to what is known about Euler's constant, y y �
another 2F1 , with parameters {1/2, m + 1, 2 m + 2} and argument 2 VT-1""1 CVT-J"" - a), actually, a Legendre function. All of its salient properties re-
0
I2 = r· e- x2 dx f" e_ uz du =
= n---> lim A n, A n = OO
( I -k1 k� 1
- In
)
n,
.57721 566490153286060651 2090082402431042 1 59335939923598805767 .
y is one of the most inscrutable con stants in mathematics. (Why couldn't the authors have tantalized the reader
3Lewin , L. Dilogarithms and Associated Functions. London, Macdonald, 1 958.
© 2006 Springer Science+Business Media. Inc . • Volume 28. Number 3. 2006
67
with just a few of its digits?) Its irra tionality is still undecided. Basically, the difficulty is that we do not have repre sentations of y that converge with suf ficient rapidity. Perhaps the solution to the problem will eventually be found by developing such representations, in other words, with progress in algorith mic mathematics. The above limit is pa thetic, for, as the authors show,
--
1 1 1 < An - y < - . - 8 n2 2n 2n The authors derive the standard inte gral representations for y and some ad ditional series representations which, to my mind, are of limited interest. Chapter 10 on the Gamma and Beta and related functions, has much stan dard material. The authors don't prove the Bohr-M0llerup theorem-rex) is the only positive logarithmically convex continuous function that satisfies rex + 1) = xrex) , re1) = 1-which is a shame. The proof is elegant and sim ple. 4 However, they do provide Totik's proof that rex) satisfies no differential equation (of a certain type). Chapter 1 1 , on the Riemann zeta function, is one of the most interesting in the book, including, as it does, some intriguing contemporary findings. Apery, in a celebrated paper, demon strated the irrationality of �(3), �(3)
oc
=I
n= l
I""( o
.0
+
x2
1 3 n ·
2ax2 +
1
)r (
x2
Philadelphia, Pennsylvania USA e-mail:
[email protected] The authors don't furnish the proof, de cidedly non-trivial, but the background information they provide is interesting in its own right. Chapter 1 2 deals with logarithmic in tegrals, and Chapter 1 3 , "A Master For mula, " studies the integral
M=
There is an appendix of great value, "The Revolutionary WZ Method." The ti tle is not hyperbole. The method, due jointly to Herbert Wilf and Doron Zeil berger, is demonstrably one of the supreme achievements of algorithmic mathematics. It continues, perplexingly, to be ignored, even though the algorithm is available online. I have used it fre quently to obtain elegant and simple dosed-form representations for certain hypergeometric functions, for instance, the associated Legendre polynomials. I gave a lecture on this method 7 years ago at a Southern university. From the questions asked, I concluded that almost no one in the audience had ever heard of it. The experience reinforced my per ception that purblind specialization, of which such ignorance is the inevitable issue, is the constant enemy of mathe matical progress. All in all, I think the present book, despite its problems and despite its stip ulated audience of naifs, may appeal to many readers. Some of the results are astonishing, and others point directions for future research. (Is �(5) irrational? �(7)? �(9)? �(1 1)? Rivoal and Zudlin have shown in Uspekhi Mat. Nauk (56 (2001), no. 4, 1 49-150) that at least one of these is.)
+ 1
x' + 1
)
Bourbaki. U ne societe secrete des mathematiciens
by Maurice Mashaal
COLLECTION LES GENIES DE
LA
LA
dx
BELIN-POUR
x2 ·
€16.00, ISBN 2-84245-046-9
M is, surprisingly, independent of s. Thus
its evaluation can be effected by replac ing s by 2, which gives an integral pre viously studied. As I have noted, this is a Legendre function, and its properties follow from known results for that func tion.
SCIENCE, 2002,
SCIENCE, PARIS, 160 P.,
�·
REVIEWED BY OSMO PEKONEN
'7\
1 \/
aurice Mashaal is a French sci. ence journalist who has com piled this lovely, richly illus trated book about the collective mathematician Nicolas Bourbaki, born
4Conway, John B., Functions of One Complex Variable, New York, Springer-Verlag, 1 978.
68
THE MATHEMATICAL INTELLIGENCER
in 1934 and possibly still alive and kick ing. It first appeared as a special issue of the French magazine Pour Ia Science and is sold at newsstands in the streets of Paris. Now it has been reissued as N° 1 of the book series 'Geniuses of Science' , which portrays Becquerel, Cu vier, Einstein, Fermi, Galilei, Kepler, Leonardo, Pauling, and Poincare, among many forthcoming others. Some of these volumes are remarkably well edited and certainly worth translating into other languages. Nicolas Bourbaki is known as the au thor of Elements de Mathematique, a treatise measuring so far some 7000 pages, whose ambition is a unified pre sentation of all mathematics deemed worthy of attention. Mashaal's book is a well-balanced synthesis of mathemat ics, biography, history of learning, and anecdotes and humor. It is aimed at the man in the street who understands-as many Frenchmen do-mathematics as an integral part of culture but doesn 't necessarily have formal training in the field. Some case studies of formal math ematics are developed in the axiomatic Bourbaki style for the benefit of those readers who remember the basics of their lycee curriculum, but the main text can be read independently of the boxed formulas. The story of the "real" Bourbaki, a general of the French-Prussian war of 1870-187 1 , has often been told. Inci dentally, a street (rue Bourbaki) in Pau, southern France, is named after him. The general's forename was Charles, whereas the fictitious forename Nicolas was suggested by Andre Weil's wife Eveline. A picaresque aspect of the tale was the appearance in Paris in 1948 of a Greek diplomat carrying the ominous name Nicolaides Bourbaki. He was courteously invited to dine with the mathematicians. Many other legendary aspects of Bourbaki, and other folklore typical of Ecole Normale Superieure, are discussed by Mashaal who has had ac cess to some old issues of La Tribu, the confidential and funny newsletter of the group. The text includes minibiographies of the following five "archicubes" : Henri Cartan, Claude Chevalley, ]ean Delsarte, Jean Dieudonne, Andre Wei!. Rare pic-
tures and many stories of other celebri ties such as Armand Borel, Pierre Cartier, Adrien Douady, Samuel Eilen berg, Alexander Grothendieck, Laurent Schwartz, and Jean-Pierre Serre are in cluded. Of particular documentary value are the two photographs, taken in 1937 and 1938, in which Andre Weil's famous sister, the philosopher and mys tic Simone Wei! (1 909-1943), is seen taking part in sessions of the Bourbaki group. The notion of structures emerged simultaneously as a key concept of mathematics and literature. Some Bour baki-inspired authors, like Raymond Queneau and Jacques Roubaud, are dis cussed. Mashaal doesn't believe, though, that mathematics influenced much the birth of structuralism as a lit erary theory; he rather views these phe nomena as interesting parallel develop ments. The story of Bourbaki is the saga of a handful of determined young revolu tionaries who wanted to reshape the way mathematics was taught in France. Their success was overwhelming, with worldwide implications. In due course, the archicubes themselves developed into clerics of a new orthodoxy which occupied all available academic space and, after a golden heyday, may have actually hampered the development of science in France. The New Math of the 1960s (to be compared with the New Economy of the 1990s . . . ) may have started with Dieudonne's exclamation in 1959: "Down with Euclid!" By the early 1970s, he rued his words and de nounced the birth of a new form of scholasticism. Benoit Mandelbrot (whose uncle Szolem Mandelbrojt was one of Bourbaki's founders) claims that he fled to the United States to escape its "suffocating influence" in France. Notable omissions in the Bourbaki program, which Mashaal doesn't fail to discuss, include probability, mathemat ical physics, and, strangely enough, many foundational issues. Bourbaki also chose to ignore category theory. Mashaal suggests that its inclusion would have implied far too devastating modifications in the whole architecture of Elements de Mathematique, which had been rooted in set theory since the 1930s. Categories and functors crept into the text nonetheless, but without being named so.
According to the rules, one must re tire from Bourbaki at the age of 50, and presumably this also corresponds to the life-cycle of the whole enterprise. When I was a student in Paris in the late 1980s, Bourbaki was already referred to as someone who had passed away but whose posthumous reputation kept lin gering on. In October 1988, the French radio channel France Culture even broadcast a formal announcement of the death of Bourbaki, due to a "non removable pathological singularity." Even so, a posthumous Chapter 10 of Commutative Algebra appeared in 1998. Mashaal's account is a rewarding one also for the professional mathematician as a source of anecdotes and rare pic tures, and as a glimpse into the French history of mathematics. It conveys very well an overall impression of the rise and fall of the polycephalic scientist. For a more scholarly account, one should rely on works by Liliane Beaulieu and others. Agora Centre 4001 4 University of Jyvaskyla Finland e-mail:
[email protected] M usic and Mathematics
edited by]. Fauvel, R. Flood, and R. Wilson OXFORD, OXFORD UNIVERSITY PRESS 2003 ISBN 0-19-851187-6 £39.50
REVIEWED BY EHRHARD BEHRENDS
everal years ago, preparing activ ities for the general public on the occasion of the ICM'98 in Berlin, the Berlin universities organized several seminars to discuss the relations be tween mathematics and music. Many different aspects were covered by the lectures, which were given by people working in these fields, among them a number of young composers. For me, it was a surprise to learn that mathe matical ideas are rather influential in certain areas of contemporary music (which, however, are not very close to my personal musical interests: classical
music and jazz). Also I had not realized before how important physiological facts are to the possibility of using math ematical structures successfully. For ex ample, most of us are unable to recog nize the absolute pitch of a tone. Only the relations between different pitches are perceptible: the interval E-G is "the same" as the interval A-C . Also, in many cases a tone may be replaced by its oc tave without noticeably changing the character of a musical piece. As a con sequence, much of the work concerned with scales can be reduced to finding the appropriate pitch ratios between one and two. "Music and mathematics" is not a well-defined area, but most of what can be said concerns one of the following queries: 1 . What are the mathematical princi ples that underlie the construction of musical scales? What are the defects of the Pythagorean scale, why is 12;::: v L. important? 2 . Are there other problems of interest to the "working musician" that can be solved by using (more or less ad vanced) mathematics? For example, of what help is Fourier analysis for the artisan who wants to construct a guitar? 3. Is mathematics helpful in analyzing musical compositions? Did certain composers have mathematical struc tures (like symmetry or remarkable relations between numbers associ ated with the composition) in mind during their work? 4 . Can mathematics be used as a tool box to produce interesting music? All of these aspects are discussed in this book. The first contribution, "Tun ing and temperament" by Neil Bibby, starts with a description of the early at tempts of the Pythagoreans to relate harmony to mathematics. They discov ered that intervals are considered to sound harmonic if the ratio of the fre quencies is a rational number with "small" numerator and denominator. The ratio 2 : 1 is not very interesting since the octave is somehow identified with the original tone. The Pythagorean scale is based on the ratio 3:2, the per fectfifth. If one starts with " C " , one first obtains "G", then "D" etc. Here, one has to apply the principle, mentioned earlier, that a note can be identified
© 2006 Spnnger Science+Business Media, Inc., Volume 28, Number 3, 2006
69
with its octave. So the fifth of the fifth, the ratio 9 :4, is replaced by 9:8 to make the ratio lie between one and two. In this way the notes corresponding to the white keys of the keyboard are gener ated. (However, to produce the "F" one has to go backwards: "C" is the fifth of "F": "F" is added to the scale for this reason.) Soon it was realized that many of the intervals which occur in this scale are far from being simple. For example, the frequency ratio of the major third ("C" to "E", say) is 8 1 : 64. Also, the Pythagorean scale is not well suited for modulation. If one considers a note al ready constructed as the keynote of a new scale it will be necessary to include new notes. For example, one has to add "F sharp" when starting with "G". This process never stops. There is no finite scale which is closed under forming Pythagorean scales if one can select any note as a basic key note. Bibby describes some of the pro posals that have been made to over come these difficulties. For example, we learn how the frequency ratios in the scale of the just intonation are defined and to what extent it is superior to the Pythagorean scale. And that Marin Mersenne designed a keyboard with 31 notes for each octave, which made it possible to distinguish between "F sharp" and "G flat", a distinction which is important in the Pythagorean scale and in the just intonation. Most readers will know that now adays nearly all instruments use the equal temperament. The frequency ra tio between two adjacent notes of the 1 2 notes in an octave on a keyboard is always the same, and in this way the twelfth root of 2 comes into play. It is a really "democratic" scale, as each note plays the same role. Nevertheless it is a rather ironical aspect of the history of music that one started with the philos ophy of small rational numbers and ended up with a scale where no inter val (up to the octave) is rational. But this is not the end of the story. Later (in Chapter 9) we learn why, be sides our 1 2-tone system, certain n-tone systems play a role in music theory. Here in particular the cases n = 13, 19, 21, 31, and 53 are of some interest. Sim ilar questions are mentioned also in some other contributions. For example, in Chapter 2-a chapter with a more
70
THE MATHEMATICAL INTELLIGENCER
historical than mathematical empha sis-]. V. Field explains how Kepler tried to find musical proportions in var ious quantities of the solar system. For me, the most interesting chap ters are those of Part Two, "The math ematics of musical sound". First, in Chapter 3, Charles Taylor describes some experiments with real instruments to demonstrate how one can hear com binations of notes. It is a strange fact that the ear sometimes "hears" the dif ference note of, e.g., 80 Hz if two notes of 500 and 580 Hz are played simulta neously. Taylor has no convincing ex planation of this phenomenon; it is ar gued that the effect is caused by a combination of physical and physio logical reasons. Next, in Chapter 4, Ian Stewart demonstrates that many interesting mathematical problems are touched upon if one wants to calculate the po sitions of the frets of a guitar correctly. He explains the difference between "construction with circle and ruler" and "construction with circle and unmarked ruler" and how simple it is to trisect an angle if one is allowed to mark a dis tinguished point on the edge of the ruler. The main part of Stewart's article is the description of Strahle's construction of the position of the frets. Strahle, a Swedish craftsman, suggested his con struction in 1743, but it was erroneously argued by Jacob Faggot, of the Swedish Academy, that the argument has a flaw. Strahle had in fact found an approxi mation of by simple geometric means, which in a sense, is optimal: the best approximation of 2x by a function of type ( ax + b)!( ex + d) is
VZ
\12)x + \12 - \12)x + \12 ,
(2 (1
and Strahle used this function; he ap proximated \12 by 1 7/12, a ratio which appears when expanding v2 as a con tinued fraction. The third article in this part (Chap ter 5) is David Fowler's essay on Helmholtz. For many readers it will be a surprise to learn that Helmholtz not only was a famous physicist but also a physiologist who worked on the phys iological basis of the theory of music. Fowler starts with a description of Helmholtz's experiments with sound generators; they were used to demon-
strate combinational tones like the dif ference notes mentioned earlier. More substantial and mathematically more in teresting, however, is the solution Helmholtz proposed for the problem of consonance. A fifth and a fourth, for ex ample, which are defined by the ratios 3 : 2 and 4:3, are perceived as a harmo nious sound, which is more pleasant for the ear than an interval selected at ran dom. What is the cause of this phenome non? Some answers, among them those of Plato, Kepler, and Galileo, are sketched (in my opinion, Euler's gradus suavitatis should also have been men tioned here). The starting point of Helmholtz's ap proach is his consonance curve. Imag ine two instruments playing a note in unison. If one of the frequencies is slowly increased, one will hear a beat ing. First it is slow, but it becomes quicker and "more unpleasant". Helm holtz quantified this sensation by asso ciating a "degree of unpleasentness": if the number of beats is x, then cp(x) mea sures the "unpleasantness". Obviously one has cp(O) = 0, and qualitatively it is clear that cp will first increase up to an maximum (which is assumed to be around x 30) and then decrease. In order to have a mathematically simple representation of cp, Helmholtz chose cp(x) = Ax/(30 + r)2, a choice which of course is somewhat arbitrary. This cp is used to explain consonance as follows. If two instruments play an in terval, one has to sum up the cp-values which belong to every pair of frequen cies from the list of all pitches which oc cur in the Fourier expansion of the two notes which constitute the interval. The result is a rather rough curve with minimum zero at the frequency ra tios 1 : 1 and 2 : 1 . But, remarkably, there are also some steep valleys in the graph at 3:2, 5 :4, and the other ratios which correspond to the Pythagorean scale. Part Three concerns "The mathe matical structure of music" . I am sure that the fascination one can feel when listening to a Schubert sonata or a Chopin mazurka will never be accessi ble to a mathematical analysis. There are, however, many "intellectual" as pects of music where a mathematical language can reasonably be applied. For example, group theory naturally comes into play when speaking about =
musical symmetries: see Chapter 6, "The Geometry of Music", by Wilfrid Hodges. But most of these symmetries are only perceptible by optical inspection of the score. (As an experiment, I suggest playing the notes from the first two bars of a popular song in reverse order. It is rather unlikely that an untrained listener will recognize the original.) Not only group theory plays a role here. In Chapter 7, in the article by Der mot Roaf and Arthur White on "Bells and mathematics", the emphasis is on combinatorics. "Ringing the changes" is the art of ringing a collection of n bells sequentially such that, at the end, all n! permutations have been heard. In ad dition, certain conditions must be satis fied, for example, from one round to the next, only transpositions between adjacent bells are admissible. This is so because, otherwise, it would be difficult to perform the sequence with really ex isting heavy bells. It is interesting to see how this problem can be solved by rather simple algorithms and how the solutions are visualized graphically. In the last chapter of this part, "Com posing with numbers" by Jonathan Cross, we are introduced to some math ematical ideas which have found their way to being used as tools for com posers. The story begins with the twelve tone row of Arnold Schonberg; a num ber of other examples are also discussed. The idea is always the same. First, one associates certain musical parameters, like pitch or duration, with numbers or more complicated mathematical objects, and then the structure of the mathemat ical part is translated to a piece of mu sic. For example, one could select a magic square and then use the rows (or the columns, or the diagonals) to define the pitches of the clarinet line or the du rations of the bassoon line. Similar ideas are found in Part Four, "The Composer Speaks" (Carlton Gamer and Robin Wilson on "Microtones and projective planes" and Robert Sherlaw Johnson on "Composing with fractals"). The titles indicate the mathematical source of the compositions. Finite pro jective planes are used to identify cer tain subsets of tones. For example, if one wants to select three notes out of seven in such a way that the selection generates a "cyclic design" , one finds everything that is needed in the geom etry of the Fano plane. (A cyclic design
in this case is a pattern such that trans lations modulo 7 give rise to subsets of (0, . . . ,6 } in which each pair of num bers is contained in precisely one of the translations.) Dynamical systems are very com mon in contemporary music. Here the well-known two-dimensional iterative patterns which lead to the Mandelbrot set generate the musical material. For example, if the channel of the synthe sizer has to be determined where the next note will be generated, then a dis cretization of the y-value of the pres ent position of the system is important: e . g . , if it lies in [7,8[, then choose chan nel 5 . I t should b e noted that the book is very carefully edited. It is a pleasure to read, and there are many interesting pictures and scores to illustrate the ma terial. Readers who are particularly in terested in the historical part of the sub ject can consult the book Mathematics and Music (edited by Gerard Assayag, Hans-Georg Feichtinger, and Jose Rod riguez, Springer 2002; reviewed in The Mathematical Intelligencer, vo!. 27, no. 3, p . 69) . There is, surprisingly, only a small overlap in the content of these two books. The generation of scales by mathematical principles naturally plays a prominent role in both of them. For me, only two aspects are miss ing. The first omission: I would have appreciated an article on Euler's work on music. He was one of the first to re late mathematics to consonance, and it would be interesting to compare his work with that of Helmholtz. And I was surprised to see that one cannot find anything substantial on "probability and music". In the music of the last century there is an abundance of examples in which the building blocks of certain compositions are generated stochasti cally, be it the pitches, the durations, or even the wave forms of the sounds. But these objections are not essen tial. Let's praise the editors that they have presented an attractive volume that covers almost all of the important aspects of the interplay between math ematics and music. Fachbereich Mathematik und lnformatik Freie Universitiit Berlin D-1 41 95 Berlin Germany e-mail:
[email protected] The Pea and the Sun: A Mathematical Paradox
by Leonard M. Wapner WELLESLEY, MA, A. K. PETERS, 2005,
xiv + 218 PP. , US $34.00, ISBN 1-56881-213-2 REVIEWED BY JOHN J. WATKINS
here is nothing quite like a good paradox. In their great comic opera, The Pirates of Penzance, W. S. Gilbert and Arthur Sullivan use a 'simple arithmetical process' to create
"a paradox, a paradox, a most inge nious paradox' that renders the entire
cast of pirates, maidens, and policemen helpless with laughter and amusement on the rocky seacoast of Cornwall. The paradox in this instance is extraordi narily silly, but Gilbert and Sullivan lightly hang the entire plot of Pirates upon it. Leonard Wapner manages a similar sleight-of-hand with the Banach Tarski paradox in his immensely en gaging book The Pea and the Sun: A Mathematical Paradox. This book may not be quite as much fun as a Gilbert and Sullivan opera, but it is pretty close. To be fair, W. S. Gilbert and Arthur Sullivan should probably be ranked as geniuses, but Wapner does have one huge advantage over them. The rather flimsy basis of the Gilbert and Sullivan opera is the paradox that their naive young hero Frederic had the misfortune to be born on February 29, and so, by counting birthdays he is but "a little boy of five" and thus must continue on in his grossly unfair apprenticeship to a band of pirates until his twenty-first birthday. This is hardly a paradox wor thy of the name. But Wapner has cho sen as the basis for his book the Ba nach-Tarski Paradox: quite simply, the finest paradox in all of mathematics. At first glance, the Banach-Tarski paradox seems so nonsensical that it might well belong in the world of Gilbert and Sullivan, for it says that it is possible to dissect a ball-that is, a solid sphere-into a finite number of pieces and then rearrange these pieces to form two balls exactly the same size as the
© 2006 Springer ScJence+ Business Media, Inc., Volume 28, Number 3, 2006
71
original ball. Yet, nothing has been cre ated or stretched in the process! All that has happened is that pieces of the orig inal ball have been moved around in space like the pieces of a jigsaw puz zle. Well, if you can form two balls, you can form three, or four, or any number you like; or, alternatively, you could re arrange the pieces to form a single ball that is twice as big as the original ball, or three times as big, or as much big ger as you like. Thus, a rather more dra matic way to restate the Banach-Tarski paradox is to say that you can trans form a ball the size of a pea into a ball the size of the sun. The Banach-Tarski paradox is such a remarkable paradox precisely be cause-as impossible as it seems-it is true. And so, as Stefan Banach and Al fred Tarski proved in 1924, it is indeed a theorem, and we should now refer to it more properly as the Banach-Tarski theorem. Leonard Wapner's goal in his book is to make this famous theorem accessible to a general, perhaps even non-mathematical, audience, and in pursuit of this goal he wisely chooses to focus on the paradoxical nature of this famous theorem-after all, who would be amused for long by "a theo
rem, a theorem, a most ingenious theo rem"?
Wapner begins his story in the first chapter with history and introduces not only his main cast of characters, Georg Cantor, Stefan Banach, Alfred Tarski, Kurt Godel, and Paul Cohen, but also many of the mathematical ideas that will be important later, such as countable and uncountable sets, the Axiom of Choice, Hausdorff's Paradox (though, strangely, Hausdorff himself is not ele vated to the main cast), as well as a nice discussion of the questions of con sistency and independence of the Ax iom of Choice and the Continuum Hy pothesis. Chapter 2 is an ill-conceived digres sion on fallacies which Wapner con fusingly calls Type 2 paradoxes; he dis cusses and resolves several such paradoxes, but this gets rather far afield even though for example the jigsaw fal lacies of Sam Loyd he presents are very amusing. I worry that while Wapner is actually trying in this chapter to alert the reader to the all-important distinc tion between a surprising theorem (a Type 1 paradox) and a false statement
72
THE MATHEMATICAL INTELLIGENCER
(a Type 2 paradox), some readers will
dius 1 . On the first circle designate an
come away with the impression that Ba nach-Tarski is nothing more than an other trick very much like Sam Loyd's Get Offthe Earth puzzle, in which a disk is rotated to make twelve Chinese war riors magically turn into thirteen Chi nese warriors. The level of the book varies consid erably. Chapter 3 begins at a very ele mentary level clearly intended for ab solute beginners and, therefore, says such things as that the proof of the im portant result that the power set of a fi nite set with n elements has cardinality 2n is by mathematical induction and will be omitted. This inductive argument is very easy to make clear to a non-math ematical reader and it is even easier to make clear the real reason for this re sult: namely, that each element of the original set of n elements is either in or not in a given subset, that is, there are two choices for each of the n elements. Wapner does an excellent job of intro ducing the notion of an isometry and the all-important idea that isometries have a group structure, but having just explained what the associative property is, what an identity element is, and what an inverse element is, he then wants to show that
arbitrary point as 0. This is the point to be removed. Then, from 0 moving counterclockwise at unit intervals mark off points 1 , 2 , 3, 4, and so on, one point for each natural number. Since the circumference is irrational, all of these points will be distinct. Let A be all of these points and let B be all of the re maining points on the circle. This de composes the first circle into two sets. The rest is easy. We reassemble the two pieces by leaving set B unchanged (the identity isometry) and by rotating set A one step counterclockwise (obviously, an isometry on the circle), thus form ing the second circle with 0 missing as advertised. In Chapter 4, called Baby BTs, Wap ner presents several similar paradoxes leading up to the Banach-Tarski Para dox, showing for example that a sphere and a sphere with one point removed (or even countably many points re moved!) are equidecomposable, but also he gets somewhat sidetracked dis cussing the cardinality of various one and two-dimensional point sets. He gets back on track with beautiful presenta tions of the Cantor set and of the ex ample found in 1905 by Giuseppe Vitali of a bounded non-Lebesgue measurable set. The statement and proof of the Banach-Tarski Theorem then forms the core of the book in Chapter 5, and Wapner then brings the paradox to a satisfying resolution in the following chapter. One thing that gets lost a bit both in the book but also in the inherently geometric nature of the statement of the Banach-Tarski Theorem is that there is a fundamental algebraic truth at its core. In 1 9 1 4 Felix Hausdorff dis covered a remarkable way to decom pose the free product Z2 * Z3. This group is just all 'words' using two gen erators, u and T, where u2 r3 = i where i is the identity of the group. Hausdorff partitioned this group into three disjoint sets
(plpz)-l
=
Pz-1PI - I
and begins his proof with
(pipz) (pz - IPI - 1)
=
P1 (pzpz -I) P1 -1
without any explanation of how asso ciativity gives you this. (It requires an intermediate step.) Continuing in this chapter on pre liminaries, he then does some very nice scissor dissections, square to equilateral triangle, octahedron to square (though then wandering off topic a good bit), as a warm-up before transitioning beau tifully to the vitally important notion of equidecomposability, showing, for ex ample, that a circle and the same circle with a single point removed are equide composable, that is, the first circle can be decomposed into a finite number of sets (in this case, two) and these sets can then be reassembled (using isome tries) to form the second circle with a point missing. Since this example is so similar in spirit to the actual proof of the Banach-Tarski Theorem given in Chapter 5, I'd like to give the details here. Let the two circles each have ra-
=
V'T, O'TV', O'TT, TTV',
. . . },
C = {TV', TT, TTV'T, TO'TTV',
. . . },
A
=
{ i,
with the defining property that B = TA and C T2 A, and so all three sets are congruent and A constitutes a third of the group. But also uA = B U C and =
so A is congruent to its complement and thus constitutes half of the group. The remarkable fact that A can be si multaneously both half and a third of the group is now known as Hausdorff's Paradox. This purely algebraic paradox is at the very heart of the Banach-Tarski Paradox. You might say the rest is just window-dressing. It was Hausdorff who applied it to the surface of a sphere. Banach and Tarski then showed how to extend the paradox to a solid sphere, that is, a ball, but Hausdorff had done the heavy lifting. More recently, Jan My cielski and Stan Wagon have applied this paradox to the hyperbolic plane where, using Mathematica, one can ac tually see the set A filling both half and a third of the standard Poincare disk [ 1 , 2 ] . Wapner does mention this lovely in terpretation, but rather strangely places it in his first chapter on history and the cast of characters. While the Banach-Tarski Theorem clearly seems to rest solely within pure mathematics and offers no hope at all for practical applications, Wapner does show us, in Chapter 7, that at least a few physicists in the world take seri ously such ideas as that the muon, a short-lived particle 200 times as massive as an electron, might in fact be a par ticle that expanded in a Banach-Tarski like way from an electron, or that cer tain well-known reactions in particle physics have Banach-Tarski interpreta tions, such as when a proton changes into a pi meson and a neutron. Such ideas have more than a whiff of fantasy about them, which Wapner freely ad mits, yet when I pause to consider how little we actually understand about the real world, it is not beyond the realm of possibility that infinite sets and even non-measurable sets might exist in some real sense that we assume to be impossible today. Wapner closes his book with a chap ter in which he bravely takes a crack at explaining seven big problems in math ematics today and also takes the op portunity to offer some of his own mus ings about the future of mathematics, guided by noted thinkers such as Yogi Berra and Niels Bohr who, as it turns out, had almost the exact same thing to say about the future. Like Gilbert and Sullivan's Major-Gen eral Stanley, Leonard Wapner is indeed
"teeming with a lot o' news with many cheerful facts" about matters mathemat ical in this thoroughly delightful book. While it is certainly true that a non-math ematical reader might well miss an im portant preliminary or two or get side tracked by one or more of the many interesting diversions along the way and, in the end, not be able to follow fully all of the details of the proof in Chap ter 5, I would be very surprised if he or she hasn't understood the general flow of ideas, learned a lot about mathemat ics, been amazed by the sheer reality of what unexpected things can be true, and had lots of fun in the process. REFERENCES
1 . Bennett, Curtis, A Paradoxical View of Es cher's Angels and Devils,
The Mathematical
lntelligencer 22(3) (2000), 39-46. 2. Wagon, Stan, A Hyperbolic Interpretation of the Banach- Tarski Paradox,
The Mathemat
ica Journal 3 (1 993), 58-61 . Department of Mathematics and Computer Science Colorado College Colorado Springs, CO 80903 USA e-mail:
[email protected] Divine Proportions: Rational Trigonometry to U niversal Geometry
by Norman Wildberger
SYDNEY, WILD EGG, 2005 (http:/jwildegg.com), 300 PP, AUD79.95 HARDCOVER, ISBN 0-9757 492-0-X
REVIEWED BY JAMES FRANKLIN
nlike "lesser" disciplines, mathe matics is not rent by disputes over what is true. What we have proved true has stayed true, give or take rare exceptions. Our argumentative en ergy has not gone to waste, however, and mathematicians debate vigorously questions on what topics are interest ing, what conjectures credible, how classical fields can be better seen in the
light of new results, and of course, how to teach. A heated debate a hundred years ago-one with close parallels to the revolution in trigonometry that Wild berger urges in his new book-resulted in major changes to linear algebra. That is a branch of mathematics which the nai've student might expect to have de veloped smoothly, rationally, and with out controversy. Axioms, span, inde pendence-what is there to become heated about in that? Yet the modern point of view is the end point of re covering from several false starts, no tably Hamilton's inept attempt to do vector geometry and physics with quaternions and Grassmann's barely in telligible foundation of the subject on what we call "flags of subspaces" . There were also difficulties i n moving beyond coordinates and matrices to the more abstract point of view of vectors and linear maps. It was only in the 1920s that British mathematicians and engineers swallowed their pride and admitted that the Germans had it right about vectors. (The story is told in Crowe's History of Vector Analysis.) Trigonometry is a much older and more settled branch of mathematics than linear algebra. It comes much ear lier in the syllabus, and every becom ing-numerate generation invests enor mous effort in the painful calculation of the lengths and angles of compli cated figures. Surveying, navigation, and computer graphics are intensive users of the results. Much of that effort is wasted, Wildberger argues. The con centration on angles, especially, is a re sult of the historical accident that seri ous study of the subject began with spherical trigonometry for astronomy and long-range navigation, which meant there was altogether too much attention given to circles. Wildberger's alternative is simple. We should avoid the concepts of length and angle as far as possible, and so do without their complicated formulas in volving square roots and transcenden tal arcsines and the like. They should be replaced with two (algebraically) simpler concepts, "quadrance" and "spread". Quadrance is just the square of length, so its formula in terms of co ordinates just involves the sum of squares of co-ordinates. Spread is a measure of separation of lines. It is (to
© 2006 Springer Science+Business Media, Inc., Volume 28, Number 3, 2006
73
slip into oldspeak for a moment, though the aim is to learn to think in the new language as fast as possible) the square of the sine of the angle(s) between the lines. The spread between the lines ax + by 0 and ex + dy 0 is a simple rational expression in =
a, b, c,d.
=
Let u s take one elementary and one more mathematical example to show case the point of doing things this way. Consider the problem, useful in such fields as railway engineering, of the re lation between slopes when climbing a hill "at an angle" . For example, if a grade of one in fifty is the maximum a train can climb and the hill has a grade of one in thirty, in what direction across the hill must one build the railway? Stan dard trigonometry would attack this problem using angles and their tangents, but the problem and its answer do not mention angles. The solution in terms of spreads (p. 231) is very simple. Mathematicians may be more ex cited by the way that the avoidance of square roots and transcendentals ren ders the results independent of the real field, and hence a true "universal geom etry". For example, at first sight the re sult that the spread subtended by a chord of a circle is a constant (p. 1 78) seems much the same kind of result as the classical one that the angle sub tended by a chord is constant. But there are subtle differences. With angles, one must consider on which side of the chord the angle lies. That is awkward in itself and prevents generalization be yond the field of real numbers. For spreads, constant really means con stant, and one may change the under lying field and retain the theorem. Wildberger develops his universal geometry at length, dealing for exam ple with the replacements of the sine and cosine rules, an alternative to spherical and polar co-ordinates with
74
THE MATHEMATICAL INTELLIGENCER
applications to moments and centers of inertia, and simplified treatments of classical surveying problems like the Snellius-Pothenot and Hansen's prob lems. Reform is intended not just of trigonometry but of the foundations of Euclidean geometry. The subject is de veloped from first principles over a general field-one cannot have "on this side of the line" in fields other than the reals, but almost all other Euclidean geometrical properties remain available (including the inside and outside of cir cles). Similarly, conics are treated from a point of view that resembles algebraic geometry but includes a metric. It is true that there is a need to re tain the "circular" or "harmonic" func tions to deal with circular motion, Fourier analysis and the like, but those wave-like functions with no natural zero would be better not called "trigonomet ric". They are not related to triangles. It is certainly convincing that we would have been better off if trigonom etry had developed this way instead of the way it did. Now to the crunch. Is it feasible for the mathematical world to junk its im mense investment in the old technol ogy and move to a new one? It is a big ask, a very big ask, but there are a few reasons to think it might just be possi ble. The first is that despite the dead hand of conservatism, it has happened before. Replacing co-ordinates, matri ces, and quaternions with abstract vec tors and linear transformations was an effort, but worthwhile in the end. The same was true of replacing sines and cosines in Fourier analysis by complex exponentials. Long before that, Arabic numerals replaced Roman because they were more rational. Revolutions are possible. One must regretfully call the author's attention to the fact that they usually take more than a single lifetime. It could be questioned also whether
launching the project from a small in dependent publisher in Australia is a good idea, but in the twenty-first century world of the Internet and blogs, perhaps that does not matter. Second, a careful examination of 3D vector geometry will reveal that a cer tain amount ofWildberger's philosophy is implicit in it already, suggesting that he is on the right track at a more ba sic level. What makes geometry with vectors so successful is that all the in formation about lengths and angles is contained in the scalar product, which is algebraically very simple. The stu dent soon learns that the way to ap proach typical problems, say on the closest distance between two non-in tersecting lines, is to stay with vectors and their scalar products as long as pos sible and only extract any needed lengths and angles at the last moment. Wildberger simply goes one step fur ther: he recommends we do the same in two dimensions, and suggests that we hardly ever have any real need for lengths and angles in any case. There has been considerable debate by interested amateurs on Internet fo rums about this book. There needs to be more mature consideration in better informed mathematics and mathematics education circles. Having things done better is one major payoff, but equally important would be a removal of a substantial blockage to the education of young mathematicians, the waterless badlands of traditional trigonometry that youth eager to reach the delights of higher mathematics must spend painful years crossing. Wildberger's book de serves very careful examination. School of Mathematics University of New South Wales Sydney 2052 Australia e-mail:
[email protected] w-1fii•i•l9•hri§l
Robin Wilson
The Philamath's Alphabet-M M
athematics
teaching Many
I
equations', predicted the existence of
stamp shows an allegorical figure rep
such phenomena as radio waves.
resenting the French metric system.
Mayan mathematics
The Mayans of
Central America (300-1000
AD) used a
counting system based on the numbers
Mobius strip
The Mobius strip was
named after the German mathemati cian and astronomer August Mobius in
20 and 18. Most of their mathematical
1858. It has only one side and one
calculations involved the construction
boundary edge,
of calendars: a 260-day ritual one with
from a rectangular strip of paper by
and is constructed
13 cycles of 20 days, and a 365-day one
identifying its ends in opposite direc
stamps feature the teaching of
with 18 months of 20 days and five ex
tions. Mobius was not the first to dis
mathematics. This stamp, issued by
tra days: combining these gave a 'cal
cover it-Johann Benedict Listing beat
Guinea-Bissau for the
International
endar-round' of 18980 days. Only a
him by a few months.
Year of the Child, illustrates the teach
handful of Mayan manuscripts have
Monge
Gaspard Monge (1746-1818)
ing of the geometry of a circle;
survived, most notably the Dresden
taught at the military school in Mez
advanced vector methods of his day,
bark and containing many examples of
for gun emplacements in a fortress, he
James Clerk Maxwell (1831-1879) syn
Mayan numbers.
Maxwell's equations Using the most
codex, painted in colour on fig-tree
ieres. While investigating positionings improved the known methods for pro
Metric system Throughout the cen
jecting three-dimensional objects on to
turies various counting systems have
a plane; this subject became known as
matical theory, confirming Faraday's
been used for weights and measures.
'descriptive geometry'. Monge's other
intuition that light consists of electro
After the French Revolution, a com
interests included 'differential geome
magnetic waves. His celebrated Trea
mission was set up to investigate the
try', in which calculus is used to study
desirability of a metric system for
curves drawn on surfaces, and he
containing the fundamental mathemat
France; the chairman of this commis
wrote the first important textbook on
ical laws now known as 'Maxwell's
sion was Joseph-Louis Lagrange. This
the subject.
thesized Faraday's laws of electro magnetism into a coherent mathe
tise on electricity and magnetism,
Maxwell's equation
Math teaching
Mayan codex
Please send all submissions to the Stamp Corner Editor,
Robin Wilson, Faculty of Mathematics,
The Open University, Milton Keynes,
Metric system
MK7 6AA, England
Monge
e-mail:
[email protected] 76
THE MATHEMATICAL INTELLIGENCER
Mobius strip
© 2006 Springer Science+Business Media, Inc.