Letters to the Editor
The Mathematical Jntelligencer
The Alternating Harmonic Series
encourages comments about the
Mengoli-Mercator Formula"
material in this issue. Letters to the
the definition of
editor should be sent to either of the editors-in-chief, Chandler Davis or
I enjoyed Friesecke and Wehrstedt's Note. ''An Elementary Proo f of th e Gregory
( Mathematical lrztelligencer 28 (2006), no. 3. 4-Sl.
While reading it. I wondered if there was a geometric argument proceeding from
e r v = � between 1 and 2. A little tinkering showed me there was. It is simply that the J(\wer Riemann sum for � dx using
ln(2) as the area u
equal intervals of width
nd
(21
1
-;;•
.
1
I1+ n
Marjorie Senechal.
II
1
1
;�1
1
=I-+ .1 n II
;� 1 11
.\'
1.
•
1
is equal to the first 2n terms of the alternating harmonic series n
1
211
1
n
1
2n
1
11
;�1
I
;�1
I
;�1
I
;�1
211
( - 1 )i
. =II�-I�=I�-2II-= 21 + 1
;�1 11
1
;�1
ll
(equation (6) in Friesecke and Wehrstedt's note). Thus the alternating series must converge to ln(2).
Jim Henle Department of Mathematics Smith College Northampton, MA 01 063 USA e-mail:
[email protected] du
Editor's
Note. Rob Uurckel informs us that H. G. Forder, in a half-page note in
Mathematical Gazette 12 0925), 390, gives a brief derivation of the formula, much
like Henle's. And Bruce Berndt points out that the idea of using Riemann sums for
the integral of� to sum series dates bJck at least to Ramanujan, who used the techx
nique to show first that
1
+ 2::-:�1
2
( 2 n) 3 - 2 n
=
2
ln (2) ("Ramanujan's Notebooks,'' Bruce Berndt, construct an identity for In (3)
Math. Mag. 1 5, 1 978) , and later to ( Ramanujan �' Notehook.s, Pm1 I. Bruce Berndt, pp.
26-27). Meanwhile, we have the following
4
THE MATH EMATICAL INT ELLIGENCER © 2007 Springer Science+Business Media, Inc.
from Down Under.
Still Shorter For n E N, let
dx.
1 +X Then
1 1l
+1
By induction
1 - - + - -+ · ··+ ( -1)"+ 1 1
1
3
2
Let rz ----> co I" ----> 0 and
1-l_+l_-···=Io= It's that simple.
2
3
{ o
1
x+1
1
n
=Io + ( -1)"+ 1 In.
dx=
J2 I
1 X
dx= In 2.
Michael D. Hirschhorn School of Mathematics and Statist ics University of New South Wales Sydney, NSW 2052 Australia e-mai l : m . hirsch
[email protected] .au
© 2007 Springer Sc1ence+ Bus1ness Media, Inc., Volume 29, Number 2 , 2007
5
•:mt1
An othe r M otivat i o n fo r the Hype rbo l i c P l ane Segments Moving on the Line MARCOS SALVAI
n interval on the real line is said to be nontrivial if it has more than one point. Let us denote :J = (nontrivial hounded and closed
intervals of the real linel.
Notice that .'J is not a subset of a Eu clidean space (although its elements. the intervals. are so) . We ask ourselves how to define a notion of distance and of best path joining two elements of .'J. For example, x < y and x' < y' \Ve can take dist C[x, }'L[x' , v'D
� vrccx )2, ' ·-).,-2 -+----,- -J--., _ (J--:, ·':-_-x -
.
that is, copying the Euclidean distance from the open set ((x,y) E IR:2 :�: < yl by the obvious bijection. The best path joining these two intervals is then the curve y: [0, 1]--'> .'J given by y(t) = [(1- t) x + tx', (1 - t) y + (v'l. Now. this notion of distance may he not the most convenient one if we con sider the nature of the elements of :J. In a certain sense, it discriminates by the size of the segments. For instance, it gives dist ([0,10-'], [1,1 + 10-'1]) = v2 = dist ([0,10-:1], [1,1 + 10-"D. Nevertheless , considering the relative size of the intervals, the first two of them seem to be much closer to each other than the last two are (let us imag ine that we ask the interval [0,10-3] to move to the position of [1,1 + 10-.:l]; it will seem to it very far away ) . Let us seek a notion of distance ·which takes this into account . We will need some basic facts about the hyperbolic plane [1].
The Hyperbolic Plane
Let� = (( u,v) E IR:211'> Ol. The hyper bolic length of a differentiable curve y : [a , b] -'>�, y(t) = (u(t), l'(t)), is de fined by �ii'Y'(t)ll dt, where
J
lly'(t)ll = 1Cu'Ct), l''cm:/l'Ct)
6
THE MATHEMATICAL INTELLIGENCER © 2007 Spnnger Sc1ence-+-Business Media. Inc.
0)
is the hvperbolic speed of y at the in stant t (here 1C'c)') = V :\·2 + y 2 denotes the Euclidean norm ) . This is a model for the upper half-plane of Lobache\·sky Poincare, the classical example of a space v.·here the fifth axiom of Euclid does not hold . By a line in� we un derstand the image of a maximal geo desic, that is. a constant-speed curve in 'J-e defined on the whole real line which mmtmizes the (hyperbolic) length between any two of its points. They are \vel! kno\vn to be the curves obtained by intersecting� with verti cal (Euclide a n ) straight lines and with circles centered at points on the hori zontal axis. The importance of the hyperbolic plane stems from the fact, among many other things, that it is essentially the unique complete simply connected sur face with constant negative curvature.
A Metric on 9 Which Does Not Discriminate by Size Recall that we are looking for a notion of distance travelled hy a segment mov ing (and changing its size as wel l ) in the real line, and we want to treat the shorter intervals fairly. We identify a segment in .'J hy its midpoint a and its length e. that is, we parametrize :J via ¢ : :A = ((a,£) E IR:2,e > Ol-" .'f.
c/J(a,€) = [a - €!2.a + €/2].
(2)
Let y(t) = [at- €1/2, at + €1/2] be a curve in :J, and let a(t) = ¢- 1 (y(t)) = (at,ft) be the corresponding curve in :A. Now we see that the hyperbolic speed (1) of a (or equivalently of y) a t t h e instant t , that is,
is appropriate if we do not want to dis criminate by size. since it "relativizes" the Euclidean speed to the size of the interval at the instant t. In the following examples we ob serve that , at the infinitesimal level, each segment of .'J takes itself as the yard-
stick to measure the size of the neigh boring �egments and the distance from them. EXAMPLE 1: If an interval moves to the
right s times its own length without changing size , then its motion will be of s units if we consider the hyperbolic metric. In fact, for the curve y ( t) = [a- £/2,a + £/2] + t€, with 0 :::s t::::; s, the speed of the associated curve aU)= (a+ t£.£) in 'Je, with respect to the hyperbolic metric, is
II :t
(a+
te,nll
= IIC£,0)ll(a+ll't) =
:(f,O)j/t = 1 .
Hence the length of y is dt= s. More concretely. a segment of length 1 000 which moves 3000 units to the right tra\·els the same distance as a segment which measures .001 and moves .003 in the same direction. (By the travelled distance \Ve understand the length of the trajectory. which may be brger than the distance bet\veen the end points . )
J;�
EXAMPLE 2: Let y be the curve in j
defined
by
y(t) = [-e',e'],
and
let
a(t)= W.2e1) be the associated curve
in 'Je. Then the Euclidean speed at the instant t, 1a'( f) = 1(0,2e1), = 2e1, coin cides with the size of the interval at f. Since the segment perceives itself as be ing the yardstick at any time. \\'e are not surprised that y has constant (hyper bolic) speed equal to one: lla'(t)JJ = (0,2e' )'/2e1 = 1 .
Geodesics i n !P
Since the metric we are considering on j is copied through the function ¢ from that of the hyperbolic plane, for which the vertical straight lines are trajectories of geodesics, it turns out that the curve y of Example 2 is a geodesic in j. We also observe, considering the other geodesics of the hyperbolic plane, that in particular the trajectory of the
The segment of length L centered at h is in the trajectOJy of the best path joining the segments of length £ centered at a and c. respectively. best path in j joining two segments of the same length , say £, consists of seg ments of length greater that £. This can he explained by noticing that since each segment takes its own length as the standard by which to measure travelled distances or changes in its size, a longer segment will cover great distances more easily. Instead of simply moving to its final position keeping its size constant, as it would under the Euclidean metric on .1'. it does better to make an extra effort at the beginning and increase its size. so it \\·ill perceive distances as shorter, and finally make the effort of reducing its length at the end.
lsometries of !P
= z + h,
Higher Dimensions If one considers balls or spheres in IR", one obtains in the same way models for the hyperbolic space of dimension n + 1, by identifying the ball or sphere of center a E IR" and diameter d with the point (a,d) in the upper half-space of �n+I ACKNOWLEDGMENT
It is well known that the orientation preserving isometries of the hyperbolic plane '3£ are the Mobius transformations of the complex plane which preserve '3£. Some of them are
.ft,(z)
ment of equal length with initial point moYed hy b units. As we could have expected. after that transformation the segments do not notice any change. The same happens for Gc: the sizes of the segments and their relative positions all change in the same proportion c.
gc(z) cz, and h(z) = - 11z, =
with b, c E �. c > 0. The correspond ing isometries of j, induced by the iden tification ¢, which I denote with capital letters, are the following. The mapping Fh takes each segment of j to the seg-
I thank Pablo Roman for his help in draw ing the picture. REFERENCES
[1] Anderson , James W . , Hyperbolic geome try , Springer Undergraduate Mathematics Series. London (2005) . FaMAF-CIEM Ciudad Universitaria 5000 Cordoba Argentina e-mai l : salvai@mate. uncer.edu
© 2007 Springer Science+ Business Media, !nc
.. Volume 29. Number 2. 2007
7
•:Mti
A Sh o rt De r ivat i o n of Lord Bro u nc l<e r's C o nti n u ed F ract i o n for 7T PAUL LEVRIE
If we define Yn =
1
2n-
2n-3
1
--2n-'5
+
... + (-1)11-21+ (-1)11-1 + (- 1)11 '\
it fol lows immediately that
= 2n+ 1
Yn+1 + Yn
Yn+2 + Yn+I =
1 ,
1T
4'
1
2n+3
We can use these two equations to generate a second-order linear homogeneous recurrence relation by dividing the second one by the first Yn+2 + Yn+l
2n+ 1 2n+3
Yn+1 + Yn
and rewriting the result as .YH+2 + Yn+I =
2n + 1 + 2n+3 (Yn+I y,z).
(1)
Some rearranging leads to Yn+2 Yn+l
+1
=
2n+ 1 2n+3
( +�) 1
(2n+ 3).Y11+2 + 2
Yn+ 1
.Yn+l
1
=
-(2n+ 1)--Yn+l J'n
and finally we get
2n+ 1
Yn+I
Yn+Z.
2+ (2n+3)
Yn
Yn+1
This recurrence generates a continued fraction: 1
:...3-2+3· -__: 2+ 5. Yl
(2)
2 +3. -----"--3-2+ 5 . ---'-5 2+7·--'--72+
- ---
Y2
which converges to the value
.E: 4
Yo
Hence
i_ 1T
4
1 - .E:
.Yl
= 1
+
4
1=-z
___
2+
1.
1T
__ _
32
52
2+�,
and this is Lord Brouncker's famous continued fraction (see, for example, [1]).
SOME REMARKS. 1. That Yn =t- 0 for a l l n is a consequence of the standard proof of Leibniz's rule for alternating series applied to Leibniz's series for .E:. 2. To prove the convergence of the continued fraction, we note t hat z�,n .Yn and z�2J = ( -l)n are both solutions of the following recurrence relation:
=
Zn+2 + Zn+l =
8
THE MATHEMATICAL INTELL IGENCER © 2007 Springer Science+Business Media, Inc.
+ -2n -2n+3 1
(Zn+l + Zn)
¢:::>
z
n+ 2 _
_
2 z 1 + 2n+ 1 z n· 2n+3 n+ 2n+ 3
The continued fraction associated with this recurrence is 1
-
2 + :\
:\
-----
-� + · . .
and its partial numerators A 11 and denominators B11 are those linear combinations of 1, A1 = 0, B0 = 0 , B1 = 1 ( see for instance [2]). It is easy and z;;' that satisfy A0 to verify that
z;/'
=
A11
=
(
)'11+ 1 -
.
;)
·
( -ll11
RII = rII -.!!. 4 ... < -u��
From
A11
are given hy:
=
B 11
+
+ (-I
=
< -n��-�
(-1)11
- !311
All
!311
=
1-
lim .:.:h
IJ�X
we
.:;
-
_!_ 3
+
1
_ _
')
-.
. . +
(- 1 )11 - I
1
2n -
1
)
.
)11 it follows that the approximants of the continued fraction ( 3)
Taking the limit for 11 � x \\·e get
where
. (1
B ,,
-----1
- J. + J.- . . . + ( -1)11-1 s :\
I
2n-l
1
1-11"'
4
have used Leihniz·s series for.!!._. Hence the continued fraction ( 3) converges
to 1 - . By constmction, the continued fraction (2) is equivalent to ( 3 ) , so (2) also converges. and by comparing ( 3) with (2), we see that convergence is to the value
-(1-
2). 7r
REFERENCES
[1 ] J . Arndt and Chr. Haenel, 71"-un/eashed, Springer-Verlag , Berlin, 2001 . [2] L. Lorentzen and H. Waadeland, Continued Fractions with Applications, Studies in Computational Math. , Vol. 3, North-Holland, New York, 1 992. Department of Computer Science K. U. Leuven
Celestijnenlaan 200A B-3001 Heverlee, Belgium Departement IWT Karel de Grote-Hogeschool Salesianenlaan 30 B-2660 Hoboken, Belgium e-mail: paul .
[email protected] tf) 2007 Springer Sc1ence+Bus1ness Media, I n c . , Volume 2 9 , Number 2 , 2007
9
•:Mti
Remark on Stirling's Formula and on Approximations for the Double Factorial F. L. BAUER
In his fascinating article 1 'e: The Master cifAll', Brian ]. McCartin refers to Stirling's formula of 1730 (1)
and to its history. Tabulating this formula shows that the relative error is asymptotically - 1 -
( -;" )" · \!,2- 1r·n -
n!
n
10
3628800
1 00
9.33262 1 544 . X 1 0 1 57
x
12n
8.295960443 . X 1 0-3
0.99551 5253273
9.324847625 . X 1 0 1 57
8.329834321 . . X 1 0-4
0.9995801 1 8588
4.023872600 .
4.023537292 .
1 0000
102567
2.846259680 .
2.846235962 .
1 00000
X 1035659
2.824229408 .
10456573
2.824227054 . X 1 04565 73
X
relative error
relative error
:
3.598695618 . X 1 06
1 000
X
1211
X
X
102567
8.332985843 . X 1 0-5
0.999958301 1 60
8.333298608 .
0.99999583301 2
10-7
0.999999583330
X 10-6
1035659
8.333329861 X
The formula is, apart from its beauty, known for its good approximation,2 and thus it was not necessary to make it widely known that even a tiny correction to Stirling's formula still improves the approximation . ;.,-----,------:::--:Let us first consider the equation n! = e-" n" V 27T ( n + o nJ which can be 11 solved for On: o11 = ( n' ( e/ n) ) 2!(27T) - n. A tabulation of o 11 for some values of n, ·
·
·
·
n
1 . 1 76004802 . . .
0.176004802 .
10
1 0. 1 68006975 .
0.1 68006975 . .
1 00
1 00.1 66805076 . .
0.1 66805076 .
1 000
1 000.1 66680550 .
0.1 66680550 ..
1 0000
1 0000.166668055 . .
0.166668055 . .
1 00000
1 00000.1 6666805 .
0.1 66666805 . . .
(n! · (e/n)")2/27T
On
not only checks Stirling's formula experimentally: it also gives reason to guess that limn->"' 011 = The conjectured modified Stirling formula reads
-2;.
(1*)
n!
�
e-11
•
n''
·
J
27T
·
( n + -2;)·
Tabulating this formula gives: n
n!
10
3628800
1 00
9.332621 544 . . X 1 0 1 57
1 000 1 0000 1 00000
102567
4.023872600 . . X
1035659
2.846259680 . . X
10456573
2.824229408 .. X
{;)" · � 21r {n + ·
i)
9.332615096 . . X 1 0 1 57
1035659
2.846259680 . . X
10456573
X
1Mathematica/ lntel ligencer 28(2006) , no. 2, 11}-2 1 .
2J. Arndt and Ch. Haenel, in their pretty book
0.0 1 % from n
10
2o
89.
THE MATHEMATICAL INTELLIGENCER © 2007 Springer Science+ Business Media, Inc.
6.908958656 . . X 1 0-7
0.994890046523
6.944089506 . X 1 0-11
0.999948888993
10-13
0.999994888889
X
1r : Algorithmen,
2007) , mention that the relative error is less than 1% from n
1 44n2
0.949 1 08 1 1 2341
0.999488899422
6.944408950 .
2.824229408 .
x
6.591 028557 . X 1 0-5
6.940895 1 34 . X 1 0-9
102567
4.023872572 . . . X
rei error
relative error
3.628560824 . . . X 1 06
r
Computer, Arithmetik (3 d ed . , Springer, Berlin
2o
9, less than 0.1 % from n
2o
28, and less than
A The approximation is now much better3 ; the relative error is asymptotically I_, l-�c4nstrict proof for the modified formula can be based on the well-known 1 asymptotic expansion _
= e-n .
n'
n". � . ( 1 +
+
1 __ 1211
For the double factorial (2n- 1 )!!
def =
_1_, 2R8n-
+ .. ·).
(2n - 1) · ( 2n - 3)· . . . ·3 · 1 =
( 2 n)! 2'' · n''
an approximation can be obtained from the Stirling formul a , eliminating (2)
( 2n
1 )!!
_
( 2n) ! 2" · n!
=
_
-, ,
( )
2 n " Vz. .
7r:
e
But perhaps we should use instead the improved Stirling formula, leading to
(----;-)
2n "
( 2n- 1 )11-
(2*)
(12n + 1 )
·
(6n + 1 )
·
The result of ( 2 * ) for n = 5, 9!! 945 . 1 7377 10, is distinctly better than with the un modified Stirling formula (2) , which gives 91! 952.8896030 (Table 1) 1 1 - for ( 2 ) , , for ( 2*): The relative error is asymptotically =
=
24 11
192 rr
--
-
(2) relative error
n
10
4.172926252 .
100
4.167510527 .
1000
4.166753230 .
10000
4.166675344 .
100000
4.166667534 .
. x
. x
rei error
w-3
x
10-4
1.000202526691
. x
10-·5
1.000020775286
w-6
1.000002082752
. x
10-7
1.000000208327
. x
(2*) relative error
24n
1 .00 1502300523
rei error
x
192n2
4.899537334 .
. . x
w-s
0.940711168172
5.205227728 .
x
10-9
0.999403723811
5.177287373 . . . X 10-7 5.208022763 .
. . X
10-11
5.208302276 . . . X 10-13
0.994039175688
0.999940370570 0.999994037039
Moreover, from the folklore5 of the growth of binomial coefficients:
( ) 2n n
d�f
-
( 2n)! n ! · n!
-
22"
\l7iYz
comes another approximation6 for (2n - 1 ) 11 which is free of ( 2n - 1 ) ! !-
(3a)
e:
2" · n ' �
Multiplying the left-hand side by (2n - 1 ) ! ! and the right-hand side by the equivalent (2n)! 2" · n ' (2n)! ) . one obtams ((2n - 1 ) 11)-- , � , i . e . , the slightly better variant 2" · n ! v 1r · n 2" · n! '
(2n - 1 )!'-
( 3b )
n 10
rei error
(3a) relative error 1.25731 9341.
. x
100
1.250776360 . . .
1000
1.250078076 . . .
10000
1.250007812.
100000
1.250000781 .
X
X
X
. x
V(2;1)! �
x 8n
(3b) relative error . . X
rei error
x 16n
10-2
1.005855472905
6.266959316 .
10-3
1.002713490580 1.000308395968
10-3
1.000621088744
6.251927474 . . . x 10-4
10-4
1 .000062460932
6.250195056 . . .
10-5
1.000006249609
6.250019528 .
10-6
1.000000624996
6.250001953 . .
x
. . X
.
x
10-5
1.000031208982
10-6
1.000003124590
10-7
1.0000003 j 2496
3The relative error is less than 0.1% from n 2: 3, less than 0.01% from n 2: 9, less than 0.001% from n "'= 27, and less
than 0.0001% from n 2: 84.
4See for example: George Marsaglia and John C. W. Marsaglia, A New Derivation of Stirling's Approximation to
n!.
American Mathematical Monthly, 87, (1990), 826-829. 5Max Koecher, K/assische elementare Analysis. Birkhauser, Basel 1987, p. 76 (Korollar 2); Reinhold Remmert et a/., Zah/en, Springer-Verlag, Berlin 1983,
p. 117.
6As soon as even small handheld computers like the Hewlett-Packard hp the double factorial, this formula became interesting.
11C
had a key for the factorial, but none for
© 2007 Springer Science+ Business Media, Inc., Volume 29. Number 2. 2007
11
Table 1 shows that (3b) is better than (3af (the relative error is asymptotically _l_ for H11 1 6 for (3b)) and that both are not very different from (2), but that the improved (3a) , 1
II
(2*) is dramatically better:
Table 1. Comparison of (2*), (2), (3a), (3b)
(2n)n. Vj(12n+1}
(2n- 1)!!
n
(6n + 1 )
e
(2ne) .
v;:-:n
v'2
��--------
--
-------===- ------
V(2rlj!
2n. n!
�
1.128379167
1.062251932- \11.1.128379167
2
3
3.002820329
3.062287889
3.191538243
3.094287435
=
V3
3
15
15.00701675
15.20846222
15.63528038
15.31434640
=
\115 . 15.63528038
4
105
105.0291997
106.0955133
108.3244000
106.6492475
=
V105
5
945
945.1737710
952.8896030
968.8828884
956.8669341
=
\1945 . 968.8828884
1.002670344
1.040520190
3.191538243
·
·
108.3244000
However, the folklore formulas may also be improved. Before doing so, we observe that the samples both for (3a) and (3b) give upper bounds. Lower bounds are obtained in the following way: The Wallis product reads as follows: .
.
. .
the elements of the sequence are 2 1 times the square of I l. -'- S2. n ·
thus asymptotically
y:;:
( n + _1__2 )
·
�
2 1
2 +1
·
2. 3
(2n- 1)11
(4a)
·
S2. �
.. .
·
·
\I
__liz_ and H 2n-1
·
:3
·
S
)
' ·
• ·
·
·
)
2 n-l ' 211
·
2" n' ·
�
J
( n + 1).
7T.
Multiplying, as above, the left-hand side by (2n- 1)!! and the right-hand side by the
· I ent eqmva
(2n)! . . a s 1·tgI1t1y I )etter vanant . , we o IJtam agam 2" n!
--·
(2n)!
(2n- 1)!!
(4b)
�
Table 2 shows that we have indeed lower bounds now. Table 2. Comparison of
n
(2n- 1)!!
(4a) I
2n
and (4b) ·
n!
(2n)!
I
�7T· (n + �)
v7T·(n+�)
0.921317732
0.959852974
=
\11.0.921317732
2
3
2.854598586
2.926396374
=
\13. 2.854598586
3
15
1 4.47545684
14.73539455
=
\115.14.47545684
4
105
102.1292238
103.5546643
=
\1105 . 102.1292238
5
945
923.7935873
934.3366310
=
\1945.923.7935873
The relative error is now again asymptotically _l_ for (4a), -1- for (4b ) : 16n
Hn
n
(4a) relative error 10-2
. . X
1 .249992968 . . . X 10-6
100
1.243002794 .
1000
1.249397216 .
. . X
100000
, \l(2rlj! J
v;-:-n
---
=
(2n- 1 )!!
8The folklore simplified the (n +
12
THE MATHEMATICAL INTELLIGENCER
x
8n
(4b) relative error
0.946379734734
5.932470444 . .
10-3
0.994402235964
6.216946495 .
10-4
0.999437773329
6.246681188 . .
0.999943752734
6.249667983 . .
0.999994375027
6.249966797
1.249929690 . . . X 10-5
10000
7Actually,
rei error
X
1.182974668 . . .
10
·
2n n!
·
v;-:n
--
.
�) in the denominator of (4a) and (4b) to n.
rei error
x
16n
. X
10-3
0.949195271180
10-4
0.994711439354
. X
10-5
0.999468990150
10-6
0.999946877402
10-7
0.999994687524
. X . X
. . . X
We could now use the arithmetic means of (3a) and ( 4a ) or of ( 3 b ) and (4b) . ob taining for n = 5 in the case ( 3a l / ( 4a ) 946.3382380, in the case C3b) / ( 4b ) 945.601782') quite respectable when compared vvith 945 . 1 7377 1 0 from ( 2*) . Note that ( 3a )/(4a) yield a n inclusion interval 946.3382380':::: 2 1 .2064 1 270, (3b)/(4b) yield an inclusion interval x- 945.6017825 1 :::: 1 1 . 26')1') 1 5 . However . \Ve are now going t o obtain even slightly better results b y improving the formulas ( :3a ); ( 4a J and ( :3b ) / ( qb J . looking for the best On· To this end . we observe that the ( a )-variant and the ( b )-\·ariant give the same oil. We consider for the (a)-\·ariant the equation:
lx-
(2 n - 1 )!1 and solve it for oil:
o ��-
(
( 2n n! )2 (2n-
YTT· (n + o;J
2 1l . n! (2 1 ) !!
n-
A tabulation of oil for some values of
n
2 1 l · n!
=
n.
)2
.,.
1)!!
. _l._ 7T
- n.
On
1.2732395447351
0.2732395447351
10
10.2530447201518.
0.2530447201518 ..
100
100.2503117163367 .
Q2503117163367 .. 0.2500312421850 . .
1000
1000.2500312421850 .
10000
10000.2500031249218 .
0.2500031249218 . .
100000
100000.2500003124992 .
0.2500003124992 . .
1000000
1000000.2500000312500 .
0.2500000312500 . .
gives reason to guess that lim11_,x 011 =_!_ ( more precisely: oil-_!_ + 1 J Thus � . � 3211 -
�
( 2n - 1 l1'-
(3a*)
'VITT·(n+±)'
.
and
( 2n)!
( 3b*)
(2n-ll!!-
/.
1
'J'TT (n + -:;-)
improves the accuracy considerably . as shown in Table :39 Table 3. Comparison of (3a*) and (3b*)
n (2n- 1)!!
2n n! (n + �) ·
\ rr .
rei error
rei error x
64n2
(2n)! (n + �)
\/ n-.
rei error rei error
x
128n2
1.009253009
0.009253009
0.592192576
1.004615851
0.004615851
0.590828928
2
3
3.009011112
0.003003704
0.768948224
3.004502178
0.001500726
0. 768371712
3
15
15.02189149
0.001459433
0.840633408
15.01094175
0.000729450
0.840326400
4
105
105.0901043
0.000858136
0.878731264
105.0450425
0.000428976
0.878542848
5
945
945.5328815
0.000563896
0.902233600
945.266403
0.000281998
0.902393600
Reversal of ( 3a *) leads to
(Bauer 1 994). 9F. L.
Bauer, Decrypted Secrets-Methods and Maxims o f Cryptology, 4th edition. Springer, Berlin 2007, p . 48. ( I n
cryptography, i t i s important t o have good and nevertheless simple approximations for (2n- 1)!!, the number o f prop erly self-reciprocal permutations, of 'genuine reflections' of 2n elements).
© 2007 Springer SCience- Bus1ness Media, Inc., Volume 29, Number 2, 2007
13
The relative error is asymptotically
64\2 1
-2 28 17 1
as can be seen here: (3a·) rei error
n
for (3a*),
rei error
x
for (3b*),
64n2
(3b') rei error
1 0-4
0.950476200605
1 0-6
0.99500399221 8
7.773465667 .
. . . X
1 0-8
0.999500039148
7.808594025
1.485 1 1 9063 .
1 00
1 .554693737 .
1 000
1.56 1 7 1 881 1
1 0000
1 .562421 875 .
1 00000
1 562492 1 87 .
. . X
. . X . . x
1 0-10
1o-· 12
rei error x 1 28n2
7.42531 9640 . . . X 1 0-s
. . X
10
. . X
. . . X
0.999950000391
7.8 1 21 09377 .
0.999995000004
7.81 2460937 .
. . X
. . X
0.9504409 1 3967
1 0-7
0.995003605487
10-9
0.999500035246
1 0-11
0.999950000352
1 0-11
0.999995000004
The different approximations are summarized in Table 4 . For n 1, n 10, a comparison of the numerical results (correctly rounded) is given. =
Table 4. Summary: Calculation of Double Factorial for 1!!
=
1, 5!!
=
945, 10!!
=
=
5, and n
=
654729075; with
asymptotic relative errors.
(4a)
( 4b)
(2•)
( 3b.)
(3a·)
(2)
(3b)
(3a)
0.921 31773
0.95985297
1 .00267034
1 .00461585
1 .00925301
1 .0405201 9
1.06225 1 93
1 . 1 28379 1 7
923.793588
934.336631
945. 1 73771
945.266403
945.532882
952.889603
956.866934
968.882888
646983796. 1 8n
65084491 4. 1 1 6n
654761 1 54. 1 1 92n2
654777691. 1 1 28n2
654826310. 1 64n2
6574612 1 1 . 1 24n
658832235. 1 1 6n
662961 1 1 0. 1 8n
Moreover, Table 4 shows upper bounds: a group of three with asymptotic relative er rors proportional to .l and a group of three with asymptotic relative errors proportional to
\; and lower bound� : a group of two with asymptotic
,r
relative errors propo1tional to .l.
ACKNOWLEDGMENTS
Thanks go to Christoph Haenel for computational support, using MATHEMATICA. Nordliche Villenstrasse 1 9 D-82288 Kottgeisering Germany
Opinion Survey The Best \lathematical Books of the Twentieth Century
(MathematicallutelliRencer.
volume 29. number 1, p;.�ge 2 1)
R ·aders ;.�re reminded rhatthe dosing date for contribmions to thut an epkyde m<xld {"hieh. in
cid ·malty. gh es a 'er� rea,onahle cstimat
·
for th
·
dis
,.
t.mu:s of the plam:ts from the r�llios of the epic}de radii.)
3.
A qualitativ
recording of the planetaf} orbits I tim
ing. �a}. (io ) ears for la�tronomical "} mho! for the pbnl:t •
aturn) and lasrronomical
"llpports on
trongl} tht•
e
}mhol for the plan ·t Jupnerl>
btence of epicycles, s
·n edge
nd moving along a slightl) indined defer ·nt c cf.Hg ·
J';') p 12'1-1 in Ill} "lhst. .\nc \I,Hh. Astron. ) .t. th •
of g.1l • I.
·
\ic''
1es in Kant-Laplace!
\-; a r '"lilt of \ Cf) sophisticated obscn·ation._ (p •r .
haps in part based on Bah} Ionian arithmctiral mcxlels!)
Hipparchus and Ptol 'Ill) found de\ iations from the um
formity of motion. It turned out that tht• ...implc
de,·ke
of a">:-.uming ec. Since the founding members
Yie'\\· of Hildesheim from !VIoritzberg. Photo counesv of ETH 13 i bl io t hek . Zurich.
brilliant i\onvegian. Sophus Lie. his older friend and collaborator. Soon thereafter. in November 1 872. Clebsch suddenly died. Most of his students in Gottingen decided to join his protege in Erbngen. and they soon found that Klein knew how to keep them busY. One of these Clehsch pupils. Ferc.li nand Lindemann. vva s given the task of editing the master's lectures. The first edition of the resulting volu me. \Yhich came to he known as Clebsch-Lincle mann. appeared in 1 H76: at least that \\·as the official date of publication. Ap parently the book came out earlier. as Lindemann sent a copy to the Gottin gen mathematician, !VI . A. Stern. \\·ho \\'rote back on 27 November 1 87 5 thanking h i m : "I have t i l l n o w on ly looked at the book very superficially. but I believe I can say that the presen tation is very illuminating. perhaps more than it would have been under Clebsch's own editorship. For despite all appreciation for the richness of his intellect. one must admit that his pres .. entations -;vere often quite obscure . , Moritz Abraham Stern. who had stud ied under Gauss, \Yas evidently still n?ry sharp when he wrote this at 68 years of age; indeed, he would not retire for 5M.
A
Stern to Ferdinand Lindemann,
6Felix Klein to M .
A
Stern,
26
July
another ten years. \X-'hen he did finally take his leave in 1 8H 5 . his chair \Yas as sumed by Felix Klein. As we shall see . Stern. Klein . and Lindemann were all destined to phl\· a major part in Hur \\·itz's career. Klein also mainta ined close ties \Yith t\vo older members from Clebsch's cir cle. Paul Gorda n and Max Noether. both of \\ hom-like Stern-were Jewish . I n lir4 Klein had t h e opportunity to fill a ne\v post in Erlangen as associate pro fessor. I !e consulted with Stern about this. and Gordan soon thereafter ac cepted (' Klein's second choice was l'e
and talk mathemat ics-�eldom anything ebe-u nt i l he fin
was the cominuation of another. on anal} tic fu nctions,
i-.h ·d hb dgar. Then we would go for
which I had not attended: I therefore had some diffi< ulty
in� t h · math ·matical discussion. H i. health '' as not too
in fol lowing and had to " ork hard. read in� man} hooks. Once when I mbsed a point in a lelture I '' ent to l l ur '' itz aftef\\ ;uds and a..,ked for a prh at
· e
planation. l ie
im ited me and another student from Br '!-.lau. K} na!-.t , . . .
to his house and ga\ e us a .,eries of pri\ ;He lett u re!-. on sonK· ch.tpter., of the th
Balancing position
With the uniqueness of the equal-hovering position for a given y, and with the continuous dependence of this po© 2007 Springer Science +Business Media. I n c. . Volume 29. Number 2 . 2007
53
sition of the table upon y, we conclude that the distance that
B and D are hovering above the ground is also a con
tinuous function of y. If we are dealing with a square table,
we can now finish the proof using IVT one more time, as described in the intuitive table-turning argument presented at the beginning of this section. For a general rectangle, let the
initial position
be an
equal-hovering position for which the z-coordinate of the center of the table is a minimum, and let the
end position
be an equal hovering position for which the z-coordinate of the center of the table is a maximum. Note that the hov ering vertices in the initial position must be on or below the ground: if not, we could create a lower equal-hovering position, contradicting minimality, by pushing vertically
Figure 4.
The four conic sections and the table top in the
case of a s q uare table with legs of length
horizontally.
� that is
balanced
down on the table until the hovering vertices touch the ground. Similarly, in the end position the hovering vertices
tains the whole table top. It is clear that the four conic sec
are on or above the ground. Now, IVT can be applied to
tions are congruent and that any two of them can be brought
guarantee that among all the equal-hovering positions there
into coincidence via a translation. Furthermore, given any
is at least one balancing position.
point of one of these conic sections, this point and the re spective points in the other three conic sections form a
Balancing real tables
To balance a real table of side lengths ratio r, we determine a balancing position of the associated mathematical table,
a s described above. We now show that legs of length at least
� guarantee that, balanced in this position, none
of the
pcii�ts of the real table is below the ground. We give
square that is congruent to our table top. Finally, since the legs have slope of at least
�, the end point of a leg of
our table on the plane is cont iined in the conic section as
sociated with the other end point of this leg.
�· �. This
To show that we may need legs of length at least we concoct a special ground with Lipschitz constant
�ircle,
the complete argument for a square table, and then de
ground coincides with the xy-plane outside the unit
scribe how things have to be modified to give the result
and above the unit circle it is the surface of the cone with
for arbitrary rectangular tables. I n the following, we will re
vertex (0,0,
fer to the four vertices of the table top as corresponding to the vertices
A ' , B' , C' , D' ,
A, B, C, D. respectively, of
the mathematical table.
�) and base the unit circle . As the diagonals
of our table a fe of length 2 , the mathematical table will bal ance locally on this ground iff its vertices are on the unit
circle. This means that the legs have to be at least as long
We first convince ourselves that no matter how long the
as the cone is high if we want to ensure that no point
legs of our table are, no part of a leg of the balanced table
of the table top is below the ground; it follows that we
will be below the ground. Let's consider the orthogonal tri
have to choose the length of our legs to be at least
pod consisting of
length of the legs is equal to
AB, AD, and the leg at A. Since the Lip
schitz constant of our ground is at most
�, t h e slopes of
AB and AD are less than or equal to this
talue;
thus, ar
�· If the � and the table is bilanced
on this ground, then the four co'h ic sections are circles that
intersect in the center of the table top as shown in Figure
guing as above, we see that the leg must have slope at
4.
least
union of these four circles. If we make the legs longer, the
.Jz· This implies that no l e g of o u r balanced table will
dip bel�w the ground.4
It remains to choose the length of the legs such that no
point of the table top of our balanced table will ever be below the ground. First, fix the length of the legs and con sider the inverted
souo
circular cone, whose vertex is one
of the vertices of our mathematical table, whose symmetry axis is vertical, and whose slope is
J:z.
Intersecting this
As you can see, the table top is indeed contained in the
circles will overlap more. If we make the legs shorter, the circles will no longer overlap in the middle. Now consider any ground, and take the legs to be of length
�; clearly, if we can show that this table does not
dip bela� the ground, then the same is true for any table
with longer legs. When we tilt the table away from the hor izontal position, the intersection pattern of the conic sections
cone with the plane in which the tabfe top lies gives a
gets more complicated. The critical observation is that tilting
conic section which is an ellipse, a parabola, or a hyper
the table results in the conic sections getting larger; it can
bola . 5 Note that since we intersect the plane with a solid
be shown that each conic section contains a copy of one of
cone this conic section will be "filled in". We can be sure
the circles in Figure
that a point in this plane is not below the ground if it is
these conic sections, it and the corresponding points in the
46
Since, given any point of one of
contained in the conic section. Therefore, what we want to
other three conic sections form a square that is congruent to
show is that the union of the four conic sections associ
our table top, the union of these conic sections will contain
ated with the four vertices of our mathematical table con-
a (possibly translated) image of the union of the circles, that
4For certain grounds with Lipschitz constant
0· it is possible that a leg of a balanced table may lie along the ground.
5Parabolas and hyperbolas can occur because the plane that the table top is contained in can have maximum slope greater than
0·
6To see this, note that what we are looking at are the possible intersections of a given cone with planes that are a fixed distance from the vertex of the cone.
54
THE MATHEMATICAL INTELLIGENCER
Figure 5.
If the table is not horizontal, the conic sections are larger than the circular (gray)
sections in the horizontal case (left). Their union is simply connected (right) .
A'
A
B'
B'
CAJ
B
Figure 6.
A
The points of intersection of the two l ines of slope
tionings of the rectangle AA' BE ' are on a circle .
A'
B'
A
B
~
---k for the different possible vertical posi 'v 2
we encountered before ; see Figure 5. Our previous picture
everything that we have done so far to end up with another
has, so to speak, just grown a little bit and been translated.
circle segment. However, the apex of this circle segment will
(Note, however, that it is not immediate that the conic sec
be closer to
tions together cover the table top rather than some transla
fact, the more we tilt, the closer we will get; see Figure
tion of the table top). Using this fact and the simple possi
A' B' than the one we encountered before. In 7.
Since t h e apex of o n e of these circles is t h e point clos
ble convex shapes of the conic sections that we are dealing
est to
with, we can conclude that no matter how we tilt the table,
horizontal, we now calculate just how close this apex gets
A' B ' , and since the apex corresponds to AB being
the union of these conic sections will always be a simply
when we tilt around
connected domain. This means that we can be sure that the
This maximal possible angle is attained if
table top is contained in this union if we can show that the
thogonal to
boundary of the table top is contained in it. We proceed to show that for all possible positions of
AB though a maximal possible angle. AD (which is or
�
AB) has slope · It i s a routine exercise to 2 check that in this position the slope of the line connecting
A with the midpoint of A' B' i s
� · This means that in this
our table in space the sides of the table top never dip be
position the apex will be contained i n
low the ground. Clearly it suffices to show this for one of
that if we choose the legs of our square table to be at
the sides of the table top, say
A' B ' . For this we consider
the possible positions of the rectangle with vertices A, A ' , B, and B' i n space. W e start with the rectangle vertical and AB horizontal. Draw lines of slope ending in A and B;
�
see Figure 6 (left) . Since the point of int� rsection of these two
lines is not above
least
A' B' . We conclude
� long, then the boundary of the table top, and hence
also th � table top itself, will not dip below the ground.
For tables that are not square, the same arguments ap
ply up to the point where we start tilting the rectangle
A' B ' , no point of this segment can be
B'
below the ground when the rectangle is positioned in such a way. Now rotate the rectangle around its center, keeping it in a vertical plane, and keeping the slope of AB less than or equal to
�· Again, draw lines of slope � ending in
A
and B; see FigJre 6 (middle) . Again, the position of the point
at which these two lines intersect tells you whether
A' B '
can possibly touch the ground with the rectangle in this position. Since the two lines always intersect in the same angle, we know that the points of intersection are on a cir cle segment through
A and B; see Figure 6 (right) .
Now tilt the original rectangle around AB. We repeat
Figure 7.
The more we tilt, the closer the apex of the circle
segment gets to A 'B ' . As long as the apex is below or on A'B ' , we can be sure that A'B' does not dip below the ground.
© 2007 Springer Science + Business Media, Inc., Volume 29, Number 2, 2007
55
AA' B' B around AB. We now have to worry about two dif
a sphere that is large enough to ensure that all legs of our
ferent rectangles corresponding to the longer and shorter
table end up on this part of the sphere whenever the table
r is the ratio of the lengths of the
is locally balanced on this ground. Then intersecting this
sides of the table, then it is easy to see that the critical
sphere with the plane that the leg points are contained in
sides of the table top . If
length of the legs that we need to avoid running into the
gives a circle that all leg points are contained in. Hence the
ground is the length that makes the longer of the two rec
leg points of the table are concircular. Now, let's consider a
tangles similar to the rectangle that we considered in the
ground that includes part of an ellipsoid which does not con
square case. This critical length is
tain a copy of the circumcircle of the leg points of the table;
�.
V l + r2
Other Balancing Acts
D
moreover. we choose the ellipsoid large enough so that all leg points of our table end u p on the ellipsoid whenever the table is locally balanced on this ground. Then intersecting
Horizontal balancing
the ellipsoid with the plane containing the leg points gives
When we balance a table locally, the table will usually not
an ellipse that is different from the circumcircle of the leg
end up horizontal, and a beer mug placed on the table may
points. However, this is impossible if the table contains more
still be in danger of sliding off. It would be great if we could
than four leg points because five points of an ellipse deter
arrange it so that the table is not only balanced but also hor
mine this ellipse uniquely. We conclude that an always lo
z
cally balancing table must have three or four leg points and
axis and balancing it somewhere else on the ground. Just
that these points are concircular. Note that requiring concir
imagine the ground to be a tilted plane, and you can see that
cularity in the case of three points is not superfluous, for we
this will not be possible in general. However, Fenn [4] proved
need to exclude the case of three collinear points.
izontal, maybe by moving the center of the table ofi the
the following result:
lf a continuous ground coincides with the xy-plane outside a compact convex disk, and if the ground never dips below the xy-plane inside the disk, then a given square table can be balanced horizontally with the center of the table lying above the disk. Let's call the special kind of ground described here a Pen n ground and the part of this ground inside the distinguished compact disk its hill. The problem of horizontally balancing tables consisting of plane shapes other than squares on Fenn grounds has
Livesay's theorem, which made the proof of Theorem 1 so easy, has a counterpart for triangles, due to Floyd [6]. It is a straightforward exercise to apply this result to prove the following theorem:
THEOREM 3 (Balancing Triangular Tables) rr
the ground function is continuous, a triangular table whose three leg points are contained in a !>phere around its center can he balanced local£y.
also been considered. Here "horizontal balancing on a Fenn ground" means that in the balancing position some interior points of the shape are situated above the hill. It has been
Of course, one should be able to prove a l ot more when it comes to balancing triangular tables'
shown by Zaks [29] that a triangular table can be balanced
In the case that the center and the (three or four) leg
on any Fenn ground. In fact, he showed that if we start
points of an always locally balancing table are coplanar,
out with a horizontal triangle somewhere in space and mark
we can say a little hit more about the location of the cen
a point inside the triangle, then we can balance this trian
ter point with respect to the leg points. Begin by balanc
gle on any Fenn ground, with the marked point above the
ing the table in the xy-plane and drawing the circles around
hill, by just translating the triangle. Fenn also showed that
the center that contain leg points; if one of the leg points
tables with four legs that are not concircular and those form
coincides with the center, then also consider this point as
ing regular polygons with more than four legs cannot al
one of the circles. Now it is easy to see that there cannot
ways be balanced horizontally on Fenn grounds. Zaks men
be more than two such circles. Otherwise a ground that
tions an unpublished proof by L. M. Sonneborn that any
coincides with the xy-plane inside the third smallest circle
polygonal table with more than four legs cannot always be
and that lies above the plane outside this circle would
balanced horizontally on Fenn grounds. It is not known
clearly thwart all local balancing efforts. Therefore, if we
whether any concircular quadrilateral tables other than
want to check whether our favorite set of three or four con
squares can always be balanced horizontally on Fenn
circular points is the set of leg points of a locally balanc
grounds. See [ 1 4] , [ 17] , [ 1 8] , and [ 1 9] for further results re
ing table, there are usually very few positions of the cen
lating to this line of research.
ter relative to the leg points which need to be considered.
Local balancing of exotic mathematical tables
ter of the circle that the leg points are contained in. As a
Perhaps the most natural choice for the center is the cen Taking things to a different mathematical extreme, we can
corollary to the above theorem for triangles, we conclude
n 2: 3 leg points in 3-space together with a n additional center point. We then ask
that a triangular table with this natural choice of center is always locally balancing. In the case of four concircular
whether, given any continuous ground, it is always possi ble to balance this table locally, that is, move this config
whether any tables apart from the rectangular ones are al
uration of
ways locally balancing. However, a result worth mention
consider a table consisting of
n + 1 points into a position in which the n leg
points are on the ground, and the center is on the z-axis. The example of a plane ground shows that the leg points of an always locally balanceable tabl e have to be coplanar. Let's consider the example of a ground that contains part of
56
THE MATHEMATICAL INTELLIGENCER
points with this natural choice of center we do not know
ing in this context is Theorem 3 in Meyerson's paper [ 1 7] (see also the concluding remarks in Martin's paper [ 1 6]). It can be phrased as follows:
Given a continuous ground and one of these special fou r-legged tables in the xy-plane, the
tahle can he rotated in the xv-plane around its center to a posit ion ll 'here the jciltr points 0 1 1 the f!,f"C){ { nd ahol'e the leg points are coplanar. The quadrilateral formed by the copla nar points on the ground will he congruent to the table if and only if the plane containing it is horizontal . in \Vh ich case we have actually found a balancing position for our table. In all other cases, the quadrilateral on the ground is a deformed version of the table. Still , if the ground is not too wild . the quadrilaterals will he very similar. and lifting the table up onto the ground should result in the table not wobbling too much. Livesay's theorem is a generalization of a theorem by Dyson [ 2] , which only deals with the square case. A higher dimensional counterpart of Dyson's theorem arises as a spe cial case of results of Yang [2')]. Theorem 3 and Joshi [ 1 2 1 , Theorem 2 : Gicen a contimwus rea!-t •cilued jimction de fined on the n -.,phere, there are 11 mutual!)' 011hof!,Oilal di ameters of this sphere such that the ji m ctio n takes on the same t •aluc at al/ 2 n endpoints of" these dimneten;. Note that the endpoints of 1 1 mutually orthogonal diameters of the // sphere are the \·ertices of an //-dimensional orthoplex. one of the regular solids in 1 1 dimensions. ( For example, a 1 -dimensional orthoplex is just a line segment and a 3-dimensional orthoplex is an octahedron . ) esing the same simple argument as in the case of Livesay·s theorem . we can prove the follcnving theorem: THEOREM 4 (Balancing Orthoplex-Shaped Tables) An ( 1 1 - 1 ) -dimensional m1hoplex-shaped table in IR '1 can be balanced /ocal{l' on any growzd git •en h1· a con
rectangul a r tables the ends of whose legs do not form a perfect rectangle are not uncommon and , as our simple ex ample shows. those uneven legs may conspire to make our anti-wobble tactics fail. Considering our examples of a discontinuous ground at the beginning of this article, it should be clear that a wob bling table on a tiled t1oor may a lso defy our table-turning efforts. How to turn tables in practice
In practice, it does not seem to matter how exactly you turn your table on the spot, as long as you turn roughly around the center of the table. Notice that you needn't ac tually establish the equal hovering: as you rotate towards the correct balancing position. there will be less and less wobble-room until, at the correct rotation, the balancing position is forced. \X'ith a square table . you can even go for a little hit of a journey , sl iding the table around i n your ( continuou s ) backyard . As long as you aim to get hack to your starting position, incorporating a quarter-turn in vour overall mm·ement. you can expect to find a balanc ing position. REFERENCES
[ 1 ] de Mira Fernandes, A. Funzioni continue sopra una superficie sfer ica. Portugaliae Math 4 (1 943), 69-72.
[2] Dyson , F. J. Continuous functions defined on spheres. Ann. of Math. 54 (1 95 1 ) , 534-536.
[3] Emch, A. Some properties of closed convex curves in a plane,
tinuousjitnctimz: [Ril- l � R
[41 Fenn, Roger. The table theorem. Bull. London Math. Soc. 2 ( 1 970),
For other closely related results see [ l l . [24] . [ 26] . [27 ] . [ 281. [ ') 1 , and [ 1 0] .
[51 Fenn, Roger. Some applications of the width and breadth of a
Balance everywhere
Imagine a square table \Vith diameter of length 2 suspended horizonta lly high above some ground, vvith its center on the z-axis. Rotate it a certain angle about the z-axis . release it, and let it drop to the ground. It is easy to identify con tinuous grounds such that all four legs of the table \\ ill hit the ground simultaneously. no matter what release angle you choose. Of course, any horizontal plane \\·ill do, and so will any ground that contains a \·ertical translate of the unit circle . We l eave it as a n exercise for the reader to con struct a ground that is not of this type but admits horizon tal balancing for any angle. Also. readers may wish to con \'ince themselves that the fol l o\\·ing is true: Consider a ground as in Theorem 2. If the center of the table has the same z-coordinate in all its equal-hovering positions ( posi tions in which A and C touch the ground and B and D are at equal vertical distance from the ground) , then in fact the table is balanced in all these positions.
Amer. J. Math. XXXI/ (1 91 3), 407-4 1 2 . 73-76.
closed curve to the two-dimensional sphere. J. London Math. Soc.
(2) 10 (1 975). 2 1 9-222 .
[61 Floyd , E. E. Real-valued mappings of spheres. Proc. Amer. Math. Soc. 6 ( 1 955), 957-959.
[71 Gardner, Martin. Mathematical Games column in Scientific Amer ican (May 1 973), 1 04 . [8] Gardner, Martin. Mathematical Games column in Scientific Amer ican (June 1 973), 1 09-1 1 0.
[9] Gardner, Martin. Knotted Doughnuts and Other Mathematical En tertainments. W . H . Freeman and Company, New York, 1 986. [1 OJ Hadwiger, H . Ein Satz uber stetige Funktionen auf der Kugelflache. Arch. Math. 1 1 ( 1 960), 65-68.
[ 1 1 1 Hunziker, Markus. The Wobbly Table Problem . In Summation Vol. 7 (April 2005), 5-7 (Newsletter of the Department of Mathematics, Baylor University). [1 21 Joshi, Kapil D. A non-symmetric generalization of the Borsuk-Uiam theorem. Fund. Math. 80 (1 973), 1 3-33.
[ 1 31 Kraft, Hanspeter. The wobbly garden table. J. Bioi. Phys. Chem. 1 (200 1 ) , 95-96.
Some Practical Advice
[ 1 4] Kronheimer, E. H. and Kronheimer, P. B. The tripos problem. J.
Short legs and tiled floors
[ 1 5] Livesay, George R . O n a theorem of F. J. Dyson . Ann. o f Math.
Note that if you shorten one of the legs of a real-life square table, this table will wobble \vhen set clown on the plane, and no turning or t ilting will fix this problem. In real life.
[ 1 6] Martin, Andre. On the stability of four feet tables. http://= 20
London Math. Soc. 24 ( 1 98 1 ) , 1 82-1 92 . 59 ( 1 954), 227-229. arxiv.org/abs/math-ph/051 0065
© 2007 Springer Sc1ence- Bus1ness Med1a. Inc
Volunle 29 Number 2 . 2007
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[ 1 7] Meyerson, Mark D. Balancing acts. The Proceedings of the 1 98 1
bles: feasting from a rnathsnack. Vinculum 42(4) , 4 November
Topology Conference (Blacksburg, Va. , 1 98 1 ) , Topology Proc. 6
2005, 6�9 (also available at www.rnav.vic.edu .au/curres/math
snacks/mathsnacks. html).
( 1 98 1 ), 59�75.
[ 1 8] Meyerson, Mark D. Convexity and the table theorem. Pacific J.
[24] Yamabe, Hidehiko and Yujob6, Zuiman . On the continuous func
[ 1 9] Meyerson , Mark D . Remarks on Fenn's "the table theorem" and
[25] Yang, Chung-Tao. O n theorems of Borsuk-Uiam, Kakutani-Yam
[20] Polster, Burkard ; Ross, Marty and OED (the cat). Table Turning
[26] Yang, Chung-Tao. On theorems of Borsuk-Uiam, Kakutani-Yam
tion defined on a sphere. Osaka Math. J. 2 (1 950), 1 9�22 .
Math. 97 ( 1 98 1 ) , 1 67�1 69.
Zaks' "the chair theorem" . Pacific J. Math. 1 10 ( 1 984), 1 67�1 69.
abe-Yujobo and Dyson I. Ann. of Math. 60 ( 1 954), 262�282 . abe-Yujob6 and Dyson. I I . Ann. of Math. 62 ( 1 955), 2 7 1 �283.
Mathsnack in Vinculum 42(2) , June 2005. (Vinculum is the quar terly rnagaz1ne for secondary school teachers published by the
[27] Yang, Chung-Tao. Continuous functions from spheres to euclid
ean spaces. Ann. of Math. 62 ( 1 955), 284�292.
Mathematical Association of Victoria, Australia. Also available at www. rnav. vic .edu. au/curres/mathsnacks/mathsnacks. html).
[28] Yang, Chung-Tao. On maps from spheres to euclidean spaces.
[2 1 ] Royden, H. L. , Real Analysis . Prentice-Hal l , 1 988. [22] Vinculum, Editorial Board. Mathematical inquiry- from a snack to a
Amer. J. Math 79 ( 1 957), 725�732.
[29] Zaks, Joseph. The chair theorem. Proceedings of the Second
meal. Vinculum, 42(3), September 2005, 1 1 1 2 (also available at
Louisiana Conference on Combinatorics, Graph Theory and Com
www.mav.vic.edu.au/curres/mathsnacks/mathsnacks.html).
puting (Louisiana State Univ . , Baton Rouge, La. , 1 97 1 ) , pp.
�
[23] Polster, Burkard; Ross, Marty and QED (the cat). Turning the Ta-
CAMB RIDGE
557�562. Lousiana State Univ., Baton Rouge, 1 97 1 .
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58
T H E MATHEMATICAL INTELLIGENCER
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53 rue de Mora 95880 Enghien les Bains
France e-ma1l:
[email protected] Mere Mathematician?
J'rom [uga r Allen Poe's The Pttrloilled l.eller. as printed in t he l leritage Edit ion
of Poe':-. :-.tories. at paj.�e 2Ho.� :
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cian. he \\'ould r ·a son '' e l l , as m ere: mathematinan. he coulu not ha\ e rea soned at a l l . and t h u s '' ould h;n c.: h ·�.:n at the merq of the Prcfe t t . " Th
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Ralph A. Rrum
Un1vers1ty of Rochester Rochester, NY 1 4627
USA
e-ma1l:
[email protected] © 2007 Springer Science -Business M edia. Inc .. Volume 29. Number 2. 2007
63
la§l)l§l.'fj
Osmo P ekonen , Editor
i
Ramsey M ethods i n Analysis Spiros A . A lg)'ros and Stel'o Todorccl'ic BASEL, BIRKHAUSER, 2005, PP 257, €38, ISBN 3-7643·7264·8
REVIEWED BY HANS-PETER A. KUNZI
Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.
C o l u m n Editor: Osmo Pekonen, Agora Centre, 40014 U n iversity of Jyvaskyl a , Fi n l a n d e-m a i l :
[email protected] 64
I
'""'"'C"'"l he book introduces the reader to sophisticated Ramsey-theoretic methods that have recently been used in the theory of Banach spaces. Before describing its contents in some detail. I discuss the combinatorial and analytic background of the presented results. In a first course of comhinatorics one is taught that Ramsey theory [ 1 1] deals with the insight that many large structures-no matter how disor dered-necessarily contain highly or dered substructures. Its simplest in stance, the pigeonhole principle, asserts that n + 1 pigeons cannot sit in n holes so that every pigeon is alone in its hole. This principle is non-con structive in the sense that it states the existence of a pigeonhole with more than one pigeon in it, hut it says noth ing about how to find it. One usually meets the first nontriv ial applications of the theory when studying graphs. Unavoidably, one learns, in any collection of six people either three of them mutually know each other or three of them mutually do not know each other. Theorems of similar type hold for infinite graphs. If G is a graph with infinitely many ver tices. then G or its complement contains a complete suhgraph on infinitely many vertices. The exact formulation of the usual Ramsey theorem [13] for infinite sets re quires some terminology. Given a set X, let [XJA' stand for the set of all sub1A
sets of X of size k. If r is a positive integer. an ,�colouring of a set D means a function 7r from D to the set l l , . . . , r } . If a set D bas an r-colour ing 7T and C 0 and all rn * II. ll x"1 - xull > 1 + E. Probably the best-knmvn result of this kind is Rosenthal's theorem that a Banach space X does not contain an isomorphic copy of f 1 if and only if every bounded sequence in X has a \Yeakly Cauchy subsequence. The use of infinite-dimensional Ramsey theory in Banach spaces was developed fur ther by Gowers during investigations which were part of the work for �·hich he �·as awarded the Fields Medal in 1 998. As an example, we mention his positive solution to the homogeneous space problem of Banach. Gowers es tablished that if a Banach space is iso morphic to all of its infinite-dimensional closed suhspaces. then it is isomorphic to a Hilbert space. The contribution of Ramsey-theoretic ideas to this theorem is covered by his dichotomy result [7,Rl. which asserts that every Banach space contains a subspace vvhich either has an unconditional basis or is hereditarily in decomposable. This dichotomy. com bined with some analytical results due to Komorowski and Tomczak-]aegem1ann. implies a positive solution to the homo geneous space problem. It is natural to attempt to use con cepts from Ramsey theory to obtain di chotomy results in mathematics ( sec [9] ) . The basic idea is to colour appro priate objects according to whether they support some good property or not . However. first. suitable objects have to he identified and a reasonable conjec-
ture has to he formulated. I n Banach space theory one useful a pproach turned out to he the stat IR is called oscillation stable on X if for all infinite-dimensional closed subs paces Y of X and E > 0 there exists a closed infinite-dimensional sub space Z of }' such that sup{l /(x) f(y) : x.y E S( Z)) < E. A combination of difficult results of several mathemati cians yields the following remarkable conclusion ( compare, e . g . , [5] ) : F'or an iz?fi'n it!!-dimensional Banach spac!! X !!1'!!1)' Lipschitzjiozction f : S( X) --> IR is
oscillation stahl!! if and on�v tf every closed inj!zzite-dimensional suh:,pace Y q/X contains an isomo1ph of c0.
The hook under review contains m·o sets of notes that were originally pre pared for an Advanced Course on Ram sey J\lethods in Analysis gi\·en at the Cen tre de Recerca Matematica. Barcelona. in January 2004. Part A is titled "Saturated and Con .. ditional Structures in Banach Spaces ( with t\vo Appendices) and is due to Spiros A. Argyros. It describes a general method of building norms with desired properties and presents in particular the theory of H I Banach spaces. The approach to HI extensions of a ground norm shares many ideas with the extension of models in set theory. The ground norm can be considered the initial model and its HI extension a new Banach space which is H I and at the same time presen·es properties of the initial space. Part A also presents a non sepa rable ret1exive Banach space con-
© 2007 Spnnger Setence -r- Bus1ness Med 1a . Inc , Volume 29. Number 2 . 2007
65
tammg no unconditional basic se quence (compare [ 1 ]) . Part B is titled "High-Dimensional Ramsey Theory and Banach Space Geometry·· and is due to Stevo Todor cevic. It explains in fou r sections Ramsey-theoretic methods relevant to modern Banach space theory: Finite dimensional Ramsey theory. Ramsey theory of finite and infinite sequences. Ramsey Theory of finite and infinite block sequences, and approximate and strategic Ramsey theory of Banach spaces. In particular, Nash-Williams's methods are used in the proof of Rosen thal's theorem stating that every weakly null sequence ( x,z) in a Banach space contains either a subsequence ( x,,) all of whose subsequences are Cesaro sum mahle, or a subsequence ( x,,,) whose spreading model is isomorphic to e j . Furthermore Part B gives a detailed exposition of the block-Ramsey theory developed hy Gowers. If F I N denotes the collection of all finite nonempty subsets of N. a finite or infinite se quence (xz) of elements of F IN is called a block sequence if xi < :c1 whenever i < j. ( For x. y E F I N. x < y denotes the fact that max(x) < min(_y) . ) Todorcevic argues that while the space N [x J has numerous interesting ap plications to Banach space theory, the block spaces such as F I N 1"'l of all in finite block sequences of finite sets seem to he more relevant to the deeper problems of that theory. The basic Ram sey-type result about block sequences is a pigeonhole principle for F I N due to Hindman which says that if F I N is coloured with finitely many colours, then there exists an infinite block se quence (arz) such that all nonempty unions of finitely many of the sets a 11 have the same colour. Among other things, Todorcevic dis cusses how F I N [xJ , endowed with some appropriate topology, satisfies analogues of results that hold for N lxJ equipped with the Ellentuck topology. In the last section of the book, he then shows that Gowers's dichotomy theorem suggests a corresponding Ramsey theory of finite and infinite block sequences in Banach spaces with Schauder bases. He notes that in this setup an unexpected new phenomenon occurs: the classes of ap proximately and strategically Ramsey sets are, in general, no longer closed un der the operation of complementation.
66
THE MATHEMATICAL INTELLIGENCER
The lecture notes should serve their purpose to give a first condensed in troduction to some of the most recent advanced investigations in its area fairly \Vel!. To this end. it might he advisable to reverse the alphabetic order and read the second part of the book first. After having mastered the basic techniques in the four discussed version.s of abstract Ramsey theory . the reader will then be able to appreciate fully the sometimes necessarily highly technical and delicate methods in the first part of the volume. which often are related to the recent re search of the authors. In the light of all the rather compli cated constructions outlined above. one is likely to hegin to wonder-like Gow ers at the end of his address [6] whether there might he a theory of .. easily described .. Banach spaces, ,-ery different from the general theory. that would eliminate many of the peculari ties discussed in this review. At present, however. it seems unclear hem· such a theory can be built. The approach taken by the authors of the presented hook is remarkably dif ferent. They propagate the message that apparently unpleasant spaces form an integral and interesting part of classical Banach space theory and are something we have to get used to.
nach spaces, Geom. Funct. Anal. 6 (1 996), 1 083-1 093. [8] W. T. Gowers, An infinite Ramsey theorem and some Banach-space dichotomies, Ann . of Math. (2) 156 (2002), 797-833.
[9] W. T. Gowers, Ramsey methods in Ba
nach spaces, Handbook of the Geometry of Banach Spaces, vol. 2, North-Holland, Amsterdam, 2003, pp 1 07 1 - 1 097.
[ 1 0] W T. Gowers and B. Maurey, The un conditional basic sequence problem , J .
Amer. Math. Soc. 6 (1 993), 851 -874. [1 1 ] R. L. Graham, B. L. Rothschild , and J. H. Spencer, Ramsey Theory, (2nd ed.), Wiley lnterscience, New York, 1 990. [ 1 2] J. B. Kruskal, The theory of well-quasi ordering: a frequently discovered concept, J. Combinatorial Theory Ser. A 13 (1 972), 297-305. [1 3] F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 (1 930), 264-286. [1 4] B. S. Tsirelson, Not every Banach space contains an imbedding of fp or c0, Funct
Anal. Appl. 8 (1 974), 1 38-1 4 1 . Department of Mathematics and Applied Mathematics University of Cape Town Rondebosch 7701 South Africa e-mail: kunzi@maths. uct.ac.za
REFERENCES
(1 ] S. A. Argyros, J. Lopez-Abad, and S. Todorcevic, A class of Banach spaces with few non-strictly singular operators, J . Funct. Anal. 222 (2005), 306-384. [2] S. A. Argyros and A. Tolias, Methods in the theory of hereditarily indecomposable Banach spaces, Mem. Amer. Math. Soc. 1 70 (2004), no. 806. [3] J. Elton and E. Odell, The unit ball of every infinite-dimensional normed linear space contains a (1 + E)-separated sequence,
Gode l's Theorem: An I ncomplete G u ide to Its U se and Abuse h.v Torkel Franzen
Colloq. Math. 44 (1 98 1 ) , 1 05-1 09.
A. K. PETERS, WELLESLEY, MASSACHUSETIS,
( 1 973), 1 93-1 98.
t the Godel Centenary Confer ence, ·'Horizons of Tr�nh, " held I at the University of Vienna in April 2006. Solomon Feferman paid tribute to the work of the late Torkel Franzen. Feferman's comments, printed on the back cover of Code! :> Theorem:
[4] F. Galvin and K. Prikry, Borel sets and Ramsey's theorem, J. Symbolic Logic 38
[5] W. T. Gowers, Lipschitz functions on clas sical spaces, European J. Combin. 13 (1 992), 1 4 1 -1 5 1 . [6] W . T . Gowers, Recent results in the the ory of infinite-dimensional Banach spaces, Proceedings of the International Congress
2005, 172 pp, ISBN 1-56881-238-8, $24 . 9 5
REVIEWED BY GARY MAR
L
Birkhauser, Basel, 1 995, pp. 933-942 .
A n Incomplete Guide to Its U1·e and A huse, succinctly pinpoint Franzen's
(7] W. T. Gowers, A new dichotomy for Ba-
distinctive achievement: "This unique
of Mathematicians, Vol. 1 ,2 (Zurich, 1 994),
exposition of Kurt Geidel's stunning in completeness theorems for a general audience manages to do what none other has accomplished: explain dearly and thoroughly just what the theorems really say and imply and correct their di,erse misapplications to philosophy, psychologv. physics . theology. post modernist criticism and what have you . " Franzen's hook will h e o f interest to three audiences: ( 1 ) beginning logic stu dents who want a concise and self contained explanation of what Gi'Jdd's theorems do say: ( 2 ) non-mathematically trained scholars and educated layper sons who want a logically correct ex planation of what Geidel's theorems do not say: and (:3 J professional logicians \Vho \\·ant a comprehensive, and criti cal. sun ey of the philosophical per spectin�s opened up by Giidel"s \York. 0 0 0
Logic students now have access to manv popular accounts of Geidel's life and \vork . among them '\fagel and New man's classic exposition Coder, Proq/ ( 19"i9) and Douglas Hofstadter's Puli tizer-Prize-winning G'6de/. Escher. Bach 0979 ) . and, more recently, John Casti and Werner DePauli"s G6del: A L!f"e ()/ Logic ( 2000 ) , based on an Austrian na tional television documentary, as wdl as Rebecca Goldstein"s novelistic biog raphy. Incompleteness: The Pro()/ and Paradox c;j"Kzt 11 Gc!del ( 2005 ) ( revie\ved in The "tfathematical IntelliRencer. \ ol 2H. no. 4. 2006 ) . Hmve\·er. these books tend to sacrifice technical correctness for public comprehensibility: none of them comment in detail on the many misstatements and missapplications of Geidel's theorem. and some commit the very errors Franzen exposes. Steering the beginning student clear of some common confusions. Franzen explains technical terms and poses instructive questions: Godel published the completeness theorem 0930) for his doctoral dis sertation and then in the following year published his celebrated in completeness theorem 0 93 1 ) . The latter is not the negation of the for mer. What are the t\\·o quite distinct meanings of completeness in these two landmark theorems hy Gi:idel the former concerning first-order logic and the latter concerning Pea no Arithmetic? •
Although it is common to speak of the incompleteness theorem, there are ac tually tzro incompleteness theorems, known as Geidel"s First and Second Incompleteness Theorems. Contem porary formulations of both theorems talk about formal systems that "con tain a certain amount of arithmetic. " What two different requirements are meant by this single phrase? One important simplification of Gi'Jdel's first incompleteness theorem was discovered by J. Barkley Rosser 0 936 ) . What is the difference be tween Rosser's notion of simple con sistency and Gi.'l del"s original formu lation of his first completeness theorem in terms of w-consistency ? Goldbach"s famous unproven con jecture states that every e\·en num ber greater than 2 is the sum of t\vo primes. How is Rosser's simplifica tion related to the fact that Gold bach-like statements ( i .e . . statements with the same logical form a s Gold bach's conjecture. known as TI-0-1 statements) that are undecidable must be true? GiJdel 's incompleteness theorem, con trary to some misstatements, does not imply that euerv consistent fom1al sys tem is incomplete. The Theory of Real Numbers. for example, is complete. Hmv is this possible since the Real Numbers include the Natural Num bers of arithmetic' Moreover, certain subtheories of Peano Arithmetic such as Presberger Arithmetic 0928 ) , are decidable. Four years after the publication of Godel's incompleteness results. Ger hard Gentzen ( 1935 ) published a proof of the consistency of elementary arithmetic making use of a generalized version of mathematical induction, known as transfinite induction. Why doesn't Gentzen·s result conflict with Gi'ldel's Second Incompleteness The orem, which concerns the unprov ability of consistency for a wide spec trum of fonnal systems? Chapter 2, "The Incompleteness Theo rem: An Overview, " introduces the reader to the First Incompleteness The orem, its relation to Hilbert's Non Ig nora himus view of mathematics, and its irrelevance with regard to explaining the "Postmodern condition.'' Chapter 3 . "Computability, Formal Systems. and In completeness," explains the conceptual •
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connections among the logical notions of computability , formal systems, and incompleteness. These initial chapters of Franzen's hook, then, give the he ginning logic student a correct and con cise account of what the Giidel incom pleteness theorems actually do say. 0 0 0
Readers who are not mathematically in clined hut are intrigued by the many claims about the implications of Godel"s work will find Franzen a sober and re liable guide in explaining what Geidel's theorems do not say. For example, does Godel"s theorem show that a Theory of Everything ( TOE) in theoretical physics is impossible? Do Gi'ldel"s theorems re fute the strong Artificial Intelligence ( A I ) thesis that the human mind c a n be mod eled by a computer? ·'No mathematical theorem," Franzen notes. "has aroused so much interest among nonmathemati cians as Gi'Jdel's incompleteness theo rem. '' Indeed, Franzen's book grew out of taking on the exhausting task of com menting on the seemingly inexhaustible erroneous references on the Internet to Godel"s incompleteness theorems. Franzen discusses misuses of the in completeness theorems in theoretical physics and theology (Chapter 4), in skeptical arguments about mathematical knowledge (Chapter 5 ) , and i n the Lu cas-Penrose arguments about the limita tions of Artificial Intelligence (Chapter 6 ) . He dispatches his task with great clarity and a little self-ref1ective humor. After ac knowledging his colleagues in the pref ace. Franzen drolly comments: "For any remaining instances of incompleteness or inconsistency in the book, I consider myself entirely blameless, since after all, Godel proved that any book o n the in completeness theorem must be incom plete or inconsistent. Well, maybe not." ··Godel's theorem is an inexhaustible source of intellectual abuses, " note Alan Sokal and Jean Bricmont i n Fashionable
:Von sense: Postmodenz Intellectuals ' A buse ()/ Science 0 997) , a continuation of the discussion raised by the famous hoax in which Sokal's parody of a postmodern article was accepted for publication in a literary journal . Had Franzen limited his sites to debunking postmodern, political, or poetic invoca tions of Godel's theorem that were "ob viously nonsensical, · · this book could easily have settled into a smugness that
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comes from dispatchi ng strawm an ar guments. Franzen aims higher. In Chapter 4 Franzen discuss es the claim that, because of Gi.idd 's theorem. the physicist's dream of a Theory of Everything is not only unattained, hut theoretically unattainable. In his essay. "" The World on the String"" in the 1\cu • York Reuiew q/ Books ( 2004 ) , Freeman Dyson argued: ""Another reason Yvhy I believe science to he inexhaustible is Giidel"s theorem . . . . His theorem im plies that pure mathematics is inex haustible. No matter ho\\" many prob lems we soh·e. there \Vill always he other problems that cannot be soh·ed within the existing rules. :'\ow I claim that because of GC">del"s theorem. . physics is inexhaustible too. . In his talk. "GC">del and the End of Physics. " Stephen Hawking has argued similarly: ""In the standard positivist approach to the philosophy of science. physical the ories live rent-free in a Platonic heaven of ideal mathematical models . . . . But we are not angels who view the uni verse from the outside. Instead, we and our models are both part of the universe we a rc describing. Thus, a physical the ory is self-referencing. like in Giidel"s theorem. One might therefore expect it .. to be either inconsistent or incomplete . Do GC">del"s theorems ha,·e such uni versal implications' Drawing on Feter man's reply to Dyson in the Neu• York Reuieu• of Books (www.nybooks.com/ articles/1 7249 ) . Franzen explains: "The basic equations of physics. whatever they may he. cannot indeed decide every arithmetical statement, but whether or not they are complete considered as a description of the physical world, and what completeness might mean in such a case, is not something that the in completeness theorem tells us anything abou t . " In other words. the incomplete ness of the arithmetic component of a physical theory need not imply any in completeness in the description of the physical world. In Chapter 5 Franzen critically dis cusses the claims advanced by J R. Lu cas ( 1 961 ), and updated more recently by Roger Penrose in his Emperors New Mind ( 1 989) and Shadows c�l the Mind 0 994 ) . Lucas argued that no matter how complicated a machine we constru ct, it will correspond to a formal system, which, in turn, will be subject to a Godelian construction for finding a for-
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mula unprovable in that system. De fending Lucas's conclusion. Penrose up dates the argument in an attempt to show that the aspirations of strong Ar t ificial Intell igence ( AI ) are doomed to failure . going on to conjecture that a non-computational extension of quan tum mechanics will someday provide a theory of consciousness. Geidel's own remarks on the subject ( in his unpublished 1 9 5 1 Josiah Willard Gibbs Lecture at Brown Cni,·ersity, see vol. III, Collected Works r:l Kl/11 Gr)del. edited by Feferman et a/. ) are more cau tious and nuanced: The human mind is incapable of formulating ( or mechanizing) all its mathematical intuitions. I . e . : If it has succeeded in formulating some of them, this very fact yields new in tuitive knowledge. e.g . . the consis tency of this formalism. This fact may be called the "incompletahility" of mathematics. On the other hand. on the basis of what has been proved so far, it remains possible that there may exist ( and even he empirically discoverable ) a theorem proving machine \vhich in fact is equivalent to mathematical intuition, hut cannot he fJn!l 'ed to he so . nor even proved to yield only correct theorems of finitary number theory. The second result is the follow ing disjunction: Either the human mind surpasses all machines ( to he more precise: it can decide more number-theoretic questions than any machine) or else there exist number-theoretic questions unde cidable for the human mind. Criticizing Lucas and Penrose, Franzen argues that ""we have no basis for claim ing that we ( ' the human mind') can out prove a consistent formal system " be cause Gi.i del"s theorem only implies the equiL•alence of the consistency of the formal system and the Goclel statement asserting its own unprovahility. In gen eral, however, we have no guarantee that the formal system in question is consistent. an assumption required for u s to draw the conclusion there is a truth u nprovable in the formal system. And what about the weaker claim that there could not be any formal sys tem that exactly represents the human mind as far as its ability to prove arith metical theorems is concerned? Franzen criticizes Hofstadter·s reflections to this
eflect from Gudel. E-;cher, Bach. noting Hofstadter's informal remarks have at least '"the virtue of making it explicit that the role of the incompleteness theorem is a matter of inspiration rather than implication ·· : The other metaphorical analogue to GC">del"s Theorem which I find provocative suggests that ultimately. v.:e cannot understand our own minds/brains. . . . All the limitative theorems of mathematics and the themy of computation suggest that once your ability to represent your own structure has reached a certain critical point. that is the kiss of death: it guarantees that you can never represent yourself totally. In such metaphorical statements. Franzen notes. the inability of a formal system to prove its own consistency is interpreted as the inability of the sys tem to "analyze or justify itself. or as a . kind of blind spot . . The problem with such a view is that ""the metaphor un derstates the difficulty for a system to prove its O\vn consistency. . [T]he unprovability of consistency is really the unassertihility of consistency. A sys tem cannot truly postulate its own con sistency. quite a part from questions of analysis and j ustification . although other systems can truly postulate the .. consistency of the system . 0 0 0
As noted above, Franzen's first two goals were to explain accurately what Geidel's theorems do say to the begin ning logic student and to curb the en thusiasm of the nonmathematically in clined who have heard exaggerated claims about the philosophical and mathematical implications of Gi'l del"s theorem by pointing out what they do not say. Franzen's book will also he of interest to logicians who \vant a model of sober clarity for explaining the philo sophical perspectives opened up by Gbdel's work. Goclel's theorems are stunning and significant enough "with out any exaggerated claims for the[ir] revolutionary impact." In Chapter 7 Franzen discusses the conceptual connections among GC">del's Completeness Theorem, non-standard models of arithmetic, and the Incom pleteness Theorems. Chapter 8 covers misleading fonnulations of incomplete ness in terms of Kolmogorov-Chaitin
complexity. Gregory Chaitin is known for his information-thecJretic interpretation of Giidel's theorem ( 196S ) and for his dis covery of the Halting Probability !l ( also known as Chaitin"s number). As Chaitin touts his results in 1be l/nknowable 0999 ) : "'In a nutshell. Giidel discovered incompleteness. Turing discovered un computahility. and I discovered random ness-that"s the amazing bet that some mathematical statements are true for no reason, they're true by accident. " How ever, Chaitin's informal explanation that ·· . . . if one has ten pounds of axioms and a twenty-pound theorem. then the theorem cannot he derived from those axioms"' is misleading. I n a recent book "V!cta/1/ath ( 200') ), Chaitin expands upon this informal account: we·re really going to get irreducible mathematical bets. mathematical facts that ·are true for no reason. · and \Vhich simulate in pure math. as much as is possible. indepen dent tosses of a fair coin . . . . " The prob lem with Chaitin's informal explanation. as Franzen points out, is that Chaitin " s version of the Gi'ldel theorems does not deal with the complexity of the theorems themselves but instead with theorems that are statements about complexity. There is. moreover. an intriguing con nection between Giidel's incompleteness theorem and axioms of infinity: postu lating the existence of various infinite sets has formal consequences for ele mentary number theory that cannot he proved by elementary means. Most of the mathematics done today can he for malized within Zermelo-fraenkel set the ory with the axiom of choice ( ZFC ) . ZFC minus the axiom of infinity. ZFC- '", is equivalent in its arithmetic part to Peano Arithmetic, and so the Geidel incom pleteness theorems apply. Therefore, zrc - w is incomplete and does not im ply its own consistency. It turns out that ZFC (which includes the axiom of infin ity) can prove the consistency of zrc··w. So here we have a n example in which adding an axiom of infinity to a theory ( in p�llticular, ZFC -w) yields ne\V arith metical theorems ( the consistency of ZFC- "i) not provable within that original theory. Stronger axioms of infinity ex tending versions of ZFC also yield ne\Y arithmetical theorems not provable in the theories they extend. Franzen remarks: ·· rrom a philosophical point of view, it is highly significant that extensions of set theory by axioms asserting the existence
· NmY
of very large infinite sets have logical consequences in the realm of arithmetic that are not provable in the theory that they extend . " As yet. no arithmetical problem of tra ditional mathematical interest is known to be among the new arithmetical theo rems of extensions of ZFC by axioms of infinity. However, a step in this direction was taken with the Paris-Harrington The orem ( 1977 ) . The Paris-Harrington The orem is related to Ramsey's theorem that. for each pair of positive integers k and l greater than 2, there exists an integer R( k, / ) ( known as the Ramsev number) such that any graph with R( k. / ) nodes whose edges are colored reel or green will ei ther have a completely green subgraph of order k or a completely red suhgraph of order !. for example. at any party with at least six people. there are either three people who are all mutual acquaintances or mutual strangers. The Paris-Harrington Theorem. a combinatorial strengthening of Ramsey's theorem, was the first "nat ural" statement found to be true hut un provable in Peano Arithmetic. At the 1930 " Epistemology of the Ex act Sciences" Conference in Konigsberg, Gi'lclel quietly announced his First In completeness Theorem. Among confer ence participants were such eminent logicians as Rudolf Carnap and Arend Heyting. but only John von Neumann ap preciated the profound significance of Goclel's I ncompleteness Theorem. Not long afterward, von Neumann realized that the Second Incompleteness Theo rem could be obtained by formalizing the argument for the first. Communicating his discovery to Gi'ldel in a letter. von Neumann graciously declined to publish \vhen Godel informed him that this stun ning theorem was already discussed in his forthcoming "On Formally Undecid able Propositions in Principia Mathe matica and Related Systems I " 093l l. which would become a celebrated achievement of twentieth centmy logic. What was Gi'ldel's Second Incom pleteness Theorem and what effect did it have on Hilbert's program? In addition to constructing the Godel statement G for the formal system S, the argument es tablishing the implication "if S is consis tent. then G is not provable in S" could be carried out within S itself. Moreover. the property of being a Godel number of a proof in S is a computable one. and is consistent' is a Goldbach-like so
·s
statement. a statement which if blse, can he shown to be false by a computation. Thus. Giidel's Second I ncompleteness Theorem follows: if S proves the state ment Con( S"') expressing · S is consistent' in the language of S, then S proves G, and hence S is in t�ICt inconsistent. Hilbert's metamathematical program call ing for consistency proofs for formal sys tems such as arithmetic in which all fini tistic arguments can be formalized \\ as etlectively clashed by the Second In completeness Theorem. Franzen carefully points out three common misconceptions about the Sec ond Incompleteness Theorem. "first. it is often said that Geidel's proof shows G to be true. or to be ' in some sense · true. But the proof does not sh v G to be true. What we learn from the proof is that G is true if and only if S is consis tent. In this observation, there is no rea son to use any such formulation as 'in some sense true' . . . " Second. Godel's theorem does not rule out consistency proof'i using methods not formalizable within Peano Arithmetic. Third. ·'[a]nother aspect of the second incompleteness the orem that needs to be emphasized is that it does not show that S can only be proven consistent in a system that is stronger than S" For example. Gentzen proved the consistency of Peano Arith metic ( PA ) in 1 936 by application of an arithmetically expressible instance of transfinite induction up to Cantor's ordi nal e0 ( the least fixed point under orcli nal exponentiation to the base w), while otherwise using arguments that can be formalized in a Yery weak subsystem of PA. So the consistency of P A is proved in a system that overlaps in part with P A hut is not an extension of it. On the other hand, it has been ar gued that if a system S like PA has been accepted as intuitively true, then one ought to accept the consistency state ment Con( S ) for S. That will give rise to a new formal system S' obtained by ad joining Con( S ) to S Now S' is also in tuitively true, so the process can he it erated. in t�Kt . through the constructive transfinite. Alan Turing ( 1939) showed that one could obtain completeness for Goldhach-like statements for ordinal logics obtained by iterating consistency statements into the constructive transfi nite starting \Vith PA. Later, Fefennan 0 96-4 ) showed that one could obtain a progression that is complete for all arith-
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metical statements by iterating certain reflection principles . Franzen's other book, Inexhaustihilitv: A Non-exhaus tive Treatment (ASL Lecture Notes in Logic "'16, 2004) contains an excellent exposition of the incompleteness theo rems. and the reader is k:d step-by-step through the technical details needed to establish a significant part of Feferman·s completeness results for iterated ref1ec tion principles for ordinal logics. Torkel Franzen's untimely death on April 1 9 . 2006 came shortly before he "\v as to attend. as an invited lecturer. the Godel Centenary Conference, "Hori . zons of Truth, . held at the L'niversity of Vienna later that month. This. and his invitations to speak at other conferences featuring a tribute to G()del. testifies to the growing international recognition that he deserved for these \\·orks . ACKNOWLEDGMENTS
I thank Solomon Feferman for substan tive and insightful correspondence dur ing the preparation of this revie\\·. and Robert Crease. Patrick Grim, Robert Shrock, Lorenzo Simpson, and Theresa Spork-Greemvood for their intellectual and material support for my participa tion in the Godel Centenary ·'Horizons of Truth'' Conference in Vienna.
Department of Philosophy Stony Brook University Stony Brook, New York 1 1 794-3750 USA e-mail:
[email protected] The Art of Conjecturing
together with Letter
to a Friend on Sets in Court Tennis by jacob Bemoulli
translated with an introduction and notes hy Edith Dudley .�vlla BALTIMORE, THE JOHNS HOPKINS U N IVERS ITY PRESS, 2006, xx + 430 PP. £46.50 ISBN 0-80188235-4.
REVIEWED BY A . W. F. EDWARDS
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n 1 9 1 5 the young stat1st1c1an R. A. Fisher. then 2 5 . and his former stu dent friend C. S. Stock wrote an ar ticle [ 1 1 bewailing the contemporary ne glect of The Origin q/ Species: So melancholy a neglect of Darwin's work suggests reflections upon the use of those rare and precious pos sessions of man-great books. It \vas. \\ e believe. the custom of the late Pro fessor Freeman [2] to warn his stu dents that mastery of one great hook \\·as worth any amount of kno\\·ledge of many lesser ones. The tendency of modern scientific teaching is to ne glect the great hooks, to lay far too much stress upon relatively unimpor tant modern work. and to present masses of detail of doubtful truth and questionable weight in such a way as to obscure principles . . . . Hmv many biological students of today have read The Origin? The majority know it only from extracts, a singularly ineflecti\·e means. for a work of genius does not easily lend itself to the scissors: its unity is too marked. Nothing can re ally take the place of a first-hand study of the work itself. With her translation ofJacob Bernoulli's A r:s- Conjectandi in its entirety Edith Sylla now makes available to English speakers without benefit of Latin another great book hitherto known mostly from extracts. As she rightly observes, only thus can we at last see the full context of Bernoulli s theorem. the famous and fundamental limit theorem in Part IV that confirms our intuition that the propor tions of successes and failures in a sta ble sequence of trials really do converge to their postulated probabilities in a strict mathematical sense, and therefore may be used to estimate those probabilities. However, I must resist the tempta tion to review A rs Cm�jectandi itself and stick to Sylla's contribution. She thinks that it ·deserues to he considered the
founding document qj' mathematical probability ', hut I am not so sure. That honour belongs to Bernoulli's prede cessors Pascal and Huygens, who math ematized expectation half a century ear lier: Bernoulli's own main contribution was 'The Use and Application of the Preceding Doctrine in Civil, Moral, and Economic Matters· (the title of Part IV) and the associated theorem. It \vould he more true to say that Ar:s- Conjectandi is the founding document of mathemati-
cal statistics. for if Bernoulli's theorem were not true. that enterprise would be a house of cards. ( The title of a recent hook by Andres Hale! says it all: A His
tmy q/ Parametn'c Statistical h?/erence from Bemoulli to Fisher. 1 713-- 193 5 [3] . ) When I first became interested in Bernoulli's hook I was very fortunately placed. There v;as an original edition in the college libra1y ( Gonville and Caius College, Cambridge) and amongst the other Fellows of the college was Pro fessor Charles Brink. the University's Kennedy Professor of Latin. Though I have school Latin I was soon out of my depth, and so I consulted Professor Brink about passages that particularly interested me. Charles \vould fill his pipe. settle into his deep wing-chair and read silently for a while. Then, as like as not, his open ing remark would be ·Ah, yes, I remem ber Fisher asking me about this passage'. Fisher too had been a Fellow of Caius. I'\ow, at last, future generations can set aside the partial. and often amateur. trans lations of An; Conjecta11di and enjoy the whole of the great work professionally translated, annotated. and introduced by Edith Sylla. in a magisterial edition beau tifully produced and presented. She has left nu stone unturned, no correspon dence unread, no secondaty literature un examined. The result is a work of true scholarship that \viii leave every serious reader weak with admiration. Nothing said in criticism in this review should be construed as negating that. The translation itself occupies just half of the long book 2 1 3 pages. An other 146 pages are devoted to a pref ace and introduction. and 22 to a 'trans lator's commentary'. Next come 4 1 pages with a translation of Bernoulli's French Letter to a Friend 011 Sets in Cow1 Tennis which was published with A rs Conjectandi and which contains much that is relevant to the main work; a translator's cmnmenta1y is again ap pended. Finally. there is a full bibliog raphy and an index. In her preface Sylla sets the scene and includes a good survey of the secondary literature (Ivo Schneider's chapter on Ar:> Conjectandi in Landmark Writings in Western Mathematics 1640- 1940 [4] ap peared just too late for inclusion). Her introduction 'has four main sections. In the first, I review briefly some of the main facts of Jacob Bernoulli's life and its so cial context. . . . In the second, I discuss
Bernoulli's other vvritings insofar as they are relevant . . . . In the third, I describe the conceptual backgrounds . . . . Finally . in the fourth. I explain the policies I fol lowed in translating the work . · The first and second pans are extremely detailed scholarly accounts \Yhich will be stan dard sources for many years to come. The third, despite its title ' Historical and Conceptual Background to Bernoulli's Approaches in A1:s- Cmzjectandi · . turns into quite an extensive commentary in its own right. Its strength is indeed in the discussion of the background, and in par ticular the placing of the Llmous 'prob lem of points' in the context of early busi ness mathematics, but as commentary it is uneven. Perhaps as a consequence of the fact that the book has taken manv years to perfect, the distribution of material be tween preface, introduction. and trans lator's commentary is sometimes hard to understand. with some repetition. Thus one might have expected com ments on the technical problems of translation to be included under 'trans lator's commentary', but most-not all-of it is to be found in the intro duction. The distribution of commen taJy between these two parts is con fusing. but even taking them together there are many lacunae. The reason for this is related to Sylla 's remark at the end of the intro duction that 'Anders Halcl, A. W. F. Eel wards, and others, in their analyses of A rs Conjectandi, consistently rewrite what is at issue in modern notation . . I have not used any of this modern no tation because I believe it obscures Bernoulli's actual line of thought. · I and others have simply been more interested in Bernoulli's mathematical innovations than in the historical milieu, whose elu cidation is in any case best left to those. like Sylla. better qualified to undertake it. Just as she provides a wealth of in formation about the latter, she often passes quickly over the former. Thus C pp . 73, 345 ) she has no detailed comment on Bernoulli's table (pp. 152-153) enumerating the frequencies with which the different totals occur on throwing n dice. yet this is a brilliant tab ular algorithm for convoluting a discrete distribution, applicable to any such dis tribution. In 1 865 Todhunter [5] ·espe cially remark[ eel]' of this table that it was equivalent to finding the coefficient of x'"
in the development of (x + x2 + .x .3 + x ' + x� + .xr, ) 11, where n is the number of dice and m the total in question. Again C pp . 74-75. 345 ) . she has nothing to say about Bernoulli's derivation of the bino mial distribution ( pp . 1 65-167), which statisticians rightly hail as its original ap pearance. Of course, she might argue that as Bernoulli's expressions refer to ex pectations it is technically not a proha hili(V distribution. but that would be to split hairs. Statisticians rightly refer to 'Bernoulli trials' as generating it, and might have expected a reference. Turning to Huygens · s Vth problem C pp. 76. 345 ) , she does not mention that it is the now-famous 'Gambler's ruin' problem posed by Pascal to Fermat, nor that Bernoulli seems to he t1oundering in his attempt a t a general solution ( p . 1 9 2 ) . And she barely comments ( p . 80) on Bernoulli's polynomials for the sums of the powers of the integers, although I and others have found great interest in them and their earlier derivation by Faulhaber in 1 6 3 1 , including the 'Bernoulli numbers'. Indeed. it was the mention of Faulhaber in A rs Con jectandi that led me to the discovery of this fact ( see m y PascaL'> A rithmetical Triangle and references therein l6L Sylla does give some relevant references in the translator's commentary, p . 347). I make these remarks not so much in criticism as to emphasize that A rs Conjectandi merits deep study from more than one point of view. Sylla is probably the only person to have read Part III right through since Isaac Todhunter and the translator of the German edition in the nineteenth century. One wonders how many of the solutions to its XXIV problems contain errors. arithmetical or otherwise. On p. 265 Sylla corrects a number wrongly transcribed, but the error does not af fect the result. Though one should not make too much of a sample of one, my eye lit upon Problem XVII (pp. 275-8 1 ) , a sort o f roulette with four balls and 3 2 pockets, four each for the numbers 1 to 8. Reading Sylla's commentary (p. 83) I saw that symmetry made finding the ex pectation trivial, for she says that the prize is 'equal to the sum of the num bers on the compartments into which [the] four balls fall' (multiple occupancy is evidently excluded). Yet Bernoulli's calculations cover four of his pages and an extensive pull-out table.
It took me some time to realize that Sylla's description is incorrect, for the sum of the numbers is not the prize it self, but an indicator of the prize, ac cording to a table in which the prizes corresponding to the sums are given in two columns headed nummi. Sylla rea sonably translates this as ·coins', though ·prize money' is the intended meaning. This misunderstanding surmounted, and with the aid of a calculator. I ploughed through Bernoulli's arithmetic only to disagree with his answer. He finds the expectation to be 4 349/3596 but I find 4 1 53/1 7980 ( 4 . 097 1 and 4 .0085) . Bernoulli remarks that since the cost of each throw is set at 4 ' the player's lot is greater than that of the peddler' hut according to my calcula tion. only by one part in about 500. I should be glad to hear from any reader \Vl10 disagrees with my result. Sylla has translated Circulator as 'peddler' ( ' ped lar' in British spelling) but 'traveler' might better convey the sense, espe cially as Bernoul l i uses the capital A n d s o w e are l e d to t h e question of the translation itself. How good is it? I cannot tell in general, though I have some specific comments. The quality of the English is. however. excellent. and there is ample evidence of the care and scholarly attention to detail with which the translation has been made. I may remark on one or two passages. First, one translated in m y Pascal 's A nlhmetical Triangle thus: This Table [Pascal's Triangle] has truly exceptional and admirable properties; for besides concealing within itself the mysteries of Com binations, as we have seen, it is known by those expert i n the higher parts of Mathematics also to hold the foremost secrets of the whole of the rest of the subject. Sylla has (p. 206): This Table has clearly admirable and extraordinary properties, for beyond what I have already shown of the mystery of combinations hiding within it, it is known to those skilled in the more hidden parts of geom etiy that the most important secrets of all the rest of mathematics lie con cealed within it. Latin scholars will have to consult the original to make a judgment, but, set tling down with a grammar and a dic tionary 25 years after my original trans-
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lation ( with which Professor Brink will have helped) , I think mine better and closer to the Latin. I might now change 'truly' to '\vholly' and prefer ·mystery' in the singular ( like Sylla l , as in the Latin. as well as simply · higher mathematics'. But her ·geometry' for G'eometria is surely misleading. for in both eighteenth century Latin and French the word en compassed the whole of mathematics. Second. there is an ambiguity in Sylla 's translation ( p . 1 9 1 l of Bernoulli's claim to originality in connection \Vith a ' propertv of figurate numbers'. Is he claiming the property or only the demon stration? The latter, according to note 20 of chapter 1 0 of Pascal�' A rithmetical
have come to light: p. xvi, lines 1 and 2. De Moivre has lost his space; p. 73 . line 1 4 . Huygens has lost his ·g'; p. 1 ')2, the table headings are awkwardly placed and do not ret1ect the original in which thev clearly label the initial columns of Roman numerals; p. 297 n, omit diario ; and in the Bibliography . p. 40H. the reference in Italian .� houlcl ha\·e 'Accademid. and Bayes's paper \Yas published in 1764; p. 4 1 '), Kendall not Kendell; and, as a Parthian shot from this admiring re\·ie\ver. on p . 4 1 4 the ti tle of my hook Pascal 's Arithmetical Triangle should not be made to suffer the Americanism 'Arithmetic ·.
Triangle.
Gonville and Caius College
Third. consider Sylla's translation ( p . 329) of Bernoulli's comment on his great theorem in part IV: This. therefore . is the problem that I have proposed to publish in this place. after I have already concealed it for twenty years. Both its nm·elty and its great utility combined \Vith its equally great difficulty can add to the weight and value of all the other chapters of this theory. Did Bernoulli actively ccmceal it' In col loquial English I think he j ust sat on it for twenty years ( pressi ' l ; De Moivre [7] writes 'kept it by me· . And does it add weight and value. or add to the weight and value? De Moivre thought the former ( actually · high value and dignity' ) . This is also one of the pas sages on which I consulted Professor Brink . His rendering was: This then is the theorem which I have decided to publish here. after considering it for twenty years . Both its novelty and its great usefulness in connexion with any similar diffi culty can add weight and value to all other branches of the subject. I n one instance Sylla unwittingly pro vides two translations of the same Latin. this time Leibniz's ( p . 48n and p. 92 ) . The one has 'likelihood' and the other 'verisimilitude' for ·uerisimilitudo � And j ust one point from the French of the ' Letter to a Friend' ( p. 564 ): surely 'that it will finally be as probable as any given probability', not ·all'. Finally, in vie\v of the fact that this irreplaceable hook is sure to remain the standard translation and comme ntary for many years to come, it may be help ful to note the very few misprint s that
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T H E MATHEMATICAL INTELLIGENCER
Cambridge, CB2 1 TA UK e-mail:
[email protected] REFERENCES AND NOTES
[ 1 ] R. A. Fisher, C. S. Stock, "Cuenot on preadaptation. A criticism, " Eugenics Re
view 7 ( 1 9 1 5) , 46-61 .
[2] Professor E . A. Freeman was Regius Pro fessor
of
Modern
History
at
Oxford,
1 884-92. [3] A. Hald, A History of Parametric Statistical Inference from Bernoulli to Fisher, 1 7 13-
1 935, Springer, New York, 2006. [4] I. Grattan-Guinness (ed.), Landmark Writ ings in Western Mathematics 1 640- 1 940,
Elsevier, Amsterdam, 2005.
[5] I. Todhunter, A History of the Mathematical Theory of Probability, Cambridge, Macmil
lan, 1 865. [6] A. W. F. Edwards, Pascal's Arithmetical Triangle , second edition, Baltimore, Johns
Hopkins University Press, 2002. (7] A. De Moivre, The Doctrine of Chances, third edition, London , Millar, 1 756.
James Joseph Sylvester: Jewish M athematician in a Victorian World by Karen Hzmga Parshall ------- ·-----
T H E JOHNS HOPKI N S U NIVERSITY PRESS, BALTIMORE. 2006. xiii + 461 PP. $69.95, I S B N : 0-8018-8291-5.
REVIEWED BY TONY CRILLY
l
ames Joseph Sylvester OH14-1897l 1s well known to mathematicians. Was he not the scatter-brained ec centric who wrote a poem of four hun dred lines. each rhyming with Rosalind' And , lecturing on it . spent the hour nav igating through his extensive collection of footnotes, leaving little time for the poem itself:' Another story told by E . T. Bell is of Sylvester's poem of regret titled "A missing member of a family of terms in an algebraical formul a . " Such scraps inevitably ewJke a smile today. hut is his oddity all there is-stories and tales to spice a mathematical life? I3ell·s essays in A1en of /V!athematics have been int1uential for generations of math ematicians. hut his snapshots could not claim to be rounded biographies in any sense. This. then, is a review of the first full-length biography of the extraordi nary mathematician J. J. Sylvester. How can we judge a mathematical biography' On the face of it, writing the life of a mathematician is straightfor ward: birth. mathematics, death. Thus t1ows the writing formula: describe the mathematics, and top and tail with the brief biographical facts and stories. A possible variant is the hriet1y written life . followed by the mathematical her itage. There are many approaches, hut these are consistent \Vith William Faulkner's estimate of literary biogra phy, "he wrote the novels and he died . " According t o this hard-line view, biog raphy should not even exist. Yet Sylvester deserYes to be rescued from Bell's thumbnail sketch of the "Invariant Twins" in which he lumped Cayley and Sylvester together in the same chapter. Perhaps only in the genre of math ematical biography do possible subjects outnumber potential authors. A stimu lating article on writing the life of a mathematician. and an invitation to con tribute, has recently been published by John W. Dawson, the biographer of Kurt Godel. 1 Writing about another per son's life is a voyage of discovery about one's own life, and surely the biogra pher is different at the end of such a project. Writing about a period of his tory different from one's own also in volves some exotic time travelling. A central problem for writers whose subjects' lives were bounded by tech nical material is to integrate technical developments with the stories of those lives. This is almost obvious, hut it is
worth reconsidering in the l ight of Par shall ' s S)'!t •estcr. The mathematician turning to biography faces the challenge of not giving an exposition of the cur rent state of the subject's mathematical field in isolation from the milieu in \\·hich it was created. At the other end of the spectrum. a writer concentrating on the l ife may give no more than a side\\·ays glance at the mathematical de velopment and thereby descend into a refined form of gossip mongering. E. C. Bentley's catchy clerihe\v does not solve this central problem of integration: The art of Biography Is different from Geography. Geography is about maps. But Biographv is about chaps. Nor does Lance Armstrong offer a so lution in the succinct title of his ( auto lhiographv: " It's not about the . hike . . Greater weight cou ld he placed on both these dictums if the exclusiv ity \\ ere removecl. Biography is about chaps and maps. and the title of Lance Armstrong's biography should he " It's . not only about the hike . . alt hough ad . mittedly to lesser effect. Parshall' � title of her book about the mathematician Sylvester places him in the Victorian world. revealing her intention to make the l ife an undivided whole. Professor Parshall has spent the last twentv years in Sylvester's company and knows him wel l . She has brought her self to the point where she understands the breadth of his mathematics and its detail. itself no mean feat. More than this, she has come to u n derstand the historical background to Sylvester's mathematical problems and most im portantly his background to these prob lems. While she cannot know every thing about him, she knows a great deal. even aspects of his l ife he did not know himself. All this takes time-hut one should not be put off by the apho rism "that it takes a l ife to know a l ife . .. Mathematical biography can prof itably be compared with literary biog raphy because mathematicians and lit erary people are members of groups concerned with producing original work. James Boswell described the craft of [literary! biography as not only relat ing all the important events of a l ife in their order. hut interleaving what the subject privately wrote, and said. and
thought. In Dr Johnson, Boswell had a life coterminous w ith his own. and he \\·as familiar with the day-to-day cir cumstances which shaped the life of his subject. Sylvester's biographer lives in a different century from her subject and has had to face the additional challenge of becoming familiar with events and customs from a "foreign country" so nec ess;uy to faithfully immerse her subject in his own time and space. And while Boswell was in daily contact with John son, Parshall only ever met Sylvester in her imagination. In common with all bi ographers of the long-since dead. all she had were the surviving sources. among them Sylvester's own writings ( poetical and mathematical ) . ne\vspa pers. diaries. notebooks. and ahm·e all. the extant Sylvester letters-thousands of them scattered in archives all over Europe and America. Handling these documents. manv touched hy Svh-ester himself. brought him alive to the biog rapher and thence to the readers. Ac cording to .\lichael Holroyd. biography allows readers to read bet�·een the lines of a person's work: and in this case. mathematicians may he encouraged to read Sylvester's papers in their original \·ersions. In the end a completed biographv is what the writer makes of it-there are no general rules of what it should con sist of. apart from being constrained by the sources. The sources suggest a tale. hut it is the artist who selects and shapes the material. pumps the blood back into the subject. and ultimate!\� tells the storv . In some branches of literary bi ography it seems permissible to invent dialogue. hut this trend has not yet reached mathematical biography. For Parshall. the sources cannot he trans gressed, and time and place are to be respected. Thus we have a birth-to death biography of the traditional kind. There is still the question of tone, how ever; while Parshall is sympathetic to Sylvester's plight she is no hagiogra pher-her subject is certainly not a great man placed on a pedestal whose plinth is strewn with mathematical results. ;\lor is she an E . T. Bell. whose essays \Yere drawn from only a few sources-chiet1y the principal obituaries. I nstead she takes us through Sylvester's life, alert ing us to its pivotal points and to his mathematical contributions as he made them. Parshall lays before us Sylvester's
anxieties and predicaments and oh seiYes how he deals \\ ith them. I\ot all the way stations would have been cho sen hy Sylvester-indeed an autobiog raphy by Sylvester would have been \·et)' different. The structure of the hook is strictly chronological . Sylvester was the youngest son of a large \Veil-to-do Jewish family li\'ing in England. The family moved around the country hut settled in Lon don's East End by the time the hoy reached school age. Adopting life's . "slippet)' path . as a leitmotif. Parshall hegins the l ife of the young Jewish hoy living in a class-ridden land ruled by Anglicans for Anglicans. Quite early on, teachers recognised his mathematical t1air which combined \\'ith his Jewish ness to separate him from classmates and expose him to exclusion. A recur ring theme i n this portrait is Syh·ester the outsider yet the hov. the man. and the old man always ready to he in volved in battles and stand up to any kind of injustice. After a private education at primary schools in London. at the age of four teen Sylvester enrolled at University College London ( l'CU. \\·here Augustus De 'Vlorgan \Yas so int1uential . There oc curred the first of Sylvester's stumbles on the slippet)' path . in \vhich he de fended himself against school bullies. and his school education was com pleted at Liverpool's Royal Institution School. To his teachers, his mathemat ical talent was obvious, and hi.s family \\·as advised that their son shou ld at tend university. As mathematics was to he the subject of study, this meant Cam . bridge . the " holy city of mathematics . . Practising Jews coul d not graduate from Cambridge University. since they could not in conscience affirm the thirty-nine articles of the Anglican religion. but they were allowed to follmv the u niversity courses as any undergraduate would . Cambridge d id not close its doors o n talent, a n d Sylvester w a s free to enjoy a ful l university life except the right to put B.A. after his name. In 1 H37. he achieved the position of being second in the order of merit. a feature of the Cambridge system which ranked the hundred or so students according to their performance in the famed !'vlathe matica l Tripos examination. Each year the students were listed from the "Se nior Wrangler" at the top to the
© 2007 Spnnger Sctence+ Business M edta, Inc., Volume 2 9 , Number 2 . 2007
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''Wooden Spooner" at the bottom, the rankings being grouped into three classes: Wranglers, Senior Optimes , and Junior Optimes. ( Here Parshall's list of the leading students in 1 837 is slightly amiss. A. J Ellis graduated sixth wran gler in 1 837 to Sylvester's second wran gler, whereas R. L. Ellis ( no relation) was the Senior Wrangler in 1 H40. ) His undergraduate career behind him, Sylvester applied for the vacant professorship of natural philosophy at UCL, \\·here he had been a pupil. Par shall reveals him as an astute young man, gathering an impressive list of sponsors willing to write him testimo nials. He was successful in gaining the appointment and set about fitting him self into a natural philosophy mould and learning the necessary lecturing skills-the latter, a considerable chal lenge. As De Morgan noted of Sylvester's performance in this period, ·'[w)hen he was with us he was an entire failure: whether in lecture room or in private exposition, he could not keep his team of ideas in hand. " Ever restless, Sylvester became embroiled \Vith the university authorities in London, \\·hich prompted his move to America , where it was re ported that a ·'little, bluff, beef-fed En . glish cockney . appeared in Virginia as the new professor of mathematics. Here Parshall is on home territory, herself a professor at the University of Virginia, the site of Thomas Jefferson's inspired educational experiment at Charlottesville. In America, Sylvester's path became more slippery and after only a short pe riod he was hack in London, this time without a job, his previous tenured po sition at UCL had been filled by another, and he was thrown back on his own devices. How could Sylvester earn a living in London during the tumultuous 1 840s? Unsettled in his personal life and in the doldrums mathematically, he needed a respectable position which paid a salary. As Parshall informs us, he found a niche in the Equitable Law Life Com pany, and a career in the embryonic in surance industry. Installed as a man of business in the City of London, he also found time and money to study for the legal Bar. A barrister had gentlema n sta tus in the stratified Victorian world of subtle class distinctions. At this time , he met the great prop in his life, Arthur Cayley, and so began their famous as-
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THE MATHEMATICAL INTELLIGENCER
sociation . The idea that this collabora tion was uninhibited teamwork, as might he inferred from Bell, is dispelled by Parshall. From Sylvester's perspec tive, it \Yas an alliance built on both co operation and competition-the coop eration enriched by mutual support and daily discourse on mathematics, the competition ingrained by the education at Cambridge, \Vhich thrived on intensi\·e rivalry. Sylvester wrote only one joint mathematics paper (with James Ham mond ), towards the end of his life, hut Cayley never did, much to his regret. The 1 8')0s was an exciting time for Sylvester. Now qualified as a barrister and established as a well-known figure in the actuarial world. he felt that some thing was missing. He had done little mathematics in the previous decade; no\\· he aimed to set this right. In con cert with Cayley, he made new discm· eries in invariant theory, the modern al gebra of the day. This is the scene for the fairly well-known image of Sylvester ''sitting , with a decanter of port wine to sustain nature's flagging energies, in a . hack office in Lincoln's Inn Fields, . while for Parshall he ''was catching fire mathematically with Cayley fanning the . flames . . As he made inroads in invari ant theory \vith a train of hastily writ ten papers ( and driving his editors to despair) , he turned his attention to seek ing a career in the academic world. The painful reality was that there were few openings, especially for Jews. He tried for a position at the Woolwich Military Academy in London hut failed. A year later the death of the recently appointed professor at Woolwich gave him an un expected opportunity. Sylvester mar shalled his testimonial writers once again and was again successful. Many would regard Sylvester as nmY settled he had a springboard from which to re launch his mathematical career while treading water with regard to the teach ing of army cadets. But we are talking about] ames Joseph Sylvester. It was not long before he was at loggerheads with the Woolwich military authorities. These quarrels took their tolL and on the slippery path, Sylvester veered be tween the ecstasy of mathematical dis covery and the gloom of acute depres sion . Life at Woolwich for Sylvester was never easy. At the end of the 1 860s, a governmental enquiry into the educa tional operation at the Academy re-
suited in the amalgamation of profes sorships, and Sylvester, on the wrong side of the new retirement age, found himself without a job. He could do lit tle about it, but he fought the authori ties for a fair pension: no less a figure than William Ewart Gladstone came to the rescue in correcting a " lamentable departmental error'' and setting his pen sion to rights. Nevertheless, Sylvester was still in his fifties and needed to fill his days with useful activity. He set to work on his poetry, took singing lessons, made contributions to Penny Readings ( high-flown educational amusements for the working classes), and took an in terest in the broader issues of secondary school education . In between. occa sional enthusiasms for current mathe matical topics took hold of him, and he promulgated these with fervour. While all these activities buoyed him, they were not enough, and he knew it. A stroke of good fortune, after five years of intellectual tourism, gave Sylvester a chance to return to the aca demic mainstream. The new Johns Hop kins University was founded and its guiding light, Daniel Coit Gilman, was looking for European talent to establish a research reputation at Baltimore. Thus in 1 876 Sylvester was gainfully em ployed once again as a professor of mathematics, in this new graduate in stitution (with a salary he insisted should be paid in gold). As Parshall points out, Sylvester now found himself with a new set of problems. He had not taught at graduate level-ever-yet was expected to inspire students in research level mathematics. He had to give reg ular lectures, albeit to small audiences, hut students would now look to him for research guidance; the stuff of Penny Readings and the latest theories on the writing of poetry would cut no ice in this environment. What should Sylvester do? His research had always been sub ject to whim; now he needed a bulk of organised lecture material on which to draw. Sylvester fell hack on his great est success: invariant theory. In that sub ject in the 1 850s he had made gains, was recognised by the mathematical community as an authority, and could see that there were outstanding prob lems to solve. From the beginning of her book, Par shall reveals her enthusiasm for her un dertaking, and what better time to he-
gin the story than on a bright day: "In Baltimore, 22 February 1877 was a day of celebration. Bright. cloudless, un characteristically springlike, the birth day of America's first president was feted in fine style. Bunting hung from the bal conies. . . " His inaugural address. in Baltimore, suggests that Sylvester relished the chance of starting again in a coun try without the prejudices that he had endured in England. He addressed his newly found audience in 1 877: "Happy the young men gathered under our wing, who, u nfettered and untram melled by any other test than that of diligence and attainments. have here af forded to them an opportunity of filling . up a complete scheme of education . . This \Yas a pivotal point in Sylvester's life. and Parshall uses it as the opening scene of the book very effectively to in troduce a flashback. Without enthusi asm. the biographer cannot succeed. and here and throughout the book, her enthusiasm for writing the l ife matches Sylvester's own for mathematics. As with any of Sylvester's current re searches there was now, in the year 1 877, no field more worthy of study than invariant theory. It had everything-it was topical , there were routine calcula tions for students. and there were also challenging problems for himself to con sider. One desideratum was plugging the gap in ''Cayley's theorem" which purported to count the number of lin early independent invariants and co variants of a binary form. Both Cayley and Sylvester believed it true, and in deed, their calculations were based on its truth. But Cayley had made a crucial assumption, and while this caused no flurry in the 1 8 50s . the heightened im portance of "mathematical proof" in the 1 870s suggested it was a result which definitely needed proof. It was indeed a proud moment in Baltimore when Sylvester proved it. and. in announcing his success to the mathematical world. he blew his trumpet with all the Sylves trian puff he could muster. Less suc cessful were his attempts to reprove Gordan's theorem in which the German mathematician Paul Gordan had demon strated that the number of irreducible in variants and covariants was finite in the case of an algebraic binary form. A proof using only plain algebra would vindi cate the English methods of algebra and put the German semi-abstract calculus
in the shade. The quest took on a na tionalist dimension because Sylvester also believed that the so-called symbolic methods of Alfred Clebsch and Gordan had been appropriated from one of Cay ley's early discoveries in the theory of hyper-determinants. These objectives were theoretical, but of equal value in invariant-theory circles of the nineteenth century were the find ing and recording of the actual algebraic expressions for the invariants and co variants. For this Sylvester started up an ambitious calculatory program for the bi nary forms of the first ten orders (in mod ern language . for polynomials up to de gree ten ) . For the first four. the task was simple and had been recognised since the 1840s: the case of the binary quintic was thought to have been recently com pleted-though details in that case still awaited attention. Sylvester and his band of students took up the calculatory chal lenge with alacrity, and if a "truth" in the theory had not yet been confirmed by mathematical proof, they plunged on in the hasty heat of calculation. This was the inductive approach par excellence, and it is a measure of how Sylvester dif fered from mathematicians today ( at least he was able to publish work based on pure surmise') Sylvester did much to create a re search atmosphere at Johns Hopkins. The students were invigorated by hav ing a mathematician with a sound Eu ropean reputation among them, and if his teaching methods were idiosyncratic, stil l , he was a stimulating presence. Sylvester was instrumental in bringing the American journal of Mathematics into being, the first successful research journal in the country. Nevertheless, af ter seven successful years in Baltimore, Sylvester increasingly missed his former l ife in England, panicularly his London social circle based at the Athenaeum Club and at the Royal Society. The last act in Sylvester's dramatic career began with his appointment to the Savilian Chair of geometry at Ox ford in 1883. Even at the age of seventy he was not prepared to retire to the sidelines and rest on his laurels. He staned a mathematical society in the university and set out further grand the ories. One was a theory of reciprocants, a theory akin to invariant theory (he was to discover it had been substan tially developed elsewhere) . He became
entranced with matrix algebra, which he attacked by considering a plethora of low-dimensional cases-for example, his system of nonions involving 3 X 3 matrices. It was again the inductive ap proach which permeated all his mathe matical endeavours. Sylvester was fortunate that, like Cay ley's, his attainments were recognised in his lifetime. To some extent, his huge ego was soothed. As the Oxford years ran on. his health deteriorated, his eyes gave him trouble, and he retired to May fair, London's fashionable West End. His last years were spent writing poetry and delving into problems in number theory. In Professor Parshall, Sylvester has found a worthy chronicler of his life. Through the exercise of a great deal of care and diligence, Parshall has over come the practical difficulties of a scat tered archive of thousands of notes and letters . some of them almost indeci pherable. Her earlier work, james
Joseph Sylvester: L{le and Work in Let ters ( Oxford University Press, 1 998), provided some of the raw materials, but writing a biography is a different propo sition. In this quest, now completed, she has succeeded admirably. Sylvester is once more before u s , and rejuvenated he excites our enthusiasm for the great mathematical enterprise. Tony Crilly
[email protected]. uk
A3 & H is Algebra : H ow a Boy from
C hicago's West Side Became a Force i n American M athematics by Nancy E. Albert i U N IVERSE I N C . , LINCOLN, NEBRASKA, 2005, 352 PP., ISBN-13:978-0-595-32817-8 U S $23.95.
REVIEWED BY JOSEPH A. WOLF
L
I
.
drian Albert was one of the most important American algebraists of the twentieth century. He was
© 2007 Spnnger Sc1ence +-Busin ess Med1a, Inc., Volume 29, Number 2 , 2007
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a remarkable person on many acco unts. Most obviously. he was a first-rate math ematician and bore much responsibility for bringing modern algebra into the current mathematics curriculum . He \\·as especially famous for his work on nonassociative algebras. on divisio n al gebras. and on Riemann matrices. Per haps less known. he: bore major re sponsibility for persuading the U . S . government to support basic research in mathematics. Even before the na tional importance of science and math ematics was underlined by Russia ' s launch o f the first satellite. "sputnik," i n 1957, Adrian Albert played a key role in establishing and increasing the amount of ONR and NSF research sup port for active mathematicians at all lev els of seniority. Some of this int1uence certainly must have been based on his defense work during and after World War I I , in particular his effective use of algebraic methods in cryptology. Perhaps known mostly to those who had direct personal contact with him. Adrian Albert put a lot of time. thought. and effort into encouraging and smoothing the \Yay for students and young researchers. On many occasions. some of which are mentioned in this biography, he arranged financial sup port for students to enable full-time study. He encouraged women in the study of mathematics, and to the extent then possible. he facilitated their career paths. The biography contains several instances of this . and there were in fact quite a few others. Adrian Albert usu ally did this sort of thing in a very warm hearted way: he would arrange the ben efit and then gleefully surprise the recipient with the good news. (Abraham) Adrian Albert, often nick named "A Cubed , " was born in 1 905 and grew up in Chicago, received his B . S . in mathematics from the University of Chicago in 1926. and earned his P h . D . from Leonard Dickson at the Uni versity of Chicago in 1 928. After two years as Instmctor at Columbia Univer sity he returned to the University of Chicago as Assistant Professor in 1 93 1 , Full Professor i n 1 94 1 . Math Department Chair 1 958- 1 962, and Dean of Physical Sciences 1962-1 97 1 . He passed away in 1 972. Adrian Albert married Frieda Davis in 1927. They had two sons, Alan and Roy, and a daughter, Nancy, who is the
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THE MATHEMATICAL INTELLIGENCER
author of the hook under review. Roy died shortly after earning a B .A . in an thropology. Alan earned a B.S. in physics, worked as an engineer. and has since passed away. Nancy earned a J.D. and \vorks as a Ia \vyer. I am reviewing the June 2006 revised edition of this biography. \X!ith modern publishing methods it is l'asy to correct typographical errors and to take into ac count newly available source materials . Nancy Albert has done j ust that. In ad dition . it is remarkable j ust how Nancy Albert was able to transpose her legal abilities and produce such a precise and accuratl' description of Adrian Albert's mathematics . of the honors he received, and of his interactions with the federal government and the other members of the scientific establishment of his clay. Her description of the history of the University of Chicago. especially its De partment of Mathematics, is accurate and to the point. She is precise and in cisive when she discusses the social set ting of both her father and her mother. his connections with the University of Chicago and the other members of the Department of !l,lathematics there, and ( rather delicately) the way he managed despite anti-Semitic attitudes that lasted into the late l 9'50s. She was well ac quainted \\·ith the many mathematicians of her father's age and a bit younger. and that certainly contributed to the quality of the book. Also, recently she had access to some relevant AMS archives. This biography has already been re vie\ved by Lance Small in the Al1S So tices ( December 200'5) and by Philip Davis in the SIAJf Xeu·s (June 2006) . Lance Small is a n algebraist who started his graduate studies at the University of Chicago toward the end of Adrian Al bert's career. I defer to his description of Adrian ·s mathematical results be cause I work in areas (geometry and analysis) more or less orthogonal to Adrian's research. There is one non trivial point of historical disagreement between Lance Small's review and Nancy Albert's book. It concerns Adrian Albert's efforts to generate an offer from the University of Chicago to Nathan Jacobson and the possibility that these efforts were initially impeded by anti Semitic attitudes. Nancy Albert has clar ified this situation, with better docu mentation and a better arrangement of
the text, in the June 2006 revised edi tion of her hook. Adrian Albert and my father were close friends since their undergraduate days in the 1 920s. When Adrian Albert died he left a list of young mathemati cians to be invited to each take some books from his library. and I was on the list. Somehow that last gesture was typical of his generosity toward his stu dents and younger colleagues. As is clear at this point, this review is \vritten from the viewpoint of a friend. student, and colleague, rather than from the viewpoint of a historian of mathe matics or a mathematical critic. With that caveat, I definitely recommend this biography to all mathematicians inter ested in the interplay between mathe matics and public policy, and especially those in pure or applied algebra and those \\·ho had contact with the De partment of Mathematics at the Univer sity of Chicago any time from the 1920s through the 1 960s. Department of Mathematics University of California Berkeley, CA, 94720-3840 USA e-mail: jawolf@math. berkeley.edu
Saunders Mac Lane. A M athematical Autobiography hy Saunders Mac Lane WELLESLEY, MASSAC H USETIS, A. K. PETERS, 2004, 358 P., $39.00, HARDCOVER. ISBN 1-56881-150-0
REVIEWED BY HENRY E. HEATHERLY
�aunders Mac Lane ( 1909-2005) was one of the epoch-making � ;: mathematicians of the 20th cen tury. He knew and interacted with many of the outstanding figures of 20th cen tury mathematics. Add to this the well known lucidity of his expositions, his profound insight into the nature of mathematics, and his experience with scientific organizations and mathemati cal centers of excellence, and one comes to the pages of his a utobiogra phy with high expectations. These ex-
� W
pectations are-for the most part-\vell met. I never met :\lac Lane, but having read all or parts of severa I of his hooks and many of his expository papers and letters to the Notices. I feel I am ac quainted with Mac Lane the mathe matician and Mac Lane the commenta tor on the state of the mathematics profession. The hook consists of fifteen parts ( 64 chapters ) , ranging over Mac Lane's long and active life. It hegins with family background, early years, and his formal education. After undergraduate years at Yale, Mac Lane began graduate studies in the mathematics department at the University of Chicago. where he had a fellowship. Overall he found his year there to he disappointing. especially he cause he saw no possibility of doing a dissertation on logic at Chicago . He did write a master's thesis. learned how to play bridge. and met Dorothy Jones ( a graduate student i n economics whom he later married ) in that year. He won an Institute for International Education fellowship and decided to continue his studies in Giittingen, where he could write a thesis on logic. which had be come his main area of interest. Gottin gen in 1 9 3 1 was the premier mathe matical center in the world . There Mac Lane began a thesis under the direction of Paul Bernays. Hilbert's chief assistant. and became active in the mathematical life of late Weimar Germany. Regulations issued by the new Nazi regime in 1 933 forced Bernays to leave Gottingen and Mac Lane officially fin ished his doctoral thesis under Her mann Weyl. In July 1 933 Mac Lane took his doctoral exams and two days later got married-then back to the U.S.A. with a postdoctoral Sterling Fellowship at Yale. Professor Oystein Ore super vised all postdoctoral fellows in math ematics at Yale then, and Ore worked in algebra. So Mac Lane did research in valuation theory under Ore's direction. . "with considerable enthusiasm . . as Mac Lane recalls. Over the next several years Mac Lane held faculty positions at Har vard (where he was a Benjamin Peirce instructor) . Cornell. the University of Chicago, and then back to Harvard as an assistant professor. in 1 938. There he collaborated with Garrett Birkhoff to produce their influential book, A Sur uev qfModern Algebra ( 1 94 1 ) . I t was the first American undergraduate textbook
that presented the then new abstract ideas of Emmy Noether and B. L. van der \Vaerden. At Han·ard. Mac Lane began to di rect Ph.D. dissertations. His first student was Irving Kaplansky. who completed a dissertation. " Maximal Fields With Val u�ttions. ·• in 194 1 . Over the next sixty years Mac Lane directed Ph.D. disserta tion research for at least forty students. most of them at the University of Chicago. The dissertations were in many different areas: algebra, logic, al gebraic topology, category theory, topos theory, and theoretical computer science. He also co-directed two Ph.D. dissertations on the history of mathe matics. Among his many students was a future Fields Medal \Yinner. John Thompson. Another of his students. David Eisenhud . provided the preface for this hook. A list of forty of his doc toral students is given in the Mathe matics Genealogy Project website. with Steven Awodey listed as Mac Lane's last student. in 1 997. A photograph of Awodey and Mac Lane is given on page 3 1 4 of the autobiography. It is clear that Mac Lane greatly enjoyed his experi ence in the guidance of graduate stu .. dents. calling it "a splendid activity . While on a one-semester research leave in 1 94 1 Mac Lane started a col laboration with Samuel Eilenberg. This began with work on group extensions. homology. and cohomology that even tually led to their seminal work on cat egory theory. The latter is probably the mathematical work for which Mac Lane is best known . Mac Lane's collaboration with Eilenberg resulted in twenty-four joint papers in the period 1 94 1-19'5'5 . These papers ranged over several ma jor mathematical ideas: homotopy struc ture of spaces; homological algebra ( e. g . , the cohomology of groups) ; cate gory theory; and simplicial sets. The basic notions of category, functor, and natural transformation appeared in de finiti\·e form in their 1 945 paper. In the spring of 1 943 Mac Lane be came invoh'ed in mathematical work re lated to the war effort. He joined the Applied Mathematics Group at Colum bia . There he worked on problems that came from the Air Force, e . g .. gunnery guidance and pursuit cutves. The Co lumbia group had many outstanding mathematicians besides Mac Lane, in cluding Kaplansky, Eilenherg, Hassler
\Vhitney, Marston Morse, and Magnus Hestenes. At night Mac Lane and Eilen herg would \York on their own mathe matical problems in Eilenberg's apart ment. Category theory took shape during this period. A Guggenheim Fellmvship allc)'wed Mac Lane to spend the academic year 1 947-4H in Europe. In Paris he attended Leray's lectures on algebraic topology and got to know Armand Borel and Serre. Next Mac Lane went to ZOrich. where he stayed for a longer period of time to work with Heinz Hopf at the Swiss Federal Technical Institute. Mac Lane says it had been Hopfs ideas that originally started him and Eilenberg working on the cohomology of groups. After ZOrich. Mac Lane ,·isited several other mathematical centers in Germany . the Netherlands. Belgium. and Great Britain . In Oxford he began collabora tion \Vith J H. C. Whitehead on alge braic topology . the results of which were published in 1 9'50. Mac Lane and Whitehead had tentative plans to do fur ther joint research. hut to Mac Lane's regret this " never came to fruition . " Returning from Europe h e did not go back to Harvard, but instead accepted a professorship at the University of Chicago. This was the Chicago of what . has been called "The Stone Age . . he cause Marshall Stone had recently he come chairman there and was building the mathematics department to what ar guably became the premier one in the U . S.A. and perhaps the \Yorld . The se nior faculty were Albert . Chern. Stone . Wei!. Zygmund. and Mac Lane. Junior members included Kaplansky. Halmos . Segal . and Spanier. In 1 9 '5 2 Stone stepped down as chairman and Mac Lane took his place, serving from 19'52 to 1 958. He states that "the chairman's job was a troubled one, with its major policy issues and assorted bureaucratic .. regulations . He was frustrated by ad ministrative troubles ranging from keep ing an efficient department secretary to losing Andre Wei! from the faculty he cause the administration gave Wei! nothing to counter an offer from the I n stitute for Advanced Study. During the period 1 9'5 2-1 9'58. no new tenured a p pointments were made to the depart ment. although some excellent non tenured and tenure track appointments were made. Mac Lane considered his term as chairman "a prolongation of the
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Stone Age that continued , but did not expand, that tradition. " In 1 949 M a c Lane was elected to the National Academy of Sciences (NAS), leading to several decades of service with that organization. In 1 95H he was elected a member of the Council of the NAS and in 1959 he was named chair man of the editorial board for the Pro ceedings of the Academy, a position he held for eight years. He served two terms, 1 973-198 1 , as vice president of the NAS. His main activity during that term in office was in managing the Re port Review Committee. This commit tee reviewed reports on matters of high level science policy for various government agencies. He was elected president of the Mathematical Associa tion of America (MAA) for 1 9 52-1 953. and of the American Mathematical So ciety CAMS) in 1 972 for a two-year term. He also was an editor of the Bulletin and the Transactions of the AMS . In 1 973-1983 he served as a member of the National Science Board. Mac Lane had a long and distinguished career in the making of science policy at the highest level. This activity and his math ematical research continued until late in his long life. Mac Lane collaborated on research with many mathematicians over the span of his career. Besides the ones already mentioned, some of his co authors were Otto Schilling, Alfred Clif ford, and V. W. Adkisson. He co authored seven papers with Schilling in the period 1 939-1943, on algebras and number fields. With Adkisson he worked on geometric topology, and with Clifford, on group theory. From early in his career Mac Lane·s interest in mathematical logic led him to concerns about the philosophy of mathematics. He found "most of the subject misdirected and felt that one should be able to describe more accu rately what really is there in mathemat ics . " In 1 983 he gave a series of lectures at the University of Minnesota, which he later organized into a book, Mathe matics, Form, and Function ( 1 986) . He also wrote several papers on the gen eral subject of the philosophy of math ematics and what mathematics is-or should be-about. Throughout his autobiography Mac Lane weaves in personal items and details about his personal life. He gives
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THE MATHEMATICAL INTELLIGENCER
lively descriptions of his wide-ranging travels, of people met, and of sights seen. There are many photos of math ematicians and of Mac Lane's family. Mac Lane gives us several examples of his well known humorous verses. Here is a sample from one he read at the banquet for the International Confer ence on Category Theory in Coimbra , Portugal, in 1 999. Sam Eilenberg said just one paper will do To introduce categorical notions so new We'll write it so \veil these ideas for to sell And publish it promptly the story to tell . He ends the book with a poignant thought: "All told, mathematics was a great career choice for me. , , There are subjects on which I wish Mac Lane had said more, e.g., his per sonal relationships with Eilenberg and Stone and the period while he was chairman at Chicago. While Mac Lane makes brief mention of his brother Ger ald, he does not mention that Gerald was also a mathematician, an analyst who was a professor at Rice University and Purdue. There is no mention of any mathematical interplay between the two brothers. The general reader will find a few chapters of the autobiography to be tough going mathematically. This is es pecially true for chapters twelve through fourteen, where some mathe matical sophistication is needed to di gest the concepts introduced and dis cussed-e. g . , crossed product algebras, covariant functors, Hom, and Ext. This caveat aside, a general scientific audi ence should find Mac Lane·s book both informative and delightful. On a tech nical note, for those wishing to use this autobiography as a source of informa tion , the format of the index will be dis appointing. Only the names of people are listed in the index. So, looking up items such as category theory, the Na tional Science Foundation, or Harvard, is made more difficult. For these rea sons, and to gain an external viewpoint, one might hope that a talented and mathematically knowledgeable biogra pher will now come along and give us another book-length view of this extra ordinary man, Saunders Mac Lane. As a supplement to the autobiogra-
phy under review, the references below should prove of interest to the reader. Note that the two AMS volumes contain five articles by Mac Lane , each well worth reading. REFERENCES
P. Duren (ed.), A Century of Mathematics in America, Parts II and Ill, Providence, R . I . ,
Amer. Math. Soc. , 1 989.
D. Eisen bud, Encountering Saunders Mac Lane, Focus 25 (2005), 5-7. Kaplansky (ed.), Saunders Mac Lane Se lected Papers , New York, Springer-Verlag,
1 979. J . MacDonald, Saunders Mac Lane, 1 9092005, Focus 25 (2005), 4. S. Mac Lane, Mathematics at G6ttingen Under the Nazis , Notices Amer. Math. Soc. 42(1 995),
1 1 34-1 1 38 . Mathematics Department University of Louisiana at Lafayette Lafayette, LA 70504- 1 0 1 0
U.SA
e-mail:
[email protected] [email protected] Alfred Tarski : Life and Logic by A nita Burdman Fefennan and Solomon Feferman CAMBRIDGE U N IVERSITY PRESS (OCTOBER 2004) ISBN: 0521802407, HARDCOVER, VI + 425 PP, $34.99
REVIEWED BY KRZYSZTOF R, APT
l·.
f you ask a well-educated person for the names of the three most promi . nent logicians in the twentieth cen tury, he will undoubtedly come up with Godel, Tarski, and Turing. While well known biographies of the first and last have been published-and their tragic deaths attracted peoples' sympathetic attention-no account of Tarski's l ife has appeared until recently. The book under review fills this gap excellently by providing a marvelously readable, informative, and gossipy account of his life and work. The book is far from be ing a dry account of Tarski's achieve ments: on the contrary. Tarski was a bon-vivant par excellence, and by delv·
ing into this part of his personality the authors transcend the genre of a cus tomary scientific biography. Before I proceed further, let me clar ify my admittedly very feeble connec tion with both Tarski and the second author of the book. This may explain my position of an interested yet impar tial bystander. In 1974 I defended a the sis in Mathematical Logic in Warsaw as the last PhD student of Andrzej Mostowski, who in turn \Vas the first PhD student of Alti·ed Tarski . A year earlier the renowned Banach Center in \varsaw organized a Logic Semester which brought to Warsaw several lu minaries in Mathematical Logic. One of them was Solomon Feferman. He prob ably does not remember the student who guided him on his first day from the Center to the neighbouring Mathe matical Institute of the Polish Academy of Sciences. I never met him afterwards and never met his wife. the first author. When I began reading this book I never expected to find so much sur prising material in it. It is a fascinating history of a huge fragment of mathe matical logic in the twentieth century. The reason for this is the enormous in fluence that Tarski wielded on the sub ject. After the Second World War he built in Berkeley, almost single-hand edly, an extremely successful school of logic that attracted and educated many of the best and smartest logicians in the world. In contrast to Godel and Turing. Tarski had many PhD students . 24 to he exact. Also, he influenced a large number of logicians throughout his ca reer. One can honestly say that the strength of mathematical logic in the United States is, to a very large extent, due to his efforts. The authors trace in detail Tarski"s life from his birth in 1901 in Warsaw, then in the part of Poland ruled by Rus sia, to his death in 1 983 in Berkeley. Tarski's original name was Tajtelbaum . I n 1 924 he changed it to Tarski and of ficially converted to Catholicism. even though he remained an atheist. In this way he hoped to circumvent the diffi culties facing scientists of Jewish origin in the newly reestablished Poland. Tarski was quickly recognized as a brilliant scientist. At the age of 23 he received a PhD degree from Warsaw University. Soon after that he and Ba nach proved the famous result on the
sphere decomposition, now called the Banach-Tarski paradox. In 1 929 he married Maria Witkowska. with whom he had t\YO children. Their marriage suf\·ived a six-year separation during the war ( the hook includes a re markable reproduction of a short note about the family reunion from the Oak land Tribune from 6 January 1 946 ) and several crises caused by his numerous love affairs with other women. As a PhD student in Warsaw, I heard that Wanda Szmielew, a renowned expert in the logical analysis of geometry, was emo tionally involved with Tarski. Little did I know that this was just the tip of an iceberg about which the authors make no secret. Indeed, they go to great lengths in describing Tarski "s numerous romances with various co-authors. PhD students, and secretaries . and the book features photos of many women who fell under his apparently irresistible charm. Wanda Szmielew even l ived for a year in the Tarski's family home in Berkeley. while having a relationship with him. Of course, all this did not contribute to a successful family life. Tarski"s wife emerges from the book as an almost angelically patient person who sacrificed her life to help her hus band's career. Eventually she moved out of their house to another place in Berkeley where she rented rooms to . . . logicians visiting her husband. Tarski's l ife-style also did not help much in fos tering warm contacts with his two chil dren. The authors discuss a striking scene in the lobby of the imposing Eu ropejski Hotel in Warsaw in which Tarski was to meet his son Jan during a brief visit to Poland in 1964. The meet ing was spoiled by the unexpected ap pearance of Wanda Szmielew. In spite of his name change and excellent mathematical record, Tarski never succeeded in getting a professor ship in Poland. He left Poland for a short visit in the United States on 1 1 August 1939. on the same ship as the brothers Adam and Stanislaw Ulam. Adam be came a brilliant historian of Russia and communism, while Stanis-law became one of the most famous American math ematicians involved in an essential way in the Manhattan Project. Because of the outbreak of the Second World War on September 1 Tarski ( and the Ulams) re mained in the United States. In 1 942, after holding a number of
temporary positions, Tarski joined the Department of :\lathematics of the Uni \·ersity of California . Berkeley which re mained his home institution until his re tirement in 1 968. Retirement did not prevent him from remaining scientifi cally active . including supervision of several PhD students. till his death in 1 983 . His fame steadily grew. In 1 96 5 , b e was elected a member of t h e Na tional Academy of Sciences. and other high honours followed. The book abounds in delightful anecdotes revealing Tarski"s magnetic personality. He was a famously brilliant teacher and fast thinker, as well as an exceptionally demanding and persistent supervisor. It suffices to say that t\YO of his students took nearly t\venty years to finish their PhD thes es. One of the stories concerns Dana Scott . a most renowned logician and re cipient of the Turing Award in computer science. who, as a PhD student, was fired by Tarski for procrastinating with the corrections of an amateurish trans lation of a compendium of Tarski"s early Polish texts. Scott subsequently finished his PhD thesis with Alonzo Church at Princeton. Eventually Tarski and Scott mended fences. and once Scott"s repu tation grew Tarski even suggested that he. Tarski, v-;ould call Scott his student. Throughout his life Tarski was a workaholic who slept little and worked till the wee hours, often relying on ben zedrine (a variant of amphetamine) to stay awake. He was also a heavy smoker and occasionally indulged in marijuana. While in Berkeley he regularly threw lav ish parties at which alcohol ( notably his homemade variant of slivovitz) would t1ow freely and higos, a Polish cabbage based stew. woul d be served . The au thors convey the jolly atmosphere of these parties by mentioning that John Myhill, a logician, once sang the statis tics on male homosexuality from the just published The Kinsey Repm1. Tarski"s entourage was regularly ex posed to his dictatorial behaviour. Chen-Chung Chang, who suffered from asthma. recalls in the pages of this book v,:hat his work as a PhD student of Tarski was typically like. They would start working about 9 p . m . , continuing till 4:30 a . m . , sitting in clouds of ciga rette smoke in an unventilated room at Tarski"s house . Around 2 a . m . Tarski would inquire whether Chang would
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like a coffee; following his positive re ply he would yell his wife's name to wake her to make it for them. Tarski left a huge scientific legacy covering several areas of mathem atical logic. After his death the Journal of' Svmholic Logic published. in 1 9H6 and 1 9HH. more than ten surveys discussing his life and research in \ arious areas of logic. It is generally agreed that his most fundamental contributions are his for mal definition of truth, his theorem on the decidahility of the first-order theory of reals, and his work on relational and cylindric algebras. The hook allows the reader to learn in reasonable detail about Tarski's contributions by means of six 'Interludes· interspersed through out the text. These are highly readable accounts of the relevant background in mathematical logic and Tarski's contri butions. Tarski's work also had a notable im pact on computer science. especially his research on decision procedures and his approach to semantics. Interested read ers are referred to a very recent article by Solomon Feferman; see l l l . His \York on the definition of truth. meta-mathe matics. and generalizations of first-order logic was influential in philosophy and linguistics. Also. the renmYned Tarski's fixed-point theorem ( a ctually its weaker version. known as the Knaster-Tarski theorem) became a stanc!arc! tool in Mathematical Economics: see. e . g .. the classic lvficroeconomics Themy of Mas Collel. Whinston, and Green ( where Tarski is misspelled as Tarsky ) and ivfathematical Economics of M. Carter ( where he is misspelled as Tarksi). The authors took the trouble to check the spelling of the often confus ing ( for ·westerners· ) Polish names. I was impressed to fine! that all Polish words, except two. \\·ere spelled cor rectly. including those that contain non ASCII characters. The hook even con tains a Polish Pronunciation Guide. In summary, this is an excellent hook from which one can learn a lot about the history of mathematical logic in the twentieth century. the remarkab le in fluence of Tarski on this discipline . and, especially, about Tarski himself. REFERENCE
[1 ] Solomon Feferman. Tarski's influe nce on computer science.
Manuscript available
from http://arXiv.org/abs/cs/06080 62.
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T H E MATHEMATICAL INTELLIGENCER
Krzysztof R. Apt CWI , Amsterdam The Netherlands e-mail: K.R.Apt@cwi .nl
Toward a Ph i losophy of Real M athematics h)' Dal 'id Co �fleld CAMBRIDGE U N I V ERSITY PRESS, 2003, 300 PP. $US 70.00, ISBN 0521817226
REVIEWED BY ANDREW ARANA
��
l:' h ilosophy of mathematics" L
} brings ,
to mind endless dehate about what numbers are and whether they exist. Since plenty of mathematical progress continues to he made without taking a stance on ei ther of these questions, mathematicians feel confident they can work without much regard for philosophical ref1ec tions. In t his sharp-toned . sprawling hook. Da\·id Corfield acknowledges the irrele\·ance of much contemporary phi losophy of mathematics to current mathematical practice. and proposes re forming the subject accordingly. Reading the introduction. it is hard not to he swept up by Corfielcl's revo lutionary fervor. Most contemporary philosophical writing on mathematics focuses on elementary arithmetic or logic, hut that is not a representative sample of mathematical practice today or at any time since Euclid. Corfiekl's push to \Viden the investigati\ e reach of philosophers of mathematics will he \Velcomed hy readers whose lo\·e of mathematics extends broadly. But is this \Videning important merely because it may he more attractive to lovers of mathematics? Or are there important philosophical questions about mathe matics that cannot he answered well without the widening? Corfield suggests that there are such questions. For instance. we can ask why some concepts ( such as groups and Hilbert spaces ) have received a great deal of attention while others have not. It could he that our interest in those concepts is just a fad . like wearing blue jeans; or is tied to matters of current
l
cultural and scientific interests ( or prac tices) that may change dramatically in time. Or it could be that there are some parts of mathematics that we can't help hut run into when our inquiry gets se rious enough. We can put the question this way: are some mathematical con cepts inel'itah!e? This question is far from a new one: defending a "yes" answer to it was one of Plato's chief goals, not just for math ematics but for every knowledge-seek ing activity. Furthermore, the view de ril'ed from Plato's that we now call "Platonism"-that mathematical objects exist. independently of human minds continues to attract followers. Notice how the question that "Platonism·· an swers differs from Corfield's. Corfield's question asks us to account for the in evitability that, mathematical practice suggests, groups and Hilbert spaces possess; "Platonism " addresses the question of whether mathematical ob jects exist ohjectiYely or are human con . structs. "Platonism . could he true yet not tell us why certain of these allegedly .. "objective features of reality. such as groups and Hilhe 11 spaces. turn out to he important. Corfield wants to turn the philoso phy of mathematics toward what is im portant for mathematics. I will consider one example in some detail. both to show what Corfield does and what he does not do. Chapter Four is a study of analogies in mathematics. cases where two apparently distinct domains seem to he related. His chief example. which I will survey in a moment, is the anal ogy between algebraic numbers and al gebraic functions that emerged from Dedekind and Weber's work in the 1 8HOs . Their work engendered an alge braic approach to the theory of alge braic curves that. as developed in the twentieth century by Chevalley and then Grothendieck. would rival Rie mann's geometric and Weierstrass's function-theoretic approach. Corfield motivates this discussion by quoting several famous mathematicians on the importance of this analogy, remarking that analogies might indicate a "deeper structural similarity " between the do mains that might in some sense he in evitable. Though Declekind and Riemann had a relatively dose personal relationship, stemming from their time together in
Giittingen, their approaches to mathe matics \Yere quite different. This is es pecially dear in their attitudes toward what we today call the Riemann-Rocl1 theorem. In lectures in l 8"i "i- 18"i6 that Dedekind attended. Riemann presented work on meromorphic functions over Riemann surfaces that he would pub lish in his 1 8"i 7 paper Tbeurie der A bel schell Fzmctiunen Among other things . Riemann considered the ques tion: given m points on f5, a Riemann surface of genus p, how many linearly independent meromorphic functions are there that have at worst simple poles at the m specified points? Riemann showed that there are 11 such functions, where 11 � m - p + 1 . ( His student Roch later identified the error term . in corporating Riemann's inequality into a more general equality. ) To prove it, Rie mann used topological considerations, in particular what \Ve no\\· call the "Dirichlet principle" . which states the existence of a function minimizing a particular integral involving that func tion. Dirichlet's principle was contro versial in the years following Riemann's work: it was unproved, and w·eierstrass sho\\·ecl in 1 870 that it failed in certain cases. Ho\vever . the Riemann-Roch the orem \\·as recognized as fundamentally important, and people tried to eliminate the use of Dirichlet's principle in its proof. Dedekind \\·as one of them: he hoped to avoid not only the Dirichlet principle hut also any " transcendental" , topological considerations whatsoever ( in practice. this meant avoiding conti nuity) . In 1882 , Dedekind and his colleague Heinrich \veber expressed the Rie mann-Roch theorem in algebraic terms involving fields of algebraic functions defined on a Riemann surface. What is striking about the Dedekind/Weber pa per (and of importance for Corfield's project) is the analogy Dedekind and Weber located between fields of alge braic numbers and fields of algebraic functions. In the 1 870s Dedekind had made great progress in algebraic num ber theory, developing his the01y of ideals in his numerous " supplements" to Dirichlet's lectures on number the ory. In rings of algebraic integers, ideals enjoy unique factorization into prime ideals and thus , said Dedekind. the al gebraic integers obey the same "laws of divisibility'' as the ordinary integers.
This seems to have confirmed his view, presented in his acclaimed 1 888 essay on the foundations of arithmetic:, "Was sind und was sollen die Zahlen?'', that "every theorem of algebra and higher analysis, no matter how remote, can be expressed as a theorem about natural numbers-a declaration I have heard repeatedly from the lips of Dirichlet. ·· This vie\v was further confirmed in Dedekind's 1 882 work with Weber. What was needed. and what they found, was an analogue of ideals of al gebraic integers in fields of algebraic functions. Consider. for simplicity. al gebraic functions defined on the Rie mann sphere-the complex plane to gether with a point at infinity-which is a surface of genus zero. But. follow ing Dedekind and Weber, let's start with the field of algebraic functions IC< (l =
{'[; .
l : fg E ICl(l . g
=I= o .
}
Dedekind and \veher took the ring of algebraic integral functions IC[(l . to he the analogue of /L and the algebraic in tegers in this setting. and took the field 1[:( (! to he the analogue of ()) and the algebraic numbers. They used this anal ogy because ideals in the ring IC[(] en joy unique prime bctorization. like the integers and algebraic integers: and as noted, it was this "law of divisibility" that Dedekind had identified as critical . \'\'e can see the prime factorization in this setting by noting that IC[(] ( a nd hence each ideal of IC[(] ) consists of el + c1( + ements C1 1 ( 1 1 + C1 1 - 1 ( 1 1 1 + c0. \\ ith each c1 E. IC. and that . by the fundamental theorem of algebra, each such expression factors into products of linear terms ( ( - z1) for z1 E. C Ac cordingly . the prime ideals of this ring \\'ill be those generated by these linear factors, ( ( - z,), in addition to the zero ideal, which will turn out to be very im portant in the later development of scheme theory in algebraic geometry. All of the non-zero prime ideals are maximal, and these maximal ideals yield all points z1 E. IC, and thus all points of the Riemann sphere, except for the point at infinity, which cor responds to the maximal ideal ( g - 0 ) in the polynomial ring IC[g] under the ·
·
·
-;;
1 . identification g = · The maximal ideals ((
- z)
(with
z =I=
0 ) in IC[(] and the
( �)
maximal ideal g -
in ICl(J a re iden
tified. As a result . we ha\·e that the maximal ideals of ICl(J and
1Cl1J
are in
one-to-one correspondence with all points on the Riemann sphere. This ob servation allowed Dedekind and Weber to shift talk of the Riemann surface to talk of the correlated ideals, u ltimately giving a proof of the Riemann-Roch the orem ( and others ) using purely alge braic considerations. ( Detlef Laugwitz's text Bernhard Rieman n 1826- 1 866, l3irkhauser . 1 999. especially p . 1 "i9. is quite helpful in understanding these parts of the Dedekind;Weher paper. ) This long detour into the details of the Dedekind/Weher paper shows how interesting Dedekind and Weber's work was, as they defended their vievv that fields of algebraic integers and of alge braic functions could and should be treated as obeying some of the same laws. On the significance of this, Cor field quotes Dieudonne: [T]his article by Dedekind and Weber drew attention for the first time to a striking relationship be tween t\\·o mathematical domains up until then considered very re mote from each other. the first mani festation of what was to become a " leitmotif ' of later \vork: the search for common structures hidden u n der at times extremely disparate ap pearances. ( p . 96 ) This detour also illustrates one of the problems with this book: Corfielc.l's ac count of this material only skims the surface of this deep topic. Corfield gives his own sketch of the DedekincV Weber analogy. discusses its connec tion with the modern-day notion of ramification, and then turns to another approach to Riemann's work, the " val uation-theoretic" approach developed first by Kronecker, later by Hensel, and extended in later work o n p-adic num bers. All this in six breathtakingly con cise pages' What Corfield intends to do with this sprawling case study is to d is cuss what is important about analogies in mathem atics . hut his purpose would have been better served if he had pro vided a more in-depth analysis of even j ust one aspect of this analogy. What we get instead is a series of Ho u rha kiste quotes, from Dieudonne . \veil, and Lang, extolling the virtues of analo-
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gies in revealing the ·structures· u nder lying the mathematics v,:e ordinarily ex perience. The quotes are interesting , but I am left wondering \vhat they (and hence, Corfield, who mostly lets them speak for themselves ) mean hy 'struc tures'. Corfield's indicated aim was to discuss how an analogy between two domains might indicate a '·deeper struc tural similarity" that could he said to be inevitable. However, we are not given any guide to what "structural similarity"" might be. aside from being shown an admittedly impressive analogy and a series of quotations from famous math ematicians commending this work. In deed . Corfield spends fewer than three pages analyzing the case study ( with four significantly sized quotations left largely unanalyzed ) , fewer than half the pages dedicated to the details of the case study itself. Mathematicians read ing Corfielc.l's book may get the wrong impression that the allegedly ""revolu tionary"" philosophy of real mathemat ics is primarily the narration of existing mathematics, and thus is by no means revolutionary. I say this is the " \vrong"" impression because I think Corfield has helped clear space for a variety of projects go ing far beyond both narration of exist ing mathematics and the types of ques tions ordinarily dealt with in the philosophy of mathematics-though he himself seems unwilling to occupy that space . I'll be more specific, hy turning back to the Riemann/Dedekind case al ready discussed. Corfield's heavy re liance on Bourhakiste quotations is no coincidence. As he makes clear else where in the book, he is sympathetic to the categmy-theoretic development of mathematics that followed Bourbaki. That gives one answer to the question of what are the structures revealed by analogies: they are categories. This plays right into the ongoing controversy between advocates of category theory and of set theory as to the "proper foun dation"" for mathematics. This is surely an interesting controversy, but one with opposing sides as entrenched as the sides of the Cold War. It is hard to see how taking sides in this debate is go ing to help Corfield's promotion of a new philosophy of mathematics. when he comes across looking like a mere partisan in an old battle. But worse , he brushes this debate u nder the rug, pro-
82
THE MATHEMATICAL INTELLIGENCER
viding little defense of the category theoretic view, and in fact marking such " " fou ndational" debates as having usurped the attention of philosophers for too long. What is needed here are new ideas. I would like to suggest two. one even hinted at by a quotation of \X7eil"s in Corfield's text. First. instead of talking of analogies as revealing "structures"". Dedekind talked instead of "" laws"" be ing obeyed in different mathematical settings. It's not a far leap from this ,·ie,,· to Hilbert's axiomatic view of mathe matics. Corfield describes Hilbertian ax iomatics as merely one step in an ""increasingly sophisticated"" series. in which Noether's algebra and Eilenberg and Mac Lane's category theory are fur ther developments ( p . R3) . I think Cor field is underappreciating the ,-ie\Y that by focusing on "laws" rather than on "objects" such as categories. Dedekind and Hilbert were able to focus their at tention on the statements-on the "laws" themselves-thus opening up metamathematical avenues of progress . In addition, it allows one ( though nei ther Dedekind nor Hilbert did this con sistently) to avoid talking of mathemat ical objects a ltogether, talking instead only of the statements we want to as sert. In practical terms, this makes little difference, since we ordinarily write mathematics in statements ( though this could change with the development of new notation or media, as Corfield dis cusses in Chapter Ten) . In philosophi cal terms. though, it means we no longer have to discuss vvhether certain mathematical objects ·'exist" , since we are no longer talking about objects. Since this is an outcome Corfield gives glimmers of favoring at times, I think it deserves our consideration. The other idea for thinking about analogies that I would like to suggest follows Weil's idea that in working out analogies, we are trying to decipher statements in the language of one do main into the language of other do mains. In a 1 940 letter to his sister Si mone ( pu blished in Sotices of the AJ1S 5 2 : 3 ( March 200 5 ) . pp. 334-341 ) . he de scribes himself as having worked in the ""Riemannian·· tongue for some time, hut wishing for the "translation·· of all the ideas of that work into the language of function theory (as developed by Weierstrass and his followers) and the
language of number fields ( in either the Dedekind/Weber ideal-theoretic dialect or the Kronecker-Hensel valuation-the oretic dialect). He saw work with <malo gies as attempts to fill in a "translation table" between the three languages , constructing a "Rosetta Stone ·· for math ematics. In explaining analogies this \Yay. Wei! made no appeal to "struc tures . " Instead, he emphasized learning to move t1uidly back and forth between these different ways of presenting math ematical things. Tme. each language of fers distinctive benefits: for instance . working in the "Riemannian" language may free us to think more visually or physically. At least as important, though, is the benefit of being able to switch languages freely which, in addi tion to granting us the advantages of each language whenever \Ye choose. lets us work simultaneously in several languages at once. Doing so lets us an ticipate results in one domain that we have not yet discovered but that we should expect, given the otherwise suc cessful translation (Wei! mentions cases of this in his letter). Weil's case of the Riemannian language is by no means the only example of this translation project in mathematics: a great deal of work since Descartes has been spent in geometry translating between analytic and synthetic languages. and in im proving the translation. I admit that my defense of the advantages of this ap proach is still quite tentative. My pur pose i n offering these two ideas is to show how interesting a problem Cor field has framed for us-and how much more remains to be said. For all the revolutionary talk given in the introduction, Corfield's views end up quite continuous with the usual top ics of the philosophy of mathematics. I have already highlighted his interest i n t h e quite traditional topic o f conceptual inevitability in mathematics, and of his advocacy of the category-theoretic side in the ongoing struggle over the "foun dations" of mathematics. When he con tinues discussion of the Riemann/ Declekind analogy in Chapter Eight ( otherwise dedicated to Lakatos's work), he turns our attention to Kronecker's role in its development, and highlights how later mathematicians such as Weyl took sides in the choice of an Dedekin dian ideal-theoretic or a Kroneckerian valuation-theoretic approach. Corfield
emphasizes that they based their deci sions in part on how "constructive" they perceived each approach to he. The value of constructive reasoning is yet another traditional topic in the philos ophy of mathematics. Here again. Cor field's project is not nearlv as radical as he would have us think. Stil l , for all my frustration with the hooks limitations . I think Corfield does a nice job of showing how the philos ophy of mathematics can begin to en gage areas of mathematics besides arith metic and logic-a shift I strongly favor. There are interesting chapters on auto mated reasoning. Bayesian reasoning. and Lakatos's work for those \vho are interested in these topics. I ha\·e tried to focus on the parts of the text that I think are the most daring. and proba bly of the widest interest among read ers of this magazine. Corfield deserves to be supported for his daring. and for his hope that the philosophy of math ematics will he revolutionized. even if his book is not the revolution \\"e might have hoped for. I thank my colleagues Zongzhu Lin and Scott Tanona for their helpful comments on earlier drafts. Department of Philosophy Kansas State University Manhattan, KS 66506-2602 USA e-mail:
[email protected] The 15 P uzzle:
H ow It Drove the
World C razy hv jerry
Slocu m an d Die Son nel 'eld
THE SLOCUM PUZZLE FOUN DATIO N , 2006. HARDCOVER. 144 PP, $30.00. ISBN-1890980153
REVIEWED BY AARON ARCHER
S"""''\ o me
�
years ago. I wrote an article about the 15 puzzle that began, -"'-·•·. · " In the 1 H70's the impish puzzlemaker Sam Lc>Vcl caused quite a stir in the United States. Britain, and Europe with his now-famous 1 5-puzzle·· [ 1 ] . I have always been pleased with myself for managing to slip the word ' 'impish" into a published mathematics article. As
it turns out, this devilish descriptor was the most accurate part of that sentence, as Slocum and Sonneveld document in their nev-.' hook 'lhc 1 S Puzzle: Hou.• it Drove the World Crazy. The 1 5 puzzle did once cause an intense craze that spread like \Yildfire across America and overseas. and it is indeed famous to this day. However, the initial fad did not oc cur until l imO. and Sam Loyd had noth ing to do with it until eleven years later, when he started to claim in print that he hac\ invented the puzzle. Through meticulous research using primary sources. Slocum and SonneYelcl not only expose Sam Loyd's fraudulent claims but also argue convincingly that the actual inventor was Noyes Chapman. the post master of Canastota. New York. The 15 puzzle is a sliding block puz zle consisting of 1 5 numbered square blocks placed inside a frame large enough to accommodate 16 blocks in a 4-by-4 grid. The empty space allows the solver to slide any of the adjacent blocks into the open space. Given a starting configuration, the puzzle is to reach some specified target configuration via a sequence of such moves. The in structions written on the cover of the original puzzle read . ·'Place the l3locks in the Box irregularly. then move u ntil in regular order" ( p . 8 ) . It did not spec ify exactly \\·hat is meant by "' regular or der."' but it did contain a picture of the blocks arranged as shown here. Later posers of the puzzle were more careful to explicitly state this as the tar get state. For the purposes of our dis cussion. let us call this the canonical state. Early fans of the puzzle discov ered that they could often reach a con figuration that differed from the canon ical state only in that the 14 and 1 5 were swapped, but tty as they might, they could not complete the puzzle. Thus. the problem of solving the puzzle from this starting state became the standard challenge, which we will call the 14-15
puzzle.
Beginning i n 1 880, numerous cash prizes of up to $ 1 000 American ( a princely sum i n those clays) were of fered for the solution of the 1 4- 1 5 puz zle. There have been numerous pub lished accounts of people spending endless hours engrossed in the puzzle, but nobody ever successfully claimed these prizes. An intriguing mathemati cal fact about the 15 puzzle is that for
exactly half of the 16! possible initial configurations. the puzzle is impossible to solve. It should come as n o surprise that the 1 4- 1 5 puzzle starts from one of these impossible configurations.
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