A Eulogy for Jack Good

A Eulogy for Jac k Go o d Dor on ZEILBER GER 1 Kadish Irving John Goo d , what a goo d n ame! So high-class sounding, “w aspy”, gen trified, and gen tile. I h a ve alw ays en v isioned Jac k Go o d- one of my great est h ero es and influenc ers - wh o studied in Cam bridge Univ ersit y , and work ed at Bletchley Park along with Alan T urin g and other gian ts, as the protot yp ical English (gen tile) gen tleman. Imagine m y surpr ise, when I finally met him, on No v. 14, 2004 (when he was almost 88-y ears-old) at the nice S habbat dinner at the house of Gail Letzter and Daniel F ark as, after my collo quium talk at Virginia T ec h , and he told me that he was b orn Isadore Jacob Gudak . No w this name is as Jewish as it gets! No wonder h e had to c hange his name, since in th ose d a ys , and p erh aps ev en to day , yo u can’t get ahead in life w ith su c h a name, even if y ou hav e th e genius of Jac k Go o d. Go o d also told me that his father, who escap ed to England from th e stiflin g shtetl and mandatory military service, starting out as a watc hmak er’s apprentic e, later b ecoming the o wn er-manager of a fashionable jew elry shop in Lon d on, was a notable Yidd ish wr iter wh o pu blished under the p en- name of M osheh Ove d . And indeed the name rang a b ell, and I am sure that I ha v e heard that name b efore. Later I got a hold of a translation of Oved’s masterpiece Visions and Jewels (F ab er and F ab er, London, 1952) that is a very unusual autobiography/memo ir (“deserved to b e read for its strangeness” acco rding to the dust-jac k et blurb). P art of that b o ok d escrib es Jac k’s father’s adv en tures, starting out w ith his genealogy , and it turn ed out that Jac k’s paternal grandfather was a sho chet (ritual slaughte rer), that in the Jewish p ec king order of the shte tl is just one notc h b elo w a rabbi (and, as it turn ed ou t, Jac k’s grandf ather would ha v e b ecome a r abbi if not for a fi re that destro y ed all their wealt h). Of course, already J ack’s father wa s not observ ant, and definitely Jac k himself was as secular as one can get, b ut I am su re that he would not ob ject to hav e someone sa y kadish for h im. Unfortunately , Jac k n ev er married, and didn ’t h av e an y children, so there was no one to d o it. Since I feel lik e his spiritual son, let me tak e th is opp ortunit y to sa y kadish for the elev ation of his soul. 1 Departmen t of Mathematics, Rutgers Univers ity (New Brunswick), Hill Center-Busc h C ampus, 110 F relingh uysen Rd., P iscataw ay , NJ 0885 4-8019, USA. z eilber g at ma th dot rut gers dot edu , http:/ /www.m ath.rutgers.edu/~zeilberg / . Dec. 3, 200 9. E xpanded version of a ta lk delivered Oct. 25, 2 009, 3:00-3:23pm DST, at th e Algebraic Combinatorics Special Session, the 1052nd A merican M athematical So ciety (sectional F all) meeting, Pennsylv ania State Unive rsit y , Thomas B uilding, R o o m 216. Accompanied by Ma ple pack age JA CK downloadable from the webpage of this article http:/ /www.m ath.rutgers.edu/~zeilberg /mamarim/mamarimhtml/jack.html , where one can also find sample input and o utput. (The pack age can also b e downloaded directly from http:/ /www.m ath.rutgers.edu/~zeilberg /tokhniot/JACK ) . This article is exclusiv ely published in the Personal Journal o f E k had a n d Zeilberger ht tp://ww w.math .rutgers.edu/~zeilberg/pj.html and a rx iv.org . Supp orted in part by the U nited States of America National Science F oundation. 1 [I r ecite, from memory , the mourner’s kadish : yitgadal ve- yitkadash shmay r ab a, b e-alma de e br a ci-r eutei, veyamlich malchutei b e- khaye chon u - b e-yome chon u - b ekhayei de-c ol b eit yisr ael. b e- agala u-bizman kariv ve-imru amen. yehe shme r ab a mevor ach le-olam u-le-olmei almaya . yitb ar ach v e-yishtab ach, ve-yitp a-ar ve-yitr omam ve- yi tnashe ve- yithadar ve-yitale ve-yithalal shme dikudsha brich ho o le-ela min c ol bir c hata ve-shir ata, tushb ekhata ve-nekhematam da-amir an b e- alma ve-imru amen. titkab al tslothon u- b a-o othon de-c ol b eit yisr ael ke dam avuhun di bishmaya ve-imru amen. yehe shlama r ab a min shmaya ve-khayim tovim aleinu ve-al c ol yisr ael ve - imru amen. ose shalom bi-mer omav ho o ya-ashe shalom aleynu ve-al c ol yisr ael, ve-imru amen. yehe shlama r ab a min shmaya ve-khayim aleinu ve-al c ol yi sr ael ve-imru amen. ] Jac k, as a b o y , is ment ioned in h is father’s b o ok [I w a v e the b o ok Vision and J ewels and read from it]. On p. 158 it says: “ On arriving home ther e r an towar ds me, to me et me, our fiery little b oy, his little thumbs stuck into the arm-holes of his waistc o at, like a national r abbi. Pul ling out his elastic br ac es, thrusting his little chest forwar d, lik e R amsay MacDonald, he said: ‘Daddy, I wil l yet b e the Honour able Master Isador e Go o d.’ . . . ” W ell, Jac k never b ecame “Honorable” (in the s ense of a p olitician or a judge), and he stopp ed b eing Isad ore, but he did b ecome on e of the greatest pr obabilists and statisticians of his time, and a pioneer of the once fringe Bayesian approac h to statistics. That Bay esian method s are no w so accepted is due, in so sm all p art, to h is p reac h in gs. But I only learned of Jac k’s seminal w ork on the foundation of pr obabilit y m uc h later. My fi rst encoun ter with Jac k’s work wa s circa 1977, when Dick Askey challenge d m e to prov e the so-called Andrews’ q -Dyson conjecture, th at lead me to lo ok up pro ofs of the original Dyson conjecture, and inevitably to Jac k Go o d ’s classic pro of th at c hanged m y mat hematical life . The Best 1-P age Mathematical Pro of of All time The b est w a y to eulogize a mathematician is to recite one of h is pro ofs. Better still, I will now give y ou eac h a cop y that y ou sh ould kee p in y our wa llet. Whenever you are feeling d o w n, y ou sh ould lo ok at it, and I am su re that you would chee r u p. [I n o w distrib u te to eve ry p erson in the audience a cop y of Go o d’s h alf-page pro of ]. Let me repro du ce it here in its entir ety . 2 Journal of Mathematical Physics V olume 11, Number 6 J u ne 1970 Short Pro of of a Conjecture by Dyson I.J. Go o d Dep artment of Statistics, Vir ginia Polyte chnic Institute, Blacksbur g, Vir gi ni a (Receiv ed 26 Decem b er 1969) Dyson made a mathematical conjecture in his work on the distribution of energy lev els in complex systems. A pro of is given, whic h is muc h shorter than tw o that have b een pu b lished b efore. Let G ( a ) denote the constant term in the exp an s ion of F ( x ; a ) = Y i 6 = j  1 − x i x j  a j , i, j = 1 , 2 , . . . , n , where a 1 , a 2 , . . . , a n are nonn egativ e integers and wh ere F ( x ; a ) is expanded in p ositiv e and n egativ e p o w ers of x 1 , . . . , x n . Dyson 1 conjectured th at G ( a ) = M ( a ), wh ere M ( a ) is the m ultinomial co efficien t ( a 1 + . . . + a n )! / ( a 1 ! · · · a n !) . Th is wa s pro v ed b y Gun son 2 and by Wilson 3 . A muc h shorter pro of is giv en here. By applying Lagrange’s in terp olation f ormula (see, for example, Kopal 4 ) to the function of x that is id en tically equal to 1 an d then p utting x = 0, we see that X j Y i  1 − x j x i  − 1 = 1 , i 6 = j . By m ultiplying F ( x ; a ) b y this fun ction we see that, if a j 6 = 0 , j = 1 , . . . , n , th en F ( x ; a ) = X j F ( x ; a 1 , a 2 , . . . , a j − 1 , a j − 1 , a j +1 , . . . , a n ) , so that G ( a ) = X j G ( a 1 , . . . , a j − 1 , a j − 1 , a j +1 , . . . , a n ) . (1) If a j = 0, then x j o ccurs only to negativ e p o we rs in F ( x ; a ) so that G ( a ) is th en equal to the constan t term in F ( x 1 , . . . , x j − 1 , x j +1 , . . . , x n ; a 1 , . . . , a j − 1 , a j +1 , . . . , a n ) , that is G ( a ) = G ( a 1 , . . . , a j − 1 , a j +1 , . . . , a n ) , if a j = 0 . (2) Also, of course G ( 0 ) = 1 . (3) Equations (1)-(3) clearly uniquely defin e G ( a ) recur siv ely . Moreo v er , they are satisfied b y putting G ( a ) = M ( a ). T herefore G ( a ) = M ( a ), as conjectured by Dyson. 1 F.J.Dyson, J. Math. Phys. 3 , 140, 157, 1 66 (1 962) 2 J. Gunson, J. Math. Phys. 3 , 752 (1962 ). 3 K.G. Wilson, J. Math. Phys. 3 , 1040 (196 2) 4 Numeric al Analysis (Chapman and Ha ll, London, 195 5), p. 21 3 Jac k Go od: A very Go o d Man The brillian t id eas in Go o d’s p ro of immediately lead to my article The algebr a of line ar p artial differ enc e op er ators and its applic ations , publish ed in SIAM J. Math. Anal. 11, 919-93 4 (1980 ). I w as v ery proud of that pap er, and sen t it out to sev eral p eople, including Jac k Go o d. I n m y y outhful naivet ´ e, I ask ed four exp erts, after reading the article and b eing duly impr essed, to pic k up the phone, and call u p the c hair of the Illinois Math Department, and tell him to giv e me a job. I needed that j ob desp erately , since m y th en -girlfriend (and no w wife, Jane) w as a graduate student at Urbana-Champaign. Th ree out of the four p olitely replied that they are v ery busy etc., and more to the p oin t, they don’t kn o w me! (o ne of th em (John Riordan) w rote: “ you c an ’t sel l a pig in a p oke ”). But only one of the f ou r solicited exp erts , Jac k Go o d , actually did call! No w that I am muc h older, and p ossibly a bit wiser, I realize ho w prep osterous m y request w as, and ho w kind it w as on Jac k ’s p art to p ic k up th e phone. As it turned out, the chairp erson, Paul Bateman, refused to giv e me th at job that year (1978-1 979), but h e finally came around -in large p art thanks to Jac k -the follo wing y ear, and I w as able to reu nite with J an e, who by then b ecame my wife. Jac k Go od: A Visionary prophet of ( strong!) AI Jac k’s pr o of of Dyson’s conjecture, that I ha v e just distr ib uted to yo u, is a m asterpiece of t erseness . I exp erimen ted with it, trying to see if I can delete an y w ord without r u ining it. I failed. Jac k’s pro of is not only a mathematical masterpiece, but a liter ary one! Y et there are times to b e terse and there are times to b e ve rb ose. Jac k w as one of the most prolific s cien tists of all time, and wrote man y wonderful, leisurely , essa ys ab out the philosophical foundations of probabilit y and the p robabilistic found ations of p hilosoph y . He w as also one of the earliest pr op onen ts of artifi cial in tellige nce, and, if y ou b eliev e the wikip edia article, was a consultan t to Stanley Ku b ric k w hen he made 2001: A Sp ac e Odyssey that featured HAL. Jac k’s m ost fa v orite articles were collected in the b o ok Go o d Thinking: The F oundat ion of Pr ob- ability and Its A pplic ations , Univ ersit y of Minnesota Pr ess, 1983. In one of these p ap ers (in p p. 106-1 16) “ Dynamic al Pr ob ability, Computer Chess, and the Me asur ement of Know le dge ” (that orig- inally app eared in “ Mac hine Inte lligence 8 (E.W. Elco ck and D. Mic h ie, eds. (Wiley 1977) 139-150) one find s th e follo win g lo v ely qu otatio n(p. 106), th at I whole-heartedly agree w ith: “ T o paro d y Wittgenstein, w hat can b e said at all can b e said clearly and it c an b e pr o gr amme d .” Tw o pages later, one can fin d an ev en b etter quote: “Believing, as I d id (and still do), that a mac hine w ill ultimately b e able to simulate all in tellectual activities of an y m an ...” But when I lo ok again at Jac k’s one-page pr o of fr om the b o ok of Dyson’s conjecture, I am not so sure. Will a computer eve r b e able to come u p with such a gorgeous pro of ? But then aga in, 4 ma yb e it will! Computerized Deconstruction of Jac k Go o d’s Lov ely Human Pro of Let’s try and see h o w a computer (once it is su itably programmed w ith general p urp ose algorithms) w ould tac kle Dyson’s conjecture. The crux of Go o d’s pro of is the fact that F ( x ; a ) satisfies the p artial line ar r e curr enc e e qu ation with c onstant c o efficients F ( x ; a ) = X j F ( x ; a 1 , a 2 , . . . , a j − 1 , a j − 1 , a j +1 , . . . , a n ) , where the co efficien ts neither dep end on a n or on x . No w the fact that there is s u c h a simple recurrence is indeed a miracle, bu t the fact that, for an y sp ecific dimension, there is some suc h linear r ecur rence (with constant co efficien ts) is guaran teed a priori, and a computer can fi nd that recurrence (for small n ) r ather fast. This wa s the m ain observ ation of m y ab o ve-men tioned 1980 pap er. Indeed, let R 1 , . . . , R n are h omogeneous Laurent p olynomials of d egree 0 in n v ariables, (or equiv- alen tly , arbitrary p olynomials of n − 1 v ariables) an d consider the Laurent p olynomial F ( x ; a ) = n Y i =1 R i ( x ) a i . In tro ducing the shift-op erators A i in the discrete v ariable a i ( i = 1 , . . . , n ), defined by A i f ( a 1 , . . . , a n ) = f ( a 1 , . . . , a i − 1 , a i + 1 , a i +1 , . . . , a n ), w e get that F is annihilate d by the n op erators A i − R i , ( i = 1 . . . n ) . By us in g the Buc h b erger algorithm (Gr¨ obner b ases) or otherwise, the compu ter can eliminate all the x ’s and get a pu re recurr en ce op erator P ( A 1 , . . . , A n ) , annihilating F ( x ; a ). Since such an op er ator is free of th e x ’s it also an n ihilates eac h and every co efficien t, in p articular the c onstant term (the co efficien t of x 0 1 · · · x 0 n ). No w , if y ou tak e random R i ’s, P ( A 1 , . . . , A n ) w ill b e usually very complicated, and the complexit y gets higher with higher d imensions. The miracle of the Dyson p ro duct was that it turn ed out, that for every n : P ( A 1 , . . . , A n ) = 1 − n X i =1 A − 1 i . A computer can disco v er it (and p ro v e it!), r outinely for eac h sp ecific n , sa y 1 ≤ n ≤ 8, bu t, at present one still needs the “human” identit y X j Y i  1 − x j x i  − 1 = 1 , i 6 = j , 5 that is purely routine for an y sp ecific numeric n , but seems to need a h uman b eing to pro v e it for symb olic n (i.e. “all” n > 0). Jack Go o d inv ok ed the L agrange Interpolation F orm ula (w h ose h uman pr o of is one-line: t w o p olynomials of degree < n that coincide in n d ifferen t v alues must b e the s ame). But even if yo u nev er ha v e heard of Lagrange or Int erp olation, y ou can s till easily pro v e this iden tit y b y in duction on n , and I am s u re that a computer can b e taugh t ho w to find suc h a pro of (in the style of the Z eilb erger algorithm, but in the cont ext of symmetric fun ctions). The Maple pack age JACK has p rograms to automatically fin d su c h recurr ences for an y giv en set of Lauren t p olynomials R 1 , R 2 , . . . . It can b e gotten dir ectly fr om http:/ /www.m ath.rutgers.edu/~zeilberg /tokhniot/JACK , or v ia a link from the webpage of th is article http:/ /www.m ath.rutgers.edu/~zeilberg /mamarim/mamarimhtml/jack.html , where one can also fin d some sample inp ut and output. In particular, automatic Go o d-styl e pr o ofs of Dyson’s conjecture for n ≤ 8, from which one can clearly see the pattern of the recurrence. But Jac k Go od May ha v e b een wrong on one p oin t ... Let’s go bac k to the ab o ve -quoted Jac k’s p rophesy , w ith my added emphasis : “Believing, as I d id (and still do), that a mac hine will ultimately b e able to sim ulate all intel le ctual activities of any man ...” I whole-heartedly agree with Go o d if y ou replace “ intel le ctual ” by “ mathematic al ” bu t I am not so sure ab out “ intel le ctual ”. Will a computer ever b e able to write so b eautifully and so elo quent ly? Isadore J acob Gudak w as muc h more than a mere mathematician and statisticia n, he was a true in tellectual, a visionary , and a p o et , and I estimate that the (curr en t, Ba y esian!) probabilit y that al l his activit ies would b e one d a y sim ulated b y mac hine-kind is rather low. Of course, Ba yesian probabilities are alwa ys s ub ject to change with more evidence, so let’s wa it and see (and hop e!). 6