Celebratio Mathematica

I could have chosen to speak about the pro­gress and changes in ap­plied math­em­at­ics that have taken place in the years since the MAA was foun­ded. I have chosen in­stead to speak about these par­tic­u­lar gi­ants of ap­plied math­em­at­ics not only be­cause they rep­res­ent a cer­tain peri­od and a cer­tain in­flu­ence but be­cause they at­tained their dis­tinc­tion in very dif­fer­ent ways. These people di­vide in­to two groups. Those who did noth­ing or nearly noth­ing but ap­plied math­em­at­ics and those who di­vided their time between pure and ap­plied math­em­at­ics. The first kind is ex­em­pli­fied par ex­cel­lence by Sir Geof­frey Taylor and Theodore von Kar­man (he might be an­noyed to have the la­bel math­em­atician) and the ex­amples of the second kind are John von Neu­mann, Norbert Wien­er and Kurt Friedrichs. I will also say if time per­mits a few words about my fath­er, John L. Synge since there is no ques­tion that I learned a lot from him about at­ti­tude and ac­tion in ap­plied math­em­at­ics. I might add that he was chair­man of math­em­at­ics on this cam­pus from 1943 to 1947.

Be­fore de­scrib­ing the first two of these men — I would like to say a word or two about the sub­ject ap­plied math­em­at­ics. This is a term that has dif­fer­ent mean­ings at­tached to it by both its friends and en­emies. Some people like to call it math­em­at­ics of the real world (an un­at­tract­ive ex­pres­sion but at least fairly gen­er­al), oth­ers think of it as be­ing use­ful and still oth­ers use the term as equi­val­ent to lack of rig­or.

I wish I could avoid the ex­pres­sion “ap­plied” al­to­geth­er, but it’s there and the mean­ing that I at­tach to it is:

1. It is math­em­at­ics.

2. It is con­nec­ted to some oth­er sci­ence in­clud­ing en­gin­eer­ing sci­ence.

I then pro­ceed to strip it down and I ex­clude stat­ist­ics. If I did not want to talk about v. Neu­mann I would also ex­clude com­puter sci­ence. And I think com­puter sci­ent­ists would most def­in­itely agree.

The oth­er sci­ences range all over: medi­cine, cryp­to­graphy, eco­nom­ics, and I think we can be happy to em­brace as much as we can.

What dis­tin­guishes pure math­em­at­ics is that it is ex­plor­ing math­em­at­ics for it­self. But I have yet to see in the flesh a pure math­em­atician who is not ec­stat­ic with de­light if someone can ap­ply his res­ult to some oth­er sci­ence.

So I have picked my gi­ants mainly on the basis of this dis­tinc­tion between pure and ap­plied. But I have also picked them on the basis of my own know­ledge and I con­fess the pos­sib­il­ity of re­min­is­cing.

I must have been in­ter­ested at an early age in the struggles of the little de­part­ment of ap­plied math­em­at­ics in Toronto that my fath­er chaired. As an un­der­gradu­ate I re­mem­ber ask­ing him how many ap­plied math­em­aticians there were in North Amer­ica and he replied with the ques­tion “You mean ex­clud­ing those who just do Laplace Trans­forms.” I have for­got­ten how many con­sti­tuted the re­mainder but it was an in­sig­ni­fic­ant minor­ity of the math­em­at­ics com­munity.

Let me start with the old­est of my group and I’ll bet an un­sung hero in most math­em­at­ic­al halls. Geof­frey In­gram Taylor, born in 1886, was the grand­son of George Boole, so math­em­at­ics can­not have been strange to him. He also had a math­em­at­ic­al aunt, Boole’s young­est daugh­ter who pub­lished her first pa­per in geo­metry in her old age. (I ima­gine that her fath­er was her teach­er). Geof­frey Taylor took the nat­ur­al sci­ence tri­pos not the math­em­at­ic­al tri­pos at Cam­bridge. I have wondered if he was not dis­cour­aged from pure math­em­at­ics by the situ­ation of his dis­tin­guished grand­fath­er who got his first po­s­i­tion at the ad­vanced age of 36 in that out­post of the Brit­ish em­pire, the Uni­versity of Cork in Ire­land. Taylor re­ceived a fel­low­ship at Cam­bridge in 1908 and there he stayed the rest of his life. In 1911 he was sci­entif­ic crew for a trip to the Arc­tic on the H.M.S. Sco­tia. There he not only en­joyed the mak­ing of meas­ure­ments in the middle of nature but he got star­ted on his study of tur­bu­lence. He was fas­cin­ated by the out­pour­ing of smoke from the ship’s fun­nel.

I re­cently had the op­por­tun­ity to get equally fas­cin­ated. Twice a day a loc­al steam­er plies its way past my sum­mer cot­tage. And this is what I see on a wind­less day, Fig­ure 1, and I sup­pose that is what Taylor saw.

In the be­gin­ning you have a plume of smoke, a lay­er of air and un­der it a lay­er of hot air and smoke which has lower dens­ity and un­der that again air.

Let us just look at one sur­face sep­ar­at­ing gas at dif­fer­ent dens­it­ies as in Fig­ure 2a. Just think of these two lay­ers between two walls sep­ar­ated by a sur­face \( S \) as in Fig­ure 2b. If we dis­place the sur­face by chan­ging its level then the in­creased pres­sure on the sur­face is the dif­fer­ence in the weight of the wa­ter above the sur­face and thus pro­por­tion­al to \[ \delta y(\rho_2-\rho_i) . \] where \( \delta y \) is the dif­fer­ence in level. The ac­cel­er­a­tion of the sur­face will be pro­por­tion­al to this force. \begin{equation*} \frac{d^2}{dt^2}\delta y =k(\rho_2-\rho_1)\delta y. \end{equation*} If \( \rho_2 > \rho_1 \) there is ex­po­nen­tial growth in time pro­por­tion­al to \[ \sqrt{k}(\rho_2-\rho_1) .\] If \( \rho_2 < \rho_1 \) things are stable.

So we are not sur­prised that the up­per lay­er of the smoke plume is un­stable. But it is really worse than that. Dis­turb the sur­face by tilt­ing it as in Fig­ure 2c. Then the pres­sure will form a torque if \( \rho_2 > \rho_1 \) that makes the sur­face tilt still more. So it is more un­stable. I won’t bore you with the large num­ber of equa­tions you have to write down to do this prob­lem fully. You as­sume there is a ripple of some pre­scribed wave length in the sur­face, again as in Fig­ure 2a and you find the cor­res­pond­ing ex­po­nents in time growth. There is no growth if \( \rho_2 < \rho_1 \) but if we have the Taylor situ­ation, every wave length (suf­fi­ciently small) pro­duces ex­po­nen­tial growth at a rate pro­por­tion­al to the in­verse of the wave length.

So this pair of lay­ers is not just un­stable. The ini­tial value prob­lem is ill-posed (but that is a long story whose an­swer is com­ing slowly now). And so the smoke be­comes tur­bu­lent very soon.

Hav­ing settled the in­stabil­ity in this case (Rayleigh–Taylor), Taylor stud­ied many oth­er prob­lems of sta­bil­ity. Then he went on to for­mu­late a the­ory for what hap­pens after the flow be­comes tur­bu­lent.

He in­tro­duced the fun­da­ment­al concept of mix­ing length and vari­ous cor­rel­a­tions. Work that in­flu­enced Wien­er’s work and Taylor him­self in turn was in­flu­enced by Wien­er. Primar­ily his in­terest in ap­plied math­em­at­ics was in un­der­stand­ing nature by math­em­at­ic­al means and for him much of nature was flu­id.

I met him in his lab in 1953 in Cam­bridge. He de­lighted in show­ing me a big trough with a huge kind of paddle for mak­ing waves. (I should add he had an in­cred­ibly able tech­ni­cian to help him make things.) He also had an un­canny abil­ity to find the right math­em­at­ic­al mod­els al­ways re­du­cing his an­swers to something cur­rently com­put­able.

One of the amus­ing tales of my vis­it is that he could not in­vite me as a wo­man, to lunch in the Com­mons, a com­pletely mas­cu­line strong­hold, so in­stead he ar­ranged a lunch for a small group in a stu­dent’s room — something that in Amer­ica would have at that time been com­pletely for­bid­den.

I saw him again, I think in 1972, in Po­land at a con­fer­ence in his hon­or. To please him he was taken sail­ing, his lifelong hobby, and be­ing my­self an afi­cion­ado of that sport I was par­tic­u­larly im­pressed that a man of 84 could jibe so el­eg­antly.

By the way, many people who know all about Taylor in­stabil­ity do not know that he de­signed an an­chor, still in com­mon use; it fol­ded bet­ter and could be stored less awk­wardly. And it also held. He left a leg­acy not only in his own work but in the work of his sci­entif­ic chil­dren and grand­chil­dren es­pe­cially G. K. Batch­el­or. There are not many like him and al­though in prin­ciple one can do fan­ci­er ex­per­i­ments fol­lowed by mod­el­ling fol­lowed by very fancy com­pu­ta­tions — it is hard to ima­gine that any one be­ing will be able to span it all as he did.

I turn next to Theodore von Kar­man born in 1881 in Hun­gary where he was trained as an en­gin­eer and when I claim him for ap­plied math­em­at­ics it is be­cause of the role he played in bring­ing math­em­at­ics in­to aero­naut­ic­al en­gin­eer­ing. In his auto­bi­o­graphy he de­scribes the poor en­gin­eer­ing edu­ca­tion he re­ceived and how he de­cided to study mech­an­ics with Prandtl in Got­tin­gen. But there he also stud­ied math­em­at­ics and phys­ics. He did some fun­da­ment­al work with Born on crys­tal lat­tices but in the end went to Aachen as a pro­fess­or of aero­naut­ic­al en­gin­eer­ing fi­nally go­ing to Cal­tech in 1930. Any­body who has stud­ied flight or vor­tices or many oth­er as­pects of flu­id dy­nam­ics knows von Kar­man’s work.

For ex­ample, I learned about the von Kar­man vor­tex street as an un­der­gradu­ate in Toronto. It had been dis­covered by Hiemenz in Got­tin­gen that no mat­ter how smoothly a cyl­in­der was honed to be cir­cu­lar, flow past it was al­ways os­cil­lat­ory and the cyl­in­der os­cil­lated too. One nev­er found the clas­sic­al 2D flow of the com­plex vari­able ap­plic­a­tion, Fig­ure 3a. Von Kar­man pos­tu­lated that vor­tices were shed al­tern­ately on each side and a wave could not be avoided as in Fig­ure 3b. He then pro­posed a simple mod­el of a vor­tex street with reg­u­lar spa­cings and equal strength vor­tices, Fig­ure 3c.

He found the very el­eg­ant res­ult that if the spa­cing and the width of the street do not sat­is­fy a very simple ra­tio con­di­tion then the ar­ray is un­stable.

Thus this pat­tern is the one that is seen. All oth­ers be­ing un­stable will not be seen.

En­am­ored by the el­eg­ance of this the­ory I wanted to go to Cal­tech my­self but since Cal­tech did not ad­mit wo­men in 1945 I landed up in Wien­er’s class at MIT in­stead. But I met von Kar­man later. I had writ­ten a pa­per on the so-called lim­it­ing line which had been a pro­posed way of ex­plain­ing why most tran­son­ic flow has shocks. I at­trib­uted the idea to von Kar­man in part without check­ing a ref­er­ence and then pro­ceeded to show that it could not be the ex­plan­a­tion. I was ac­tu­ally fol­low­ing up some work of Friedrichs. The next time von Kar­man came to N.Y. he in­vited Cour­ant and Friedrichs, Lax as a rep­res­ent­at­ive Hun­gari­an and me to lunch in his hotel. Every few minutes he turned to me and asked where I had seen the lim­it­ing line pro­pos­i­tion. I felt ter­rible and learned my les­son but from then on was treated very well by von Kar­man.

His biggest role was as the fath­er of our space pro­gram and the de­veloper of rock­etry. But wherever he could, he brought math­em­at­ics to bear. I think Friedrichs, or per­haps Cour­ant told me that once when an ad­mirer asked him about his suc­cess­ful mas­tery of a prob­lem he pat­ted a large pile of cal­cu­la­tions mean­while mut­ter­ing “Phys­ic­al in­tu­ition, phys­ic­al in­tu­ition.”

Let me turn now to the “oth­er kind” of a gi­ant and let me be­gin with Norbert Wien­er. Prob­ably most of you have read his auto­bi­o­graphy. There is no ques­tion that his form­at­ive years were his child­hood years and that his tastes and am­bi­tions were a product of his struggle to be in­de­pend­ent of his fath­er.

After gradu­at­ing at the age of 14 from col­lege, he tried gradu­ate school in zo­ology but his clum­si­ness and bad eye­sight made him real­ize that the ex­per­i­ment­al sci­ence of that time was not for him. He tried philo­sophy then lo­gic and fi­nally ended in math­em­at­ics. His enorm­ous tal­ent took time to de­vel­op and it was not un­til he had made many starts, that he settled down by his own ac­count to really study math­em­at­ics at the age of 24. Fi­nally, he was ap­poin­ted to the fac­ulty of MIT. In the late twen­ties he went for a second time as a young post-doc to Got­tin­gen where his, I was go­ing to say ar­rog­ance but per­haps it’s bet­ter to say his par­tic­u­lar mix­ture of self-es­teem and lack of self-es­teem led Hil­bert to try to “cure” him with scorn. The Hil­bert en­tour­age fol­lowed suit and Wien­er un­der­stand­ably re­tained a lifelong dis­like of many of them es­pe­cially Richard Cour­ant. When I told Wien­er at a Sunday lunch at his house that I was leav­ing MIT to join Cour­ant’s group as my new hus­band was in the New York area, I was not aware of this bad situ­ation and couldn’t un­der­stand why Wien­er re­fused to carry on the con­ver­sa­tion. Thus, to say the least, I had very little con­tact with Wien­er. But as a fresh gradu­ate stu­dent I had briefly tried out his course in Noise and Ran­dom Pro­cesses. It was clearly for those who knew more math­em­at­ics than I had learned in Toronto in ap­plied math­em­at­ics and I dropped out. In spite of the Got­tin­gen story Wien­er ad­mired Hil­bert and con­tin­ued to see him as, to quote his auto­bi­o­graphy, “the sort of math­em­atician I would like to be­come, com­bin­ing tre­mend­ous ab­stract power with a down-to-earth sense of phys­ic­al real­ity.”

To quote Mark Kac writ­ing in 1964 on Wien­er’s work:

“The simplest and most cel­eb­rated ex­ample of a stochast­ic pro­cess is the Browni­an mo­tion of a particle. Wien­er con­ceived in (1921) the idea of basing the the­ory of Browni­an mo­tion on a the­ory of meas­ure in a set of all con­tinu­ous paths. This idea proved enorm­ously fruit­ful for prob­ab­il­ity the­ory. It breathed new life in­to old prob­lems.”

This work drew strongly on Taylor’s work that came from pic­tures of “plumes of smoke” and which had led Taylor to in­tro­duce his spe­cial cor­rel­a­tions. As Nor­man Lev­in­son put it there were two reas­ons for Wien­er’s in­terest in Taylor’s work — one was that it in­spired him to try tur­bu­lence as a mod­el for his prob­lem of in­teg­ra­tion in func­tion spaces. And the oth­er was that it sug­ges­ted his own auto and cross cor­rel­a­tion func­tions for his gen­er­al­ized har­mon­ic ana­lys­is.

I re­read the in­tro­duc­tion to Wien­er’s “Cy­ber­net­ics.” Looked at from a hu­man point of view one per­ceives the in­cred­ibly high sci­entif­ic as­pir­a­tions of Wien­er. Rest­ing, I would like to say com­fort­ably (but even I knew as a stu­dent his un­eas­i­ness), on a long his­tory of suc­cess­ful ac­com­plish­ment Wien­er wanted to con­quer the brain with math­em­at­ics. In the course of it he worked very hard with able phys­i­cians and ex­perts in neur­o­logy to be­come not only as know­ledgable as he could but able to in­ter­act. His great am­bi­tion was to put to­geth­er the ex­tant know­ledge of com­puters, feed­back sys­tems and physiology coupled with the emer­ging sub­ject of sig­nal pro­cessing — all sub­jects he had con­trib­uted to fun­da­ment­ally. The course I went to in 1945 was a spinoff of these in­terests.

In his in­tro­duc­tion, read now 43 years later, he takes a rather high po­s­i­tion for the role of his new sub­ject. Still cy­ber­net­ics has stood the test of time with en­gin­eers. His view of the re­la­tion of sci­ence and so­ci­ety is in­ter­est­ing if pess­im­ist­ic.

“The best we can do is to see that a large pub­lic un­der­stands the trend and the bear­ing of the present work, and to con­fine our per­son­al ef­forts to those fields, such as physiology and psy­cho­logy, most re­mote from war and ex­ploit­a­tion. As we have seen, there are those who hope that the good of a bet­ter un­der­stand­ing of man and so­ci­ety which is offered by this new field of work may an­ti­cip­ate and out­weigh the in­cid­ent­al con­tri­bu­tion we are mak­ing to the con­cen­tra­tion of power (which is al­ways con­cen­trated, by its very con­di­tions of ex­ist­ence, in the hands of the most un­scru­pu­lous). I write in 1947, and I am com­pelled to say that it is a very slight hope.”

There is no good source of bio­graph­ic­al in­form­a­tion for my next gi­ant, John von Neu­mann. This situ­ation is be­ing re­paired and we can look for­ward to a full bio­graphy in the next couple of years. v. Neu­mann was, like Wien­er, a math­em­at­ic­al prodigy as a child. His fath­er, however, was an en­lightened banker and some­how or oth­er when v. Neu­mann was ready to go to uni­versity a com­prom­ise between bank­ing (big busi­ness) and math­em­at­ics (purest of sci­ences) was worked out. v. Neu­mann went to Zurich to study chem­istry (pos­sibly ap­plic­able in the eyes of the fam­ily).

I nev­er met von Neu­mann. Oc­ca­sion­ally I saw him in Richard Cour­ant’s com­pany. And that af­fected my life a lot be­cause as I un­der­stand it, it was at v. Neu­mann’s re­com­mend­a­tion that the first big uni­versity com­puter was placed with Cour­ant’s group at N.Y.U. That was prob­ably not dis­con­nec­ted from the fact that Cour­ant and those around him shared v. Neu­mann’s view that math­em­at­ics would be­come an ar­id sub­ject if it lost con­tact with sci­ence and en­gin­eer­ing.

Those who knew v. Neu­mann al­ways re­mark on the speed of his brain. He grasped things im­me­di­ately. His in­terests ranged over everything. I don’t know wheth­er one should call his early work in quantum mech­an­ics ap­plied. One might say he set it up as a part of pure math­em­at­ics. His early el­eg­ant work in game the­ory re­ceived little at­ten­tion un­til after the Second World War. But even be­fore the Second World War broke out he be­came in­volved in bal­list­ics in an­ti­cip­a­tion.

von Neu­mann, as every­one knows played a big role in the de­vel­op­ment of the atom­ic bomb. And it was in that con­nec­tion that he made his mark in flu­id dy­nam­ics.

Des­pite some strik­ing con­tri­bu­tions to the field (lots of us are hard at work these days on the para­doxes he un­covered in shock re­flec­tion) it led him quickly in­to big com­pu­ta­tion (big for its day) and hence in­to the whole area of large scale com­put­ing: The uni­ver­sal ma­chine, the cod­ing, the pro­gram­ming. Some ideas were around but he cleaned them up and set the whole thing on a lo­gic­al and ex­pand­able foot­ing. As Peter Lax has sug­ges­ted he would have de­veloped par­al­lel com­put­ing if he had lived long enough. I have really no time to bring up his many con­tri­bu­tions, to eco­nom­ics, to Monte Carlo meth­ods etc. It might be said of v. Neu­mann that his sweep was so broad that it in­cluded most of ap­plied math­em­at­ics.

I would only like to re­it­er­ate his philo­sophy that math­em­at­ics would be­come an eso­ter­ic ar­id branch of sci­ence if it lost its con­nec­tions. I think he would be happy to see today how mod­ern math­em­at­ics is knit­ting bonds with­in its many branches and even more with oth­er sci­ences.

I turn now to my teach­er Kurt Friedrichs, or as he was known to those around him, Frieder.

Born in Kiel in 1901 he entered uni­versity in Dus­sel­dorf. Fol­low­ing the Ger­man prac­tice he stud­ied a vari­ety of top­ics in a vari­ety of places (in­clud­ing the philo­sophy of Husserl and Heide­g­ger). Fi­nally, he came to “the Mecca of math­em­at­ics,” Got­tin­gen in 1922.

His re­la­tion to math­em­at­ics, pure and ap­plied, is best de­scribed by his own ex­pres­sion that “he was like a dan­cing bear on a stove, first hop­ping on his pure foot till it got too hot and then on his ap­plied foot.” In fact, if one looks over his work one finds a pretty ran­dom dis­tri­bu­tion. His first of­fi­cial ap­plied math­em­at­ic­al work was as von Kar­man’s as­sist­ant in Aachen. He took the po­s­i­tion ac­cord­ing to his own ex­plan­a­tion to Con­stance Reed be­cause Cour­ant thought that in the late twen­ties Friedrichs be­ing so shy and with­drawn would have a hard time com­pet­ing against pure math­em­aticians for an aca­dem­ic po­s­i­tion in Ger­many. He be­came shortly the young­est pro­fess­or at the Uni­versity of Braun­sch­weig. He left Ger­many to join Cour­ant in Amer­ica partly from dis­gust with the Nazis but also to be able to marry Nel­lie Bruell, for­bid­den un­der the Nazi ra­cist rules.

From then on he worked in elasti­city, flu­id dy­nam­ics, quantum field the­ory, plasma phys­ics al­tern­ately with par­tial dif­fer­en­tial equa­tions, asymp­tot­ics, spec­tral the­ory and oth­er sub­jects too pure to be ap­plied but too ap­plied to be quite pure. Friedrichs liked to say that ap­plied math­em­at­ics was whatever the phys­i­cist had dis­carded as no longer ex­cit­ing.

As a gradu­ate stu­dent at New York Uni­versity I first worked at edit­ing the book on “Su­per­son­ic Flow and Shock Waves” by Cour­ant and Friedrichs. That was my good luck. I learned the main ideas from Cour­ant and the ex­cep­tions and the ne­ces­sity to be ac­cur­ate from Friedrichs.

After passing my or­als after my first child I went to Friedrichs for a thes­is top­ic. I thought it would be in flu­id dy­nam­ics but he showed me a whole bunch of top­ics mainly I think on spec­tral the­ory. He asked me if I could get ex­cited about one; that was an es­sen­tial part of my tak­ing it on. I could not but we agreed I would work on one. But when my second child was on the way, the gods (Cour­ant, Stoker and Friedrichs) de­cided my con­tract work could be de­veloped quickly in­to a thes­is. It was on sta­bil­ity of im­plo­sions (used as neither I nor Friedrichs knew for det­on­at­ing the atom­ic bomb. It was con­nec­ted to the col­lapse of su­per­novae un­der self-grav­it­a­tion.) Beau­ti­ful spe­cial solu­tions can be found us­ing the group in­vari­ance of the equa­tions. Friedrichs was hop­ping on his pure foot and at times it was very hard to get him to think about flu­ids. In­cid­ent­ally the idea of an im­plo­sion had been con­sidered by v. Neu­mann, G. I. Taylor and by the Ger­man aero­naut­ic­al sci­ent­ist Guder­ley and for all I know the Rus­si­ans. My sta­bil­ity res­ult was very mod­est but it did give me a thes­is and the asymp­tot­ic the­ory in­volved gave me a good start for oth­er prob­lems. I kept on learn­ing from Friedrichs but I nev­er did get in­volved in either quantum mech­an­ics or spec­tral the­ory.

Once Friedrichs got a phys­ic­al prob­lem prop­erly and clearly math­em­at­ized (as say with either flu­id dy­nam­ics or mag­neto-hy­dro­dynam­ics), he then went after it with every tool he knew and wrestled it to the ground. When I was help­ing him with his Se­lecta just a few years be­fore he died he kept say­ing “oh let’s not pick that one. So and so did it much more cleanly later.” He some­how had trouble real­iz­ing how im­port­ant his in­nov­a­tions had been.

One of the things that sur­prised me about Friedrichs was his in­dif­fer­ence to the role of the big com­puter and even his own con­tri­bu­tions to dif­fer­ence schemes as a use­ful tool for find­ing an­swers as op­posed to ex­ist­ence the­or­ems. I tried once to draw him out on that sub­ject but got nowhere — which is the way it was when Friedrichs did not want to fol­low a par­tic­u­lar line of thought. I wish now that I had asked him a lot more.

My fath­er is an­oth­er ex­ample of the ap­plied math­em­atician of this cen­tury. Born in 1897 and trained in Trin­ity Col­lege Dub­lin he came to Canada as a young man and star­ted work­ing in mech­an­ics. He was di­ver­ted in­to dif­fer­en­tial geo­metry and the ex­cit­ing new sub­ject of re­lativ­ity by the in­flu­ence of Veblen. His lifelong in­terest was in the in­ter­sec­tion of geo­metry and phys­ics. He was fas­cin­ated like Taylor at the way nature worked and he fought hard for the turf of ap­plied math­em­at­ics through the thirties. The war threw a lot of math­em­aticians in­to ap­plied prob­lems (in Canada that was 1939) and he fit­ted nat­ur­ally. None of us should for­get how frightened our world was at the thought that Hitler would win and his ter­rible ideas would pre­vail. Today some may look back and ask how we could have helped with weapons of de­struc­tion. But by and large there was very little pa­ci­fism and ap­plied math­em­aticians for the most part were heav­ily en­gaged. It was in that peri­od I first stud­ied math­em­at­ics and its util­ity was of para­mount in­terest. Only later did I cap­ture the sense from my fath­er of the beauty of nature trans­formed in­to math­em­at­ics and from Friedrichs the beauty of proof of the res­ult­ing math­em­at­ics.

My fath­er’s work has ranged from ideal steer­ing mech­an­isms to gen­er­al re­lativ­ity with the lat­ter be­ing his main stomp­ing ground. A par­tic­u­lar phys­ics-geo­metry ap­proach led him to in­vent the first fi­nite ele­ment meth­od ac­com­pan­ied by es­tim­ates. But the item I would like to tell you about is his ex­cur­sion in­to dentistry and his feel­ings about it, since they have a uni­ver­sal ap­plic­a­tion. In the thirties he was ap­proached by a dent­ist, H. K. Box, about the prob­lem of trau­mat­ic oc­clu­sion caused by bit­ing. What’s that? In Fig­ure 4, we have a ri­gid tooth ly­ing in its ri­gid sock­et and sep­ar­ated from each oth­er by the peri­od­ont­al mem­brane which is trans­mit­ting the force of the bite and also the pain of trau­mat­ic oc­clu­sion if the mem­brane de­fects. Clearly a prob­lem in elasti­city and, mind­ful of George Bern­ard Shaw’s say­ing that even the Arch­bish­op of Can­ter­bury is 90% wa­ter, my fath­er de­cided to tackle the prob­lem with a mod­el of a thin in­com­press­ible elast­ic mem­brane to rep­res­ent the peri­od­ont­al mem­brane. He worked hard and got some res­ults but in 1972, 40 years later when he re­ceived the Boyle medal in Dub­lin he re­flec­ted:

“I have a so­cial con­science of sorts. When Dr. Box told me about trau­mat­ic oc­clu­sion, I lacked the strength of mind to tell him that I had oth­er things to do. So I en­gaged on this work as a so­cial duty. But as the math­em­at­ic­al ar­gu­ment took shape, my pro­fes­sion­al­ism took over and I was fas­cin­ated by this prob­lem in which the geo­metry of the tooth and the phys­ics of the mem­brane were com­bined. The fi­nal res­ult calls for a sar­don­ic laugh. On the one hand, you have a pa­per of over forty pages, pub­lished nearly forty years ago, full of in­tric­ate for­mu­lae de­veloped (if I may say so) with con­sid­er­able skill. On the oth­er hand, you have hu­man­ity suf­fer­ing still, I pre­sume, from trau­mat­ic oc­clu­sion.”

So one must be wary of try­ing to do good in math­em­at­ics.

I’d like to close with a part­ing shot in the dark. Of all the sci­ences math­em­at­ics has had the least im­pact on bio­logy. v. Neu­mann died be­fore the spec­tac­u­lar de­vel­op­ments of mo­lecu­lar bio­logy had star­ted and that is really even true of Wien­er. Both were chal­lenged by prob­lems of bio­logy but as it turned out some­what peri­pher­al prob­lems. Can we look for­ward in the next dec­ade to new gi­ants: to a new G. I. Taylor or a new von Kar­man thor­oughly im­mersed in bio­logy as they were in mech­an­ics who will bring to the v. Neu­mann or Wien­er of the day the deep and as yet not for­mu­lated math­em­at­ic­al prob­lems of bio­logy? I think that’s an ex­tremely in­ter­est­ing fu­ture to look for­ward to.

Thank you.