I have created this small corner of my web site to give a small insight into the warm, generous human side of a very great mathematician, Dirk Jan Struik (1894-2000). TESTING TESTING

Near the end of my second year as a schoolboy (1962-65) in Cowley, Oxford, I read Dirk J. Struik's A Concise History of Mathematics. The version I read could have been the original first edition of 1948; my own copy is the Fourth Revised Edition of 1986-87.

At that time, 1964, Euler was something of a hero of mine, and, because DJ Struik had written about Euler, I wrote to DJ Struik, using the address from his book. In those days - as an innocent youth - I hadn't the remotest idea as to what the MIT was, and was not deterred...

DJ Struik replied to me on one of those old fold-up airmail letters: letter#1address, letter1a and letter1b.

I must have been 'over the moon' to have heard from him, and I responded. He replied as follows: letter#2address, letter2a and letter2b.

I further replied, and I heard from him yet again: postcard.

Despite what DJ Struik wrote at the end of his postcard, I never wrote to him again, something which I very much regret. After all these years it is difficult to recall why I didn't. It could be that I had picked up a sense that 'MIT' was a really famous place and that I shouldn't be bothering such a famous person... I really cannot say.

But although that seemed to be the end of it, it wasn't. A very wonderful moment for me was to meet Dirk Jan Struik himself (at a dinner marking the Centenary of the American Mathematical Society (A.M.S.), Cincinnati, January 1994), in the one hundreth year of his life. At the dinner I was seated next to the Canadian mathematician John Coleman .

(John C. had sought me out as he had once worked with the Irish mathematician John L. Synge , and J. C. wanted me to call on J. S. on my return to Dublin to pass on his best wishes and greetings. On my return to Dublin I did call on Synge, twice in fact - he was then in a nursing home near to where I live - and wrote detailed accounts of my visits to J. C.. Once, in Cincinnati, J. C. remarked to me that he had suffered a house fire in which he had lost his mathematical library - which included a signed (to him, by Synge) copy of Synge's Relativity: The General Theory - and I was able to send to John Coleman a copy of that very book, one which had been the personal copy of Cornelius Lancoz, signed for Lancoz by Synge. I had originally obtained it from a Dublin bookdealer called Rodney Danker.)

At the end of the dinner the A.M.S. President Ronald Graham made a speech in which he particularly welcomed so many long-standing members of the Society, and said he would call out the names of all those present whose membership exceeded fifty years, starting with the longest-standing member (of over seventy years), none other than D. J. STRUIK - a thoroughly delightful surprise for me - (the second longest-standing member - at that time - as I recall, was J. L. Synge) ... , and he ended by calling on D. J. Struik to say a few words.

D. J. was at a table some distance away from where J. C. and I were sitting, and he delivered a witty speech, one which ended with something like this:

" ... Recently the Dutch Mathematical Society made me a life member - which means I no longer have to pay annual membership fees - I've since thought it would be nice if the A.M.S. did something similar. "

whereupon Ron Graham leapt to his feet and said something like:

" I hereby declare by executive decree that Professor Dirk Jan Struik is henceforth a life-long member of the American Mathematical Society, and will no longer have to pay any annual membership subscription. "

This was greeted wich much laughter and many calls along the lines of I no longer wish to pay my annual fees as well!! Then I remarked to J. C. that I had once been in correspondence with D. J. S. (and related much of what I've related earlier), and he suggested I should go over and speak with him, which I did, as related in the following Dec. 2000 email to his daughter Ruth Rebekka:

This is the email that I sent to D. J. Struik's daughter, the mathematician Ruth Rebekka Struik in December 2000, after I had heard of her father's death. I heard back from R.R.S., and I was happy that she did want to have copies of her father's letters and postcard.

Note of 8th June 2017. Last night, having dinner with neighbours J. J. and Cliona O'Connor, Cliona remarked at one point that she had a relative, a priest, who at one time had been a Professor of Mathematics in Maynooth (the department there is now Maynooth Mathematics Department). Ah!! Would that have been J.J. McMahon (see letter2a above)? And indeed it was... (and we had a long chat about him). Cliona's mother and J.J were first cousins (the connection was as follows: Cliona's maternal grandmother was Annie Walsh, while J.J.'s mother was Sis Walsh; Annie and Sis were sisters, and they came from Woodford in County Galway). Small world...

Here is an obituary notice and three appreciations (three pages) for J. J. McMahon that appeared in the December 1982 issue of the Bulletin of the Irish Mathematical Society (it was the Newsletter of the IMS until 1986). The author of J.J.'s obituary notice - "R. Timoney" - is also a neighbour, Richard Timoney of Trinity College Dublin. From Richard Timoney's obituary notice it will be seen that J. J. McMahon obtained his PhD under the supervision of Synge (see the Mathematics Genealogy Project page).

Despite hearing from D.J. Struik in August 1964 about J. J. Mahon, I didn't get around to meeting J.J. until September 1965 (just days before I began my university studies at Royal Holloway College). I was at home in Cavan at the time, took a bus up to Dublin, and then a bus out to Maynooth. On arrival, and meeting J.J., he told me that he had been assigned a car-drive duty: to transport some visiting U.S.A. cleric (a 'monsignor', as I recall) to a meeting with the (renowned) Irish politician W. T. Cosgrave. J. J. suggested I sit with him in the front and we could talk Mathematics (while the visitor sat in the back doing whatever he did), and we arrived at Cosgrave's residence (was it in Rathfarnam? I do not know... In those days 'Dublin' was an almost entirely unknown place to me) and were led into a 'sitting room'. W. T. and the cleric sat talking, while J. J. and I sat to the side - in the same room - having our conversation...

After some time a young woman (it could have been his grand-daughter Mary, though I wouldn't have known at the time; I should add that in those days I was so utterly ignorant of my country's recent history - indeed its history - that I knew next to nothing about 'W. T. Cosgrave') brought in a tray of tea, buns, cakes, ... and J. J. and I were invited to join them. W. T. asked after my name, and hearing that I was a 'Cosgrave', W. T. wondered if we could be related. I offered that we probably weren't, he being from Leinster, while my father was from Gort in County Galway.

Later we returned to Maynooth, where J. J. showed me his mathematical library, and then I returned to Cavan.

Months later (in Nov.) a fellow Mathematics student (Pamela Smith, now Idelson. Pamela's father-in-law - Vladimir Idelson - was a legal advisor to Kerensky's post First World War Russian Provisional Government) told me she'd just heard on the radio that some famous Irish politician called 'Cosgrave' had just died - it was W. T. - and I was able to tell her that I'd had afternoon tea with him just a couple of months earlier (for an English person that would have been unthinkable, I imagine).

Note of 11th Nov. 2020. Rooting through some boxes here at home I came upon these two lost letters written to me by J. J. McMahon: JJ McMahon 17 Sept 1965, JJ McMahon 19 April 1967_one, and JJ McMahon 19 April 1967_two.

The re-discovery of the second letter was especially interesting for me, for I learned from it something which I had long forgotten: that I had evidently entertained the idea then (nearing the end of my second year at Royal Holloway) of going to Oxford to do a PhD in Mathematics. In the (unlikely) event that some young mathematician might one day read this - and to let her/him see how different the whole study culture was then - I remark that the notion of going somewhere to do a PhD was somewhat of an accident: no one, and I mean no one, ever discussed the possibilities facing one.

No one ever said to us something like: you need to think about what you might do after you graduate (indeed think about it before you graduate); you, you seem especially interested in Quantum Mechanics, Group Theory, Number Theory, Analysis, whatever... You should look to this, that or the other person with a view to doing research (we never thought of doing research, we just thought of doing Mathematics) ... . We were given no advice whatsoever.

So, how did I get it into my head that I might go to Oxford to do a PhD? I have absolutely no idea... As it happered I didn't apply there because I was struck down by Cupid's arrow... I began 'going out' (as we used to say down the country where I came from in Ireland) with a friend on the 16th of June that year (Mary, who was also in her Royal Holloway second year - French major, Italian minor), and Mary was about to spend a year as an assistant in a school in France (in Béziers), while I remained at Royal Holloway for my final undergraduate year ('67-'68).

And, not only did I not try to do an Oxford PhD, but it didn't even occur to me to attempt to find a supervisor at another London college (the great Klaus Friedrich Roth - one of the nicest mathematicians I ever met - would have been an obvious choice - and I do believe he would have accepted me), and so I stayed at Royal Holloway and did my PhD there. I should say (as far as my Mathematics is concerned) I have absolutely no regrets that I stayed - yes, I would have had an entirely different mathematical life with Roth (a truly historic mathematician), I would certainly have turned out to be a better mathematician (whatever that might be), but I would not then have had the life that I've had, would not have done the Mathematics that I did (the best, and most certainly the most beautiful, in my seventh decade). I would have done quite different things, but I know with absolute certainty that - given a choice between what I did and what I might have done - I would want the former before anything else.

A note on Mary Bradburn. Mary was one of the five members of the Royal Holloway Mathematics department who interviewed me in January 1965 for a place as an undergraduate; it wasn't just for me; every applicant for Mathematics had the same interview board, and an extra one with the remarkable, and loveable Dame Marjorie Williamson, the college's Principal. Perhaps she didn't interview every applicant, but she did interview me.

The board consisted of the renowned Professor William McCrea, a Professor F. R. Keogh (he worked in some branch of Analysis, and had left RHC by the time I went there in Sept. later that year. I have managed to track down that he moved to the University of Kentucky, but there my search goes cold), Mary Bradburn, and two other irish mathematicians, Gearóid De Barra and Barbara Yates.

At Royal Holloway - especially in my first year - Mary Bradburn was especially supportive, and I mention that in May (as I recall) of '66 she invited me to accompany her to a meeting of the London Mathematical Society. There she introduced me to three renowned number theorists (all together in a group): Harold Davenport (I was too much in awe of him to say I'd read his wonderful The Higher Arithmetic (in later years - 1995 - I obtained a first edition copy in a run-down bookstand in Blackrock Market, down the road from where I live)), Heini Halberstam (the greatest speaker I've heard in my mathematical life), and Klaus F. Roth (the mathematician from whom I later (early 1970) learned more in a single afternoon's conversation than from any other one I have known; Roth simply exuded brain power).

(In fact I wrote about that afternoon with Roth in this lecture of mine on transcendental numbers in this corner of my web site - just go down the page to the Klaus Friedrich Roth sub-section near the end, click on it, and scroll down to the 'Personal anecdote' part. The home page for my transcendental numbers page is here.)

(How I would have loved had Davenport lived to have seen the vast extension that Karl Dilcher and I gave (Theorem 7 of our second paper here) of Gauss's celebrated binomial coefficient congruence - which I first learned of from Davenport's Higher Arithmetic.)

Aside. In late March 1980 I attended a joint meeting of the London and French Mathematical Societies in Exeter, and - at an end-of-the-week dinner (Fri. 28th) - I found myself seated beside Heini Halberstam. Discovering that he was soon to visit Dublin to give the annual TCD Donegall Lecture I asked if he would also give a talk to my students (who had a good basic grounding in Number Theory) at Carysfort College, and he readily agreed. His visit to Carysfort was the mathematical highlight of my years there; a Sister Bernard (who could have had a career in any top-level hotel) laid on a splendid dinner for him (before his talk) and invited heads of Dublin Mathematics departments, and later we had a supper for him and guests here at our home.

Mathematicians with whom I have corresponded over the years. I began this Struik tribute page away back in... I'm not exactly sure, but it seems to have taken off a life of its own. He was the very first mathematician to whom I wrote, and it's given me the thought that I should try to recollect as many names as I can of other mathematicians (I don't include mathematicians who are personal friends, nor do I include any Irish mathematicians - not that I have anything against Irish mathematicians, on the contrary - but I make an exception of one whom I especially admire (Tom Laffey), and another - a very dear friend (Mark Daly), with whom I enjoyed an especially fruitful time at an important moment in my life) with whom I enjoyed a correspondence over the years.

In the early days correspondence was of course by letter (I hope no younger readers don't have to look up the meaning of that word!), and I have inserted an 'L' in parentheses after a name when that was the case (for some Ls we also engaged in email correspondence).

D.J. Struik (L) and J.J. McMahon (L), Klaus F. Roth (L), Jean Pierre Serre (L), John Coates (L), Alan Baker (L), Naum Il'ich Feldman (L), David Masser (L), Emil Grosswald (L), David Klarner (L), Michel Waldschmidt (L), Kurt Mahler (L), Wolfgang Schmidt (L), Tom Laffey, Tony Gardiner, John Cremona, Hendrik Lenstra, Alexander A. Zenkin ("I have read your WEB pages with a great pleasure... " Spasibo!), Carl Pommerance, Andrew Granville, Bruce Berndt, Kenneth Williams, Ron Evans, Hugh Williams, Chris Caldwell, Richard Crandall, John Selfridge, John Brillhart, Richard Guy (L), Neal Koblitz (L), Ken Ribet, Gunter Ziegler, Paul Halmos (L; see the Halmos corner of my web site), Mark Daly, Yves Gallot (L), Wilfrid Keller, Constance Reid (L), François Morain, Mel Nathanson, Doron Zeilberger, Frits Beukers, Jon Borewin, Franz Lemmermeyer, N.J.A. Sloane, Keith Matthews, David Bailey, Barry Mazur, Anand Pillay, David Broadhurst, Erhard Glötzl, Peter Larcombe, Stefanie Krivsky, David Joyner, Kevin Buzzard, Béla Bollobás, Tim(othy) Gowers, János Pintz, József Szabados, Alf van der Poorten, Sam Wagstaff, Robert Baillie, Nicholas Katz, ... INCOMPLETE (RETURN SOMETIME)

One particular correspondence was with the following, who wrote to me in connection with the Oxford 1969 corner of my web site:

Oxford 69: Paul Bateman, David Boyd, John Brillhart, Michal Bulant, David Cantor, Harvey Cohn (who eventually, in November 99, supplied me with a complete list), Jean Cougnard, Andrew Granville, Peter Pleasants, Daniel Lazard, Odile Lecacheux, Phil Leonard, Keith Matthews, Jim Milne, Hugh Montgomery, Richard Pinch, Eric Reyssat, John Stillwell.

Contact details. jbcosgrave at