Archive for the 'Matherati' Category

The New World versus the Old

Propose to an Englishman any principle, or any instrument, however admirable, and you will observe that the whole effort of the English mind is directed to find a difficulty, defect, or an impossibility in it.  If you speak to him of a machine for peeling a potato, he will pronounce it impossible:  if you peel a potato with it before his eyes, he will declare it useless, because it will not slice a pineapple.  Impart the same principle or show the same machine to an American, or to one of our colonists, and you will observe that the whole effort of his mind is to find some new application of the principle, some new use for the instrument. ”

Charles Babbage, 1852, in a paper on taxation.  Cited on page 132 of Doron Swade [2000]:  The Cogwheel Brain:  Charles Babbage and the Quest to Build the First Computer. London, UK:  Little, Brown and Company.

Love and Math

Frenkel Edward 2008Talking of his grandfather who had overcome poverty and blindness to become a US Senator, Gore Vidal once wrote that no challenge is finally insurmountable if you mean to prevail.  I was reminded of this in reading Edward Frenkel’s superb memoir, Love and Math.  Frenkel overcame the widespread and systemic anti-semitism in Soviet Mathematics to establish himself as a world-leading mathematician at a very young age.  

Denied entry in 1984 because of his ethnicity to Moscow State University’s (MGU’s) Department of Mechanics and Mathematics (Mekh-Mat), the leading undergraduate mathematics programme in the USSR, he entered instead the mathematics program at Kerosinka, the Moscow Institute of Oil and Gas.  Anti-semitism (and anti-Armenianism, anti-Chinese racism, etc) in the admissions process at Mekh-Mat was so widespread, that other Moscow institutions, such as Kerosinka, were able to recruit very good Jewish and minority students.   One theory is that this policy was deliberate, since having all the Jewish mathematicians studying in one or two institutions made their monitoring easier for the KGB.

Frenkel had grown up in Kolomna – only 70 miles from Moscow, but well into the provinces – and had not attended a special mathematics school (as did, for example, Vadim Delone at FizMat #2), nor had an opportunity to participate in the mathematical study circles that were widespread in the larger soviet cities.  He did have the help of a local mathematician, Evgeny Petrov, a professor at a teacher training college in Kolomna.   Frenkel was very fortunate to have such help.   I recall my envy on learning on the first day of lectures in my first year at university that some of my fellow students, who had grown up near to the university, had been meeting our professors for years previously for after-school mentoring and coaching. (On the other hand, even the brightest of my fellow students so mentored ended up winning no Fields Medal, nor even becoming a mathematician.)

Good mathematical undergraduates from Kerosinka and other specialized institutes in Moscow literally scaled the fences at MGU to attend, illegally but often with the encouragement of the teachers, lectures at Mekh-Mat.  Frenkel did this and was again fortunate in being befriended by some very great mentors:  Dmitry Fuchs (now at UC Davis), his student Boris Feigin, and Yakov Khurgin.  Their generous mentoring was unpaid, time-intensive, and often brave, given the society they lived in.   As a result, Frenkel wrote his first research paper in only his second year as an undegraduate, a paper subsequently published in Israel Gelfand’s famous journal, Functional Analysis and Applications.  Gelfand was someone that even my professors, in the 1970s and in faraway Australia, spoke of with awe.

With the opening of perestroika, the Mathematics Department at Harvard University decided to invite some young Soviet mathematicians for research visits, and Frenkel was one of these:  He received his invitation in March 1989, before he had even completed his first degree.  While at Harvard, he had another Russian mentor, Vladimir Drinfeld (now at University of Chicago), and Frenkel completed his PhD there, in 1 year, under the supervision of another Russian, Joseph Bernstein (now at Tel-Aviv).  Frenkel is very generous in his acknowledgement of the support he received from his mentors and from others, and his story warms the heart.  Despite the anti-semitism he experienced, he has prevailed in the end, being now a professor at U-Cal Berkeley (and a film-maker).  Reading his account, I was reminded repeatedly of the ancient spiritual wisdom:  When the disciple is ready, the guru will appear.

Frenkel interleaves his personal story with an account of his changing research focus along the way, a focus which has mostly followed the powerful thread of the Geometric Langlands Programme.   His writing is fluent, wise and witty, and he manages to convey well the excitement and pure, joyous exhilaration that mathematical thinking can provide.  His writing makes most of the underlying mathematical ideas clear to non-experts.  That said, however, the text has a couple of weaknesses, both minor, although both I found irritating.  No one who does not already know something of category theory would understand it, even at a high level, from the single paragraph devoted to it on page 156.   Another minor criticism is that the text does not always adequately explain the diagrams, or what is being done with them.    But then I have particular views about reasoning over diagrams.

In summary, this is a superb book – wise, generous, witty, and heart-warming – and reading it will enlarge your knowledge of mathematics, of the Langlands Program, and of the power of the human spirit.   Everyone in the pure mathematical universe should read it.

An index to posts on the Matherati is here.


Edward Frenkel [2013]:  Love and Math:  The Heart of Hidden Reality. New York, NY:  Basic Books.

Mathematicians of the 20th Century

With so many blogs being written by members of the literati, it’s not surprising that a widespread meme involves compiling lists of writers and books.  I’ve even succumbed to it myself.   Lists of mathematicians are not as common, so I thought I’d present a list of the 20th century greats.    Some of these are famous for a small number of contributions, or for work which is only narrow, while others have had impacts across many parts of mathematics.

Each major area of mathematics represented here (eg, category theory, computer science) could equally do with its own list, which perhaps I’ll manage in due course.  I’ve included David Hilbert, Felix Hausdorff and Bertrand Russell because their most influential works were published in the 20th century.  Although Hilbert reached adulthood in the 19th century, his address to the 1900 International Congress of Mathematicians in Paris greatly influenced the research agenda of mathematicians for much of the 20th century, and his 1899 axiomatization of geometry (following the lead of Mario Pieri) influenced the century’s main style of doing mathematics.   For most of the 20th century, mathematics was much more abstract and more general than it had been in the previous two centuries.    This abstract style perhaps reached its zenith in the work of Bourbaki, Grothendieck, Eilenberg and Mac Lane, while the mathematics of Thurston, for example, was a throwback to the particularist, even perhaps anti-abstract, style of 19th century mathematics.  And Perelman’s major contributions have been in this century, of course.

  • David Hilbert (1862-1943)
  • Felix Hausdorff (1868-1942)
  • Bertrand Russell (1872-1970)
  • Henri Lebesgue (1875-1941)
  • Godfrey Hardy (1877-1947)
  • LEJ (“Bertus”) Brouwer (1881-1966)
  • Srinivasa Ramanujan (1887-1920)
  • Alonzo Church (1903-1995)
  • Andrei Kolmogorov (1903-1987)
  • John von Neumann (1903-1957)
  • Henri Cartan (1904-2008)
  • Kurt Gödel (1906-1978)
  • Saunders Mac Lane (1909-2005)
  • Leonid Kantorovich (1912-1986)
  • Alan Turing (1912-1954)
  • Samuel Eilenberg (1913-1998)
  • René Thom (1923-2002)
  • John Forbes Nash (1928- )
  • Alexander Grothendieck (1928- )
  • Michael Atiyah (1929- )
  • Steven Smale (1930- )
  • Paul Cohen (1934-2007)
  • Nicolas Bourbaki (1935- )
  • Stephen Cook (1939- )
  • William Thurston (1946-2012)
  • Edward Witten (1951- )
  • Andrew Wiles (1953- )
  • Richard Borcherds (1959- )
  • Grigori Perelman (1966- )

And here’s my list of great mathematical ideas.

Our impact on others

JOHN-Fritz 1953

A nice story about the possibly unknown effects of our actions, from a review by mathematician Marjorie Senechal of a book about German Jewish mathematicians:

Fritz John (1910–1994) [pictured in 1953], Jewish on his father’s side, left Germany in 1933 for England; in 1935 he was appointed assistant professor of mathematics at the University of Kentucky in Lexington.  Back in the 1930s, the University of Kentucky was small and isolated but, except for two years of war-related work, John stayed there until 1946, when he moved permanently to New York University.  Surely he was glad to rejoin his mentor, [mathematician Richard] Courant.  But he made a difference in Lexington; I don’t know if he ever knew it.

I grew up near Lexington and took piano lessons from a teacher in town named Helen Lipscomb.  Helen was a polio victim, confined to a wheelchair; her brother, Bill, was a chemist at the University of Minnesota.  I met Bill Lipscomb for the first time in 2009, two years before he died at the age of ninety-two.  By then he’d taught at Harvard for forty years and earned a Nobel prize (1976) [in Chemistry] for his work on boranes.  Unlike me, Bill had attended the University of Kentucky after a Lexington public high school; he’d had a music scholarship and studied chemistry on the side.  “Why did you decide to become a chemist instead of a musician?” I asked him. “What changed your mind?” “A math class,” he told me. “A math class taught by a German named Fritz John.” (page 213)


Marjorie Senechal [2013]:   Review of:   Birgit Bergmann, Moritz Epple, and Ruti Ungar (Editors):  Transcending Tradition: Jewish Mathematicians in German-Speaking Academic Culture.   Springer Verlag, 2012.  Notices of the American Mathematical Society, 60 (2):  209-213.  Available here.

One day in the life . . .

Khan-Tengri Mountain at sunset

. . . of Boris N. Delone (1890-1980), Russian mathematician, moutaineer, and polymath, member of a famous family of mathematicians and physicists, whose grandson was a dissident poet:

July 6, 1975, Delone spends a cold night (-25 degrees C) in a tent on a glacier under the beautiful peak of Khan Tengri (7000 m, the Tien Shan mountain system, Central Asia) [pictured, at sunset] at a height of about 4200 m.  In the morning a helicopter picks him up to take him to Przhevalsk (now Karakol), a Kyrgyz city at the eastern tip of Lake Issyk-Kul.  From Przhevalsk he takes a local flight  to Frunze (now Bishkek), the capital of Kyrgyzstan, where the heat exceeds 40 degrees C.   After queuing up for a few hours and with the help of some “kind people” and the Academy of Sciences membership card he succeeds in purchasing an air ticket to Moscow.   Late at night he arrives at Domodedovo airport in Moscow, from which he still needs to go to his country house near Abramtsevo (Moscow oblast).   Taking the last commuter train, he arrives at the necessary station at around 2 am; from there it is another three kilometers to his house, half of which are in a dark dense forest.  He loses his way and, after roaming around the night forest for a long time, leaves his heavy rucksack in a familiar secluded place.  Only in the morning does Delone succeed in getting home safely.”  (page 13).

In that year, 1975, Boris Delone was 85 years old.



N. P. Dolbilin [2011]: Boris Nikolaevich Delone (Delaunay): Life and Work. Proceedings of the Steklov Institute of Mathematics, 275: 1-14.  Published in Russian in Trudy Matematicheskogo Instituta imeni V. A. Steklov, 2011, 275:  7-21.  A pre-print version of the paper is here.


Thurston on mathematical proof

Bill Thurston

2012 saw the death of Bill Thurston, leading geometer and Fields Medalist.   Learning of his death led me to re-read his famous 1994 AMS paper on the social nature of mathematical proof.   In my opinion, Thurston demolished the views of those who thought mathematics is anything other than socially-constructed.  This post is just to present a couple of long quotes from the paper. 

Continue reading ‘Thurston on mathematical proof’

Transitions 2012

Some who have passed on during 2012 whose life or works have influenced me:

The pure mathematical universe

Somewhere on his blog, the indefatigable Cosma Shalizi has written about the disciplinary universe of mathematics – that in addition to pure mathematics itself, pure mathematics get used in (and is essential to) the disciplines of Statistics and Computer Science.  This idea struck a chord, and I began to wonder exactly what particular aspect of pure mathematics was being used in each of these other disciplines and where else such methods or approaches were being used.  Of course, having trained as a pure mathematician who turned to mathematical statistics and then eventually to computer science, I know precisely what parts or theories of  pure math were being used in these two disciplines, so this is not my question.    For example, the theory and practice of mathematical statistics draw on probability theory (which itself draws on measure theory and the theory of integration, which in turn require Cantor’s theory of infinite collections), and, in statistical decision theory,  on the differential geometry of information.    (Indeed, I recall being strongly annoyed in my introductory statistics courses that so often proofs of theorems were postponed until “after you know measure theory.”)  Rather, what interests me is what abstract processes – what we might call, mathematical styles of thinking (mathmind, as distinct from, say, the styles of thinking of anthropology or history or chemistry)  – were being used, and where.

Not for the first time, I considered an input-process-output model.  From this viewpoint, we can view pure mathematics itself as a process of (mostly) deductive reasoning that transforms facts about abstract formal objects into other facts about abstract formal objects.   The abstract formal objects may have a basis in some (apprehension of some manifestation of) some real domain or objects, but such a basis is neither necessary nor important to the mathematics.   Until the mid 20th century, people used to say that mathematics was the theory of number, although why they thought this when the key examplar of this theory was Euclidean geometry, a theory which is mostly number-free and scale-invariant, I don’t know.    Since the mid-20th century, people have tended to say that mathematics is the theory of structure and relationship, which better describes most parts of pure mathematics, including Euclidean geometry, and better describes the potential applications and utility of the subject.

Many of the disciplines in the mathematical universe use the same processes – essentially deductive reasoning and, sometimes, calculation – to transform different inputs to certain outputs.   Here is my list (to be added to, when I think of others).

Pure Mathematics

  • Input = Abstract formal structures and objects
  • Process = Manipulation based on deductive reasoning (and, occasionally, calculation)
  • Output = Knowledge about abstract formal structures and objects

Theoretical Physics

  • Input = Mathematical models of physical reality
  • Process = Manipulation based on deductive reasoning and calculation
  • Output = Knowledge about (mathematical models of) physical reality

Mainstream Economics

  • Input = Mathematical models of economic reality
  • Process = Manipulation based on deductive reasoning and calculation
  • Output = Knowledge about (mathematical models of) economic reality

Computational Economics

  • Input = Computational models of economic reality
  • Process = Manipulation based on deductive and inductive reasoning, calculation and simulation
  • Output = Knowledge about (computational models of) economic reality

Exploratory Statistics:

  • Input = Raw data
  • Process = Processing and manipulation
  • Output = Information

Computer Processing:

  • Input = Information
  • Process = Processing and manipulation, including operations derived from both deductive and inductive reasoning, and simulation
  • Output = Information

Statistical Decision Theory (quantitative decision theory)

  • Input = Information
  • Process = Processing and manipulation, both inductive and deductive reasoning
  • Output = Knowledge, Actions

Computer Science:

  • Input = Abstract formal structures and objects, intended as models of computational processes
  • Process = Manipulation based on both deductive and inductive reasoning, and simulation
  • Output = Knowledge about (abstract formal structures and objects, as models of) computational processes


  • Input = Physical objects and materials
  • Process = Manipulation based on deductive reasoning and calculation
  • Output = Physical objects and materials

(Formal) Logic

  • Input = Formal representations of statements and arguments
  • Process = Manipulation based on deductive reasoning
  • Output = Formal representations of statements and arguments

AI Planning

  • Input = Information, actions
  • Process = Manipulation based on deductive reasoning and simulation
  • Output = Knowledge, actions, plans

Qualitative Decision Theory

  • Input = Information, actions
  • Process = Processing and manipulation, both inductive and deductive reasoning
  • Output = Knowledge, actions, plans.

Musical composition

  • Input = Abstract formal structures and objects with a sonic semantics
  • Process = Manipulation, based on deductive-like reasoning or simulation-like generation
  • Output = (Plans for the production of) sounds

Some of the statements implied by these input-process-output schemas are contested.  I would argue that, for instance, any knowledge gained by mathematical economics is only ever knowledge about the mathematical model being studied, and not about the real world which the model is intended to represent.   But this is not the view of most economists, who seem to think they are talking about reality rather than their model of it.  Perhaps this view explains why economics seems peculiarly immune to the major revision or rejection of models on the basis of their failure to predict or describe actual empirical data.

A word on the last schema above:   The composition, performance and even the auditing of music may involve thinking, as I argue here.  Some of the specific modes of musical thinking involved have much in common with deductive mathematical reasoning, in the sense that they can involve the working out of the logical consequences of musical ideas, where the logic being used is not Modus Ponens or Reductio ad Absurdum (as in pure mathematics), but a logic of sounds, pitches, rhythms, timbre and parts.

RIP: Ernest Kaye

While on the subject of Britain’s early lead in computing, I should mention the recent death of Ernest Kaye (1922-2012).  Kaye was the last surviving member of the design team of the LEO computer, the pioneering business and accounting machine developed by the Lyons Tea Shop chain in the early 1950s.  As with jet aircraft, computers were another technological lead gained and squandered by British companies.

Kaye’s Guardian obituary is here.   A post on his LEO colleague John Aris is here.  An index to Vukutu posts on the Matherati is here.

Michael Dummett RIP

The death has just occurred of the philosopher Michael Dummett (1925-2011), formerly Wykeham Professor of Logic at Oxford.    His writings on the philosophy of language and the philosophy of mathematics have influenced me, particularly his thorough book on intuitionism.   Having been educated by pure mathematicians who actively disparaged intuitionist and constructivist ideas, I found it liberating to see these ideas taken seriously and considered carefully.  The precision of Dummett’s writing and thought clearly marked him out as a member of the Matherati, as also his other formal work, such as that on voting procedures.

POSTSCRIPT (2012-01-21):  The logician Graham Priest remembers Dummett as follows:

It is clear that Dummett was one of the most important — perhaps the most important — British philosopher of the last half century. His work on the philosophy of language and metaphysics, inspired by themes in intuitionist logic, was truly groundbreaking. He took intuitionism from a somewhat esoteric doctrine in the philosophy of mathematics to a mainstream philosophical position.

Perhaps his greatest achievement, as far as I am concerned, was to demonstrate beyond doubt the intellectual respectability of a fully-fledged philosophical position based on a contemporary heterodox logic. Philosophers in the United Kingdom, even if they do not subscribe to Dummett’s views, no longer doubt the possibility of this. Dummett had an influence in Australia, too. It was quieter there than in the U.K., but the relevant philosophical lesson was amplified by logicians who endorsed heterodox logics of a different stripe (for which, I think, Dummett had little sympathy). The result has been much the same.

In the United States, though, Dummett had virtually no significant impact. Indeed, I am continually surprised how conservative philosophy in the United States is with regard to heterodox logics. It is still awaiting a Dummett to awaken it from its dogmatic logical slumbers.

Graham Priest, City University of New York Graduate Center, and the University of Melbourne (Australia)


His Guardian obituary is here.  An index to posts about members of the Matherati can be found here.

M. Dummett [1977/2000]: Elements of Intuitionism. (Oxford: Clarendon Press, 1st edition 1977; 2nd edition 2000).