Archive for the 'Mathematics' Category

Rigour and speculation in pure math

On the categories email list on 5 March 2006, Ronald Brown quoted the following paragraph on mathematical speculation from a 14 June 1983 letter he had received from Alexander Grothendieck:

Continue reading ‘Rigour and speculation in pure math’




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.

Reference:

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




String theorists in knots

Last week’s Observer carried a debate over the status of string theory by a theoretical physicist, Michael Duff,  and a science journalist, James Baggott.  Mostly, they talk past each other.   There is much in what they say that could provoke comment, but since time is short,  I will only comment on one statement.

Duff’s final contribution includes these words:

Finally, you offer no credible alternative. If you don’t like string theory the answer is simple: come up with a better one. “

This is plain wrong for several reasons.  First, we would have no scientific progress at all if critics of scientific theories first had to develop an alternative theory before they could advance their criticisms.   Indeed, public voicing of criticisms of a theory is one of the key motivations for other scientists to look for alternatives in the first place.  So Duff has the horse and the cart backwards here.  

Secondly, “come up with a better one“?   “better“?     What means “better“?  Duff has missed precisely the main point of the critics of string theory!  We have no way of knowing – not even in principle, let alone in practice – whether string theory is any good or not, nor whether it accurately describes reality.  We have no experimental evidence by which to assess it, and most likely (since it posits and models alleged additional dimensions of spacetime that are inaccessible to us) not ever any way to obtain such empirical evidence.    As I have argued before, theology has more empirical support – the personal spiritual experiences of religious believers and practitioners – than does string theory.    So, suppose we did come up with an alternative theory to string theory:  how then could we tell which theory was the better of the two?   

Pure mathematicians, like theologians, don’t use empirical evidence as a criterion for evaluating theories.  Instead, they use subjective criteria such as beauty, elegance, and self-coherence.   There is nothing at all wrong with this.  But such criteria ain’t science, which by its nature is a social activity.




Mathematics in Britain

From the music critic of The Times, writing in 1952:

At Redbrick [University] they treat mathematics as an instrument of technology; at Cambridge they regard it as an ally of physics and an approach to philosophy; at Oxford they think of it as an art in itself having affinities with music and dancing.”

Cited by Ida Winifred Busbridge, in a 1974 history of mathematics at Oxford University, here.

Oxford University was a strong supporter of Catholicism in Elizabeth I’s time (eg, Thomas Campion), while Cambridge and the Fens, due to their proximity to the Netherlands, was the centre for an extreme Protestant sect, called the Family of Love, or the Familists.    Elizabeth I’s religious policy often sought to find a middle ground between these two extremes.   These religious differences persisted, so that Oxford was again, in the mid 19th-century, a centre of Catholic, and, within the Anglican Church, Anglo-Catholic (“High Church”) ideas.   The Redbrick Universities (Liverpool, Birmingham, Leeds, Victoria University of Manchester, etc), mostly founded in the North and Midlands of England in the late 19th century or early 20th century, were the result of money-raising campaigns by local business people and civic worthies, who were often of a Nonconformist or Jewish religious background.   The name Redbrick arose from novels written by a professor of Spanish at the University of Liverpool, Edgar Peers, about a fictional northern university modeled on Liverpool.

I don’t think the distinct differences between Nonconformist, Protestant and Catholic world views could be better expressed than those here between the philosophies of mathematics of Redbrick, Cambridge and Oxford:   Nonconformism as pragmatic utilitarianism; Protestantism as serious reflection on life’s higher ends; and Catholicism as enjoyment of life!
 



What do mathematicians do?

Over at the AMS Graduate Student Blog, Jean Joseph wonders what it is that mathematicians do, asking if what they do is to solve problems:

After I heard someone ask about what a mathematician does, I myself wonder what it means to do mathematics if all what one can answer is that mathematicians do mathematics. Solving problems have been considered by some as the main activity of a mathematician, which might then be the answer to the question. But, could reading and writing about mathematics or crafting a new theory be considered as serious mathematical activities or mere extracurricular activities?”

Not all mathematics is problem-solving, as we’ve discussed here before, and I think it would be a great shame if the idea were to take hold that all that mathematicians did was to solve problems.  As Joseph says, this view does not account for lots of activities that we know mathematicians engage in which are not anywhere near to problem-solving, such as creating theories, defining concepts, writing expositions, teaching, etc.

I view mathematics (and the related disciplines in the pure mathematical universe) as the rigorous study of structure and relationship.   What mathematicians do, then, is to rigorously study structure and relationship.  They do this by creating, sharing and jointly manipulating abstract mental models, seeking always to understand the properties and inter-relations of these models.

Some of these models may arise from, or be applied to, particular domains or particular problems, but mathematicians (at least, pure mathematicians) are typically chiefly interested in the abstract models themselves and their formal properties, rather than the applications.  In some parts of mathematics (eg, algebra) written documents such as research papers and textbooks provide accurate descriptions of these mental models.  In other parts (eg, geometry), the written documents can only approximate the mental models.     As mathematician William Thurston once said:

There were published theorems that were generally known to be false, or where the proofs were generally known to be incomplete. Mathematical knowledge and understanding were embedded in the minds and in the social fabric of the community of people thinking about a particular topic. This knowledge was supported by written documents, but the written documents were not really primary.

I think this pattern varies quite a bit from field to field. I was interested in geometric areas of mathematics, where it is often pretty hard to have a document that reflects well the way people actually think. In more algebraic or symbolic fields, this is not necessarily so, and I have the impression that in some areas documents are much closer to carrying the life of the field. But in any field, there is a strong social standard of validity and truth.

. . .

When people are doing mathematics, the flow of ideas and the social standard of validity is much more reliable than formal documents. People are usually not very good in checking formal correctness of proofs, but they are quite good at detecting potential weaknesses or flaws in proofs.”




Progress in computing

Computer science typically proceeds by first doing something, and then thinking carefully about it:    Engineering usually precedes theory.    Some examples:

  • The first programmable device in modern times was the Jacquard Loom, a textile loom that could weave different patterns depending on the instruction cards fed into it.   This machine dates from the first decade of the 19th century, but we did not have a formal, mathematical theory of programming until the 1960s.
  • Charles Babbage designed various machines to undertake automated calculations in the first half of the 19th century, but we did not have a mathematical theory of computation until Alan Turing’s film-projector model a century later.
  • We’ve had a fully-functioning, scalable, global network enabling multiple, asynchronous, parallel, sequential and interleaved interactions since Arpanet four decades ago, but we still lack a fully-developed mathematical theory of interaction.   In particular, Turing’s film projectors seem inadequate to model interactive computational processes, such as those where new inputs arrive or partial outputs are delivered before processing is complete, or those processes which are infinitely divisible and decentralizable, or nearly so.
  • The first mathematical theory of communications (due to Claude Shannon) dates only from the 1940s, and that theory explicitly ignores the meaning of messages.   In the half-century since, computer scientists have used speech act theory from the philosophy of language to develop semantic theories of interactive communication.  Arguably, however, we still lack a good formal, semantically-rich account of dialogs and utterances  about actions.  Yet, smoke signals were used for communications in ancient China, in ancient Greece, and in medieval-era southern Africa.

An important consequence of this feature of the discipline is that theory and practice are strongly coupled and symbiotic.   We need practice to test and validate our theories, of course.   But our theories are not (in general) theories of something found in Nature, but theories of practice and of the objects and processes created by practice.  Favouring theory over practice risks creating a sterile, infeasible discipline out of touch with reality – a glass bead game such as string theory or pre-Crash mathematical economics.   Favouring practice over theory risks losing the benefits of intelligent thought and modeling about practice, and thus inhibiting our understanding about limits to practice.   Neither can exist very long or effectively without the other.




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.




Gradgrinds de nos jours

Peter Hitchens reviews the latest jeremiad from Anthony Grayling in The Spectator, here:

‘Atheism is to theism,’ Anthony Grayling declares, ‘as not collecting stamps is to stamp-collecting’.  At this point, we are supposed to enjoy a little sneer, in which the religious are bracketed with bald, lonely men in thick glasses, picking over their collections of ancient stamps in attics, while unbelievers are funky people with busy social lives.

But the comparison is flatly untrue.  Non-collectors of stamps do not, for instance, write books devoted to mocking stamp-collectors, nor call for stamp-collecting’s status to be diminished, nor suggest — Richard Dawkins-like — that introducing the young to this hobby is comparable to child abuse.  They do not place advertisements on buses proclaiming that stamp-collecting is a waste of time, and suggesting that those who abandon it will enjoy their lives more.

Professor Grayling is too pleased with himself to have realised this. Intoxicated with amusement at his own dud metaphor, he asks: ‘How could someone be a militant non-stamp-collector?’  I rather think he has written the manual for anyone who might like to take up this activity.

It strikes me, once again, that the new atheists – particularly Grayling, Dawkins, Dennett, and C. Hitchens - want to show the world that they can think for themselves, that they are beholden not to churchly prince nor earthly potentate, neither to ideology nor tradition, in the formulation and articulation of their own views.   But thinking for themselves seems somehow insufficient to each of these people:  they want also to think for everyone else, to impose on the rest of us their own personal view.    Perhaps it’s that common affliction of people lacking in self-confidence:  by getting others to adopt their judgments (or beliefs, or consumer purchases, or career path), they can validate their own choices.

As I have remarked before, in my experience most people who profess religious belief and/or undertake religious practices do so because of some personal, transcendant experience they have had with non-material realms, or what they perceive to be such realms.  Grayling et al. tell all these people that they are each wrong:  Here’s what you should believe, not the evidence of your own lying eyes!   Who knows, our little Graylings may even be correct, and objective science may even – one day long in the future – show the delusion or invalidity of these many billions of subjective experiences.   But, in the meantime, it surely takes industrial-strength levels of presumption to assume that your own personal subjective viewpoint takes precedence over the subjective lived experience of others. 

Noticeable too, is the complete absence of any sense of wonder or awe at the unknown and the not-yet-explained in the writings of these folks (Sam Harris excepted).   It is remark-worthy, I think, that none of these people are poets, or artists, or musicians, or mathematicians, all people who seek regularly to wrangle the transcendant.   Gradgrinds the new atheists seem to be, as well as arrogantly presumptious.




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)

Reference:

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.




The afterlife of 19th-century mathematics

packed circles

I posted earlier some thoughts of mathematician Bill Thurston, here and here.  I have just encountered a reminiscence of Thurston and his specific contribution to mathematics by Philip Bowers of Florida State University.   What is noteworthy is Bowers’ view of Thurston’s style of doing mathematics – a reversion to the particularist and concrete style of 19th century mathematics, something very much out of place in the abstract and generalist world of 20th century mathematics.

It is 1978 and I have just begun my graduate studies in mathematics. There is some excitement in the air over ideas of Bill Thurston that purport to offer a way to resolve the Poincare conjecture by using nineteenth century mathematics – specifically, the noneuclidean geometry of Lobachevski and Bolyai – to classify all 3-manifolds.  These ideas finally appear in a set of notes from Princeton a couple of years later, and the notes are both fascinating and infuriating – theorems are left unstated and often unproved, chapters are missing never to be seen, the particular dominates – but the notes are bulging with beautiful and exciting ideas, often with but sketches of intricate arguments to support the landscape that Thurston sees as he surveys the topology of 3-manifolds.  Thurston’s vision is a throwback to the previous century, having much in common with the highly geometric, highly particular landscape that inspired Felix Klein and Max Dehn. These geometers walked around and within Riemann surfaces, one of the hot topics of the day, knew them intimately, and understood them in their particularity, not from the rarified heights that captured the mathematical world in general, and topology in particular, in the period from the 1930′s until the 1970′s.  The influence of Thurston’s Princeton notes on the development of topology over the next 30 years would be pervasive, not only in its mathematical content, but even more so in its vision of how to do mathematics. It gave a generation of topologists permission to get their collective hands dirty with the particular and to delve deeply into the study of specific structures on specific examples.

What has geometry to do with topology? Thurston reminded us what Klein had known, that the topology of manifolds is closely related to the geometric structures they support.  Just as surfaces may be classified and categorized using the mundane geometry of triangles and lines, Thurston suggested that the in finitely richer, more intricate world of 3-manifolds could, just possibly, be classified using the natural [page-break] 3-dimensional geometries, which he classified and of which there are eight. And if he were right, the resolution of the most celebrated puzzle of topology – the Poincare Conjecture – would be but a corollary to this geometric classification.

The Thurston Geometrization Conjecture dominated the discipline of geometric topology over the next three decades. Even after its recent resolution by Hamilton and Perelman, its imprint remains embedded in the working methodology of topologists, who have geometrized not only the topology of manifolds, but the fundamental groups attached to these manifolds. Thus we have as legacy the young and very active field of geometric group theory that avers that the algebraic and combinatorial properties of groups are closely related to the geometries on which they act. This seems to be a candidate for the next organizing principle in topology.

The decade of the 1980′s was an especially exciting and fertile time for topology as the geometric influence seemed to permeate everything. In the early part of the decade, Jim Cannon, inspired by Thurston, took up a careful study of the combinatorial structure of fundamental groups of surfaces and 3-manifolds, principally cocompact Fuchsian and Kleinian groups, constructing by hand on huge pieces of paper the Cayley graphs of example after example. He has relayed to me that the graphs of the groups associated to hyperbolic manifolds began to construct themselves, in the sense that he gained an immediate understanding of the rest of the graph, after he had constructed a large enough neighborhood of the identity.  There was something automatic that took over in the construction and, after a visit with Thurston at Princeton, automatic group theory emerged as a new idea that has found currency among topologists studying fundamental groups. In this work, Cannon anticipated the thin triangle condition as the sine qua non of negative curvature, itself the principal organizing feature of Thurston’s classification scheme.  He studied negatively curved groups, rather than negatively curved manifolds, and showed that the resulting geometric structure on the Cayley graphs of such groups provides combinatorial tools that make the structure of the group amenable to computer computations. This was a marriage of group theory with both geometry and computer science, and had immediate ramifications in the topology of manifolds.” (Bowers 2009, pages 511-512).

 

References:

Philip Bowers [2009]:Introduction to circle packing: a review.   Bulletin of the American Mathematical Society (New Series), 46 (3): 511–525.

Circle packing has surprising connections to the theory of complex functions.   For introductions to the  mathematics of circle packing, see:

Kenneth Stephenson [2003]:  Circle packing:  a mathematical tale.  Notices of the American Mathematical Society, 50 (11): 1376-1388.  Available here (PDF).

Kenneth Stephenson [2005]:   Introduction to Circle Packing:  The Theory of Discrete Analytic Functions.  Cambridge, UK:  Cambridge University Press.

The image shows a surface packed with circles of varying radii.