Archive for the 'Mathematics' Category

Mathematics in Britain

Published
by
peter
on 2013-04-14
in Mathematics, Music and Religion
. Closed

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?

Published
by
peter
on 2013-04-01
in Human intelligence and Mathematics
. Closed

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

Published
by
peter
on 2013-03-20
in Computer Science, Computer technology and Mathematics
. Closed

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

Published
by
peter
on 2013-03-13
in Concats, Mathematics and Matherati
. Closed

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

Published
by
peter
on 2013-03-03
in Mathematics and Religion
. Closed

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

Published
by
peter
on 2013-03-03
in Mathematics and Matherati
. Closed

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

Published
by
peter
on 2013-03-01
in Mathematics
. Closed

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.




One day in the life . . .

Published
by
peter
on 2013-02-03
in Mathematics, Matherati and Post-Industrial Nomads
. Closed

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.

Delone-Boris-Nikolaevich

Reference:

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.

 




Hard choices

Published
by
peter
on 2013-01-27
in Decision theory, Literature and Mathematics
. Closed

Adam Gopnik in the latest New Yorker magazine, writing of his former teacher, McGill University psychologist Albert Bregman:

he also gave me some of the best advice I’ve ever received.  Trying to decide whether to major in psychology or art history, I had gone to his office to see what he thought.   He squinted and lowered his head.  “Is this a hard choice for you?” he demanded.  Yes! I cried. “Oh,” he said, springing back up cheerfully.   “In that case, it doesn’t matter.  If it’s a hard decision, then there’s always lots to be said on both sides, so either choice is likely to be good in its way.  Hard choices are always unimportant. ” (page 35, italics in original)

I don’t agree that hard choices are always unimportant, since different options may have very different consequences, and with very different footprints (who is impacted, in what ways, and to what extents).  Perhaps what Bregman meant to say is that whatever option is selected in such cases will prove feasible to some extent or other, and we will usually survive the consequences that result.  Why would this be?    I think it because, as Bregman says, each decision-option in such cases has multiple pros and cons, and so no one option uniformly dominates the others.  No option is obviously or uniformly better:  there is no “slam-dunk” or “no-brainer” decision-option.  

In such cases, whatever we choose will potentially have negative consequences which we may have to live with.  Usually, however, we don’t seek to live with these consequences.  Instead, we try to eliminate them, or ameliorate them, or mitigate them, or divert them, or undermine them, or even ignore them.  Only when all else fails, do we live in full awareness with the negative consequences of our decisions.   Indeed, attempting to pre-emptively anticipate and eliminate or divert or undermine or ameliorate or mitigate negative consequences is a key part of human decision-making for complex decisions, something I’ve called (following Harald Wohlrapp), retroflexive decision-making.   We try to diminish the negative effects of an option and enhance the positive effects as part of the process of making our decision.

As a second-year undergraduate at university, I was, like Gopnik, faced with a choice of majors; for me it was either Pure Mathematics or English.    Now, with more experience of life, I would simply refuse to make this choice, and seek to do both together.  Then, as a sophomore, I was intimidated by the arguments presented to me by the university administration seeking, for reasons surely only of bureaucratic order, to force me to choose:  this combination is not permitted (to which I would respond now with:  And why not?); there are many timetable clashes (I can work around those);  no one else has ever asked to do both (Why is that relevant to my decision?); and, the skills required are too different (Well, I’ve been accepted onto Honours track in both subjects, so I must have the required skills).   

As an aside:  In making this decision, I asked the advice of poet Alec Hope, whom I knew a little.   He too as an undergraduate had studied both Mathematics and English, and had opted eventually for English.  He told me he chose English because he could understand on his own the poetry and fiction he read, but understanding Mathematics, he said, for him, required the help of others.  Although I thought I could learn and understand mathematical subjects well enough from books on my own, it was, for me, precisely the social nature of Mathematics that attracted me: One wasn’t merely creating some subjective personal interpretations or imaginings as one read, but participating in the joint creation of an objective shared mathematical space, albeit a space located in the collective heads of mathematicians.    What could be more exciting than that!?

More posts on complex decisions here, and here. 

Reference:

Adam Gopnik [2013]: Music to your ears: The quest for 3D recording and other mysteries of sound.  The New Yorker, 28 January 2013, pp. 32-39.




Listening to music by jointly reading the score

Published
by
peter
on 2013-01-22
in Human intelligence and Mathematics
. Closed

Another quote from Bill Thurston, this with an arresting image of mathematical communication:

We have an inexorable instinct to convey through speech content that is not easily spoken.  Because of this tendency, mathematics takes a highly symbolic, algebraic, and technical form.  Few people listening to a technical discourse are hearing a story. Most readers of mathematics (if they happen not to be totally baffled) register only technical details – which are essentially different from the original thoughts we put into mathematical discourse.  The meaning, the poetry, the music, and the beauty of mathematics are generally lost.  It’s as if an audience were to attend a concert where the musicians, unable to perform in a way the audience could appreciate, just handed out copies of the score.  In mathematics, it happens frequently that both the performers and the audience are oblivious to what went wrong, even though the failure of communication is obvious to all.” (Thurston 2011, page xi)  

Reference:

William P. Thurston [2011]:   Foreword.   The Best Writing on Mathematics: 2010.  Edited by Mircea Pitici.  Princeton, NJ, USA:  Princeton University Press.




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