with or without proofs? a worthwhile activity -------- Original Message -------- Subject: du Sautoy Date: Sat, 15 Apr 2006 14:49:02 -0400 (EDT) From: Peter Freyd <pjf@saul.cis.upenn.edu> To: MathPeople@saul.cis.upenn.edu Most people's idea of what I do as a research mathematician is long division to lots of decimal places. But fundamentally, mathematics isn't about numbers - it's about finding structure and logic and connections that help us negotiate the complex world we live in. Copyright 2006 TSL Education Limited The Times Higher Education Supplement April 14, 2006 SECTION: OPINION; No.1738; Pg.14 LENGTH: 831 words HEADLINE: Scales Fall Short Of Grand Symphonies In Maths BYLINE: Marcus du Sautoy BODY: Pique children's interest in maths with elegant epics, enigmatic mysteries and cold hard cash, says Marcus du Sautoy. World pi Day was marked at 1:59 on March 14 - 3.14159 being the beginning of the decimal expansion of pi. Although I am appreciative of any publicity mathematics can get, I found that most people were interested in how many decimal places I knew of this important number. They were disappointed that five was my limit. To me, that response revealed the deep misconception people have of what mathematics is really about. Most people's idea of what I do as a research mathematician is long division to lots of decimal places. But fundamentally, mathematics isn't about numbers - it's about finding structure and logic and connections that help us negotiate the complex world we live in. The belief that mathematics is no more than long division is fuelled by the way most pupils are taught the subject at school. Imagine a student learning a musical instrument by playing only scales and arpeggios and never even hearing a symphony. No one would judge them for giving up. Yet all too often in pupils' mathematical education, this is all they are exposed to. Pupils I talk to are surprised to learn that there are complex mathematical equations controlling the evolution of their PlayStation games or that the sine waves that they learn about in trigonometry are the building blocks used by their MP3 players to recreate the sound of the Arctic Monkeys in their headphones. Practical applications are a powerful way to awaken people to the importance of the subject. But beauty and elegance can also attract many to the subject. It is the great stories of mathematics, many of them unfinished, that I believe have the potential to capture pupils' imaginations when they doubt the value of mathematics. Therefore, it is the responsibility of those who create these stories, the research mathematicians, to bring the subject alive. There is no escaping the hard graft of doing your arithmetic scales and arpeggios. But if these are set in the context of the big mathematical symphonies they help write, students may feel more inclined to apply themselves. The story of the primes is one of the sagas that I have found can pull young people on to the mathematical bandwagon. They are the building blocks of all numbers. And as you play with them, they very soon draw you into one of our biggest mathematical mystery stories. The great challenge is to understand how nature chose these enigmatic numbers. The search for a pattern behind the primes goes to the heart of what it means for me to be a mathematician. Yet intriguingly, our subject seems to be built out of numbers with no patterns to them at all. The biggest prime we know has more than 9 million digits - a number that would take more than a month and a half to read aloud. But bigger primes will always be discovered - there is a prize of $100,000 (Pounds 57,000) waiting for the first person to break the 10 million digit mark. The records to date are not held by boffins with big computers but amateurs with desktops. Money is a great incentive for getting kids' eyes to light up. And one can use it to introduce the deeper meaning behind the headline. Once they have won $100,000, then they can move on to the million-dollar prize of finding the underlying structure that makes these numbers tick, which involves solving the Riemann hypothesis. The National Centre for Excellence in Teaching Mathematics, to be launched in May, has the potential to communicate some of the big stories of mathematics to teachers who can, in turn, spread the word in our schools and colleges. But it is important for those at universities to play their part in keeping alive the narrative tradition. In our conferences and journals, we are all engaged in telling the tales of our mathematical adventures. If we want more young explorers to join us on the hard treks across the mathematical mountains, then research mathematicians have a part to play in telling those outside the ivory towers our best stories. Scientific research consists of two important components: discovery and communication. Without one, the other will die. Oswald Veblen, in his opening address to the International Congress of Mathematicians in 1952, expressed well this need to perform our theorems: "Mathematics is terribly individual. Any mathematical act, whether of creation or apprehension, takes place in the deepest recesses of the individual mind. Mathematical thoughts must nevertheless be communicated to other individuals and assimilated into the body of general knowledge. Otherwise they can hardly be said to exist." Marcus du Sautoy is professor of mathematics at Oxford University and author of The Music of the Primes, published by Harper Perennial, Pounds 8.99. This article is based on his inaugural Drapers lecture on teaching and learning at Queen Mary, University of London.
The story of the primes is one of the sagas that I have found can pull young people on to the mathematical bandwagon. They are the building blocks of all numbers. And as you play with them, they very soon draw you into one of our biggest mathematical mystery stories. Marcus du Sautoy is professor of mathematics at Oxford University and author of The Music of the Primes
Challenge would appear to be a key ingredient here. To continue the recent thread on bringing categories to the masses, is there a short list of such sagas whose challenges big and small might pull young people on to the category theory bandwagon? Abelian categories? Toposes? Monads? Synthetic differential geometry? n-categories? All would seem to be fairly easily accessed from very accessible parts of respectively topology (coffee cups, Betti numbers), constructive logic (Brouwer vs. Hilbert, proofs as programs), number systems (Galois and unsolvability by radicals), analysis (infinitesimals according to Cauchy, Weierstrass, Robinson, Kock), and cosmology (the organization of strings). What other challenges, big and small, met and unmet, might young people find a compelling lead-in to categorical thinking? In all these areas, bringing the novice to the mathematics is surely a less promising strategy than bringing the mathematics to the novice. If home delivery can radicalize the pizza business, why can't it do the same for category theory? Vaughan Pratt
Challenge would appear to be a key ingredient here. To continue the recent thread on bringing categories to the masses, is there a short list of such sagas whose challenges big and small might pull young people on to the category theory bandwagon? Abelian categories? Toposes? Monads? Synthetic differential geometry? n-categories?
I have given introductory courses in Category Theory (including Monads), Toposes, Locales, Synthetic Differential Geometry in Mexico (UNAM), Spain (University of the Balearic Islands, Spain), and Egypt (Cairo University). The background material can be incorporated intro the lectures as needed. Bdest wishes, Marta
Dear Vaughan, I meant to write a more substantial reply to your question, but I was interrupted by an important telephone call and accidentally I sent a partial reply. I meant to say that there are many attractive results in classical mathematics than can be shown to advantage using category theory, and that I found that emphasizing those in my courses (which of course I have given also repeatedly here at McGill, not just in Spain, Mexico and Egypt) is the key to interest students who do not even intend to work in categories. After all, we want to educate future analysts, topologists, algebraists, computer scientists, logicians to feel that knowing a bit of categories can help in their fields. To me, this is the goal in teaching categories. I only take (or have taken so far) students with a broad mathematical culture and who can get motivated to do categories with a view to better understand and relate different mathematical fields. This is how Gorthendieck pursued mathematics and of course, as it must happen, often going off tangent to develop a theory suggested by obstructions in ordinary work. I feel happier when that happens and do not necessarily think that one ought to aim at forming (often poor) students in category theory. Only the very best, if they can be lured to do so, should work in category theory. Of course, I would myself have been eliminated at the onset had my "rules" been applied in those days. But in the 60's it was different and I now see that catgegory theory must come after the "mathematical experience", not before. I can take the time some time this summer to make a list of such attractive results in the fields I know within category theory. I am too busy now. Best wishes, Marta
Dear Marta, I couldn't agree more. Usually I find myself disagreeing with some picky point or other but somehow your message managed to completely avoid my (too many) hot buttons! Your two points (broad publicity for the general benefits of the subject but only taking the best students to actually work in it) are of course applicable to any subject. Executing well on both brings a new subject up to the stature of the established subjects. CT has done very well on the latter but might be judged as having fallen short on the former so far, though perhaps not for want of trying but rather the manner of presentation. When in Rome speak Italian (and don't mention home delivery pizza). On the concern you raised a while back about perceptions of crankiness, physics runs the gamut from well-publicized spectacular advances to more cranks than just about any other scientific discipline; in that respect it nicely brackets both CT and chemistry on both sides. Whether CT has accumulated more cranks than chemists is an interesting question, which brings to mind the category theory professors from the Mahareshi Yogi's TM university in Fairfield buttonholing Bill Lawvere at an AMAST meeting in Iowa a while back. Wish I could have video'd that. Best, Vaughan Marta Bunge wrote:
Dear Vaughan,
I meant to write a more substantial reply to your question, but I was interrupted by an important telephone call and accidentally I sent a partial reply.
I meant to say that there are many attractive results in classical mathematics than can be shown to advantage using category theory, and that I found that emphasizing those in my courses (which of course I have given also repeatedly here at McGill, not just in Spain, Mexico and Egypt) is the key to interest students whlo do not even intend to work in categories. After all, we want to educate future analysts, topologists, algebraists, computer scientists, logicians to feel that knowing a bit of categories can help in their fields. To me, this is the goal in teaching categories. I only take (or have taken so far) students
with a broad mathematical culture and who can get motivated to do categories with a view to better understand and relate different mathematical fields. This is how Gorthendieck pursued mathematics and of course, as it must happen, often going off tangent to develop a theory suggested by obstructions in ordinary work. I feel happier when that happens and do not necessarily think that one ought to aim at forming (often poor) students in category theory. Only the very best, if they can be lured to do so, should work in category theory. Of course, I would mysef have been eliminated at the onset had my "rules" been applied in those days. But in the 60's it was different and I now see that catgegory theory must come after the "matrhematica;l experience", not before.
I can take the time some time this summer to make a list of such atractive results in the fields I know within category theory. I am too busy now.
Best wishes, Marta
On 18 Apr 2006, at 14:59, Marta Bunge wrote:
... I now see that catgegory theory must come after the "mathematical experience", not before. ...
Dear Marta, I think this is exactly the key to the success of Mac Lane's book. Throughout, he shows how working mathematicians are applying category theory already without realizing it. One of the basic expositional problems for teaching CT in computer science is that our students do not have the body of mathematical experience that Mac Lane presumed. Regards, Steve.
Dear Steve,
I think this is exactly the key to the success of Mac Lane's book. Throughout, he shows how working mathematicians are applying category theory already without realizing it. One of the basic expositional problems for teaching CT in computer science is that our students do not have the body of mathematical experience that Mac Lane presumed.
Of course, by "mathematical experience" one need not assume that it should be the same for everybody. MacLane was thinking of the pure mathematicians only, because that was what motivated him all along. I think that "Conceptual Mathematics" by Lawvere and Schanuel, though seemingly too elementary, is a great introduction to categorical thinking that can be widely appreciated, since the examples chosen therein to illustrate new concepts are simple. I say this in more detail in a review (in Spanish) that can be found in my home page (http://www.math.mcgill.ca/bunge/LS.pdf (.ps)). Even so, you must agree that computer scientists ought to have learnt a certain amount of pure mathematics, or else how are they going to appreciate the more sophisticated developments in their field, or even less contribute to it? I used "Categories and Computer Science" by Bob Walters twice when teaching "Computability and Linguistics" at McGill. Although I have heard some negative comments about it (sorry to mention it, Bob), I liked it a lot. The exercises are often quite demanding, and the exposition clear. I do not know what you think about it. Of course, with a book like that, as with any other, it is up to the instructor to use it to his advantage, and to complement it as needed by the particular audience he has to face. Nice hearing from you, Marta
Dear Vaughan,
On the concern you raised a while back about perceptions of crankiness, physics runs the gamut from well-publicized spectacular advances to more cranks than just about any other scientific discipline; in that respect it nicely brackets both CT and chemistry on both sides. Whether CT has accumulated more cranks than chemists is an interesting question, which brings to mind the category theory professors from the Mahareshi Yogi's TM university in Fairfield buttonholing Bill Lawvere at an AMAST meeting in Iowa a while back. Wish I could have video'd that.
The thread I unintentionally initiated (with mixed results) did not express any concern about cranks, but about crackpots, whom I view as dangerous only if not spotted in time. I think that "cranks" means "eccentric" and, in it itself, it means nothing to me -- crankiness (if that is the correct adjective) can be: (a) the result of genuine absent-mindedness and total commitment to their activities as mathematicians/scientists, or (b) it can also be a pose by an insecure person who may have nothing else but his crankiness to be distinguished from the others. Some fields (like Physics) have both. Chemists are too serious (boring) to tolerate any cranks in their midst. CT? Yes, there are a few, but in my view, that is the least of our worries. Maybe by "crank" you meant something else ("crackpots"?), as the incident you recall (first time I hear about it) seems to indicate. In any case, the last thing anybody wants right now is to go back to discuss this sensitive issue. Best, Marta
ah, linguistic problems not sure about British/Canadian English but in American cranks ae slightly worse than crackpots and not at all the same as being cranky Jim Stasheff jds@math.upenn.edu Home page: www.math.unc.edu/Faculty/jds On Wed, 19 Apr 2006, Marta Bunge wrote:
Dear Vaughan,
On the concern you raised a while back about perceptions of crankiness, physics runs the gamut from well-publicized spectacular advances to more cranks than just about any other scientific discipline; in that respect it nicely brackets both CT and chemistry on both sides. Whether CT has accumulated more cranks than chemists is an interesting question, which brings to mind the category theory professors from the Mahareshi Yogi's TM university in Fairfield buttonholing Bill Lawvere at an AMAST meeting in Iowa a while back. Wish I could have video'd that.
The thread I unintentionally initiated (with mixed results) did not express any concern about cranks, but about crackpots, whom I view as dangerous only if not spotted in time.
I think that "cranks" means "eccentric" and, in it itself, it means nothing to me -- crankiness (if that is the correct adjective) can be: (a) the result of genuine absent-mindedness and total commitment to their activities as mathematicians/scientists, or (b) it can also be a pose by an insecure person who may have nothing else but his crankiness to be distinguished from the others. Some fields (like Physics) have both. Chemists are too serious (boring) to tolerate any cranks in their midst. CT? Yes, there are a few, but in my view, that is the least of our worries. Maybe by "crank" you meant something else ("crackpots"?), as the incident you recall (first time I hear about it) seems to indicate. In any case, the last thing anybody wants right now is to go back to discuss this sensitive issue.
Best, Marta
Dear Steve,
One of the basic expositional problems for teaching CT in computer science is that our students do not have the body of mathematical experience that Mac Lane presumed.
I don't think that this is the problem. There are quite a few areas in CS (mainly semantics) where it is even impossible to formulate the problem when not having the language of CT available. Paradigmatic example being solution of recursive domain equations. In my regular course on semantics I introduce category theory by need and some of those people then attend my course on category theory and categorical logic (all available on my home page if you want to look). One certainly need not know a lot about algebra of geometry for these purposes. The problem rather is that most students of CS are not open to any theory whatsoever be it categorical or not. BTW another example are socalled "effects" (i.e. something fairly applied and "impure" if you want). For modelling them appropriately one needs either monads or cpo-enriched Lawvere theories. Maybe what you deplore is the absence of SIMPLE examples from CS. Well, I think one can use posets, graphs, monoids, abelian groups, fields etc. What's more problematic is the usual ignorance of simple topological examples. Maybe a bit of analysis (done properly) would do them good? Thomas
participants (6)
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James Stasheff -
jim stasheff
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Marta Bunge -
Steve Vickers -
Thomas Streicher -
Vaughan Pratt