Re: [Fwd: du Sautoy]

Re: [Fwd: du Sautoy]

[Thomas Streicher]

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

Re: [Fwd: du Sautoy]

[James Stasheff]

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
>
>

Re: [Fwd: du Sautoy]

[Marta Bunge]

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

Re: [Fwd: du Sautoy]

[Marta Bunge]

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

Re: [Fwd: du Sautoy]

[Steve Vickers]

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.

Re: [Fwd: du Sautoy]

[Vaughan Pratt]

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
>
>

Re: [Fwd: du Sautoy]

[Marta Bunge]

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

Re: [Fwd: du Sautoy]

[Marta Bunge]

>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

Re: [Fwd: du Sautoy]

[Vaughan Pratt]

> 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

[Fwd: du Sautoy]

[jim stasheff]

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.