Re: WHY ARE WE CONCERNED? I

Re: WHY ARE WE CONCERNED? I

[Peter Selinger]

> F W Lawvere wrote:
> > WHY ARE WE CONCERNED? I
> >
> > 	"Dumbing down" is an attack not only on school children and on
> > undergraduates, but also one taking measured aim at colleagues in
> > adjacent fields and at the general public. The general public is
> > thirsty for genuinely informational articles to replace the
> > science fiction gruel served constantly by journals like the
> > Scientific American and the New York Times "Science" section.
> > Those journals have never published anything resembling a
> > mathematical proof and hence have rarely actually explained any
> > scientific subject in a usable way.

jim stasheff wrote:
>
> a math proof is hardly necessary to explain a scientific subject in a
> usable way.
>
> now for a mathematical subject a math proof is sometimes but not always
> necessary

I agree with Bill that the prevailing style of expository writing,
especially in newspapers, is often of poor quality. It would be nice
if such articles more often gave a glimpse into the nature of
research, rather than serving, as Bill puts it, entertainment.

However, I disagree on the role of proofs in expository writing.
Clearly, proofs are central in mathematics. But to say that
mathematics is only about proofs is a bit like saying that dentistry
is only about clinical research. Of course, the research is important,
and most of us who have root canals are very glad that it is being
done. However, I would like to believe that mathematics is ultimately
about solving problems that *matter*, and the reason they matter often
has nothing to do with their proofs.

I am of course not advocating replacing proofs by conjecture. I am
only speaking of expository writing, where I believe it is often more
important to explain the results than their proofs. And sometimes, it
can even be justified to give an "approximate" proof, i.e., a proof
idea, or even an "approximate" definition, if it is stated clearly
that there has been some simplification.

The poor state of mathematical exposition is not confined to articles
about mathematics. The following quote, from an ordinary new article
in yesterday's Times, send my logic-circuits spinning:

 French lawmakers, for example, gave preliminary support this month to
 a measure that would require the company to open the iPod to play
 music purchased from any online music service; currently, songs
 purchased from iTunes can be played only on iPods.

 New York Times, 2006/03/29, "Apple vs. Apple in Dispute Over Trademark"

This is of course not a logical contradiction; but I would be very
surprised if it is what the writer really meant to say. Sadly, most
readers probably won't know the difference one way or the other.

-- Peter

Re: WHY ARE WE CONCERNED? I

[jim stasheff]

Proofs may be of ultimate importance
but a lot can be accomplished at the penulitmate level
or even sooner

jim


Colin McLarty wrote:
>> There's a saying about Lefschetz that he "never wrote a valid
>> proof, and never made a false conjecture". Now it's not an attitude
>> that want to encourage, but if you have great mathematicians who
>> are like that (and Lefschetz was not just a good mathematician, but
>> a great mathematician, without whom a good deal of modern algebraic
>> geometry would be unimaginable), then this ought to tell us something.
>
>
> This, and much else about Lefschetz has to tell us a lot.  As to proof,
> Lefschetz also never published a theorem without a purported proof, and
> he often came to feel very strongly that his proofs were not good
> enough.  He wrote two long books on topology in the attempt to repair
> the bad proofs in his influential booklet on cohomology in algebraic
> topology, L'Analysis situs et la Topologie Algebrique.  It was so
> important to him that he enlisted many others.  Notably for us, he
> asked Eilenberg and Mac Lane to contribute an appendix to his 1942
> TOPOLOGY.  This was their first published collaboration "On homology
> groups of infinite complexes and compacta" and pursued the questions
> that quickly led to category theory.
>
> Lefschetz had encouraged work on solving specific problems just over
> the edge of what well-understood foundations for homology could
> handle.  Apparently he believed such solutions would lead to
> significantly deeper understanding.  He had encouraged Steenrod to work
> on p-adic solenoids because existing methods did not seem adequate to
> it.  But whatever his motive, he was determined to see rigorous
> solutions to quite specific problems.
>
> Colin
>
>

Re: WHY ARE WE CONCERNED? I

[Colin McLarty]

> There's a saying about Lefschetz that he "never wrote a valid
> proof, and never made a false conjecture". Now it's not an attitude
> that want to encourage, but if you have great mathematicians who
> are like that (and Lefschetz was not just a good mathematician, but
> a great mathematician, without whom a good deal of modern algebraic
> geometry would be unimaginable), then this ought to tell us something.


This, and much else about Lefschetz has to tell us a lot.  As to proof,
Lefschetz also never published a theorem without a purported proof, and
he often came to feel very strongly that his proofs were not good
enough.  He wrote two long books on topology in the attempt to repair
the bad proofs in his influential booklet on cohomology in algebraic
topology, L'Analysis situs et la Topologie Algebrique.  It was so
important to him that he enlisted many others.  Notably for us, he
asked Eilenberg and Mac Lane to contribute an appendix to his 1942
TOPOLOGY.  This was their first published collaboration "On homology
groups of infinite complexes and compacta" and pursued the questions
that quickly led to category theory.

Lefschetz had encouraged work on solving specific problems just over
the edge of what well-understood foundations for homology could
handle.  Apparently he believed such solutions would lead to
significantly deeper understanding.  He had encouraged Steenrod to work
on p-adic solenoids because existing methods did not seem adequate to
it.  But whatever his motive, he was determined to see rigorous
solutions to quite specific problems.

Colin

RE: WHY ARE WE CONCERNED? I

[Marta Bunge]

Dear Bill,

Congratulations on your posting, particularly in what refers to Mac Lane,
which is very revealing.

>	When Saunders Mac Lane penned his hard-hitting 1997 Synthese
>article, he was defending mathematics from an attack many of us hoped
>would just go away. But Saunders was aware of the seriousness of the
>threat, which indeed is still here with greater determination.
>Although the title of that article was "Despite physicists, proof is
>essential in mathematics", he was not opposing physics, nor even that
>immediate handful who, assuming the mantle of "mathematical physicists",
>gave themselves license to insult generations of scrupulously serious
>physicists and to demand that mathematics adopt a culture that considers
>conjecture as nearly-established truth. In essence it was an attack on
>science itself, as the highest form of knowing, that Saunders was
>opposing.


In case there may be somebody not acquainted with MacLane's excellent
article, here is a link to it:


http://www.math.nsc.ru/LBRT/g2/english/ssk/proof_is_necessary.pdf


>	The contempt for Mac Lane's fight, recently expressed in articles
>supposedly memorializing him, takes the form of the claim that category
>theory itself is a "cool" instrument for deepening obscurantism. Not only
>Harvard's "When is one thing equal to another thing?" and the Cambridge
>"morality" muddle, but also a 2003 article aimed at teachers of
>undergraduates, quite explicitly support that claim.

I suppose that you cannot (or do not want to) be more explicit.  I do not
know (for the most part) which articles you are referring to.

Best wishes,
Marta

************************************************
Marta Bunge
Professor Emerita
Dept of Mathematics and Statistics
McGill University
805 Sherbrooke St. West
Montreal, QC, Canada H3A 2K6
Office: (514) 398-3810
Home: (514) 935-3618
marta.bunge@mcgill.ca
http://www.math.mcgill.ca/bunge/
************************************************

Re: WHY ARE WE CONCERNED? I

[Vaughan Pratt]

In response to Peter Johnstone (and those who responded privately), my
point about the Fundamental Theorem of Algebra was not that this
particular proof (based on the limiting behaviors of small and large
circles) was not known to anyone, but that it had not emerged, instead
being effectively sat on by those in the know, even if not
intentionally.  At this risk of sounding like an Abu Ghraib
interrogator, "who knew?"

My claim is that no extant proof at all, that or any other, was
considered fit for an elementary exposition more than a couple of
decades ago.  If that estimate is right, the 1982 Pontrjagin article
cited by Nikita Danilov would be one of the earliest popular expositions
based on the circles argument, assuming the section containing Fig. 6 is
the relevant one (my Russian is even rustier than my algebra).  I'd be
very interested in seeing an earlier popular account that didn't claim
that every proof necessarily either was long or depended on out-of-scope
material.

As a case in point, just now I checked a relatively recent Brittanica
article on algebra (1987 ed.), which states flatly (p.260a) that "No
elementary algebraic proof of [the FTAlg] exists, and the result is not
proved here."  (Not even "is known" but "exists"; an expository article
should not assume that the reader knows the jargon meaning of this term
as "exists in the literature".)  The authors taking responsibility for
this claim were Garrett Birkhoff, Marshall Hall, Pierre Samuel, Peter
Hilton, and Paul Cohn.  They go into detail to show that z^n = a has n
roots, starting with the geometry of addition and multiplication in the
Argand diagram, so it's not as if their exposition was at too elementary
a level to talk in terms of mapping circles, or that "algebraic" ruled
out simple geometric arguments.

I submit their nonexistence claim as prima facie evidence for my claim
that the very few who knew this argument weren't even letting the likes
of Birkhoff, Hall, etc. in on it, let alone "the rest of us."

The general message in the literature prior to the 1980's seemed to be,
if Gauss couldn't find a simple proof in half a dozen tries, there isn't
one.  If you don't possess the necessary higher maths or the stamina for
an intricate argument, we can't help you with that result, ask us about
solvability of z^n = a.

Good for Pontrjagin for promoting FTAlg to school children!

Vaughan Pratt

Re: WHY ARE WE CONCERNED? I

[Nikita Danilov]

Vaughan Pratt writes:

[...]

 >
 > This slick proof seems only to have emerged in the past couple of
 > decades.  It is an interesting commentary on mathematics that it took
 > this long for people to come up with an argument "for the rest of us."
 > Maybe some people "knew" it all along, but in that case they were
 > keeping pretty quiet about it.

This proof was used by Pontryagin in the April 1982 issue of "Kvant"
magazine (targeting school-children!):
http://kvant.mccme.ru/1982/04/osnovnaya_teorema_algebry.htm (in
Russian, but pictures should be enough).

 >
 > Vaughan Pratt

Nikita.

Re: WHY ARE WE CONCERNED? I

[Prof. Peter Johnstone]

On Wed, 29 Mar 2006, Vaughan Pratt wrote:

> a few decades ago an elementary exposition of the Fundamental
> Theorem of Algebra would not be expected to include an elementary proof
> since the extant proofs were either lengthy arguments or nonelementary
> appeals to the minimum modulus principle, properties of holomorphic
> functions such as Liouville's theorem, or other results the reader would
> be unlikely to be on top of.  The dominant belief was that the only
> short proofs were nonelementary ones.
>
> But for an audience aware only that z^i for any nonnegative integer i
> maps circles at the origin to i-fold circles of radius r^i at the
> origin, an entirely elementary notion, an expositor today would be
> morally obligated to include a full proof since there is hardly anything
> left to explain.  The polynomial a_d z^d + ... + a_0 maps little circles
> to the neighborhood of a_0 and big circles to a loop tending to a very
> big d-fold circle of radius a_d r^d, whence the smoothly growing image,
> under the polynomial, of a smoothly growing circle is obliged to cross
> the origin at some stage.  Still a topological argument, but now an
> entirely elementary one.
>
> Except, that is, for the theorem that a loop wound d times around the
> hole in the punctured plane cannot be continuously retracted to a point,
> which was tacitly smuggled in there.   But that statement is less
> intimidating than anything based on holomorphic functions.
>
> This slick proof seems only to have emerged in the past couple of
> decades.

Has it? It seems to me no more than (an explicity homotopy-theoretic
formulation of) the (implicitly homotopy-theoretic) proof via
Rouch\'e's Theorem, which I was taught as an undergraduate (and which
I've taught to undergraduates on many occasions since then).

Peter Johnstone

Re: WHY ARE WE CONCERNED? I

[Graham White]

> jim stasheff wrote:
>
> > now for a mathematical subject a math proof is sometimes but not always
> > necessary
>

There's a saying about Lefschetz that he "never wrote a valid
proof, and never made a false conjecture". Now it's not an attitude
that want to encourage, but if you have great mathematicians who
are like that (and Lefschetz was not just a good mathematician, but
a great mathematician, without whom a good deal of modern algebraic
geometry would be unimaginable), then this ought to tell us something.

What it tells us is, of course, not easy to formulate: it's an example
that causes severe problems for almost every philosophy of mathematics
that I know. But it ought to stop us saying things of the form
"if we don't do category theory in such and such a way, then it
won't be mathematics at all".

(Of course we'll all keep saying this, because we all have a secret
fear that, if we aren't really careful about what we do, the grown up
mathematicians will kick sand in our face, but that's a
psychological problem and not a mathematical problem.)


-- 
Dr. Graham White
Lecturer
Department of Computer Science
Queen Mary, University of London
Mile End Road
London E1 4NS
http://www.dcs.qmul.ac.uk/~graham
(+44)(020)7882 5242

Re: WHY ARE WE CONCERNED? I

[Vaughan Pratt]

jim stasheff wrote:

> now for a mathematical subject a math proof is sometimes but not always
> necessary

Absolutely.  I would add publication date as a factor here.  As an
example, a few decades ago an elementary exposition of the Fundamental
Theorem of Algebra would not be expected to include an elementary proof
since the extant proofs were either lengthy arguments or nonelementary
appeals to the minimum modulus principle, properties of holomorphic
functions such as Liouville's theorem, or other results the reader would
be unlikely to be on top of.  The dominant belief was that the only
short proofs were nonelementary ones.

But for an audience aware only that z^i for any nonnegative integer i
maps circles at the origin to i-fold circles of radius r^i at the
origin, an entirely elementary notion, an expositor today would be
morally obligated to include a full proof since there is hardly anything
left to explain.  The polynomial a_d z^d + ... + a_0 maps little circles
to the neighborhood of a_0 and big circles to a loop tending to a very
big d-fold circle of radius a_d r^d, whence the smoothly growing image,
under the polynomial, of a smoothly growing circle is obliged to cross
the origin at some stage.  Still a topological argument, but now an
entirely elementary one.

Except, that is, for the theorem that a loop wound d times around the
hole in the punctured plane cannot be continuously retracted to a point,
which was tacitly smuggled in there.   But that statement is less
intimidating than anything based on holomorphic functions.

This slick proof seems only to have emerged in the past couple of
decades.  It is an interesting commentary on mathematics that it took
this long for people to come up with an argument "for the rest of us."
Maybe some people "knew" it all along, but in that case they were
keeping pretty quiet about it.

Vaughan Pratt

Re: WHY ARE WE CONCERNED? I

[James Stasheff]

I deliberately overstated the case
presenting accessible proofs should of course be done
and certainly equations should not be eschewed
pace Penrose

but there's more to math than proofs
cf. Reinhard's own reference to *thinking*
i.e. *before* a formal proof is worked out
if we could convey even that


	Jim Stasheff		jds@math.upenn.edu


On Wed, 29 Mar 2006, Reinhard Boerger wrote:

> Hello,
>
> just a few remarks.
> Jim Stasheff wrote:
>
> > F W Lawvere wrote:
> > > WHY ARE WE CONCERNED? I
> > >
> > > 	"Dumbing down" is an attack not only on school children and on
> > > undergraduates, but also one taking measured aim at colleagues in
> > > adjacent fields and at the general public. The general public is
> > > thirsty for genuinely informational articles to replace the science
> > > fiction gruel served constantly by journals like the Scientific
> > > American and the New York Times "Science" section.
> >
> > so far so good
> >
> >   Those journals have never published anything
> > > resembling a mathematical proof
> >
> > why should they?
>
> Because otherwise the readers do not learn what mathematics is about.
>
> > a math proof is hardly necessary to explain a scientific subject in a
> > usable way.
> >
> > now for a mathematical subject a math proof is sometimes but not
> > always necessary
>
> That depends on what you mean by explaining a subject. Of course, many people
> know what a prime is, and if a journal reports that some larger (Mersenne) prime has
> been found, or if the journal contains some nice pictures of fractals, they may either
> admire this or ask "so what?" In any case they do not see what a mathematical result
> is. I met several people with an academic education in another field. When I told
> them that I am a mathematician, some of them replied: "I always liked maths - except
> proofs." If this misconception is so wide-spread among educated people - at least in
> Germany, Canada and the United States - I think it is more important that these
> people see easy proofs of mathematical results (e.g. Euclid's proof for the existence
> of infinitely many primes) than that they see mysterious mathematical statements,
> which they don't understand. Mathematics is thinking rather than computation, and if
> one does not know what a proof is, one does not know what mathematics is. So for
> which subject do you think that a proof is not necessary?
>
>
>                                                                         Greetings
>                                                                          Reinhard
>

Re: WHY ARE WE CONCERNED? I

[Reinhard Boerger]

Hello,

just a few remarks.
Jim Stasheff wrote:

> F W Lawvere wrote:
> > WHY ARE WE CONCERNED? I
> >
> > 	"Dumbing down" is an attack not only on school children and on
> > undergraduates, but also one taking measured aim at colleagues in
> > adjacent fields and at the general public. The general public is
> > thirsty for genuinely informational articles to replace the science
> > fiction gruel served constantly by journals like the Scientific
> > American and the New York Times "Science" section.
>
> so far so good
>
>   Those journals have never published anything
> > resembling a mathematical proof
>
> why should they?

Because otherwise the readers do not learn what mathematics is about.

> a math proof is hardly necessary to explain a scientific subject in a
> usable way.
>
> now for a mathematical subject a math proof is sometimes but not
> always necessary

That depends on what you mean by explaining a subject. Of course, many
people know what a prime is, and if a journal reports that some larger
(Mersenne) prime has been found, or if the journal contains some nice
pictures of fractals, they may either admire this or ask "so what?" In any
case they do not see what a mathematical result is. I met several people
with an academic education in another field. When I told them that I am a
mathematician, some of them replied: "I always liked maths - except
proofs." If this misconception is so wide-spread among educated people -
at least in Germany, Canada and the United States - I think it is more
important that these people see easy proofs of mathematical results (e.g.
Euclid's proof for the existence of infinitely many primes) than that they
see mysterious mathematical statements, which they don't understand.
Mathematics is thinking rather than computation, and if one does not know
what a proof is, one does not know what mathematics is. So for which
subject do you think that a proof is not necessary?


Greetings
Reinhard

Re: WHY ARE WE CONCERNED? I

[jim stasheff]

I beg to differ - a little



F W Lawvere wrote:
> WHY ARE WE CONCERNED? I
>
> 	"Dumbing down" is an attack not only on school children and on
> undergraduates, but also one taking measured aim at colleagues in adjacent
> fields and at the general public. The general public is thirsty for
> genuinely informational articles to replace the science fiction gruel
> served constantly by journals like the Scientific American and the New
> York Times "Science" section.

so far so good

  Those journals have never published anything
> resembling a mathematical proof

why should they?

and hence have rarely actually explained
> any scientific subject in a usable way.

a math proof is hardly necessary to explain a scientific subject in a
usable way.

now for a mathematical subject a math proof is sometimes but not always
necessary


> 	In January of 2005 the Notices of the AMS announced that they had
> for a full ten years been strictly following a certain editorial policy.
> There had been a widespread demand for expository articles. To that
> demand, the response was a new definition of "expository": all precise
> definitions of mathematical concepts must be eliminated. Authors of
> expository articles were forced to compromise their presentation, or to
> withdraw their paper.

Not all of us

and notice you are talking about the NOTICES
not the Bulletin
Mathematicians, who were for several years
> becoming aware that these new expository articles are absolutely useless
> for developing a mathematical thought,

developing a mathematical thought,

depends what you mean by that
developing in the sense of enough to be active in the field - of course not

developing a sense of what the thought of the experts are so that one
might want to learn more or NOT
or
might see relevance to ones own disparate research - they work fine
  were shocked to learn that a
> conscious policy had forced that situation.
> 	A peculiar sort of anti-authoritarianism seems to be the only
> justification offered for degrading the role of definition, theorem, and
> proof; certainly, serious expositors have never considered that the use of
> those three pillars of geometrical enlightenment excludes explanations and
> examples. Others have urged, however, that those instruments be
> eliminated even from lectures at meetings and from professional papers.

Examples ? I certinaly have not seen such
In fact as an editor and referee and all the referees I've used
have never tolerated such elimination.  in fact, due to cross
fertilization, even some physics papers now have defintions

> 	That threat is part of the background for the concern expressed in
> the many messages to the categories list over the past weeks. Deeply
> concerned mathematicians ask me "How can we know?". Indeed, how can we
> know whether it is worthwhile to attend a certain meeting or a certain
> talk, and how can a scientific committee know whether a proposed talk is
> scientifically viable? If the "you don't want to know" culture of no
> proofs, no definitions, is accepted, we will truly have no way of knowing,
> and will be pressured to fall back on unsupported faith.
>
Me thinks thou doth protest too much

or you've run into some alternate universe I'm unfamiliar with

;-D  jim

WHY ARE WE CONCERNED? I

[F W Lawvere]

WHY ARE WE CONCERNED? I
	When Saunders Mac Lane penned his hard-hitting 1997 Synthese
article, he was defending mathematics from an attack many of us hoped
would just go away. But Saunders was aware of the seriousness of the
threat, which indeed is still here with greater determination.
Although the title of that article was "Despite physicists, proof is
essential in mathematics", he was not opposing physics, nor even that
immediate handful who, assuming the mantle of "mathematical physicists",
gave themselves license to insult generations of scrupulously serious
physicists and to demand that mathematics adopt a culture that considers
conjecture as nearly-established truth. In essence it was an attack on
science itself, as the highest form of knowing, that Saunders was
opposing.
	The increased determination of that attack is expressed in two
ways. To equip and organize the attack, finance capital has set up several
institutions, some of which rather openly proclaim their goal of
submitting science to the service of medieval obscurantism. Others say
that they support mathematical research, but encourage a barrage of
"popular" writings to shock and awe the public into continuing in the
belief that they will never understand mathematics and hence never be able
to actively participate in science.
	The contempt for Mac Lane's fight, recently expressed in articles
supposedly memorializing him, takes the form of the claim that category
theory itself is a "cool" instrument for deepening obscurantism. Not only
Harvard's "When is one thing equal to another thing?" and the Cambridge
"morality" muddle, but also a 2003 article aimed at teachers of
undergraduates, quite explicitly support that claim. In the MAA Monthly, a
Clay Fellow states as fact that category theory "is mathematics with the
substance removed". Mastering the technique of disinformation whereby the
readers are first told that now finally they will be informed, the article
suggests that some raising of the level of understanding of the
relationship between space and intensively variable quantity is going to
be achieved. Then the author short-circuits any such understanding via the
simplifying assumption that omits the distinction between covariant and
contravariant functors as "unwieldy". As final display of the mastery of
expositional technique, the categorical object which has, for nearly
twenty pages, been heralded as simple, is revealed in the final pages in
the most complicated and unexplained form possible. (Totally passed over
is the issue that had led Grothendieck to the considerations allegedly
being treated: not only the category of affine schemes, but also the
category of all its presheaves, where the author implicitly wants us to
work, fails to have the geometrically correct colimits needed to define
projective space.)
	Another level of attack was launched when Cornell University was
given very large sums of money to develop methods of teaching geometry
without mentioning any geometrical concepts. No proof of the desirability
of such a draconian excising of content needed to be given, beyond some
phrases from the Dalai Lama.
	"Dumbing down" is an attack not only on school children and on
undergraduates, but also one taking measured aim at colleagues in adjacent
fields and at the general public. The general public is thirsty for
genuinely informational articles to replace the science fiction gruel
served constantly by journals like the Scientific American and the New
York Times "Science" section. Those journals have never published anything
resembling a mathematical proof and hence have rarely actually explained
any scientific subject in a usable way. Nor have they even undertaken any
program to raise the level of knowledge of calculus or linear algebra
among their readers in a way which would make such explanations feasible.
Instead, they provide games and amusements to divert the
mathematically-interested public.
	In January of 2005 the Notices of the AMS announced that they had
for a full ten years been strictly following a certain editorial policy.
There had been a widespread demand for expository articles. To that
demand, the response was a new definition of "expository": all precise
definitions of mathematical concepts must be eliminated. Authors of
expository articles were forced to compromise their presentation, or to
withdraw their paper. Mathematicians, who were for several years
becoming aware that these new expository articles are absolutely useless
for developing a mathematical thought, were shocked to learn that a
conscious policy had forced that situation.
	A peculiar sort of anti-authoritarianism seems to be the only
justification offered for degrading the role of definition, theorem, and
proof; certainly, serious expositors have never considered that the use of
those three pillars of geometrical enlightenment excludes explanations and
examples. Others have urged, however, that those instruments be
eliminated even from lectures at meetings and from professional papers.
	That threat is part of the background for the concern expressed in
the many messages to the categories list over the past weeks. Deeply
concerned mathematicians ask me "How can we know?". Indeed, how can we
know whether it is worthwhile to attend a certain meeting or a certain
talk, and how can a scientific committee know whether a proposed talk is
scientifically viable? If the "you don't want to know" culture of no
proofs, no definitions, is accepted, we will truly have no way of knowing,
and will be pressured to fall back on unsupported faith.


************************************************************
F. William Lawvere
Mathematics Department, State University of New York
244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA
Tel. 716-645-6284
HOMEPAGE:  http://www.acsu.buffalo.edu/~wlawvere
************************************************************