Subject: categories: cracks and pots 93

From: Eduardo Dubuc <[email-masked]>
Date: Thu, 30 Mar 2006 15:31:17 -0300 (ART)

Hi,

The 93 is because I have by now 92 msages in my cracks and pots file.

I apologize for the length of this posting. It is intended to be a (may be
biased) partial account of the debate, and some comments.

Well, by now the "cracks and pots" debate is establishing itself as, in my
opinion, an interesting and worth-wile event. Congratulations   Marta !!

We are learning about:

a) Understand  (for many of us) better what is mathematics, and what is
physics, what is rigor and what is buccaneering, and also what is
bullshit.  

b) "Something is rotten in the state of category theory community" 

Pay attention that The Bard does not say "category theory", but he says
"category theory community"

I start from who has made the more refreshing, humorous, down to
earth, honest and intelligent contributions to this debate:

**Vicent Schmitt: that theoretical physics, computer science, phylo., a
mix of those, or whatever? , is used to justify poor "categorical" work
is, in my view, an existing problem. More or less everyone is conscious of
it (come on!...) but so far that has not been publicly debated.**

Yes Vincent!!, you point right to what it is at the center (or very near
it) the problem raised in MartaÕs original "cracks and pots" posting!. And
the "(come on!...)", beautiful !.

Now, talking about rigor, conjectures and proofs:

**Maclane : If a result has not yet been given valid proof, it isn't yet
mathematics. This however does not deny the many preliminary stages of
insight, experiment, speculation or conjecture, which can lead to
mathematics. It states simply that a conjectured result is not yet a
theorem **

It is relevant to compare this with Motl's distinction between physics and
mathematics:

**Motl: In physics, we propose different conjectures about the real world,
and it is important that we're not guaranteed that these conjectures will
be true.

String theory itself is not just a conjecture, but rather a
seemingly consistent mathematical framework. Once we accept string theory
as an objectively existing mathematical structure, a structure that we
treat as a part of "generalized physics" - which is of course what all
string theorists are doing every day - we can ask a lot of questions about
its properties.**

He does distinguish between  "physics as conjecture" and mathematics with
applications to physics. He call this mathematics  "generalized physics"

But "conjecture" to be acceptable is not unrigourous neither buccaneering.
he says:

**Motl: the statements about string theory are just conjectures, and they
need to be proved or supported by evidence, otherwise they're irrelevant
and "wrong", in the physical sense.**

He also says:

** Motl: I always feel very uneasy if the mathematically oriented
people present their conjectures about physics, quantum gravity, or string
theory as some sort of "obvious facts".

He is clearly saying that those "mathematically oriented people" are
lacking  rigor. 

Many postings in this debate confound mathematical rigor with formalism,
and push forward the idea that a formal and logically correct statement
has automatically rigor. Even if it is foolish:

**V. Pratt: In axiomatic mathematics, everything that is not forbidden is
permitted. **

**R. Dawson: If the math itself meets mathematical standards of rigor, its
application to physics need surely only meet the standards appropriate to
that subject.**

It seems to me that he is equating here "mathematical standards of rigor"
with "logically correct", and  "the standards appropriate to that subject"
(in this case, physics) with " buccaneering "

Nothing more wrong!! . In both cases, failing to convey what it should be
considered  "rigor in mathematics"  and  "rigor in physics"

But again Saunders and Lubos:

**MacLane: real proof is not simply a formalized document, but a sequence
of ideas and insights**

** Motl: the primary physical motivation is to locate the right ideas and
equations that describe the real world. Category theory has been used by
many to achieve completely wrong physical conclusions - for example, by
considering the "pompously foolish" quantization functor.**

He however seems to be pushing forward the same misconception of "rigor":

**Motl: It may be nice to be rigorous, but it's always more important to
be correct: if the specific kind of rigor leads us to stupid conclusions
in physics, we should avoid it.**

From the original Marta's "cracks and pots"

**M.Bunge:  Are we category theorists as a whole going to quietly accept
getting discredited by a minority of us presumably applying category
theory to string theory?**

**J. Baez: I had never heard anyone before suggest that category theory
could be discredited by applications to string theory. It completely
surprised me.  I'm used to the opposite complaint: that category theory is
discredited by its *lack* of applications.**

Here it is a clear and rigorous answer:

(1) **W. Lawvere: The question is not whether mathematics should be
applied. Most of us agree that it should. The concern is rather that our
subject is sometimes being used as a mystifying smoke screen to protect
pseudo-applications against the scrutiny of the general public and of the
scientific colleagues in adjacent disciplines. We need to ensure that
applications themselves be maximally effective, not clouded by
misunderstanding.**

Now, an example of superficial conclusions:

** J. Baez: Indeed, the funny thing about string theory is that while
leading to an abundant harvest of rigorous mathematical results, it has
not yet correctly predicted a single result from a single experiment,
even after more than 20 years of work on the part of many smart people.**

There is nothing funny about this. Lubos say:

** Motl: One of the fascinating features of string theory is that its
objects and investigations, even though they've been partially
disconnected from the daily exchanges with the experimentalists, remained
extremely physical in character. All of the objects that we deal with are
analogous to some objects in well-known working physical theories, to say
the least.**

Bill has made a serious, well fundamented and non-bullshit contribution to
"crack and pots" (he utilizes a different heading:" WHY ARE WE CONCERNED?")

In contrast to many passages of some contributors that it will be tiresome
to reproduce here, and where one founds an overwhelming proliferation of  

highly technical, sophisticated, difficult and impressively sounding words 

such that it becomes impossible to see what they are saying, unless you
are an expert, in which case you may find out that it is only superficial
thinking  (I am thinking specially in certain parts of  Davis Yetter's
postings).

** W. Lawvere: Professors may not consider the possibility of learning
from undergraduate text books, and some may feel bored that I have once
again repeated the above basic definitions and observations.**

If you have some real thoughts, you do not need impressive jargon.

See what an original and deep insight:

** W. Lawvere: As quantity includes zero, so structure includes the case
of no structure, which Cantor considered one of his most profound and
exciting discoveries**

Superficial thinking  (which could be malicious, but very often is simply
stupid) has manifested itself in these postings  by pushing forward the
idea that there are two different  kinds of category theory:  

"Categories as Foundation" and  "Categories as Algebra", the first
implicitly (but not explicitly said) the "bad one", and the second the
"good" one. 

** D. Yetter: All of these are part and parcel of a different face of
category theory than one saw in the old days: category theory as algebra,
rather than category theory as foundations.**

We have an excellent analysis of this fallacy in Bill's postings, which
should be read carefully and slowly.  

I imagine now to add something that Lawvere himself pointed out a long
time ago: The laws of logic are a particular instance of the categorical
concept of adjoint functors, a concept that grew out of mathematical
experience.  

There is any way some explanation to Yetter's prejudice against 
"categories as foundation".  Often very poor category theory has been
justified by people writing on foundations. Bill's quote (1) above also
applies to this and related use of category theory in theoretical computer
science.   

Somebody else that does not need either noisy language sees better:

** Dusko: I am of course saying things very clear and familiar to many
people on this list, but maybe they are worth saying nevertheless.**

** Dusko: but at the end of the day, I think, we'll all agree that the
source of the unreasonable effectiveness of categorical algebra is its
foundational content ** 

Then, he passes to consider Grothendiek's  ("the greatest of the
category theorists") work  on Topos theory as work on foundations, which
agrees with the analysis of foundations made by Lawvere.

I can not restrain myself to quote the following magnificent piece of
meaningless  hallucinogenic  discourse:

**V. Pratt: In the millions of years of evolution of primate thinking, no
productive mathematical mechanism has a higher probability of being
stumbled on than mathematics founded on the Yoneda axiom. I know of no
better explanation of how human thought could have evolved to its present
form than evolution finding and exploiting the Yoneda principle**

Now, some serious business: 

In recent years J.Baez and his followers have been occupying more and more
space in the categorical community (this fact is at the starting point of
the present debate). 

I think this is so because they have some interesting category theory to
show, but they are occupying more space than their mathematics deserves 
because they bring a refreshing air to a community until now dominated by
an old guard that has not shown signs of necessary evolution, and that has
not being able to attract very good and talented young mathematicians to
the community. There is now not other exiting body of developments within
the community. The old guard is being pushed out (prone or supine ?), but,
alas, not by better mathematicians.  

Category theory is in good shape (in particular pushed forward by the
Russian school), and it is now passing over the category community. I
have lost the information now, but recently it was in Europe an important
congress that it had two subjects: one was a prestigious subject (that I
do not remember now), the other was category theory. Not a single name
(including Baez group) that we see in the category theory community
meetings was there.

Best wishes to all   e.d.