- Contemporary messages sorted: [ by date ] [ by thread ] [ by subject ] [ by author ] [ by messages with attachments ]

From: Tim May <tcmay.domain.name.hidden>

Date: Tue, 3 Dec 2002 11:30:59 -0800

On Sunday, December 1, 2002, at 11:21 AM, Ben Goertzel wrote:

*> FWIW, I studied category theory carefully years ago, and studied topos
*

*> theory a little... and my view is that they are both very unlikely to
*

*> do
*

*> more than serve as a general conceptual guide for any useful
*

*> undertaking.
*

*> (Where by "useful undertaking" I include any practical software
*

*> project, or
*

*> any physics theory hoping to make empirical predictions).
*

You should study what interests you. If this type of math does not

interest you, you should of course study something else.

As for measures of usefulness, it's certainly the case that most

engineers, programmers, and even working physicists need little more

than some linear algebra and bit of differential equations to handle

99% of the situations they encounter. From what I see of the current AI

literature, maybe a bit of logic, a smattering of graph theory, and

whatever specialty math in the chosen subfield of AI. For physics, the

mix is different but the same applies.

And then I look to those doing the kind of physics of interest to me:

John Baez, Fotini Markopoulou, Carlo Rovelli, Chris Isham, Lee Smolin.

Then it's a lot of algebraic topology, n-category theory, homology,

noncommutative geometry, topos theory (sheaves, locales, etc.), lattice

theory, and so on. (Those interested in strings and field theories

would deal with a slightly different mix.)

Certainly I agree that most advanced math doesn't lead to specific

theories having specific predictions. And before anyone jumps on me for

saying this, this is a fact of our times. The "low-hanging fruit" of

50-80 years ago, where a few weeks or months spent brushing up on the

then-abstract area of Hilbert spaces could lead to some testable

predictions are over.

My hunch is that the insights and understandings will come from those

doing math and mathematical physics for the love of it, with occasional

new testable predictions.

*>
*

*> My complaint is that these branches of math are very, very shallow, in
*

*> spite
*

*> of their extreme abstractness. There are no deep theorems there.
*

I'd count the Adjoint Functor Theorem as very deep, and useful.

Likewise, Grothendieck's seminal work in the late 50s and early 60s on

generalized spaces (Grothendieck topologies), along with the work by

Serre, Lawvere, and others, eventually led to a category-theoretic

proof by Deligne in 1974 of the Weil Conjecture.

(All of these terms can be explored on the Web, of course, with some of

them actually explained pretty well. A basic text like Mac Lane's

"Categories for the Working Mathematician," will help. Here's one of

many Wikipedia-type articles on adjoint functors and the Adjoint

Functor Theorem. http://www.wikipedia.org/wiki/Adjoint_functors )

The early 60s saw three parallel developments, all deeply interrelated:

the aforementioned Grothendieck topologies work, the efforts by Lawvere

to provide an alternative foundation to mathematics besides set theory,

and Cohen's forcing methods to prove the independence of the Axiom of

Choice from much of the rest of mathematics. This was the birth of

topos theory. (And the connections with intuitionistic logic, that is,

the logic of Brouwer and Heyting, notably, is closely tied to these

areas. And, I believe and so do others, closely tied to issues of

partial knowledge, quantum cosmology, the holographic model (still very

hypothetical), and issues with black holes, information content, and

quantum mechanics.)

What in mathematics is "deep," anyway? To some, the proof of the

Taneyama Conjecture was deep, while to others it was just number

theory. To others, proving certain properties about pullbacks in

categories of objects and morphisms is deeper than proving something

about the zeroes of a polynomial or calculating a Mersenne prime. To

each their own.

*> There are
*

*> no surprises. There are abstract structures that may help to guide
*

*> thought,
*

*> but the theory doesn't tell you much besides the fact that these
*

*> structures
*

*> exist and have some agreeable properties. The universe is a lot
*

*> deeper than
*

*> that....
*

When I look at the expanse of mathematics, I find it useful to see that

category theory (and topos theory) tells us that a proof for

semigroups, for example, automatically applies to an ostensibly

different domain. This language allows us to "move up a level" and see

the underlying similarities, even the isomorphisms. To see that while

we may have several different names for things, they are actually the

same thing.

Robert Geroch bases his text "Mathematical Physics" around category as

a unifying approach...I hope this becomes the norm as this century

unfolds, similar to the way the differential forms version of general

relativity took over from the index gymnastics of ordinary tensors.

(And apropos of your point about the new math not leading to many new

predictions, did the Cartan-influenced differential forms approach to

GR lead to new predictions that the classical, tensor-oriented approach

did not? Probably few, if any, as most of the accessible predictions of

GR were made a long time ago. But should students learning GR learn the

methods for raising and lower indices in tensors or the more modern

differential forms approach? The time saved, and the unity gained, may

lead to new syntheses, such as in quantum gravity.

Likewise, is the Hilbert space formulation of QM dramatically different

in making predictions that the Schrodinger wave equation formulation?

Working chemists still calculate Hamiltonians and wave equations--they

don't need to think in terms of Hilbert space abstractions. (And in the

area of observables, the great Von Neumann actually got it _wrong_ in

his formulation, as Bell proved several decades later...)

Geroch says this in his introduction:

"In each area of mathematics (e.g., groups, topological spaces) there

are available many definitions and constructions. It turns out,

however, that there are a number of notions (e.g., that of a product)

that occur naturally in various areas of mathematics, with only slight

changes from one area to another. It is convenient to take advantage of

this observation. Category theory can be described as that branch of

mathematics in which one studies certain definitions in a broader

context--without reference to the particular area to which the

definition might be applied. It is the "mathematics of mathematics."

"Although this subject takes some getting used to, it is, in my

opinion, worth the effort. It provides a systematic framework that can

help one to remember definitions in various areas of mathematics, to

understand what many constructions mean and how they can be used, and

even to invent useful definitions when needed."

(p. 3)

And apropos of one of the direct themes of this list, the chart on page

248 is a better chart of the categories which are of direct (known)

relevance to modern physics than Max Tegmark's chart of what he thinks

of as the branches of mathematics. (I don't mean this to sound

snide...it's just a statement of my opinion. Further, Tegmark and

others working on All Math Models need to get up to speed on this

"mathematics of mathematics.")

*>
*

*> Division algebras like quaternions and octonions are not shallow in
*

*> this
*

*> sense; nor are the complex numbers, or linear operators on Hilbert
*

*> space....
*

*>
*

*> Anyway, I'm just giving one mathematician's intuitive reaction to these
*

*> branches of math and their possible applicability in the TOE domain.
*

*> They
*

*> *may* be applicable but if so, only for setting the stage... and what
*

*> the
*

*> main actors will be, we don't have any idea...
*

Sure, there's juicy stuff in the details of octonions. John Baez would

agree with you. Getting down to making exact calculations is almost

always necessary, and sometimes illuminating. But he also connects

quaternions, octonions, etc. to n-categories and more generalized

truths. Read his stuff for details--he writes more about both of these

areas, various algebras and various categories, and their connections

to physics, than anyone I know.

Look, I'm happy that you looked at category theory and didn't find it

to your taste. I had the opposite experience. Diversity is good.

--Tim May

Received on Tue Dec 03 2002 - 14:36:20 PST

Date: Tue, 3 Dec 2002 11:30:59 -0800

On Sunday, December 1, 2002, at 11:21 AM, Ben Goertzel wrote:

You should study what interests you. If this type of math does not

interest you, you should of course study something else.

As for measures of usefulness, it's certainly the case that most

engineers, programmers, and even working physicists need little more

than some linear algebra and bit of differential equations to handle

99% of the situations they encounter. From what I see of the current AI

literature, maybe a bit of logic, a smattering of graph theory, and

whatever specialty math in the chosen subfield of AI. For physics, the

mix is different but the same applies.

And then I look to those doing the kind of physics of interest to me:

John Baez, Fotini Markopoulou, Carlo Rovelli, Chris Isham, Lee Smolin.

Then it's a lot of algebraic topology, n-category theory, homology,

noncommutative geometry, topos theory (sheaves, locales, etc.), lattice

theory, and so on. (Those interested in strings and field theories

would deal with a slightly different mix.)

Certainly I agree that most advanced math doesn't lead to specific

theories having specific predictions. And before anyone jumps on me for

saying this, this is a fact of our times. The "low-hanging fruit" of

50-80 years ago, where a few weeks or months spent brushing up on the

then-abstract area of Hilbert spaces could lead to some testable

predictions are over.

My hunch is that the insights and understandings will come from those

doing math and mathematical physics for the love of it, with occasional

new testable predictions.

I'd count the Adjoint Functor Theorem as very deep, and useful.

Likewise, Grothendieck's seminal work in the late 50s and early 60s on

generalized spaces (Grothendieck topologies), along with the work by

Serre, Lawvere, and others, eventually led to a category-theoretic

proof by Deligne in 1974 of the Weil Conjecture.

(All of these terms can be explored on the Web, of course, with some of

them actually explained pretty well. A basic text like Mac Lane's

"Categories for the Working Mathematician," will help. Here's one of

many Wikipedia-type articles on adjoint functors and the Adjoint

Functor Theorem. http://www.wikipedia.org/wiki/Adjoint_functors )

The early 60s saw three parallel developments, all deeply interrelated:

the aforementioned Grothendieck topologies work, the efforts by Lawvere

to provide an alternative foundation to mathematics besides set theory,

and Cohen's forcing methods to prove the independence of the Axiom of

Choice from much of the rest of mathematics. This was the birth of

topos theory. (And the connections with intuitionistic logic, that is,

the logic of Brouwer and Heyting, notably, is closely tied to these

areas. And, I believe and so do others, closely tied to issues of

partial knowledge, quantum cosmology, the holographic model (still very

hypothetical), and issues with black holes, information content, and

quantum mechanics.)

What in mathematics is "deep," anyway? To some, the proof of the

Taneyama Conjecture was deep, while to others it was just number

theory. To others, proving certain properties about pullbacks in

categories of objects and morphisms is deeper than proving something

about the zeroes of a polynomial or calculating a Mersenne prime. To

each their own.

When I look at the expanse of mathematics, I find it useful to see that

category theory (and topos theory) tells us that a proof for

semigroups, for example, automatically applies to an ostensibly

different domain. This language allows us to "move up a level" and see

the underlying similarities, even the isomorphisms. To see that while

we may have several different names for things, they are actually the

same thing.

Robert Geroch bases his text "Mathematical Physics" around category as

a unifying approach...I hope this becomes the norm as this century

unfolds, similar to the way the differential forms version of general

relativity took over from the index gymnastics of ordinary tensors.

(And apropos of your point about the new math not leading to many new

predictions, did the Cartan-influenced differential forms approach to

GR lead to new predictions that the classical, tensor-oriented approach

did not? Probably few, if any, as most of the accessible predictions of

GR were made a long time ago. But should students learning GR learn the

methods for raising and lower indices in tensors or the more modern

differential forms approach? The time saved, and the unity gained, may

lead to new syntheses, such as in quantum gravity.

Likewise, is the Hilbert space formulation of QM dramatically different

in making predictions that the Schrodinger wave equation formulation?

Working chemists still calculate Hamiltonians and wave equations--they

don't need to think in terms of Hilbert space abstractions. (And in the

area of observables, the great Von Neumann actually got it _wrong_ in

his formulation, as Bell proved several decades later...)

Geroch says this in his introduction:

"In each area of mathematics (e.g., groups, topological spaces) there

are available many definitions and constructions. It turns out,

however, that there are a number of notions (e.g., that of a product)

that occur naturally in various areas of mathematics, with only slight

changes from one area to another. It is convenient to take advantage of

this observation. Category theory can be described as that branch of

mathematics in which one studies certain definitions in a broader

context--without reference to the particular area to which the

definition might be applied. It is the "mathematics of mathematics."

"Although this subject takes some getting used to, it is, in my

opinion, worth the effort. It provides a systematic framework that can

help one to remember definitions in various areas of mathematics, to

understand what many constructions mean and how they can be used, and

even to invent useful definitions when needed."

(p. 3)

And apropos of one of the direct themes of this list, the chart on page

248 is a better chart of the categories which are of direct (known)

relevance to modern physics than Max Tegmark's chart of what he thinks

of as the branches of mathematics. (I don't mean this to sound

snide...it's just a statement of my opinion. Further, Tegmark and

others working on All Math Models need to get up to speed on this

"mathematics of mathematics.")

Sure, there's juicy stuff in the details of octonions. John Baez would

agree with you. Getting down to making exact calculations is almost

always necessary, and sometimes illuminating. But he also connects

quaternions, octonions, etc. to n-categories and more generalized

truths. Read his stuff for details--he writes more about both of these

areas, various algebras and various categories, and their connections

to physics, than anyone I know.

Look, I'm happy that you looked at category theory and didn't find it

to your taste. I had the opposite experience. Diversity is good.

--Tim May

Received on Tue Dec 03 2002 - 14:36:20 PST

*
This archive was generated by hypermail 2.3.0
: Fri Feb 16 2018 - 13:20:07 PST
*