RE: Applied vs. Theoretical
Ben Goertzel <[EMAIL PROTECTED]> wrote: >Tim May wrote: >> As I hope I had made clear in some of my earlier posts on this, mostly >> this past summer, I'm not making any grandiose claims for category >> theory and topos theory as being the sine qua non for understanding the >> nature of reality. Rather, they are things I heard about a decade or so >> ago and didn't look into at the time; now that I have, I am finding >> them fascinating. Some engineering/programming efforts already make >> good use of the notions [see next paragraph] and some quantum >> cosmologists believe topos theory is the best framework for "partial >> truths." >> >> The lambda calculus is identical in form to cartesian closed >> categories, program refinement forms a Heyting lattice and algebra, >> much work on the fundamentals of computation by Dana Scott, Solovay, >> Martin Hyland, and others is centered around this area, etc. > >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). > >My complaint is that these branches of math are very, very shallow, in spite >of their extreme abstractness. There are no deep theorems there. 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 > >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... Although I would agree that there is an atom of truth in the idea that categories are shallow structures, I do think they will play a more and more important role in the math, physics and (machine) psychology of the future. 1: Shallowness is not incompatible with importance. Sets are shallow structures but are indispensable in math for example. 2: Categories are just sets, in first approximation, where morphism are taken into account, and this has lead to the capital notion of natural transformation and adjunction which are keys in universal algebra. 3: Categories are non trivial generalisation of group and lattice, so that they provide a quasi-continuum between geometry and logic. This made them very flexible tools in a lot of genuine domains. 4: Special categories are very useful for providing models in logic, like *-autonomous categories for linear logic, topoi for intuitionist logic, etc. Some special categories appear in Knot Theory, and gives light on the role of Quantum field in the study of classical geometry. Despite all this, some domain are category resistant like Recursion Theory (I read the 1987 paper by Di Paola and Heller "Dominical Categories: Recursion Theory Without Element" The journal of symbolic logic, 52,3, 594-635), but I still cannot digest it, and I don't know if there has been a follow-up. So my feeling is that category theory and some of its probable "quantum generalisation" will play a significant role in tomorow's sciences. In fact, categories by themselves are TOEs for math. Topoi are mathematical universes per se. At the same time, being problem driven, I think category theory can distract the too mermaid-sensible researcher. 'Course there is nothing wrong with hunting mermaids for mermaids sake, but then there is a risk of becoming a mathematician. Careful! ;-) Bruno
Re: Applied vs. Theoretical
ORE CONCRETE OR NOT] SELECTION CRITERIA AMONG THEM. It is somewhat like the Ocean - if an explorer worships the Ocean, then he will go off in any direction that Ocean seems to be leading him, and at least half of the time he will be wrong in terms of applications and predictions and usefulness and depth and so on. ANALYSIS, on the other hand, uses algebra and geometry and topology and number theory, but its core ideas are LIMITS [ONE- OR TWO-SIDED], MEASURES, CHANGE, and a few other things, and how these relate to geometries and topologies and algebras. Rovelli and Smolin and Ashtekar and Schwarz and Witten are slightly different types of algebraists - they are either ALGEBRAIC GEOMETERS or ALGEBRAIC TOPOLOGISTS, with PHYSICS thrown in of course. Their string and brane and knot and loop theories have about the same depth as those of MacLane and Lawvere in terms of not predicting new things. Oh, they do marvellously at REINVENTING THE WHEEL, at reconstructing QM and QFT, etc. Osher Doctorow - Original Message - From: "Tim May" <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Sent: Tuesday, December 03, 2002 11:30 AM Subject: Re: Applied vs. Theoretical
Re: Applied vs. Theoretical
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, ten
Re: Applied vs. Theoretical
>From Osher Doctorow [EMAIL PROTECTED], Sunday Dec. 1, 2002 1243 Sorry for keeping prior messages in their entirety in my replies. Let us consider the decision of category theory to use functors and morphisms under composition and objects and commuting diagrams as their fundamentals. Because of the functor-operator-linear transformation and similar properties, composition and its matrix analog multiplication automatically take precedence over anything else, and of course so-called matrix division when inverses are defined - that is to say, matrix inversion and multiplication. It was an airtight argument, it was foolproof by all that preceded it from the time of the so-called Founding Fathers in mathematics and physics, and it was wrong - well, wrong in a competitive sense with addition-subtraction rather than multiplication-division. There is, of course, nothing really wrong with different models, and at some future time maybe the multiplication-division model will yield more fruit than the addition-subtracton models. And, of course, each model uses the other model secondarily to some extent - nobody excludes subtraction from the usual categories or multiplication from the subtractive models. What do I mean when I say it was relatively wrong, then, in the above sense [question-mark]. Consider the following subtraction-addition results - in fact, subtraction period. 1. Discriminates the most important Lukaciewicz and Rational Pavelka fuzzy multivalued logics from the other types which are divisive or identity in their implications. 2. Discriminates the most important Rare Event Type [RET] or Logic-Based Probability [LBP] which describes the expansion-contraction of the universe as a whole, expansion of radiation from a source, biological growth, contraction of galaxies, etc., from Bayesian and Independent Probability-Statistics which are divisive/identity function/multiplicative. 3. Discriminates the proximity function across geometry-topology from the distance-function/metric, noting that the proximity function is enormously easier to use and results in simple expressions. It sounds or reads nice, but the so-called topper or punch line to the story is that ALL THREE subtractive items above have the form f[x, y] = 1 plus y - x. ALL THREE alternative division-multiplication forms have the form f[x, y] = y/x or y or xy. Category theory has ABSOLUTELY NOTHING to say about all this. So where are division and multiplication mainly used [question mark]. It turns out that they are used in medium to zero [probable] influence situations, while subtraction is used in high to very high influence situations. Come to your own conclusions, so to speak. Osher Doctorow - Original Message - From: "Tim May" <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Sent: Sunday, December 01, 2002 10:44 AM Subject: Applied vs. Theoretical
RE: Applied vs. Theoretical
Tim May wrote: > As I hope I had made clear in some of my earlier posts on this, mostly > this past summer, I'm not making any grandiose claims for category > theory and topos theory as being the sine qua non for understanding the > nature of reality. Rather, they are things I heard about a decade or so > ago and didn't look into at the time; now that I have, I am finding > them fascinating. Some engineering/programming efforts already make > good use of the notions [see next paragraph] and some quantum > cosmologists believe topos theory is the best framework for "partial > truths." > > The lambda calculus is identical in form to cartesian closed > categories, program refinement forms a Heyting lattice and algebra, > much work on the fundamentals of computation by Dana Scott, Solovay, > Martin Hyland, and others is centered around this area, etc. 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). My complaint is that these branches of math are very, very shallow, in spite of their extreme abstractness. There are no deep theorems there. 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 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... -- Ben Goertzel
