RE: My current readings in Category Theory
On Wed, 3 Apr 2002, Fisher Mark wrote: > Why it is that Perl has this large of a community is probably really OT for > cypherpunks, though... Cypherpunks write code. -- There is less in this than meets the eye. Tellulah Bankhead [EMAIL PROTECTED] www.ssz.com [EMAIL PROTECTED] www.open-forge.org
Re: My current readings in Category Theory
In passing about category theory and ML: * ML supports generic programming by a language feature called a "functor." I don't know enough category theory to know how close ML's notion of "functor" is to a mathematician's. this page is a small intro http://www.kingston.ac.uk/~bs_s075/MLWorkshop/unit8.html * ocaml and ML may not be as widely used as C++, but I have seen them used fairly widely in academia. Sometimes in conjunction with other fun topics. Check out this course on computational game theory http://www.eecs.harvard.edu/~avi/CS281r/ exercises include programming in ocaml. * The Fox project at CMU wants to use ML for systems programming. http://foxnet.cs.cmu.edu/HomePage.html Note that Peter Lee is also involved in "proof-carrying code." -David
Re: My current readings in Category Theory
On Wednesday, April 3, 2002, at 06:54 AM, Julian Assange wrote:
> Category theory is nice, but would be nicer if could draw in
> information theoretic and cognitive metaphors. Maths is a programming
> language selected over time for execution on (perhaps slightly
> modified) human brains. The failure to recognize this deeply held
> anthropomorphism is excusable in other areas of mathematics, but
> not in foundational studies. Human beings look at the world through
> their own mental "axioms" which have evolved to understand the
> physical world and each other. These axioms must be chased down
> and stated instead of allowing them to subtly invade the mathematical
> process. For category theory, this failing likely evolved from
> its birth as a theory of mathematical classification as opposed to
> a theory of _origin_, although to be fair no foundational theory
> seems to be free of it.
Many thanks for your comments, Julian! I have already downloaded the Mac
OS X version of O'Caml 3.04, though installation has hit a snag, which
I'll have to investigate later.
About using more human-centric methods, maybe there's hope. Though CT is
usually seen as notoriously abstract, the fact that it uses diagrams to
understand functors and commuting maps, and so on, is nice. The Lawvere
book is filled with diagrams and maps...and I mean a _lot_ of them, not
just the canonical diagrams Mac Lane, for example, uses. (I'm not
knocking Mac Lane...it isn't meant as a teaching intro.)
>
> About 15 years ago category theory leaked into programming language
> design. See http://www.haskell.org/ and http://www.ocaml.org/ (yes,
> thats me) for the two major real-world-useful category theory
> inspired languages.
I had followed ML a little bit, back when I was attending some of the
Functional Programming conferences in the mid-80s. I must've paid no
attention to the CT stuff (understanding denotational semantics was
struggle enough, a struggle I gave up on when I moved on to other
things).
>
> OO programming is classification from above. Classification
> comes first and is then acts as (an often inconsistent and poorly
> realized) behavioral constraint. For category theory inspired
> languages, classification (types) are inferred from _behavior_. For
> these languages, behavioral inconsistency is discovered by the
> inability to come to consistent classification which describes it.
This is a powerful insight. To pick an example from Smalltalk, with
which I'm more familiar, an object is more defined by the methods
(functions) it understands than anything else. To pick a list-related
example: A banker is one who understand (sends, receives, processes)
messages about deposits and withdrawals, more so than what he is called.
("You take in deposits and you hold money and you give out money in
withdrawals, so I don't care _what_ you call yourself: you're a bank.")
This does not mean using class-derived objects is not useful: if the
work has been done to define a "banker factory" class, with all of the
functions/methods already worked out, then it makes sense to instantiate
new banks from this factory class.
As I understand it, category theory actually treats the categories
(objects) themselves as secondary, almost superfluous, with the
interesting parts being the mappings or functions and then the
transformations (functors) of these mappings being where things get more
interesting.
(And if a compiler is a function or mapping from one category (source
code) into another category (compiled, or executable code), then talking
about compilers and the connections between them is an obvious area
where functors, adjoints, and so on would come in. Hence the close
connection of CT to computability theory, the lambda calculus, and so
on.)
Another interesting area seems to be using CT for linguistics. Image
English nouns as a category, French nouns as a category, and then the
mappings (isomorphisms being the "exact" translations) between them.
E.g., dog ---> chien.
(So far, high school simple functions.)
But there are other morphisms, such as the function or process of
"forming plurals" or "possessives." These are maps as well.
Then one might look at the transformations (rules) between these maps,
and between other languages. And connections to machine languages and
grammars.
Using theorems about functors, adjoints, etc., one might make better
sense of natural languages than by using conventional methods. (And the
links with the above example are suggestive as well.)
Or consider a very difficult problem, one explained in great detail by
Douglas Hofstadter in "Godel, Escher, Bach" (or possibly "Metamagical
Themas"...I'd have to check). Namely, the many forms of the letter "A"
and how humans can recognize them all as examples of "A" because of
their "A"-ness (no jokes, please), but how computer programs have
generally failed, even with all the buzzword-compliant tricks like
neural nets and ex
RE: My current readings in Category Theory
Tim May writes: > * object-oriented systems. In my view, this one _has_ > basically lived up > to its billing, largely because it works for building more complex > systems (and is arguably how Mankind has usually built > complex systems > like bridges and skyscrapers and chips). But some of the > bolder claims > about "reusable software" and "software ICs" have yet to be realized. I wouldn't go that far - the Comprehensive Perl Archive Network (CPAN, http://www.cpan.org/) has been able to provide me with code for dealing with everything from database connections to Win32 API calls to OPML (outlines expressed in XML). When I can work in Perl (which isn't all the time), I'm usually assembling software ICs rather than creating the transistors, flip-flops, etc. on my own. It can get to be painful -- it takes about 35 CPAN downloads to get Slashcode, the code behind http://slashdot.org/, running -- but that is several boatloads of code that I didn't have to write to get a weblog with authentication, discussion groups, polls, RSS feeds, and God knows what else going (I'm still in the exploration stage). Why it is that Perl has this large of a community is probably really OT for cypherpunks, though... === Mark Leighton Fisher[EMAIL PROTECTED] Thomson multimedia, Inc.Indianapolis IN The Illuminati are not dead -- they're just pining for the fnords...
Re: My current readings in Category Theory
> 4. Perhaps even more strangely, but more practically, there may be some > very interesting uses for Cypherpunks visions. Weirder than Baez, > weirder than Egan...in some ways. The basic insight is that just as > category theory is about "objects/things" (called categories) and the > "transformations/functions" (called arrows) between them, this is what a > lot of crypto protocols are. (Note that I'm not explicitly talking about > object-oriented programming, a la Java or Smalltalk, but more about > viewing crypto protocols are transformations.) Category theory is nice, but would be nicer if could draw in information theoretic and cognitive metaphors. Maths is a programming language selected over time for execution on (perhaps slightly modified) human brains. The failure to recognize this deeply held anthropomorphism is excusable in other areas of mathematics, but not in foundational studies. Human beings look at the world through their own mental "axioms" which have evolved to understand the physical world and each other. These axioms must be chased down and stated instead of allowing them to subtly invade the mathematical process. For category theory, this failing likely evolved from its birth as a theory of mathematical classification as opposed to a theory of _origin_, although to be fair no foundational theory seems to be free of it. ... About 15 years ago category theory leaked into programming language design. See http://www.haskell.org/ and http://www.ocaml.org/ (yes, thats me) for the two major real-world-useful category theory inspired languages. OO programming is classification from above. Classification comes first and is then acts as (an often inconsistent and poorly realized) behavioral constraint. For category theory inspired languages, classification (types) are inferred from _behavior_. For these languages, behavioral inconsistency is discovered by the inability to come to consistent classification which describes it. -- Julian Assange|If you want to build a ship, don't drum up people |together to collect wood or assign them tasks and [EMAIL PROTECTED] |work, but rather teach them to long for the endless [EMAIL PROTECTED] |immensity of the sea. -- Antoine de Saint Exupery
Re: My current readings in Category Theory
On Tuesday, April 2, 2002, at 02:58 PM, Sampo Syreeni wrote: > On Tue, 2 Apr 2002, Tim May wrote: > >> I've been having a lot of fun reading up on "category theory," a >> relatively new branch of math that offers a unified language for >> talking >> about (and proving theorems about) the transformations between objects. > > Baez convinced you, no? He seems to be a category freak. > >> I'll say a few words on why this is more than just the "generalized >> abstract nonsense" that some wags have dubbed category theory as. > > It seemed like that at first, of course. However, some fairly deep > observations have been made in the area, concerning the basic > assumptions > underlying math. Namely, the prevalence of sets, functions, first order > logic and the like. There might just be something to categories, after > all. Yes, I believe there's a lot more. By the way, even though category theory may be about as foundational as set theory (a la Zermelo-Frankel axiomatization), it looks to be a _lot_ more useful in other areas. We all know what sets are, and use them every day, and use things like Venn diagrams more than almost any other tool (at least I do), but the axiomatic foundations are seldom used. The Axiom of Choice? > >> I won't try to explain what categories and toposes are here in this >> e-mail message. > > Thank god. But isn't it "topoi"? I was drinking coffee out of one of my thermoi and realized you were...of that camp. As I said, I'm also using Goldblatt's "Topoi." (But it's out of print, and unpurchasable, so far, so I use UCSC's copy.) Note that McClarty's book says "Toposes." One of these authors, maybe McClarty, maybe Johnstone, points out that plurals of words which were never Latin to begin with, like Thermos bottle, may be "thermoses," not "thermoi." I find "toposes" sounds better than "topoi." It's only topoi-logical, after all. >> >> Relativity was exciting--I took James Hartle's class using a preprint >> edition of Misner, Thorne, and Wheeler's massive tome, "Gravitation." > > The Big Black Book. Tried it, didn't like it much. Somehow they manage > to > make the subject totally inaccessible to anyone used to the standard > concept of tensor spaces. I mean, if they have a basis, why not simply > talk about multilinear mappings? (They do, when talking about tangent > spaces. I'm just wondering why tensors are needed at all.) But they were able to at least eliminate the "index gymnastics" of manipulating indices in, for example, the Riemann tensor. R-sub-ijk and all that rot. My copy of Sokolnikoff and Redheffer could be safely put away. > >> The fact that we use "Alice and Bob" diagrams, with "Eve" and "Vinnie >> the Verifier" and so on, with arrows showing the flow of signatures, or >> digital money, or receiptswell, this is a hint that the >> category-theoretic point of view may be extremely useful. (At other >> levels, it's number theory...the stuff about Euler's totient function >> and primes and all that. But at another level it's about commutative >> and >> transitive mappings, and about _diagrams_.) > > I don't see the connection. Category theory mostly seems to be about > questioning the way we represent and visualize mathematics. There, it is > beginning to have some real influence. However, what you're describing > above is well below that, in the realm of ordinary sets and functions. I > seem to think categories have very little to do with such things. Look at some of the "computer science" references, as opposed ot the "theory of math" references. Barr and Wells, or Pierce, for example. They point out that people are successfully using category theory terminology as a means of clarifying the unclear, not as a means of pushing the frontiers of math. The value of looking at functors (natural transformations between categories) as opposed to ordinary sets and functions is the ability to draw conclusions from other areas of math, it seems to me. > >> * "game theory." We all know that most human and complex system >> interactions have strong game-theoretic aspects. Cooperation, >> defection, >> Prisoner's Dilemma, Axelrod, etc. But thinking that "all crypto is >> basically game theory" has not been fruitful, so far. > > Axelrod? I just started reading up on basic game theory and the theory > of > oligopoly (Cournot, Nash, price vs. quantity selection, the works), but > haven't bumped into that name, yet. What gives? Axelrod, "The Evolution of Cooperation." >> * the whole ball of wax that is complexity, fractals, chaos, >> self-organized criticality, artificial life, etc. Tres trendy since >> around 1985. But not terribly useful, so far. > > No? I seem to recall a couple of articles on how actual markets behave > chaotically, based on time-series data. Such a conclusion is quite a > feat, > I'd say, and there's bound to be more out there. I'm not saying chaos isn't real, just that it's not turning out to be very surprising or useful. >
RE: Re: My current readings in Category Theory
Sampo Syreeni said: | >* "game theory." We all know that most human and complex system | >interactions have strong game-theoretic aspects. | Cooperation, defection, | >Prisoner's Dilemma, Axelrod, etc. But thinking that "all crypto is | >basically game theory" has not been fruitful, so far. | | Axelrod? I just started reading up on basic game theory and | the theory of | oligopoly (Cournot, Nash, price vs. quantity selection, the | works), but | haven't bumped into that name, yet. What gives? Robert Axelrod did a popular/layman's study of the iterated prisoner's dilemna, and how it relates to evolution, called The Evolution of Cooperation (ISBN: 0465021212 in paperback), that I liked a lot. He's done a few other books as well, but I haven't read them, so I can't comment on his works in general. Kurt
Re: My current readings in Category Theory
On Wed, 3 Apr 2002, Sampo Syreeni wrote: > >The fact that we use "Alice and Bob" diagrams, with "Eve" and "Vinnie > >the Verifier" and so on, with arrows showing the flow of signatures, or > >digital money, or receiptswell, this is a hint that the > >category-theoretic point of view may be extremely useful. (At other > >levels, it's number theory...the stuff about Euler's totient function > >and primes and all that. But at another level it's about commutative and > >transitive mappings, and about _diagrams_.) > > I don't see the connection. Category theory mostly seems to be about > questioning the way we represent and visualize mathematics. There, it is > beginning to have some real influence. However, what you're describing > above is well below that, in the realm of ordinary sets and functions. I > seem to think categories have very little to do with such things. It is about visualizing any sort of relationship, not just mathematics. Category Theory has a lot to say about the 'simplicity' of the cosmos. It also has a lot to say (in a self-referential manner) about the way humans think about thinking. It will, in the long run, be a critical component in developing AI. > >* the whole ball of wax that is complexity, fractals, chaos, > >self-organized criticality, artificial life, etc. Tres trendy since > >around 1985. But not terribly useful, so far. > > No? I seem to recall a couple of articles on how actual markets behave > chaotically, based on time-series data. Such a conclusion is quite a feat, > I'd say, and there's bound to be more out there. Besides, I'm not quite > sure chaotics hasn't had an impact on e.g. cipher design -- current cipher > design seems to concentrate a lot on diffusion, for instance. What is > diffusion but a discretized version of a Lyapunov exponent-like > characterization of chaotic blow-up? Actualy it's very useful, it even leads into CT if you keep at it. Diffusion may be -fractal-, but that is not the same as -chaotic-. You're confusing the two. > Of course. But how is this interesting? I view objects mainly as a logical > extension of the analytic method: to-undestand-break-it-down. Not nearly > as interesting as blind learning algos or the like. ??? Object oriented programming is about memory and function consolidation. It flows from the management of effects and side-effects, not from any generalization of the analytical process. -- There is less in this than meets the eye. Tellulah Bankhead [EMAIL PROTECTED] www.ssz.com [EMAIL PROTECTED] www.open-forge.org
My current readings in Category Theory
-BEGIN PGP SIGNED MESSAGE- Hash: SHA1 Tim wrote: >* "general systems theory," a la Bertanlanffy. I knew a guy who was >majoring in this as an interdisciplinary self-study program. Whatever >became of this? >(And this is kissing cousin to Operations Research, which is mostly a >high bullshit term for linear programming, decision support tools, a >little bit of game theory, etc.) Bah. You might find the following approaches to the above a little more sophisticated and interesting: Exploratory Analysis and a Case History of Multiresolution, Multiperspective Modeling, Paul K. Davis, James H. Bigelow, and Jimmie McEver, Reprinted from Proceedings of the 2000 Winter Simulation Conference, Jeffrey A. Joines, Russel R. Barton, K. Kang, and Paul A. Fishwick (editors), December, 2000 and Proceedings of the SPIE, Vol. 4026, 2000. http://www.rand.org/publications/RP/RP925.pdf Title: Experiments in Multiresolution Modeling (MRM). http://www.rand.org/publications/MR/MR1004/ Author(s): Paul K. Davis, J.H. Bigelow Abstract: This study describes the motivation for multiresolution modeling (MRM) within a single model or a family of models. After introducing a new measure of consistency for models of different resolution, the study discusses in some depth obstacles to and methods for multiresolution modeling (also called variable-resolution modeling), illustrating issues with a detailed military example involving precision fires. The study highlights the value of visual design, array formalism, formal mathematics to identify natural aggregation fragments, integrated hierarchical variable resolution (IHVR) yielding "trees" of variables, estimation theory, alternative aggregate representations called out in a user interface, "stretcher variables," and computational methods to identify natural phase transitions and facilitate calibrations. *** Not exactly what you were getting at, but this approach certainly doesn't suffer from the "mathematicians writing on the board for it's own sake" problem you alluded to. Bridging the speculative and the practical couldn't be more central here (as everywhere.) Reminds me of something John Von Neumann once said: "The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work." This might sound a little off-the-wall, but have you considered sitting in on some graduate classes in the sorts of areas you're interested in at Berkeley, just for the sake of generating more discussion with people in the field? Ill bet bouncing everything in your post off people there would generate a lot a lot of return for a small investment of your time. None of my business but it's at least worth a thought. ~~Faustine. *** He that would make his own liberty secure must guard even his enemy from oppression; for if he violates this duty he establishes a precedent that will reach to himself. - --Thomas Paine -BEGIN PGP SIGNATURE- Version: PGPsdk version 1.7.1 (C) 1997-1999 Network Associates, Inc. and its affiliated companies. (Diffie-Helman/DSS-only version) iQA/AwUBPKpfg/g5Tuca7bfvEQL4vwCcCA98uyVl36y+61wwsHaNfwNyDZoAoIi/ 5eyWAiN07n/n+fWgidqLxupr =3FWa -END PGP SIGNATURE-
Re: My current readings in Category Theory
On Tue, 2 Apr 2002, Tim May wrote: > Category theory is basically about the common categories (sets, > topologies, algebras, recursively-computable functions, stuff even from > computer science) and the diagrams that link them. There's more to it > than just saying "it's about talking about diagrams." It's a more > unified way of looking at "patterns" (no one that I know of has made the > connection between CT and pattern languages of programming (PLOP), a la > the Gang of Four (authors of the famous book), but I will be surprised > if there is not a deep link to be found there. Read more, they have also been connected to such things as ERD's as well. > Am I making the same mistake in thinking category theory is some kind of > uber-theory which unifies and connects disparate areas? What it DOES do is allow one to make predictions from one area of experience into another area of experiece *when the second area of experience isn't as well as explored as the first*. -- There is less in this than meets the eye. Tellulah Bankhead [EMAIL PROTECTED] www.ssz.com [EMAIL PROTECTED] www.open-forge.org
Re: My current readings in Category Theory
On Tue, 2 Apr 2002, Tim May wrote: >I've been having a lot of fun reading up on "category theory," a >relatively new branch of math that offers a unified language for talking >about (and proving theorems about) the transformations between objects. Baez convinced you, no? He seems to be a category freak. >I'll say a few words on why this is more than just the "generalized >abstract nonsense" that some wags have dubbed category theory as. It seemed like that at first, of course. However, some fairly deep observations have been made in the area, concerning the basic assumptions underlying math. Namely, the prevalence of sets, functions, first order logic and the like. There might just be something to categories, after all. >I won't try to explain what categories and toposes are here in this >e-mail message. Thank god. But isn't it "topoi"? >* http://math.ucr.edu/home/baez/John Baez (a cousin of Joan) is >a mathematical physicist with wideranging interests...his site has all >sorts of good stuff I think Week's Finds are the ones best suited for nonlinear-agoric-geodesic, cypherpunkly consumption. >"Category theory is about how mathematicians draw diagrams on the >blackboard." A waggish, snide comment, but true. It becomes non-trivial once one starts to do true 2D algebra. In the absence of categories, one would never think there is anything "wrong" or "limiting" about the way algebra is now done. >Relativity was exciting--I took James Hartle's class using a preprint >edition of Misner, Thorne, and Wheeler's massive tome, "Gravitation." The Big Black Book. Tried it, didn't like it much. Somehow they manage to make the subject totally inaccessible to anyone used to the standard concept of tensor spaces. I mean, if they have a basis, why not simply talk about multilinear mappings? (They do, when talking about tangent spaces. I'm just wondering why tensors are needed at all.) >But the recent work on supergravity, loop spaces, knot theory, strings, >M-branes, etc., has re-ignited some interest...at the "amateur" level, >needless to say. It's well beyond a pipe dream, so why not? >The fact that we use "Alice and Bob" diagrams, with "Eve" and "Vinnie >the Verifier" and so on, with arrows showing the flow of signatures, or >digital money, or receiptswell, this is a hint that the >category-theoretic point of view may be extremely useful. (At other >levels, it's number theory...the stuff about Euler's totient function >and primes and all that. But at another level it's about commutative and >transitive mappings, and about _diagrams_.) I don't see the connection. Category theory mostly seems to be about questioning the way we represent and visualize mathematics. There, it is beginning to have some real influence. However, what you're describing above is well below that, in the realm of ordinary sets and functions. I seem to think categories have very little to do with such things. >* "game theory." We all know that most human and complex system >interactions have strong game-theoretic aspects. Cooperation, defection, >Prisoner's Dilemma, Axelrod, etc. But thinking that "all crypto is >basically game theory" has not been fruitful, so far. Axelrod? I just started reading up on basic game theory and the theory of oligopoly (Cournot, Nash, price vs. quantity selection, the works), but haven't bumped into that name, yet. What gives? >* the whole ball of wax that is complexity, fractals, chaos, >self-organized criticality, artificial life, etc. Tres trendy since >around 1985. But not terribly useful, so far. No? I seem to recall a couple of articles on how actual markets behave chaotically, based on time-series data. Such a conclusion is quite a feat, I'd say, and there's bound to be more out there. Besides, I'm not quite sure chaotics hasn't had an impact on e.g. cipher design -- current cipher design seems to concentrate a lot on diffusion, for instance. What is diffusion but a discretized version of a Lyapunov exponent-like characterization of chaotic blow-up? >* AI. 'Nuff said. We all know intelligence is real, and important, but >the results have not yet lived up to expectations. Maybe someday. The connectionist stuff seems interesting, here. So does silicon learning via genetic programming. >* object-oriented systems. In my view, this one _has_ basically lived up >to its billing, largely because it works for building more complex >systems (and is arguably how Mankind has usually built complex systems >like bridges and skyscrapers and chips). But some of the bolder claims >about "reusable software" and "software ICs" have yet to be realized. Of course. But how is this interesting? I view objects mainly as a logical extension of the analytic method: to-undestand-break-it-down. Not nearly as interesting as blind learning algos or the like. >* agoric, market-oriented approaches. I'm still very hopeful about this >one. It seems people never completely understand the power of emergent sys
My current readings in Category Theory
Cypherpunks, I've been having a lot of fun reading up on "category theory," a relatively new branch of math that offers a unified language for talking about (and proving theorems about) the transformations between objects. I'll say a few words on why this is more than just the "generalized abstract nonsense" that some wags have dubbed category theory as. I won't try to explain what categories and toposes are here in this e-mail message. For one thing, it would take a while. For another, there are people who've been doing it for a long time who are good teachers, and they've written good explanations. Most importantly, because the Web has plenty of good resources, even complete downloadable textbooks in PDF and PS form. For example, here a couple of good sites: * as always, Google: search on "category theory" or "topos" (beware the lit-crit meanings) * http://math.ucr.edu/home/baez/John Baez (a cousin of Joan) is a mathematical physicist with wideranging interests...his site has all sorts of good stuff * http://plato.stanford.edu/entries/category-theory/A Stanford Encyclopedia article on what category theory is * http://dmoz.org/Science/Math/Algebra/Category_Theory/ A page of links to other pages * several good textbooks, which I'll talk about later Category theory is basically about the common categories (sets, topologies, algebras, recursively-computable functions, stuff even from computer science) and the diagrams that link them. There's more to it than just saying "it's about talking about diagrams." It's a more unified way of looking at "patterns" (no one that I know of has made the connection between CT and pattern languages of programming (PLOP), a la the Gang of Four (authors of the famous book), but I will be surprised if there is not a deep link to be found there. "Category theory is about how mathematicians draw diagrams on the blackboard." A waggish, snide comment, but true. (Motivation for reading further: and what is it that we do when we draw diagrams showing Alice, Bob, and Eve with various kinds of arrows linking them?) First, a couple of motivations for why I've been looking at this stuff: 1. The pure math side. I didn't specialize in math, but have good memories of studying analysis (of the Rudin/Kelley sort, involving point sets, covers, compactness, Tychonoff's Theorem, etc.). This was back in the early 70s, and the application of abstract math to my then-interest, physics, seemed to be too much of a reach. (Applying math is not the sina qua non of studying math, of course, even to physicist folks. But seeing that Reality somehow _involves_ topology, or fiber bundles, or sheaves, or toposes...well, that adds some excitement.) 2. Recent trends in higher-order categories (K-categories) seem to have connections to knot theory, superstrings, the work of guys like Witten and Smolin. The Baez site above talks about these connections at length. One of the reasons I basically got out of pure physics was that there were two main paths: particle physics and relativity. Particle physics in the 70s was not too exciting, in terms of fundamental insights. Feynman characterized it as "building bigger accelerators to smash wristwatches together at higher and higher speeds to get more little pieces to catalog." (I'm paraphrasing.) Relativity was exciting--I took James Hartle's class using a preprint edition of Misner, Thorne, and Wheeler's massive tome, "Gravitation." Motivated me to look at differential forms (Flanders) and differential geometry. But I sensed that I wasn't going to be one of the tiny handful of people able to actually get _jobs_ doing this, so I moved into solid-state, and thence to Intel. But the recent work on supergravity, loop spaces, knot theory, strings, M-branes, etc., has re-ignited some interest...at the "amateur" level, needless to say. 3. Reading Greg Egan's recent SF novels. A lot of intriguing, though of course ultra-speculative!, stuff on "all topology models." Egan is obviously plugged-into a lot of this stuff and is making daring leaps. Highly unlikely to be true, but as "inspiring" as similar leaps were in the Golden Age of SF when Klein bottles and Moebius strips figured into stories (notably by Arthur C. Clarke) and where the wormholes and hyperspace geometries of general relativity were plot elements. (The fact that the "warp drives" described in 1950s-era science fiction, and later used in "Star Trek," weren't plausible in any serious way didn't reduce their inspirational/pedagogical value.) 4. Perhaps even more strangely, but more practically, there may be some very interesting uses for Cypherpunks visions. Weirder than Baez, weirder than Egan...in some ways. The basic insight is that just as category theory is about "objects/things" (called categories) and the "transformations/functions" (called arrows) between them, this is what a lot of crypto protocols are. (Note that I'm no
