Intuistionistic topology textbook online

[Delmac]

Includes some speculation about the relevance of foundational issues in constructive mathematics to physical law. Anyway, this may be of interest to some people here, and maybe some discussion on the subject would be interesting.

http://www.fwaaldijk.nl/natural-topology.pdf

[Baron_Corvo]

Thanks for posting this. I have to admit I struggled with topology at university and apparently got a poor mark on my Finals paper which included it, so I'll have a look at this when I've got time.

A simple book which covers a few of the key concepts is "From Geometry to Topology", by Graham Flegg. I don't know whether or not it's still available but it might be worth searching for online.

[Gedanken]

Back to the main topic:

I ahve not read the whole book yet but from the brief comments that it makes about "physics", "measurement" etc. it seems to confuse different questions.

The point is that the use of "continuity" has been the best shortcut for making progress in modelling anything whatsoever - even for objects that literally are discrete. I. Newton believed (like others) in a corpuscular theory, and yet he used the calculus of continua - the limiting process was justified by the large number of atoms in any particular substance, which allows for approximations (more on the rejection of this by "constructivism" below). The sort of measurement issues that are often brought up does not seem to me to be conceptually crucial whatsoever, for any of the technically interesting results in physics - or anywhere for that matter. The only issue of relevance is "constructibility" in another sense, that is whether you can exhibit an actual object - and for this reason ultrafilters and the like (such as in "nonstandard analysis") don't have any direct application in physics.

Whether actual continua exist or not is an open question - the dismissal of it is pure ideology, and it has no possible merits other than as ideology. And on that matter I have some doubts (other than the ones expressed above). The construction of models for smooth infinitesimal analysis (http://books.google.com/books?id=TXqpcQAACAAJ) gives us a good idea of what it might mean for time itself to be "continuous", complementing the more informal arguments to the same effect.

Very recently the work of Smale et al. have shewn that continuous models (in which computers input real numbers) are more useful than "Turing machines" for a model of computation. (A new book will be published soon by Oxford U.P. - repeatedly delayed over the years - by Moore and Mertens, that will be the definitive statement on these matters.)

Of course, mathematically the project of the OP may be of some interest.

An interesting topic in connection with this is Bishop/Markov's "constructivism" - IIRC (from my first perusal of Bishop and Bridges) they have some kind of constructive counterpart to the intermediate value theorem and such. But primarily they abolish "real numbers". This is my simplification but: instead of approximations to a real number they would just define a real number as a "program" that given any n in the positive integers, outputs a rational number x subject to the condition that | xn - xm | < or = 10^-n + 10^-m.

One of the things Bishop denounced in his opening Ch was the "synthetic" geometry (of course, "non-constructive") as developed in Euclid, Archimedes et al. (and also: Huygens, Barrow, Newton, etc.). In my view, however, their objections don't count as any "objection" to mathematics, broadly considered. For what that "conception" may be one can read: "From Absolute to Local Mathematics (http://publish.uwo.ca/~jbell/absolute.pdf)".

[Macrobius]

Thanks for the OP and also, thanks Gedanken for the paper by Bell.

How common is 'Bishop style mathematics' these days?

[Gedanken]

Constructive analysis has its niche following, and there is a nice recent book (http://books.google.com/books?id=H5BEsaM9XQcC) on it by Bridges (stressing the essential move that intuitionistic logic replaces classical logic). Something equivalent to it was developed in Russia, by A. A. Markov, b. 1903 (who is not the "Markov" of probability theory - see the book by Bridges).

The book I referred to by Moore & Mertens is this one (http://www.nature-of-computation.org/). I have been waiting since 2007 for this to come out, and finally it seems that it will come out. Hopefully. Here is the table of contents (http://www-e.uni-magdeburg.de/mertens/noc/samples/toc.pdf).

[Gedanken]

Also, the book by Smale et al.:

http://ecx.images-amazon.com/images/I/51TZpJfNj6L.jpg

[Macrobius]

Constructivism was all the rage when I was going to college -- I learned logic from the Jews: Warren G Goldfarb (a constructivist) and later from Hilary 'The Goys don't need no stinkin' foundations for Maths' Putnam. It always seemed to me there was more to Brouwer and Weyl than I was taught, but I've never had the chance to follow it up. (Honestly, I learned more reading Frege, Russell and esp. the books by WVO Quine than either of the above teachers, but that was the breaks in the late 70s). Quine was retired when I went through, or I might have learned more in class.

[Macrobius]

Constructive analysis has its niche following, and there is a nice recent book (http://books.google.com/books?id=H5BEsaM9XQcC) on it by Bridges (stressing the essential move that intuitionistic logic replaces classical logic). Something equivalent to it was developed in Russia, by A. A. Markov, b. 1903 (who is not the "Markov" of probability theory - see the book by Bridges).

The book I referred to by Moore & Mertens is this one (http://www.nature-of-computation.org/). I have been waiting since 2007 for this to come out, and finally it seems that it will come out. Hopefully. Here is the table of contents (http://www-e.uni-magdeburg.de/mertens/noc/samples/toc.pdf).

It seems to be out now, and merely looking at *who* is reviewing it moves it to the top of the short list.


“To put it bluntly: this book rocks! It's 900+ pages of awesome. It somehow manages to combine the fun of a popular book with the intellectual heft of a textbook, so much so that I don't know what to call it (but whatever the genre is, there needs to be more of it!).” —Scott Aaronson, MIT

“A creative, insightful, and accessible introduction to the theory of computing, written with a keen eye toward the frontiers of the field and a vivid enthusiasm for the subject matter.” —Jon Kleinberg, Cornell

“If you want to learn about complexity classes, scaling laws in computation, undecidability, randomized algorithms, how to prepare a dinner with Pommard, Quail and Roquefort, or the new ideas that quantum theory brings to computation, this is the right book. It offers a wonderful tour through many facets of computer science. It is precise and gets into details when necessary, but the main thread is always at hand, and entertaining anecdotes help to keep the pace.” —Marc Mézard, Orsay

“This is, simply put, the best-written book on the theory of computation I have ever read; one of the best-written mathematical books I have ever read, period. ...from beginning to end, and all the 900+ pages in between, this was lucid, insightful, just rigorous enough, alive to how technical problems relate to larger issues, and above all, passionate and human.” —Cosma Shalizi, Carnegie Mellon, Three-Toed Sloth

“A treasure trove of ideas, concepts and information on algorithms and complexity theory. Serious material presented in the most delightful manner!” —Vijay Vazirani, Georgia Tech

“A fantastic and unique book---a must-have guide to the theory of computation, for physicists and everyone else. ” —Riccardo Zecchina, Politecnico di Torino


Aaronson, Mezard, Shalizi, *and* Vazirani? All on one page? *wow* Just wow. ... I mean why not throw in a Valiant and Turing or two, or maybe Kleene, or at least Terrance Tao or Dick Lipton? I haven't heard of the other guys but just by association they move into the 'watch' category.

[Baron_Corvo]

TBH, I'm out of my depth in this company, but I don't believe in continuity either. It may be one of the reasons I sucked at topology at university - I could neither fully accept the premises of what I was taught, nor articulate what was the alternative to what was being taught. (I'd rather believe that than simply acknowledge that I might be too thick to understand the subject. :D ).

Anyway, as I understand it (taking the real line as an example); the existence or positing of continuity implies that a number such as pi has another number which is immediately bigger than it in size, i.e. that there is a "next" number to pi in order of magnitude. I don't believe any such number exists, for the simple reason that if any such number did exist it would have to be larger than pi by an irreducible infinitesimal.

No such infinitesimal does or can exist, because infinitesimals do not behave proportionately in the same way as finite numbers do. The best you can say for certain is that, for any closed interval in between any two members of the set of reals R , you can construct a bijective mapping of a closed interval between any two other members of R, and this holds true no matter what the disparity between the sizes of the two intervals; my old tutor used to call the set of reals a "SET" (a systematic elastic totality) in recognition of this fact.

It's even possible to construct such a bijective mapping between a closed interval of this kind on R, and the set R itself; the infinite ends of the set can be taken care of by something like A = arctan B, where A is a simple function of the form kx + l; both k and l would be determined by the limits of the finite interval R is being mapped onto.

[Macrobius]

A copy of Moore and Martens (http://www.amazon.de/Nature-Computation-Cristopher-Moore/dp/0199233217) was my Christmas present. It looks like a rollicking good read. I promise a review by this time next year. ;)

[Baron_Corvo]

I'm tempted by this one at the moment, money allowing;

http://www.amazon.co.uk/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809/ref=sr_1_1?s=books&ie=UTF8&qid=1325070111&sr=1-1

[Macrobius]

I'm tempted by this one at the moment, money allowing;

http://www.amazon.co.uk/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809/ref=sr_1_1?s=books&ie=UTF8&qid=1325070111&sr=1-1

You know a book is good when the ask for used is LL. 4 higher than the bid for new.

[Delmac]

As this seems to be evolving into a "what are you currently thinking about?" thread, here are two interesting sites:

This one has some intersting material about interpreting lambda calculus in binary. One motivation for looking at this stuff seems to be to find an objective measure for algorithmic complexity.
John's Combinatory Logic Playground (http://homepages.cwi.nl/~tromp/cl/cl.html)
There is also some material on S & K combinators.

Here is a (very small) wiki aggregating some interesting material about the use of copulas (copulae?) in statistics and probablility theory and in the wider scientific world:
https://sites.google.com/site/copulawiki/

[Delmac]

I may be a bit keener on gossip than some posters on this thread, but I do think this book about the formulation of the Standard Model sounds more interesting than most popularisations:

The Infinity Puzzle (at Amazon..) (http://www.amazon.com/Infinity-Puzzle-Quantum-Orderly-Universe/dp/0465021441)

Peter Woit at "Not even wrong" (http://www.math.columbia.edu/~woit/wordpress/) seems to rate it reasonably highly.