slackenerny

Oh, that's another wonderful story — of Clifford algebras: http://en.wikipedia.org/wiki/Dual_number , useful in automatic differentiation.

As to your general point of what constitutes "internal representation", someone below tries to convince people that definitions involved with Reals are arbitrary and can be made differently thus forming Hyperreals. But no. One first have to construct Reals anyways, then one arrives at infinity, which is not a number, hyperreal or otherwise, but a consequence of numbering per se. Nothing ad hoc here. Hyperreals are that or another notation, but the main thinking is still the same, just paradoxes pop in different places, the only gain is increase in confusion as hyperreals, tuns out, do not conform to that "internal representation" in many more places than the poor 0.(9) .

OMIGOD, just noticed, its a 389-page thread!

And I thought 1st page was already pretty epic..

Sorry for I might have not expressed myself clearly. "There is no number you can name between these numbers shown in this notation" is in English and appeals to intuition, it does not show that there is indeed no such number. Food for thought: http://scienceblogs.com/goodmath/2009/05/you_cant_write_that...

You have to trust it and you have to trust your teachers.

No. Mathematics is a virtue of explanation (i.e. proving). No trust involved.

It may be intuitive in English, just as 1/3 = 0.(3) thing is convincing, but you'll lose your infinit(el)y-hostile audience at a proof.

That is fortunate. I wouldn't like Turing to become another archetypal wicked scientist in tv shows puppetry inventory. Compulsive schizophrenic Nash seducing boys, autistic Newton, insane Cantor crawling on the floor whispering "infinty, infinty" are enough for me already.

I've read up on this just now and it seems that logo indeed had nothing to do with commemorating Turing's oppression. The rainbow was due to a design trend at the time, Commodore and Atari logos sporting rainbows too.

How?

As to PG's definition of mathematics as study of terms that have a precise meaning, this is wrong too, as exemplified by problems of set theory and its logic, from which mathematicians escaped into more loosy setting of category theory and categorical logic.

I think article "When is one thing equal to another" by Barry Mazur somewhat fits into the theme of this blog note http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf

I first got embarassed by not knowing of that direct way here: http://gowers.wordpress.com/2008/07/30/recognising-countable...

Ideally, a theory based on which engineers built something, but not the other way around, i.e. when something exists and some theorists are trying to build a de facto model of it. Edit: a theory that involves infinities.

Spot the contradiction in this sentence. Even if I would like to argue with you on unedefined problem you have, I can't satisfy your requirements when they're contradictory.

I see PDE's in TCP papers, I don't know what's your problem.

Of course I expect you will dismiss that as too "astral" of an example, but ask Vint Cerf if he did not need any calculus. And then ask him (or Lloyd Wood) again about his new suite of protocols designed for guaranteed connectivity in spite of enormous latencies in orbital and interplanetary internet (with cisco routers with PDE-optimized buffer sizes orbiting already on several satellites).

The real reason why it is a paradox doesn't have much to do with notion of infinity per se, but with problem of infinite decision aka Supertask.

Which reminds me of an article on physics of supertasks that among other things somewhat outlines how classical theory is more paradoxical before than after quantization http://th-www.if.uj.edu.pl/acta/vol36/pdf/v36p2887.pdf (43pp but easy reading).

Take any serious book on modeling bus and routing infrastructure (and not vocational guide on using protocols designed by someone else) and you will see plenty of partial differential equations.

Calkin and Wilf cleverly avoid duplications by arranging reals not on a lattice but on a (Stern-Borcot) tree. http://en.wikipedia.org/wiki/Calkin%E2%80%93Wilf_tree http://www.mathlesstraveled.com/?p=94

http://telescoper.wordpress.com/category/bad-statistics/

which drops state and restarts every two hours.

How you and capable people manage to keep an eye on so many fields is a mystery for me.

Oh, so sorry.

PS. I got that as consciously and purposely insulting.

What I wanted to say, and probably failed, is that beautiful patterns do not form any interesting science, that is they lead to no knowledge. This is a classical dichotomy between intuitionist/inductive and deductive philosophies of science.

1/ Aristotle's classification of forces and the idea of quintessence;

2/ Plato's idea that planets move necessarily in circles, because it's the most beautiful and simplest shape. That one witheld progress in astronomy for 2000 years;

3/ Beautiful idea that the market is a statistical system, and thus can be modeled by tools of mathematical physics. That the assumptions of these models are not met in economy, no one bothered much till just recently.

So you take some guiding rule first, and then force world around to accept it.