Monday, July 12, 2010

Future Directions

It is kind of open what I am going to do with this notebook/blog. I don't really know myself. Personally, I prefer TeX as my markup language as opposed to html, which is why I don't write as much as I should.

(Rant: TeX is just so much more convenient! It actually allows me to change notation with the flip of the wrist! For example, consider \let\propersubset=\subsetneq, then I simply use A\propersubset B and if I hate the notation...well, one line of code changed! With html, I have to change everything by hand. Or counters...html has no counter macros, grr...)

It would be nice to "categorify" Bourbaki's work. Take that with a grain of salt! What I mean by this is to be as comprehensive as Bourbaki (I'm reading through his Algebra right now), presenting definitions as e.g. a group object instead of a group, a magma object instead of a magma, etc.

That is to say, present the same material "internalized" in an arbitrary category. So for an example of this, consider the following definition:

Definition 1. A Magma Object consists of an object M in a monoidal category C equipped with a morphism μ:M⊗M→M.

I must confess that such an endeavor is appealing to me, but I might do it in LaTeX (taking advantage of its flexibility).

Short Summary

I think I will probably end up writing up a cohesive collection of notes — in the spirit of Bourbaki — that is self contained covering all of mathematics (from Set Theory and Category Theory foundations to...whatever!). However, I think I'll TeX it up, and post it online.

(As an aside, there is an interesting project called LuaTeX (there is also LuaLaTeX). There is no memory limits for it, and it has embedded Lua code. Perhaps it would be interesting to use this when writing notes involving numerical calculations?)

Monday, June 21, 2010

A Remark on Reading Bourbaki

A very brief and small remark that may be helpful to those that are studying Bourbaki. The first volume of the Elements of Mathematics (The Theory of Sets) is more or less useless.

The only useful (and in my humble opinion, coherent) parts of that book is the "Summary of Results".

Although, if one were really "hardcore", one would have a collection of e.g. composition notebooks to write notes on the series. By giving actual explanation and examples, it should expand the size of the text several fold.

Also, it helps to create a "cheat sheet" of notation. Bourbaki used bizarre notation since, I assume, they had to work with typewriters.

Unfortunately, no one uses their notation. So, we are forced to come up with a "Rosetta stone" to translate their notation into modern notation.

Some guidelines for your "Rosetta Stone for Bourbaki" might be:

  1. Include the book and page numbers where it is first introduced or defined.
  2. Include definitions, since those too are "Bourbaki-dependent".
  3. Have a separate "Stone" for each book.

I just thought that it may be helpful to someone trying to read through this ancient tome…

Addendum Tuesday August 16, 2011 at 01:10:11PM (PDT)

After some more research, I found out that the bizarre system Bourbaki uses in The Theory of Sets is really something called "Epsilon Calculus."

Math Overflow had a discussion on Bourbaki's epsilon calculus which is useful, and the Stanford Encyclopedia of Philosophy's page is instructive.

I doubt that this system was intended to be fully used, since (as some pointed out in the math overflow discussion) "even trivial proofs require an astonishing number of steps directly from axioms. Existence of the empty set can be proved with 11,225,997 steps and transfinite recursion can be proved with 11,777,866,897,976 steps."

The internet encyclopedia of philosophy states:

The growing awareness of the larger meaning and significance of epsilon calculi has only come in stages. Hilbert and Bernays introduced epsilon terms for several meta-mathematical purposes, as above, but the extended presentation of an epsilon calculus, as a formal logic of interest in its own right, in fact only first appeared in Bourbaki's Elements de Mathematique (although see also Ackermann 1937-8). Bourbaki's epsilon calculus with identity (Bourbaki, 1954, Book 1) is axiomatic, with Modus Ponens as the only primitive inference or derivation rule. Thus, in effect, we get:

(X ∨ X) → X,
X → (X ∨ Y),
(X ∨ Y) → (Y ∨ X),
(X ∨ Y) → ((Z ∨ X) → (Z ∨ Y)),
Fy → FεxFx,
x = y → (Fx ↔ Fy),
(x)(Fx ↔ Gx) → εxFx = εxGx.

This adds to a basis for the propositional calculus an epsilon axiom schema, then Leibniz' Law, and a second epsilon axiom schema, which is a further law of identity. Bourbaki, though, used the Greek letter tau rather than epsilon to form what are now called "epsilon terms"; nevertheless, he defined the quantifiers in terms of his tau symbol in the manner of Hilbert and Bernays, namely:

(∃x)Fx ↔ FεxFx,
(x)Fx ↔ Fεx¬Fx;

and note that, in his system the other usual law of identity, "x = x", is derivable.

The principle purpose Bourbaki found for his system of logic was in his theory of sets, although through that, in the modern manner, it thereby came to be the foundation for the rest of mathematics. Bourbaki's theory of sets discriminates amongst predicates those which determine sets: thus some, but only some, predicates determine sets, i.e. are "collectivisantes". All the main axioms of classical Set Theory are incorporated in his theory, but he does not have an Axiom of Choice as a separate axiom, since its functions are taken over by his tau symbol. The same point holds in Bernays' epsilon version of his set theory (Bernays 1958, Ch VIII).

Over at the nLab, the entry on Choice Operators really helps explain what that pesky τ operator is in Bourbaki's Theory of Sets.