Of Mathematics and Radical Change: Alain Badiou’s Set-Theoretical Ontology by Glenn Gomes

Hi Glenn Gomes,

I looked in vain for the radical change promised in the title. Except for who commented on set theory, Riemannian geometry and the like in which fashion, I did not find anything I was not yet aware of. Well, some quoted sentences correspond to my own attitude. However, where are concrete applications?

Some radical comments in my essay will definitely shock you. Please be nonetheless fair, try to understand and possibly refute my factual arguments. If you are an independent thinker I may even hope you will overcome what you learned.

You quoted someone unknown to me who did not mention Euclid: "The axiomatization of set theory as the foundation for mathematics completed the process begun by Descartes and the arithmetization of geometry, namely, the liberation of mathematics from all spatial or sensory intuition."

Descartes resumed ancient geometry which culminated in Euclid's axioms and his anything but intuitive definition of number, I revealed the issue of intuition mainly the other was round than set theorists, and I consider arithmetization a method that does not need illusory intuitively based (set theoretic) rigor. Admittedly there were and still are only a few opponents to set theory.

In my essay I quoted one out of 16 more of less intriguing contributions by Detlef D. Spalt. Perhaps you do not even know his name.

You wrote "... universe of Cantor's that was properly axiomatized by Ersnt Zermelo and Abaraham Fraenkel ...". Sorry, I see four bewildering details.

Eckard

Hello Eckard,

Thank you for your comments! I'm impressed that none of this was new to you, but I imagine that much of it will be for others. Let me see if I can clarify some of your specific concerns:

1) What is "radical change"? In Badiou's language, radical change would correspond to a complete rupture in the prevailing bodies (sets) and languages (predicates) of a situation through the forcing and incorporation of a generic set. In socio-scientific language, this would correspond to a traditional Kuhnian "paradigm shift" where a new discovery, or change in perspective, in response to theoretical stress may cause a complete break with the current paradigm and create a new paradigm incommensurable to the initial one. (I had wanted to include a section on how I believe Badiou to have, in a way, described a mathematical formalization of Khunian paradigms and paradigm shifts, but there was no space). The key here is a "complete" rupture with the current state of affairs, not due to some mystical intervention, but through the work of committed individuals.

I tried to argue the Einstienian revolution as an example of such a radical change/paradigm shift that may be formally descibed using the set-theoretical ontology outlined in the paper. The beauty of Badiou's philosophy is its being a philosophy of "praxis": a philosophy that claims to tie this mathemtical ideology to actual practice. I will refer you to Badiou's works for how he applies this set-theory (as well as category theory) to actual/historical mathematical, political, and artistic situations. So as not to take up too much space here, I can give more examples myself if you (or others) request them. I personally find these practical applications fascinating.

Additionally, radical change may be contrasted with what Badiou calls mere "modification" or "simple becoming" of a situation; akin to a Khunian "reinforcing" of the dominant paradigm, where a scientific innovation may occur, but still falls under the language of that paradigm.

2) The quote you mention is from Peter Hallward, a philosopher and foremost commentor of Badiou's works. I chose the quote because I felt that it perfectly expressed both the crucial philisophical stakes and mathematical stakes involved in the axiomatization of set theory. I had no intention of mentioning Euclid at that point.

3) I have not heard of Mr. Spalt , but will look him up. I am curently travelling, but will read your essay as soon as聽I聽get the chance!

4) While it was Cantor that formulated the basics of set-theory, it was Zermelo and Frankel (amongst others) who axiomatized Cantor's principles. I do not believe this to be a point of contention, unless I am missing something?

Please let me know if you have any more questions, and thank you for the comments!

- Glenn

Hi Glen,

If I was young and educated as were you and many others, I too would definitely have no chance to question that Kuhnian (you wrote twice Khunian) paradigm shifts towards set theory and Einstein's theory of relativity are necessary steps in the correct direction.

Please do not take it amiss if I suspect your knowledge of German language is at best limited. And since you wrote "Cantor which" instead of "Cantor who" and your name is Gomez, I guess your background could be Spanish. Profound command of German is a must for anybody who intends to read Spalt because his books were not yet translated into English. It is also highly recommendable reading the original papers by Georg Cantor, Zermelo, Hilbert, Fraenkel, Einstein (you wrote Einstien), Hermann Weyl in German.

Is Badiou's philosophy "that claims to tie this mathemtical ideology to actual practice" really practical? The topic of this contest is: Which of Our Basic Physical Assumptions is Wrong?".

Do you know that Fraenkel (you wrote Frankel) in [7] of my essay could not deny that Cantor's definition of an infinite set was untenable? Zermelo in 1904 and 1908 seemingly rescued Cantor's paradise by fabricating the axiom of choice based on the exhaustion of the inexhaustible. Cantor himself who had driven Kronecker in desperation got insane and died in a madhouse.

Eckard

Dear Edward and Glenn,

You are invited see the result of radical thinking in mathematics in the essay on Five Dimensions of universe .

You will see,w e question validity of arithmatic numbers to express quantity and reach a subset of transfinite numbers that can express the quantity.

Look forward to your evaluation on departure from conventional approach to mathematics.

thanks & Regards,

Vijay Gupta

Dear Glenn Gomes,

Law of sines for tetrahedral, may be applicable to describe the string dynamics of the infinite universe for quantizing in finite conditions, in that its consistency in continuum is expressional.

With best wishes,

Jayakar

Hello Glenn,

Very interesting and unusual essay for the contest. It makes us look for new ways to solve old problems in the foundations of physics and mathematics. I constantly read and ponder in this regard the paper by Zenkin «Scientific Counter-Revolution in Mathematics» and contemplate his idea: "the truth should be drawn and should be presented to" an unlimited circle "of spectators".

Sincerely, Vladimir Rogozhin

Well now, if this isn't precisely the point.

We are loathe to question the characteristics of our own shadow -- let alone even enumerate them in the first place -- because we're just too busy forcing the rest of the universe to conform to our strict beliefs.

Bravo!

A very good essay. In my opinion, the essay fits the topic perfectly.

P.S. That comment was aimed at Ekart, not Glenn.

Glenn

I noted your attitude to non- euclidean geometry

In the last essay competition my essay

http://www.fqxi.org/community/forum/topic/946

dedicated ANALOGY BETWEEN PHYSICS AND MATH.

Can you comment?

If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is [math]R_1 [/math] and [math]N_1 [/math] was the quantity of people which gave you ratings. Then you have [math]S_1=R_1 N_1 [/math] of points. After it anyone give you [math]dS [/math] of points so you have [math]S_2=S_1+ dS [/math] of points and [math]N_2=N_1+1 [/math] is the common quantity of the people which gave you ratings. At the same time you will have [math]S_2=R_2 N_2 [/math] of points. From here, if you want to be R2 > R1 there must be: [math]S_2/ N_2>S_1/ N_1 [/math] or [math] (S_1+ dS) / (N_1+1) >S_1/ N_1 [/math] or [math] dS >S_1/ N_1 =R_1[/math] In other words if you want to increase rating of anyone you must give him more points [math]dS [/math] then the participant`s rating [math]R_1 [/math] was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process. I hope the FQXI community will change the rating process.

Sergey Fedosin

Dear Glenn,

Thanks for your important contribution! After reading your essay, I looked at your bio expecting to find you an eminent mathematician in one of the world's great research institutions and was surprised to see that you are on another path entirely! Regarding your essay, some thoughts come to mind:

1. Nice discussion of Cantor, set theory, and ordinals. In my own efforts to understand quantum gravity, I have found it necessary to deal with something similar, though somewhat more general (which I call a "semiordinal;" the object in footnote 14 of my essay is an example of one). All this arises from the most naive physical ideas of cause and effect and reasonable local conditions!

2. I've run into the continuum hypothesis in thinking about physics too! It's funny how most people regard these things (and other "purely mathematical issues" like undecidability and the uncountable axiom of choice) as physically irrelevant when they are anything but.

3. Regarding ZFC, Russell's paradox, etc... as you know the ordinals form a proper class, not a set. Hence, again these issues become physically relevant from the most primitive physical considerations.

4. Regarding Godel's work and model theory... Torsten Asselmeyer-Maluga, Jerzy Krol, and Michael Goodband all have excellent essays here that would interest you.

5. Regarding geometry, as an aspiring algebraic geometer, I strongly suspect that the geometry of smooth manifolds may ultimately prove "too good to be true" in regard to spacetime structure. But it's been fantastically useful and astonishingly accurate so far.

6. Excellent endnotes!

Thanks again for the interesting read! Your work rates very highly in my opinion. Take care,

Ben Dribus