The Axiom of Choice - Conclusion
Posted May15-11 at 09:06 AM by micromass
Welcome to my final blog on the axiom of choice. This time I'm going to tell you my personal opinion on the matter. The question is a philosophical one, since Godel and Cohen proved the independence of the axiom of choice, and had thus shown that there is no mathematical problem with either accepting or refusing the axiom of choice. And the problem with philosophical questions, is that everybody is correct!
Firstly, I'm not against the axiom of choice at all, I have used the axiom of choice quite a lot of times in my work and I will continue to do so. I think that a mathematics without the axiom of choice is necessarily an incomplete, quite ugly kind of mathematics. You'll have tons of problems in general topology, analysis and algebra!
But of course, I do see problems with choice. It's counterintuitive and it yields counterintuitive results (=Banach-Tarski). Let me give a quote of Borel (who was very much against choice!):
As you may notice from this quote, Borel was not against every kind of choice. Indeed, he accepted a countable number of choices, this is the so called "axiom of dependent choices", which I could sadly not discuss. The good thing about the axiom of dependent choices is that a lot of analysis and topology can be done with it, and that it doesn't imply ugly things like Banach-Tarski and the like! So the axiom of dependent choices might seem like a good replacement...
The question for me is, whether you want a mathematics that is beautiful or a mathematics that is realistic. If you want a beautiful, elegant mathematics, then choice is more than necessary. But it doesn't make mathematics realistic, as I don't believe that the axiom of choice exists in reality. But then again, I don't believe that infinity exists in reality, does that mean that we have to do a mathematics without infinity? This would be madness, but I don't think it would be impossible. I've had the luck to talk with a "strict finitist" once who believed that infinity has no place in mathematics and who was actually trying to give a model for the real numbers without using infinity. His thoughts were that he would probably fail in inventing a kind of mathematics without infinity, but at least he would have a better understanding in where exactly infinity is necessary in mathematics. I've always remembered these words, and I think they apply in this case too. Mathematics without choice is something that is destined to fail, but it's still interesting to know where exactly choice is used in mathematics.
Again: I don't want people to stop using choice, but all I wanted to obtain with this blog is to let people know that the axiom of choice is very intriguing and has a lot of exciting consequences. Also, if people are now more aware that they are using it, then I could say that I have achieved my goal.
I hope you have enjoyed this series!
Firstly, I'm not against the axiom of choice at all, I have used the axiom of choice quite a lot of times in my work and I will continue to do so. I think that a mathematics without the axiom of choice is necessarily an incomplete, quite ugly kind of mathematics. You'll have tons of problems in general topology, analysis and algebra!
But of course, I do see problems with choice. It's counterintuitive and it yields counterintuitive results (=Banach-Tarski). Let me give a quote of Borel (who was very much against choice!):
Quote:
The axiom of choice seems to me to be entirely devoid of sense. As regards a denumerable infinity of choices, they cannot, of course, all be performed, but we can at least indicate such a procedure that, if we establish it beforehand, we may be sure that each choice will be made within a finite period of time; therefore, if two given systems of choice are different, we are sure to notice this after a fnite number of operations. When an infnite number of choices is not denumerable, it is impossible to imagine a way of defining it, i.e. distinguishing it from ananalogous infinite number of choices; thus it is impossible to regard it as a mathematical creation which can be introduced inarguments.
The question for me is, whether you want a mathematics that is beautiful or a mathematics that is realistic. If you want a beautiful, elegant mathematics, then choice is more than necessary. But it doesn't make mathematics realistic, as I don't believe that the axiom of choice exists in reality. But then again, I don't believe that infinity exists in reality, does that mean that we have to do a mathematics without infinity? This would be madness, but I don't think it would be impossible. I've had the luck to talk with a "strict finitist" once who believed that infinity has no place in mathematics and who was actually trying to give a model for the real numbers without using infinity. His thoughts were that he would probably fail in inventing a kind of mathematics without infinity, but at least he would have a better understanding in where exactly infinity is necessary in mathematics. I've always remembered these words, and I think they apply in this case too. Mathematics without choice is something that is destined to fail, but it's still interesting to know where exactly choice is used in mathematics.
Again: I don't want people to stop using choice, but all I wanted to obtain with this blog is to let people know that the axiom of choice is very intriguing and has a lot of exciting consequences. Also, if people are now more aware that they are using it, then I could say that I have achieved my goal.
I hope you have enjoyed this series!
Total Comments 7
Comments
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Great work, I really enjoyed reading it! I particularly sympathize with your metaphysical views on AC, as they are mine also.Posted May16-11 at 04:03 PM by Gingia 
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Enjoyed all the posts. So, now that you've done Choice, what's next, Micro?Posted May17-11 at 03:46 PM by jhae2.718
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Thanks both! I have some ideas for some next blogs. Some will be pure mathematics, some philosophy and some personal thingies. In particular, I might post something on fractals (but it's kind of hard because you can only post 10 LaTeX equations in blogs...). I might do a study on crackpot behaviour, etc. Wait and see!
Posted May17-11 at 05:25 PM by micromass
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Micromass, the ten-equations constraint is not necessarily a problem, as you can always attach a pdf.Posted May18-11 at 04:02 AM by Demystifier
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Great posts.
Choice is independent of the other axioms - but why does that make it particularly a philosophical question. Infinity is (I think) independent of the others, as is regularity and replacement. What makes the adoption of choice particularly philosophical as opposed to the adoption of the other axioms?Posted Jun7-11 at 01:41 PM by yossell
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Quote:Quote by yossell
Great posts.
Choice is independent of the other axioms - but why does that make it particularly a philosophical question. Infinity is (I think) independent of the others, as is regularity and replacement. What makes the adoption of choice particularly philosophical as opposed to the adoption of the other axioms?
Indeed! Why this special prejudice? Or is this a case of anything outside of logicism is for the poet and artist only? What about Peano's 5th axiom? Do you accept that any ordered proposition is true (ergo TRUE) given the veracity of some first proposition and another relating a subsequent proposition? Without that, it will get might hard to demonstrate much of anything... On the other hand, that too is strictly speaking an article of faith...
Inquiring minds want to know!
(and thanks for this series, very nice to read this stuff again after a long absence)Posted Jun19-11 at 01:40 PM by Pilot7
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Bravo!
The best part was about having a beautiful and elegant Mathematics. One can ignore "Choice" or "Infinity" for that matter, but the price is really high.
I can ignore the stars and pretend they don't exist, but in doing so I lose the pleasure of looking at the skies.Posted Mar5-13 at 10:26 AM by Bachelier
