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Can Kolmogorov's dream of '62 be realized?

Let me try however to formulate my long-time dreams:

  1. to formulate the general logical foundation of mathematics in a way that would allow them to be taught to 14 and 15 year olds

  2. to eliminate the distinction b/w the "rigorous" methods of pure mathematics & the "non-rigorous" methods of pure reason employed by mathematicians physicists engineers

these two problems are closely linked

-- andrei kolmogorov izvestiya 12.31.62

This is quoted in Karp & Vogeli Russian Mathematical Education, 2010.

I'm an engineer, neither a mathematician nor historian, but it appears that many important mathematicians from antiquity up to let's say up to World War II era were also scientists and engineers, whereas today, academic mathematicians (perhaps due to the value proposition of publish-or-perish) by and large seem to develop another variation on set theory axioms or investigate properties of semi- quasi- pseudo- or weak- something or other. It seems like a game.

Scientists and engineers are often sloppy with mathematical details, but on the other hand are constrained by the properties of real world systems.

Is it possible to bridge the two worlds? What would you do to realize AK's dream?

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The last two sentences are questions. If they're not real they're complex –  alancalvitti Jun 27 '12 at 3:22
@GerryMyerson I would say it is a real question, because he is asking something. However, the last two questions implicitly accept "Kolmogorov's dream" as an ideal for mathematics when I am not convinced that it is. I am especially skeptical of #2. You might be able to make a case for "not constructive". –  Andrew Salmon Jun 27 '12 at 3:25
@alancalvitti Why should we eliminate the distinction between rigor and what you call the "non-rigorous methods of pure reason"? What exactly do you mean by this? Proofs seem unsuitable for engineers but absolutely necessary for mathematics. –  Andrew Salmon Jun 27 '12 at 3:31
@AndrewSalmon, why are you asking me, it's a quote from AK –  alancalvitti Jun 27 '12 at 3:58
@alancalvitti I am inferring from your question "What would you do to realize AK's dream" that you would like to see this realized. However, I am asking why we should try to realize it, when there are important distinctions between what an engineer does in the real world and what a mathematician does when they consider an abstract object. –  Andrew Salmon Jun 27 '12 at 4:03
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closed as not constructive by Gerry Myerson, Potato, Brandon Carter, Sam, Zev Chonoles Jun 27 '12 at 8:27

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1 Answer

Partially, I think you have this view because you don't know the motivation are behind some these concepts. This is not a bad thing. Yes, sometimes it is more difficult to learn or even care about something, if it seems like it was just invented because someone could do it. The fact that you may not know why something was invented may even be one of the benefits of the modern mathematics approach. Often the author invents some concepts that comes naturally from solving some problems. This concept may have become quite useful, and mathematician have since abstracted to the essential properties, perhaps to the extent no longer recognize the original. However, with much of the "distraction" removed, properties became more evident, theorems were easier to prove, and results became more broadly applicable.

A nice example may be the abstract definition of a topological space. Upon the first encounter with this concept, one may feel that this is some nonsense set-theoretic fabrication. However, this led to other abstract ideas such as connectedness and topological definitions of compactness which has since given easier proofs or some sort of deeper understand of familiar results such as the intermediate value theorems or the extreme value theorems from calculus.

Another branch may be Number Theory. You should take a look at the sort of concepts invented in attempt to solve very simple equations. However, these results have since resulted in concepts like Dedekind Domains, Dirichlet Unit Theorem, Class number, etc that apparently are useful for other aspect of number theory.

So because the fact you feel the unnaturalness of some aspect of mathematics may very well be the result of method of mathematics that has since proven quite useful in solving problems. However, before you judge something to be invented, you should look at its history. However, personally, I would like to do mathematics without having to know all the history, the antiquated methods, the inefficiencies, and the haphazardness of past research. This unnaturalness may just be the fact that these results are just too polished that you don't see the sweat and groans of the mathematicians who had to produce them.

As for your questions of logical foundation ... Do you even consider logic as one of these branches of mathematics that are not "games"? A lot of people don't. Inherently logic is abstract and philosophical, especially foundations. You are trying to create a framework that can be used to formulate anything you want to say. Some aspect of this question is clearly philosophical. As to the existing foundations, I think ZFC set theory is already quite intuitive. Everything is a set. There are weaker logical systems such as second order arithmetics. However usually they give up logical expressiveness (in general harder to use in practice).

I think the existing language of mathematics is sufficient for even some of the applied areas. Physicist often do use concepts studied in mathematics and even phrase them in those mathematical terms. Perhaps set theory and other aspects of mathematics may appear abstract to engineers and others whose line of work requires only the ability to describe numbers, geometries, etc ... whereas the foundations set theorist wants to express everything mathematically.

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If there's one thing in the world I don't want to see it's the groin of a sweaty mathematician. –  Generic Human Jun 27 '12 at 3:51
@William, Here's an example of lack of uniqueness in foundations: ZFC implies Banach-Tarski decompositions, while in ZF+AD this is precluded but then so are free (ie nontrivial) ultrafilters on the natural numbers- this according to Herrlich's book on AC. So like parallel lines in Euclidean geometry, intuition seems relative. –  alancalvitti Jun 27 '12 at 4:03
Sadly, the Incompleteness Theorem is as hampering to a theoretical mathematician as the Uncertainty Principle can be to a theoretical physicist. Intuition is relative, and occasionally, relatively useless. –  Cameron Buie Jun 27 '12 at 4:21
@CameronBuie, interesting that you mention Uncertainty Principle, since the physical manifestations are explained directly from mathematical principle: Pontryagin duality. –  alancalvitti Jun 27 '12 at 4:38
@alancalvitti An engineer doubts the axiom of determinacy because there must be non-principal ultrafilters on $\omega$ .... You have to balance what you want, what you don't want, and how practical your foundation is. Without some form of the axiom of choice, you can not even prove countable union of countable sets are countable. Sometimes, you have to choose between non principal unltrafilters or the fact every set is measurable. Also if you like determinacy, an interested fact is that $ZFC$ + projective determinacy is consistent (if ZFC is). Perhaps this is where you would like to do math. –  William Jun 27 '12 at 5:22
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