Talk:Hilbert 2nd problem

I created this article (obviously still under construction) partly in pursuit of my lifelong interest in the history and philosophy of mathematics.

I have only recently discovered the work of Gregory Chaitin. He and I are similar in age and, in addition, studied only minutes from one another along the Pennsylvania Railroad: he mathematics at CCNY in New York City, I philosophy at Rutgers University in New Brunswick, New Jersey.

Chaitin's interest in Hilbert's promotion of mathematical formalism and my own interest in Carnap's et. al. promotion of logical positivism are, I feel, mirror images of a single mind.

The subject has engaged the comments of many mathematicians and philosophers of mathematics since Hilbert's 1900 lecture. The title that Chaitin gave to his own 2000 lecture allows us to see his work as a homage to Hilbert's vision.

Feasible and infeasible

Encouraged by you, I've read that text by Chaitin, and like to share some impression. Here is my outrageous claim. Results of Goedel, Turing, Chaitin and probably some others are a wonderful proof of an intuitively evident point that a mathematical theory cannot prove "the truth, the whole truth, and nothing but the truth" about natural numbers. I do not know whether or not this was Hilbert's program, but if it really was, then it is a wonder that for Hilbert, the opposite was intuitively evident; and then we observe a dramatic change of our intuition, somewhat similar to the change from "evidently, a continuous function must be differentiable in most of the points" to "evidently, a generic continuous function is nowhere differentiable".

Once upon a time I was thinking hard about probabilistic cellular automata (there was a wonderful problem attacked by me, among others, but ultimately solved in an exciting work by Peter Gacs). And I got very pessimistic, for the following reason. Imagine a deterministic cellular automation powerful enough for having "computers" and "robots" among possible finite configurations. Introduce small randomness; being small it does not prevent these computers and robots from successful functioning during reasonably long time; being non-zero, the randomness introduces "mutations", it ensures that every possible computer/robot will emerge somewhere, sooner or later. Now ask a clever question about the asymptotic behavior of this automation on large time.

In order to answer such a question, you probably need to understand what is "the optimal civilization", a finite but growing combination of computers/robots most succesfully enlarging itself and fighting all other civilizations! Or alternatively you need to understand a sequence of civilizations that tends to optimality...

Why hope that such knowledge follows from reasonable axioms?

Yes, we mathematicians have an encouraging experience: when we want to prove or disprove something, sooner or later we can. But this only shows that our intuition is able to choose only feasible tasks.

Go to a wild, find a tiger that looks at a zebra, and ask a physicist to predict the result: who will succeed this time, the tiger or the sebra? No, the physicist cannot. He is successful in answering properly chosen questions. And the same applies to us mathematicians.

And now I wonder, which one of the following two scenarios is closer to the historical truth.

The first scenario. Hilbert believed that every question formulated in the formal arithmetic is "mathematically feasible". Thus, his intuition looks quite naive nowadays.

The second scenario. Hilbert's intention was more modest: to create a formal arithmetics that proves "the truth, and nothing but the truth" but not quite "the whole truth" about natural numbers. It was enough for him, if the theory solves all "mathematically feasible" questions.

Boris Tsirelson (talk) 11:03, 6 June 2015 (CEST)

Well, from the article (as of now) I understand that the answer to my question is "the first scenario". Boris Tsirelson (talk) 18:49, 9 June 2015 (CEST)

Yes, as you say, Hilbert, a mathematician of great renown, nevertheless entertained notions that we nowadays regard as naive.

Yet, as we say this, we smile and recall that few things in mathematics (even fewer in everyday life?) come into being fully formed and wholly mature -- as did Athena, who is said to have sprung full-blown from the head of Zeus!

It is enough to note that the subtitle of the ‘’E of M’’ article Set theory is naive to remind ourselves of the difficult journey from yesterday’s ‘’Unknown Unknowns’’ to tomorrow’s ‘’Known Knowns’’ -- I'm thinking here of U.S. Vice President Dick Cheney's famous remarks about these two notions. With respect to perplexing matters in mathematics (and in everyday life!), we repeatedly find ourselves somewhere in the midst of such a journey.

Hilbert was not the first mathematician to insist that some branch or other of mathematics needed to be placed on a firm(er) foundation, that mathematical concepts needed to be defined (more) rigourously, that mathematical assumptions needed to be stated (more) explicitly, and that more formal methods needed to replace intuition.

Hilbert must have known in 1900 what we know today, namely, that in all such efforts, the formalisms (e.g. definitions and axioms) that we develop inside mathematics are a way of expressing our regard for and paying our respect to the intuitive notions outside mathematics from which those formalisms have spring and upon which they are ultimately based.

William Hayes (talk) 17:22, 11 June 2015 (CEST) Best regards. It's always a pleasure to read and reflect on your comments.

The difference between mathematics and philosophy?

Thank you for your candid and very interesting reflection. Quite obviously you, yourself, have also been a student of philosophy!

No, never! :-) Though, one of the two my most influential teachers was a logician. [1] Boris Tsirelson (talk) 18:21, 8 June 2015 (CEST)

Thinking about this subject (Foundations of Math -- Indeed, are there any?) brought to mind a long-ago event in which one of my math profs at Rutgers, in a brief conversation, worked some mentoring magic in my undergraduate life.

Prof Dekker was, as I recall, a logician, though my studies with him were of algebra and topology. One afternoon, just before or just after a class (interesting that very important events happen on the periphery of and almost in spite of the plans we make for ourselves) I mentioned to him Nagel and Newman’s book on Godel’s incompleteness theorems, through which I was working on my own -- there was, at the time, no one in the philosophy department capable of helping me with that material. Prof Dekker made a bit of a face and then shared an anecdote from his own student past:

As an undergraduate, I had wrestled with the issue of academic direction: would I pursue philosophy or mathematics. The answer came to me as a revelation during a scholarly conference to which one one of my philosophy professors had kindly invited me.

A respected scholar rose and delivered a keynote address on a subject that was of great interest to me. Immediately afterwards, another scholar, equally respected, rose as a devil’s advocate to deliver some comments to the contrary of the keynote address. Finally, the keynote speaker rose once more and made a brief reply, which he began with the words, “But you have completely misunderstood me!”

Dekker paused for a moment, then looked at me and said the words that caused him to choose mathematics rather than philosophy:

You know, those words made the decision (between math and philosophy) for me. They would never have been said by a mathematician. We don’t ever misunderstand one another!

A bit of an exaggeration (?) and yet his words helped me with my own academic indecision, although in my case, the choice I made (for mathematics) was rendered moot by other, overarching events in my life. C’est la vie!

William Hayes (talk) 16:06, 8 June 2015 (CEST)

Feeling this article was growing unmanageably large, I today determined that it should deal only with the mathematics that preceded Hilbert's Problems Address, including (of course) his statement of the 2nd problem itself. Accordingly, I have revised the introductory remarks and excised headings for the sections no longer needed. I will deal with the subsequent history of the development of Hilbert's program itself in another article, Hilbert program, which I have also created today. Trust that this seems reasonable. Thanks for your occasional, watchful, and helpful edits. --William Hayes (talk) 17:43, 25 July 2015 (CEST)