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 Hilber, 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 historic 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)