Dear Julia, Dear Yuri: A mathematical correspondence (2022)
celebratio.org
celebratio.org
> Julia thought of mathematicians “as forming a nation of our own without distinctions of geographical origins, race, creed, sex, age, or even time (the mathematicians of the past and you of the future are our colleagues too) — all dedicated to the most beautiful of the arts and sciences.”
The mathematics is way over my head, but I find this inspiring & would love to see how we might discover/co-create realms beyond such distinctions in other endeavors.
https://link.springer.com/chapter/10.1007/978-1-4471-0307-3_...
https://inference-review.com/article/doing-mathematics-diffe...
> he builds a LISP with it
cool, just found these:
The Limits of Mathematics---Tutorial Version : https://arxiv.org/abs/chao-dyn/9509010
An implementation of his Lisp, written to explore the above Tutorial : https://github.com/poppingtonic/chaitin-ait
Mathematicians speak languages non-math people can't grasp, so they gravitate toward and connect to one another.
Math simply doesn't advance without promiscuous sharing of ideas. Soviet censors notwithstanding, there's certainly a reason correspondence like this was permitted even during the Cold War.
You could say that the above is true of other sciences, but I imagine falsification of results in math is extremely difficult or just impossible. So math is mostly immune (I suggest) to the politics and protectionism that inevitably emerges around fuzzy and controversial scientific disciplines.
Just look at the good faith Julia has in her treatment of Chudnovsky's work. Even the skepticism is respectful.
Another quote I like from this piece: “No one can be a charlatan mathematician for long”.
If only that were true in certain other domains.
On one hand, the article claims that Diophantine equations are polynomials. On the other hand, it claims that when JR is true, a Diophantine equations grows faster than a polynomial.
How can a polynomial grow faster than polynomial? That seems like a contradiction to me.
Fermat: x^N + y^N = z^N. The Diophantine set only has the number N=2, because Fermat's Last Theorem tells us that no other possible N will work if x, y, z have to be integers. This is just a finite set.
Other polynomial expressions work with an infinite number of parameter settings: Pell's equation is x^2 - N*y^2 = 1. In this case, every N that is not a square will work. This is an infinite set.
Because a lot of the parameter sets are infinite like this, the size of the parameter set are instead measured using big-O notation. For Fermat's Last Theorem, {N=2} is constant, so big-O(1). For Pell's equation, {N not a square} is O(N). For other Diophantine expressions, the parameter settings are O(N^2), or O(2^N), or any other growth function you want that's achievable with Turing machines.
And this is the essence of what MRDP proved. Robinson, Davis, Putnam, and Matiyasevich over several papers proved that if you could encode a set that grew roughly like O(2^N), you also could encode any other computable set in the parameters. Approximately, demonstrating an exponentially-fast-growing family of parameters for some specific Diophantine problem was enough for Diophantine problems as a whole to be Turing complete. Matiyasevich's clinching discovery was a concrete example using the Fibonacci numbers.