[HN Gopher] Dear Julia, Dear Yuri: A mathematical correspondence...
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Dear Julia, Dear Yuri: A mathematical correspondence (2022)
Author : georgecmu
Score : 88 points
Date : 2024-07-03 01:55 UTC (21 hours ago)
(HTM) web link (celebratio.org)
(TXT) w3m dump (celebratio.org)
| quibono wrote:
| Gregory Chudnovsky is one of the Chudnovsky brothers responsible
| for the Chudnovsky algorithm for calculating digits of pi.
| trallnag wrote:
| Makes me proud to be a chud myself
| linguaz wrote:
| Such a wonderful piece. I'd not heard of Julia Robinson or Yuri
| Matiyasevich...what a touching story of two people forming a
| friendship across time, place and culture.
|
| > 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.
| Cacti wrote:
| gregory chaitin has an accessible version of the math, if
| you're interested. he builds a LISP with it.
| linguaz wrote:
| Thanks for the pointer to Chaitin, his writing seems
| approachable:
|
| 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
| mpalmer wrote:
| I liked this quote too, though I wonder if it's not more out of
| necessity and the special nature of math than anything else.
|
| 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.
| linguaz wrote:
| > I imagine falsification of results in math is extremely
| difficult or just impossible.
|
| 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.
| pierrebai wrote:
| I'm utterly confused by the description of the solution of
| Hilbert's 10th problem.
|
| 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.
| isotropy wrote:
| Yeah, this is a little bit subtle. Please ignore the lack of
| rigor below, but this is my understanding: They aren't looking
| at a particular polynomial like 4x + 3y = 2. That has a
| solution x=2, y=-2, but they don't really care about the exact
| value for (x, y), just that it's integers. They're more
| interested in the parameters: (4, 3, 2), and in general Ax + By
| = C. If you like, it's almost like hyper parameter tuning: what
| are the parameter settings for (A, B, C) where I can get this
| shape of equation to "work", e.g. give me acceptable x and y? A
| "Diophantine set" is a family of parameters that all work. It
| turns out you can think of "parameters of a given shape of
| Diophantine equation" as an exotic programming paradigm.
|
| 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.
| isotropy wrote:
| In fact, according to the Esolang Wiki
| [https://esolangs.org/wiki/List_of_ideas#Mathematics] ,
| Gregory Chaitin apparently has built a Diophantine equation
| whose parameters encode an entire Lisp interpreter.
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