[HN Gopher] Infinite series reveal the unity of mathematics
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Infinite series reveal the unity of mathematics
Author : rbanffy
Score : 53 points
Date : 2022-02-04 23:17 UTC (1 days ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| echopurity wrote:
| auggierose wrote:
| The closing sentence
|
| > But once I learned about infinite series, I could no longer see
| math as a tower. Nor is it a tree, as another metaphor would have
| it. Its different parts are not branches that split off and go
| their separate ways. No -- math is a web. All its parts connect
| to and support each other. No part of math is split off from the
| rest. It's a network, a bit like a nervous system -- or, better
| yet, a brain.
|
| is also something I recently read in similar form in Saunders Mac
| Lane's "Mathematics: Form and Function":
|
| > We cannot realistically constrain Mathematics to be a single
| formal system; instead we view Mathematics as an elaborate
| tightly connected network of formal systems, axiom systems,
| rules, and connections. The network is tied to many sources in
| human activities and scientific questions.
| refset wrote:
| Also reminiscent of Stephen Wolfram's recent thinking about the
| "Ruliad"[0], which I initially heard him discussing during the
| keynote at the recent re:Clojure conference:
| https://www.youtube.com/watch?v=FzbWAiu50MU&t=3200s
|
| [0] https://writings.stephenwolfram.com/2021/11/the-concept-
| of-t...
| zmgsabst wrote:
| jimsimmons wrote:
| Somewhat misleading title. All they talk about is infinite series
| and Euler formula. I clicked because I thought there was a new
| discovery in infinite mathematics.
| mhh__ wrote:
| Unless the title has changed that seems consistent with the
| current title.
| ogogmad wrote:
| https://en.wikipedia.org/wiki/Combinatorial_species
| goldenkey wrote:
| One thing I've been working on lately is to derive a series
| that can be used for constructing generating functions for
| superexponential sequences. Did you know that any GF, ie. OGF,
| EGF, DGF, etc.. don't exist for these sequences? Because these
| GF don't converge. The growth of the series needs to be
| balanced with the growth of the coefficients in order to
| provide convergence. We need to find a new series if we want to
| allow for Combinatorial species and Analytic Combinatorics of
| fast growing sequences.
| zmgsabst wrote:
| goldenkey wrote:
| In response to the dead poster:
|
| An ordinary generating function converges only when the
| coefficients of the sequence grow no faster than polynomial
| growth. On the other hand, exponential generating functions
| converge for sequences that grow faster than polynomials,
| including some exponential growth. Therefore, calculus of
| exponential generating functions is wider in scope than that
| of ordinary generating functions but still not large enough
| to represent superexponential sequences.
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