> 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.
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
Interesting read. Why I am not sure about "Ruliads" :-), it led me to a previous post of his [0], where he talks about how he found the minimal axiom system for classical propositional logic.
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.
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.
Generating functions are often treated as purely formal objects. You only have to worry about convergence if you need to evaluate them at some complex number: https://en.wikipedia.org/wiki/Formal_power_series
Are you trying to use complex analysis to study fast-growing sequences?
If a generating function does not condense to some small symbolic expression, it is next to useless. Yes, the radius of convergence is not very relevant for G.F., however, without any convergence, there will be no shorthand symbolic expression. And that's really what we are after with G.F. For example 1/(1-x) for the OGF of the sequence 1,1,1,1.....etc. We don't care that the interval of convergence is between (-1,1). However if there was no convergence at all, the expression 1+x+x^2+x^3..... would not simplify.
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.
> 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.