alt Hacker News> mathematics is a social construct
If you believe Wittgenstein then all of math is more and more complicated stories amounting to 1=1. Like a ribbon that we figure out how to tie in ever more beautiful knots. These stories are extremely valuable and useful, because we find equivalents of these knots in nature—but boiled down that is what we do when we do math
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In my view mathematics builds tools that help solve problems in science.
You don’t really have to believe Wittgenstein; any logician will tell you that if your proof is not logically equivalent to 1=1 then it’s not a proof.
More like 1 = 0 + 1.
Read about Lisp, the Computational Beauty of Nature, 64k Lisp from https://t3x.org and how all numbers can be composed of counting nested lists all down.
List of a single item:
(cons '1 nil)
[ 1 | nil ]
List of three items:
(cons '1 (cons 2 (cons 3 nil)))
(list '1 '2 '3)
[ 1 | --> [ 2 | -> [ 3 | nil ]
A function is a list, it applies the operation over the rest of the items:
(plus '1 '2 3')
Which is like saying:
(eval '(+ '1 '2 '3))
Eval will just apply the '+' operation to the rest of the list, recursively.
Whis is the the default for every list written in parentheses without the leading ' .
(+ 1 (+ 2 3))
(+ '1 '(+ '2 '3))
How arithmetic is made from 'nothing':
https://t3x.org/lisp64k/numbers.html
Table of contents:
https://t3x.org/lisp64k/toc.html
Logic, too:
I like the Kronecker quote, "Natural numbers were created by god, everything else is the work of men" (translated). I figure that (like programming) it turns out that putting our problems and solutions into precise reusable generalizable language helps us use and reuse them better, and that (like programming language evolution) we're always finding new ways to express problems precisely. Reusability of ideas and solutions is great, but sometimes the "language" gets in the way, whether that's a programming language or a particular shape of the formal expression of something.