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Thank you for the extensive reply. There's a hint in there about commutativity inspiring the connection, but my mental model is now simply "Euler did it", which

Thank you for the extensive reply. There's a hint in there about commutativity inspiring the connection, but my mental model is now simply "Euler did it", which somewhat like creationism relieves me from having to ask further questions. Something I used to imagine: what if we were radically different creatures, like ant colonies, and by habit we communicated non-linearly (with thousands of limbs and organs swarming in parallel all over our mathematical work, perhaps written in 3D)? That could make equations mostly trivial, since our symbols wouldn't be constrained to any particular arrangement in the first place: and that seems kind of advantageous. But then grasping the concept of "non-commutative" would be a real strain for these poor ant-hills. They'd have to deliberately reintroduce linear ordering, maybe with special symbols to mark precedence.

082349872349872 2y ago
(a) if you haven't read it, Chiang, Story of Your Life (1998) might have interesting aliens. (b) the special symbols is a good point. Reading older maths papers is cool because you get to see all sorts of things people tried before we settled on what we use now. Two works that come immediately to mind: Principia Mathematica (1910) uses various numbers of dots instead of parentheses to mark precedence, while Peano, Arithmetices principia: nova methodo exposita (1889) uses very modern-looking notation, including parens as we would use them, but its expository text is all in latin! (c) I don't think your aliens would have any more trouble with non-commutative than we have with commutative. Have you heard of the Boom Hierarchy? It starts with trees; when we add an associative law, so (AB)C == A(BC), then we only have flat lists; when we add a commutative law, so AB == BA, then we only have unordered bags; and finally when we add an idempotent law, so AA == A, then we have unduplicated sets. It turns out (exercise!) that if we have information encoded in any of these representations, we always have at least one way to represent the same information in all the other representations, such that we can "round trip" between any two levels of this hierarchy without losing any information. So for programming, where we care about time and space, picking ordered or unordered representations can be very important, but for maths, where all that matters is the existence of invertible functions between all these representations, that decision is unimportant. Does that make sense?
kjellsbells 2y ago
I dont want to get too far into a joint and doritos speculation here, but I wonder if the fact that humans have a very small capacity for holding multiple objects in their minds, a small number of physical digits, and excellent visual acuity is why we do a lot of math the way we do. Group theory comes out of symmetry for example. Algorithms come out of linear stepwise problem solving. It takes us considerable mental effort to think about problems in ways that are not like this. An example that stayed with me for years is when Adelman of RSA fame considered whether DNA could be used as a computer. (Spoiler: yes). It basically does all the computations at once, and then discards all the non optimal solutions.
User23 2y ago
> That could make equations mostly trivial, since our symbols wouldn't be constrained to any particular arrangement in the first place You might find this little tidbit[1] from C.S. Peirce by way of John Sowa interesting then. Existential Graphs (EG) are an unordered diagramatic representation of mathematical logic. And Peirce is the real deal. His more conventional notation was adopted by Peano (who substituted the familiar symbols for capital sigma and pi (which created confusion when used in the context of broader proofs, even though sigma and pi pretty much directly correspond to what existence and universality mean)) and he is credited as an independent co-discoverer of both quantifiers with Frege. For EGs, only one axiom is necessary: a blank sheet of assertion, from which all the axioms and rules of inference by Frege, Whitehead, and Russell can be proved by Peirce’s rules. As an example, Frege’s first axiom, a⊃(b⊃a), can be proved in five steps by Peirce’s rules Peirce gives us the entire predicate calculus with three rules and one axiom. And of course it's all built on NAND. And another teaser: In the Principia Mathematica, Whitehead and Russell proved the following theorem, which Leibniz called the Praeclarum Theorema (Splendid Theorem). It is one of the last and most complex theorems in propositional logic in the Principia, and the proof required a total of 43 steps ... With Peirce’s rules, this theorem can be proved in just seven steps starting with a blank sheet of paper. John Sowa himself is also no slouch, having been one of the leading lights of the earlier AI push. I expect advances in modern AI will come when we stop trying to do everything with ngrams and start building on richer models of knowledge representation. [1] https://www.jfsowa.com/pubs/egtut.pdf https://www.jfsowa.com/pubs/egtut.pdf

082349872349872 2y ago

(a) if you haven't read it, Chiang, Story of Your Life (1998) might have interesting aliens. (b) the special symbols is a good point. Reading older maths papers is cool because you get to see all sorts of things people tried before we settled on what we use now. Two works that come immediately to mind: Principia Mathematica (1910) uses various numbers of dots instead of parentheses to mark precedence, while Peano, Arithmetices principia: nova methodo exposita (1889) uses very modern-looking notation, including parens as we would use them, but its expository text is all in latin! (c) I don't think your aliens would have any more trouble with non-commutative than we have with commutative. Have you heard of the Boom Hierarchy? It starts with trees; when we add an associative law, so (AB)C == A(BC), then we only have flat lists; when we add a commutative law, so AB == BA, then we only have unordered bags; and finally when we add an idempotent law, so AA == A, then we have unduplicated sets. It turns out (exercise!) that if we have information encoded in any of these representations, we always have at least one way to represent the same information in all the other representations, such that we can "round trip" between any two levels of this hierarchy without losing any information. So for programming, where we care about time and space, picking ordered or unordered representations can be very important, but for maths, where all that matters is the existence of invertible functions between all these representations, that decision is unimportant. Does that make sense?

kjellsbells 2y ago

I dont want to get too far into a joint and doritos speculation here, but I wonder if the fact that humans have a very small capacity for holding multiple objects in their minds, a small number of physical digits, and excellent visual acuity is why we do a lot of math the way we do. Group theory comes out of symmetry for example. Algorithms come out of linear stepwise problem solving. It takes us considerable mental effort to think about problems in ways that are not like this. An example that stayed with me for years is when Adelman of RSA fame considered whether DNA could be used as a computer. (Spoiler: yes). It basically does all the computations at once, and then discards all the non optimal solutions.

User23 2y ago

> That could make equations mostly trivial, since our symbols wouldn't be constrained to any particular arrangement in the first place You might find this little tidbit[1] from C.S. Peirce by way of John Sowa interesting then. Existential Graphs (EG) are an unordered diagramatic representation of mathematical logic. And Peirce is the real deal. His more conventional notation was adopted by Peano (who substituted the familiar symbols for capital sigma and pi (which created confusion when used in the context of broader proofs, even though sigma and pi pretty much directly correspond to what existence and universality mean)) and he is credited as an independent co-discoverer of both quantifiers with Frege. For EGs, only one axiom is necessary: a blank sheet of assertion, from which all the axioms and rules of inference by Frege, Whitehead, and Russell can be proved by Peirce’s rules. As an example, Frege’s first axiom, a⊃(b⊃a), can be proved in five steps by Peirce’s rules Peirce gives us the entire predicate calculus with three rules and one axiom. And of course it's all built on NAND. And another teaser: In the Principia Mathematica, Whitehead and Russell proved the following theorem, which Leibniz called the Praeclarum Theorema (Splendid Theorem). It is one of the last and most complex theorems in propositional logic in the Principia, and the proof required a total of 43 steps ... With Peirce’s rules, this theorem can be proved in just seven steps starting with a blank sheet of paper. John Sowa himself is also no slouch, having been one of the leading lights of the earlier AI push. I expect advances in modern AI will come when we stop trying to do everything with ngrams and start building on richer models of knowledge representation. [1] https://www.jfsowa.com/pubs/egtut.pdf https://www.jfsowa.com/pubs/egtut.pdf