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I think it starts as a nagging feeling that there's a common thread between diverse stuff like transforming triangles, transforming more complicated/interesting

I think it starts as a nagging feeling that there's a common thread between diverse stuff like transforming triangles, transforming more complicated/interesting shapes, numbers, and other weirder stuff. It's a natural instinct, at least for math people, to try to nail down and name that common thread. And then it takes, as the article mentioned, a lot of thinking and trying things to figure out the best way to formalize it. At least that's how I feel it in a niche way, trying to figure out my own formalisms for some things. Maybe you're just not that type of person, though.

bbor 2y ago
This is close, but I think it's more than an instinct: it was a philosophical challenge. Math is part of a larger project to formalize thought, and "we have a bunch of tools for a bunch of different things that work in different ways" is a lot less metaphysically/intuitively satisfying than "we have a single cohesive system of formalization that can be applied to any system of quantities and qualities found in human experience, all based on the same solid foundation." This challenge/goal was best expressed by Bertrand Russell and Alfred Whitehead, I think: https://plato.stanford.edu/entries/principia-mathematica/ https://plato.stanford.edu/entries/principia-mathematica/ The present work has two main objects. One of these, the proof that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental concepts, and that all its propositions are deducible from a very small number of fundamental logical principles... will be established by strict symbolic reasoning... The other object of this work... is the explanation of the fundamental concepts which mathematics accepts as indefinable. This is a purely philosophical task.
- andrewflnr 2y ago
  Well yes, but it takes a certain kind of instinct to take on that philosophical challenge, and again to guide your particular angles of attack on it. I took the question to be, within the framework of formalized math, how do people arrive at particular abstractions like groups? How do you pick the axioms and rules?

bbor 2y ago

This is close, but I think it's more than an instinct: it was a philosophical challenge. Math is part of a larger project to formalize thought, and "we have a bunch of tools for a bunch of different things that work in different ways" is a lot less metaphysically/intuitively satisfying than "we have a single cohesive system of formalization that can be applied to any system of quantities and qualities found in human experience, all based on the same solid foundation." This challenge/goal was best expressed by Bertrand Russell and Alfred Whitehead, I think: https://plato.stanford.edu/entries/principia-mathematica/ https://plato.stanford.edu/entries/principia-mathematica/ The present work has two main objects. One of these, the proof that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental concepts, and that all its propositions are deducible from a very small number of fundamental logical principles... will be established by strict symbolic reasoning... The other object of this work... is the explanation of the fundamental concepts which mathematics accepts as indefinable. This is a purely philosophical task.

andrewflnr 2y ago
Well yes, but it takes a certain kind of instinct to take on that philosophical challenge, and again to guide your particular angles of attack on it. I took the question to be, within the framework of formalized math, how do people arrive at particular abstractions like groups? How do you pick the axioms and rules?