I used to have the same annoyance about group and category theory; learning some about the type-theory-as-foundations work (particularly homotopy type theory and cubical type theory) has helped w/ this; in that setting, you have several distinct but well-developed notions of equality: propositional equality, which group theory and most math cares about, vs judgemental equality, which is the one that's "obviously true" by the rules of the system.
e.g. 5 = 5 is true under judgemental and propositional equality, whereas x + 2 = 2 + x is only true under propositional
Re:0.999...= 1
by tom-thistime on Tuesday April 28, 2020 @12:03PM
(#23008350)
Yeah forcing myself to keep making sure I'm defining "equality" seems to be a really useful way of "breaking my brain" until I understand what an algebraic structure "really is."
Re:0.999...= 1
by pasquinelli on Tuesday April 28, 2020 @01:04PM
(#23009052)
I think a lot of people don't think of math in terms of definitions and proof. Math was just something they were taught as kids. And even if they've gotten into more advanced math, i think the 1 = .9... question hits their kindergarten brain and they just say "no" to it the same way they'd say "no" to someone singing the alphabet song in the wrong order.
Re:0.999...= 1
by karatestomp on Tuesday April 28, 2020 @06:29PM
(#23012914)
I think a lot of people see “find me a number between 0.9999... and 1” as no more or less valid (plus, perhaps, no more or less pedantic and dickish) than “ok smarty pants, why don’t you just keep adding 9s until you reach 1, then we’ll talk.”
Re:0.999...= 1
by tom-thistime on Wednesday April 29, 2020 @11:57AM
(#23020714)
I imagine that's true sometimes.
On the other hand, hypothetically, if it were the case that we were missing a key definition that's needed for some proof, and someone didn't believe that proof, then maybe we would't quite know enough yet to decide that the person is in mental kindergarten. A better first step for us might be to supply the missing definition.
I have this problem every time I play with group theory again. You get the axioms for a group, which say there is some identity but don't explicity require the identity to be unique. You can easily prove that the identity of a group is unique ... so long as you define "unique" to mean "if element e1 and element e2 are equal, then we say they are the same element."
You could count things differently and say the identity is "not unique", it would just lead to a lot of stupid and un-illuminating consequences.