Comment by math_comment_21
1 month ago
In topology, if you have a continuous surjective map X --> Y, then it might have a continuous splitting (a map the other way which is a "partial" inverse in the sense that Y ---> X ---> Y is the identity) e.g. there are lots of splittings of the projection R^2 ---> R, you could include the line back as the x-axis but also the graph of any continuous function is a splitting.
On the other hand, there's no continuous splitting of the map from the unit interval to the circle that glues together the two endpoints.
So the category of topological spaces does not have the property "every epimorphism splits."
As the article mentions, the axiom of choice says that the category of sets has this property.
So we can think of the various independence results of the 20th century as saying, hey, (assuming ZFC is consistent) there's this category, Set, with this rule, and there's this other category called idk Snet, that satisfies the ZF axioms but where there are some surjections that don't split, and that's ok too.
Then whatever, if you want to study something like rings but you don't like the axiom of choice, define a rning to be a snet with two binary operations such that blah blah blah, and you've got a nice category Rning and your various theorems about rnings and maybe they don't all have maximal ideals, even though rings do. You're not arguing about ontology or the nature of truth, you're just picking which category to work in.
Yeah, it's important to think of these axioms as choosing the rules of the game, rather than what intuitively makes sense. The real question is if playing the game produces useful results.
Axioms are also introduced in practical terms just to make proofs and results "better". Usually we talk in terms of what propositions are provable, saying that indicates the strength/power of these assumptions, but beyond this there are issues of proof length and complexity.
For example in arithmetic without induction, roughly, theorems remain the same (those which can still be expressed) but may have exponentially longer proofs because of the loss of those `∀n P(n)`-type propositions.
In this sense it does sometimes come back to intuition. If for all n we can prove P(n), then `∀n P(n)` should be an acceptable proposition and doesn't really change "the game" we are trying to play. It just makes it more intuitive and playable.
I’m not sure what you mean by “theorems remain the same”. If you take away induction from Peano arithmetic, you get Robinson arithmetic, which has many more models, including (from https://math.stackexchange.com/a/4076545):
- ℕ ∪ {∞}
- Cardinal arithmetic
- ℤ[x]⁺
Obviously, not all theorems that are true for the natural numbers are true for cardinals, so it seems misleading to say that theorems remain the same. I also believe that the addition of induction increases the consistency strength of the theory, so it’s not “just” a matter of expressing the theorems in a different way.
I would agree more for axioms that don’t affect consistency strength, like foundation or choice (over the rest of the ZF axioms).
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> If for all n we can prove P(n), then `∀n P(n)` should be an acceptable proposition
But how can you prove that P(n) for all n without induction? Maybe I misinterpret what you're saying, or I'm naive about something in formal languages, but if we can prove P(n) for all n. then `∀n P(n)` just looks like a trivial transcription of the conclusion into different symbols.
I think the crux of the matter is that we accept inductive arguments as valid, and so we formalize it via the inductive axiom (of Peano arithmetic). i.e., we accept induction as a principle of mathematical reasoning, but we can't derive it from something else so we postualte it when we come around to doing formalizations. Maybe that's what you mean by it coming down to intuition, now that I reread it...
Poincaré has a nice discussion of induction in "On the nature of mathematical reasoning", reprinted in Benacerraf & Putnam Philosophy of Mathematics, where he explicates it as an infinite sequence of modus ponens steps, but irreducible to any more basic logical rule like the principle of (non-)contradiction
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Good point. I would argue, however, that having nicer proofs is a "useful" result of the game.
Spoken like a true formalist.
It doesn't really have to mean anything when we say that the reals are a larger set than the natural numbers - that's just the conclusion of the game that we are playing.
What fraction of people who "know" that there are more reals than natural numbers, do you think really understand that this is not an eternal verity of mathematics, but only a conclusion that follows from a particular set of rules that we're playing the mathematics game with?
> What fraction of people who "know" that there are more reals than natural numbers, do you think really understand that this is not an eternal verity of mathematics, but only a conclusion that follows from a particular set of rules that we're playing the mathematics game with?
The claim that there are more reals than naturals holds given classical ZF(C) set theory. But there are alternative set theories in which the reals are countable, e.g. NFU+AxCount. These alternative set theories ensure all reals are countable by rendering Cantor’s diagonalisation argument invalid, since their axioms are too weak to validate it. But, they contain all the same reals as the high school mathematics concept of “reals”. So, there are many reals, and that some of them are countable and others are not are indeed “eternal truths” (it is an eternal truth that whatever axioms have the consequences they do), but the everyday (non-expert) concept of reals isn’t any of them in particular - and it is unclear if the dominance of classical notions in mainstream professional mathematics was historically inevitable or a historical accident - maybe, on the other side of the galaxy, there exists some alien civilisation, in which different foundations of mathematics are mainstream, because their mathematics took a different evolutionary course from ours - maybe for them, reals are classically countable, and uncountability is an exotic notion belonging to alternative foundations of mathematics
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> Spoken like a true formalist.
Doesn't seem to be a bad thing. There are some famous cranks who reject the concept of infinity, since I suppose they have problems wrapping their head around it.
> What fraction of people who "know" that there are more reals than natural numbers, do you think really understand that this is not an eternal verity of mathematics, but only a conclusion that follows from a particular set of rules that we're playing the mathematics game with?
People misunderstand mathematics all the time. It's ok, it's part of the journey.
How is it different from using ZF as a meta-theory to study ZF(C)? Is there anything special about category theory vis-à-vis ZF as a meta-theory? You're not arguing about ontology or the nature of truth, because you've picked category theory as your ontology just like you could pick ZF or ZFC.
Category theory gives a structural framework for discussing these things. The various categories live side by side and can be related with functors. This allows a broader view and makes it easier perhaps, to understand that there isn’t a right answer to “what is true” about sets in the absolute.
But then you would think there is a right answer to “what is true” about categories, and you would face AC again.
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Doesn't "continuous" make all the difference here? AC doesn't contain a comparable limitation, so the analogy doesn't work that week.
Yes, but the parent comment is trying to say "imagine the world would only be made up of topological spaces and continuous maps". Then the retraction principle would not hold.
Careful here, you might wake up the diabol (of pure algebra)