What makes mathematics factual?
In: Mathematics
It depends what you mean by factual.
All math is based off of things called “axioms”. These are essentially statements that you just assume are true, without any proof.
The axioms used for modern math are put forth by [Zermelo–Fraenkel set theory](https://en.m.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory).
Generally, axioms are picked so that they make intuitive sense, like saying that if two things are made up of the same stuff, they are equal (although some axioms needed for math might be less intuitive, like the axiom that infinite sets exist).
In any case, everything else in mathematics is proven by using these axioms.
As a quick detour, let’s go over mathematical proving. Let’s try to show that 1 + 3 == 2 + 2. Well, first we can use some addition rules (which can be proven with set theory axioms) to make 1 + 3 into 4. Now we have 4 == 2 + 2. We can apply those rules again to turn 2 + 2 into 4 to get 4 == 4.
From here, we can see that the set on the left side is made up of all the same elements as the set on the right. Using our earlier axiom, if two sets are made up of all the same elements, then they are equal. Assuming our axiom is correct, 4 == 4 is therefore a true statement, which means that 1 + 3 == 2 + 2 is also a true statement. Again, this is all assuming that we treat our axioms as true statements as well.
All of modern mathematics can be proven in this manner, just using the axioms given by set theory. Of course, it usually takes much longer. A good half of a book was dedicated just to proving that 1 + 1 == 2. **Edit:** *u/komandanto_en_bovajo has pointed out that a lot of that half-book was dedicated to setting up mathematics in general, and not just proving 1 + 1 == 2, so my previous statement is misleading.*
Anyway, this means that, if you agree the axioms from set theory are factual (go ahead and read through them and see if they make sense), the rest of math is also factual, since you can show that the rest of math is just a consequence of those axioms.
*As a side note, nothing is forcing us to use those axioms, except that most people agree they make sense, and that they haven’t led us to any contradictions or illogical outcomes yet. We used to have different axiom sets, but those eventually led to paradoxes, sort of like the mathematical equivalent of saying “this sentence is a lie”. We refined our axioms to get rid of those contradictions, but ZF set theory could still have some that we haven’t found yet. Anyway, nothing’s keeping anyone from deciding they want to make a better math and making their own axioms, except that ZF seems to make the most sense as of now.*
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