What makes mathematics factual?

In: Mathematics

The ELI5 answer is that math is more or less like a game. Mathematics is carried out by a predefined set of rules on a predefined set of objects. Because the game never changes, the allowable moves never change, and so the possible outcomes never change. Any situation in which the rules of the game are allowed is therefore subject to the outcomes mathematics applies to. So things like basic physics which work on just adding and multiplying numbers representing physical quantities are essentially universal because they are a situation in which you are playing the game, just using different words for the same things.

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Sometimes, the rules of the game do change. This can happen because we find out the game is broken (as in e.g. the crisis of naive set theory), and sometimes it happens because we find out that some other games are also cool or useful, or less controversial. Not all mathematicians agree that we *should* be playing the game that we are, and so they play slightly different games, but this doesn’t effect the main allowable moves of the game. The basic things like + and x are still the same, it’s that there are some weird edge cases involving things like ungodly huge infinities and what you can and can’t do with them that some people object to.

That’s like asking what makes the English language factual… math is just another abstract language we use to communicate and it can be factual or non-factual. Testing hypotheses and proving them out is generally how mathematical ideas are considered factual or not, at least until somebody manages to prove it’s wrong.

Mathematics takes arbitrary claims about the world in, and reasons about them. If you reason about unicorns, the math probably has less of a grounding in real world than if you reason about physics, say. Math sincerely doesn’t care how you choose your starting point for your reasoning. It just takes it in, and allows you to draw conclusions.

And as it turns out, many times we’ve only later found out that things that we reasoned about just due to mathematical curiosity, actually have some counterparts in reality. Like for example, non-euclidean spaces were thought to be fanciful nonsense for a very long time, until people realized that actually surface of a sphere can be seen as a non-euclidean space, so all that fanciful unicorn math actually relates directly to say, maps of the world and navigation.

Math doesn’t care. Cartographers might. Or physicists. Or electric engineers. Or programmers. Or locksmiths. Or doctors. Or whoever just so happens to start from somewhere and wants to start reasoning about where they are.

Or, you can go the other way around, start with facts and try to figure out where you are located in the world of math. For example, Newton thought that if you’re sitting in a train moving forward at 50mph, and on the next track you see a train pass you by at 30mph relative to you, you could tell how fast the second train moves relative to the ground is simply 50+30=80. Einstein then showed that that math, while totally okay math, does not describe reality. In reality, the second train moves slightly faster than 80mph relative to the ground. He located a better math that more closely aligns with the reality.

You start with the most basic of concepts. The number 1, which represents a unit, a discrete item. This item can be added or subtracted from others items. Multiplying and dividing are pretty much adding and subtracting but on a large scale. From there you can expand to more complex concepts. In a sense, these concepts are constructs, we invented them, but they are all based on counting numbers, which don’t change.

I feel like this question doesn’t make much sense. What does it even mean for math to be “factual”?

I would argue it isn’t. I mean if you take 3×4+1 you can have 2 different answers based on what order you do your operations. 13, or 15. Now people have decided that 13 is is the “correct” answer, but that decision was made arbitrarily, not off of some universal fact…

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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Do you mean, how do we know that mathematics is describing the actual universe we live in?

It probably doesn’t. However, some of it may come pretty close- enough to do physics and chemistry and all that stuff with.

The fact that maths interpret reality itself, which is fact. So in other words, you’re asking what makes fact factual. Maths are literally the interpretations of pure fact.

yeah, i agree.

1+1=2 according to who?

What do you mean by factual?