# What are the philosophic positions regarding the ontology of mathematical facts?

`1+1=2` and, discarding any mildly clever counter-examples that don't really matter (eg 1.4 + 1.4 = 2.8, which rounds to 3), I have a hard time imagining how the discrete quantity 1 could ever be added to the discrete quantity 1 to produce anything other than the discrete quantity 2.

Of course I'm not talking about the symbols themselves, the concepts they represent.

Now, it's my opinion that mathematical facts are more real than reality itself. That this universe and all possible univereses will rely on these types of facts. Obviously you can imagine universes with different physics from our own, different ways of dealing with space and time, and there may be certain clever ways of arranging space that make certain intuitions about quantities weird, but they aren't weird because the numbers work differently, they're weird because the numbers are put into a weird context.

So it would be my view that any reality cannot escape certain mathematical facts, it can only manipulate them into stragenesses, and so these mathematical facts are unchangable even in principle.

I know there are probably other views than that, and I may actually just be objectively wrong, so I guess my question is two-fold:

What are the names of these positions, and am I objectively wrong?

• At minimum, this would be truth-value realism. If you adopt an ontology in which mathematical objects exist as abstract objects and that's what grounds the truth of mathematical statements, this would be mathematical platonism, see the articles here and here. Jul 28 at 16:24

I have my Bachelors' degree in Math, so I can tell you what the pros think about it:

https://aimath.org/textbooks/approved-textbooks/judson/ is an example of a college-level text covering the subfield most relevant to what you're thinking about here.

Mathematics is based on a series of axioms (semi-)arbitrarily declared true. All other traits of mathematics descend from interactions of these axioms.

The way they define numbers is substantially weirder than I can describe here (See textbook linked above). (It... involves the set of nothing, and then the set of the set of nothing, and then... oh, also, what they mean when they say "set" is very important!) When you say `1+1' you're not just dealing in the concept of the number `1`, you're also enlisting the operation `+`, which has been defined to mean a particular thing.

Mathematics is in some senses more about a set of rules for determining truth or falsehood (or indeterminability!) from a set of axioms than it has anything to do with numbers. It could be, in fact, that all of mathematics is a sort of cosmic JPEG artifact of the axioms we accept as true - except, because of how observations of the world shake out, this set of axioms is applicable to the nature of 'here.'

Math doesn't really mean anything outside of how it is or is not applicable to the world; it simply is. Divorced from context, it's more a pretty and engaging toy than a philosophy with meaning?

• This is the standard and common "agnostic" view of the working mathematicians, and it is perfectly fine! There is no need to answer the question "what numbers (sets, structures) are in order to "do mathematics". But since Plato and Aristotle the question is still there, and it is an interesting philosophical question: "What numbers are?", as well as "What is good?", "What is beauty?" and so on. IMO, the fact that mathematicians "do not need" an answer to that question does not mean that the question is meaningless. Jul 29 at 9:36