Does maths exist inside or down outside the universe?

Idk what this means, the universe is just all space. So you can't really have "outside the universe". Constructs will stay the same wherever you are.

Given that abstract concepts exist within our heads, does that mean they must exist within the universe, and furthermore, if they exist only within our heads are they meaningful?

I don't know what within the universe means.

Yes - maths serves the purpose of being practically applicable. The constructions are inspired by the real world, but are consistent and make sense in their own right. Even if 2 + 2 = 4 had no physical intuition behind it, (can't think of a better example) for example because you had a universe where everything was a continuum and you can't really have separate objects to count - our construction of the integers and definition of 2, +, =, 4, would still make sense and so we could still say this is true.

Well, it depends on how you would define the term universe. There are two ways of defining it I suppose
1) everything physical - all matter and space and time itself from the the beginning of the Big Bang to the further most stretches of space and in the furthest distance of time. However this leaves abstract concepts in something of a grey area

Or

2) Anything that could, does or can exist whether physical, ideological or abstract.

Space is usually defined in the former instance.

Not really sure what you're getting at, sorry

I’m letting you define the term ‘universe’ how you want and then answering my question how you want to.

Well the universe is just all space.

For a concept to "exist" I'd merely say someone has to have thought of it, preferably written it down or communicated it otherwise.

So the universe is separate from ‘existence’ as such and and something could be said to exist even if it wasn’t in the universe. If you say a thought exists in side someone’s head such as, for example a social construct like ‘gender’ then can it be said to meaningfully exist in any objective sense?

Not sure what meaningfully exists in an objective sense means.

Yes - you can write down a definition of gender and as long as it is not self-contradictory and does not rely on/follow from false statements, it's a valid definition. However, if it's not widely used you can't expect to have others understand what you mean.

So you would say this is a meaningless conversation as there is not an abstract concept of maths that meaningfully exists, just, as you implied earlier it’s practical applications.

It meaningfully exists in the sense that we can write it down, it's logically consistent and makes logical sense, and ties in with other concepts that we've defined. Concepts usually don't come from nowhere, they'll be motivated by something else, which'll be motivated by something else, etc. Eventually, it'll tie in to some sort of practical application. (eg. into an experimental science) There's a misconception that pure maths is all abstract and not at all applicable, in some cases mathematicians basically just do the gruntwork for other sciences to eventually use. Doesn't sound too fun, but it is.

Yes, I know maths has applications in the sciences. I’m interested in what you mean by ‘concepts’ and what it means for concepts to be ‘motivated by something else’?

Mainly talking about concepts of particular objects in maths, or the concept of an object having a named property, or transformations that can be made to said objects, and so on. (this covers most definitions in maths)

Well, say we assume all objects of a particular type X have property Y, but we actually discover that we can have objects of type X that don't have property Y. We might decide that objects of type X with property Y are sufficiently interesting to be examined in their own right, we might give these objects their own name. Though arguably this isn't really "creating" a new class of object, rather naming one that already exists. An example of a construction motivated by this might be to look at what objects we can create outside of type X that might be considered to satisfy property Y in some sense. Both of these instances appear throughout fields like topology.

In terms of concepts motivated by the real world - integration for example arose from wanting to find the area and volume (possibly in higher dimensions than just 3) of regions of space. However, it transpires that integration is extremely useful in places that seem completely disconnected from the measure of area and volume, for example in solving differential equations. They're also used extensively in probability, where integrals are generalised even further. (in measure theory, where you generalise integrals to be with respect to some measure) But even if we were in a universe where there wasn't such a thing as area or volume or these things weren't unique, (we can [probably] model these mathematically but I'm not familiar enough with measure theory/geometry) we could still make sense of the construction of an integral given everything surrounding it was defined properly, it just wouldn't at all conform to the intuition of that universe.

This might be mostly tangential and not answer your question in which case sorry lol.