Not at all, it’s my pleasure to talk about philosophy of mathematics (although it might be considered off-topic for this thread?)
I might go a little overboard with my next response, but don’t blame me, eh?
So yes, this is the major dividing line in philosophy of mathematics. There is the realist view, of which probably the Platonist viewpoints is the most common, that our mathematical ideas are abstractions of existing concepts and rules (compare with the shadows on the wall of Plato’s cave). And then there is the anti-realist view, of which formalism and intuitionism could be considered common, in which mathematics has no objective truth and is merely the result of rules made up by us.
I have do make the disclaimer that consider myself to be in the formalist camp (as you might have noticed). I find it more beautiful if the thing we study is just how some rules interact with each other, than as a quest to find the truth. For one, I’m assured that it is impossible to find the truth (which I’ll explain below in a digression), and for another, one of my main interests is studying how different assumptions can give rise to different interesting mathematical models. If I were searching for the truth, most of that would become meaningless.
Perhaps it is a good point to stop and think for a moment what infinity actually is. How is infinity different from “finity”? Things like “unboundedly many”, “until forever” and “approaching zero” give some intuition to what it might mean, but they never get concrete. It does give us the idea that infinity has to do with the size of things.
So let’s start with the easy part: how do we know for a certain collection of objects, what its size is? Well, we do it by counting the objects. If I have n apples, then I could stick a unique number from 1 to n on each apple (that is, count them). How do we compare sizes? If I have a collection of m apples and a collection of n pears, I put them in pairs: for each apple, I choose a pear, and for each pear I choose an apple. If I run out of one of the two fruits, then I know that that collection had less than or equal objects than the other. In particular, if I can do it in such a way that I run out of the apples and the pears both at the same time, then I have two collections of equal size.
So now let’s lift this idea to the infinite. To begin with, we have a problem: using just finite resources (time, space, objects), I can never reach infinitely many things. This is actually something you could prove using the above description of size. The idea is that at each step x in my trying to reach infinity can only use a finite number f(x) of resources, so in total I have collected f(0)+f(1)+…+f(x) resources. But since everything is finite, this is still just a finite collection of resources: I didn’t get anywhere. Surely, if I do this for infinitely many steps, I can get infinitely many things, but who says there is enough time to do this for infinitely many steps? Could I ever claim to be at a point in my construction where I have done an infinite number of steps? It is not at all something obvious we should be allowed to do so.
So, we need to assume that there exists something that is infinite to get something infinite. This already makes it incredibly difficult to ever point to some actual thing being infinite. But if we disregard the Platonic difficulties with accepting infinity (which would need us to be convinced it actually exists), we could argue in the formal way that if such a thing as infinity existed, then we could study its properties. Usually we take the set of all finite counting numbers (0,1,2,3,…), also called ordinals, as our prototype for an infinitely large collection. We could “create” such a collection by a neverending process of taking the collection of finite ordinals up to a certain size 1,2,…,n, and then making a new collection 1,2,…,n,n+1 with just one more object. If we assume what would be the result of this neverending process, we get a collection with infinitely many objects (finite ordinals in this case).
Now that we have our infinitely many ordinals, I could describe any collection of things to have an infinite size, when I could label each object in my collection with a unique (finite) ordinal. If I run out of (finite) ordinals, then I know my collection is infinite.
In fact, I wish to digress a little yet againIn fact, I wish to digress a little yet again, to discuss that there are actually many sizes of infinity. This is what gave rise to Axiomatic Set Theory, after Georg Cantor discovered this fact. There are collections that are properly larger than some infinite collection (in fact, for any infinite collection there is a strictly larger infinite collection). This might seem obvious at first sight, since we could just add more objects to an infinite collection, but this is not the case: simply adding some objects might not change the size.
For example, if I take all the even counting numbers (0,2,4,6,…), then there are just as many of them as there are general counting numbers (0,1,2,3,…): I could pair to each general counting number a unique even counting number, by pairing the number n to the number 2×n. By the above description of when two collections would have an equal size, this means that these two collections have the same size, even though “half” of the counting numbers are missing in my collection of even numbers.
However, there are properly larger collections, such as the collection of real numbers. The proof that this is true is one of the most beautiful mathematical proofs out there, and not too difficult to understand. It’s called Cantor’s diagonal argument, and I highly recommend reading it.
Here's the promised digression about that I believe it impossible to find the truth
Mathematical proofs are in the end based on some assumptions. Simply speaking, we can’t start without something. In the early 19th century, a school of mathematics arose that was specifically interested in describing a sufficient set of these starting assumptions, called axioms that were strong enough to prove all of mathematics in a very formal, logically sound way. The philosophy of this school was called logicism, made particularly famous by the attempts of Bertrand Russel and Alfred North Whitehead, resulting in the Principia Mathematica.
Now along came the 25 year old logician Kurt Gödel, who showed using a very clever trick, that it is impossible for any collection of axioms that is strong enough to do arithmetic in (that is, counting, addition and multiplication), can never be able to prove for all mathematically expressible statements whether it is true or false. Furthermore, he showed that it is impossible for such a collection of axioms to show that it would not lead to a contradiction: this means that if the axiomatic system was reasonable in that it would not have contradictory statements, then we would never be able to use this axiomatic system to prove so.
And sure enough, there were mathematical statements that have been proved to be unprovable using “standard” mathematical approaches (I’ll leave “standard” undefined, but read it as the kind of mathematics that 95% of mathematicians are concerned with). Two famous ones are the Axiom of Choice, which states that we could pick representatives from an infinite number of nonempty collections, or the Continuum Hypothesis, which states that any infinite collection of real numbers can only have two possible sizes. It has been shown for both of these statements that they could be true, and that they could be false, without making any difference to “standard” mathematics.
The question them becomes, are these statements Platonically speaking true or are they false? It will be extremely difficult to make a good case for either of these claims, since we cannot use “standard” mathematics to determine it. In my opinion this makes a strong case to refuse to assign any truth value to such statements, and instead consider them as formally following from certain (non-“standard”) assumptions and being negated by other (non-“standard”) assumptions.
I agree with this. In the abstract, the concept of infinity is something tangible, that we could work with, and create new concepts with. In the concrete, the concept of infinity does exist, since we can think of it. However, in the concrete, I am not convinced of the existence of infinity, i.e. the thing the concept talks about. To be clear: I’m ambivalent about it, I think it is very well possible that it exists (and would love it if it did), but it could equally well be that our universe is finite in nature.
Here I feel there is a flaw in the distinction between the concept and the thing itself. The space of “all the possible products of the human imagination” might be infinite (although I am not convinced of this either), but I think only the concept of this state space is part of the actual state space of our universe. Meaning that it might be that not every possible idea is in itself an existing things.
In my view, the thing that is “an idea that a human has” is an existing part of our universe, while “the idea” that a human has, is not an existing part of our universe.
Here you describe a process of creating something larger. As with the problems above, I find it not naturally obvious that such a process would have a result. There is a difference in the things a process can produce if we let it run forever, and the things a process could actually produce.
I also object to the state space of the universe expanding when ideas are born. In my view, those the “thinking of those ideas” are still part of the same state space that already existed. The object that is the idea itself, does not make part of the state space (unless it already was).