There's a lot to unpack here, so first I'm going to summarize my understanding of your question:
- You disagree with Platonism.
- Based on what you have written, you sound like a nominalist (i.e. you think mathematical entities are representations of "real" or physical entities).
- Some people who support Platonism make an argument of the form "If aliens (or humanity's successors) invent or discover math, it will look similar to our math. Therefore, mathematical concepts must exist independently of humans."
- You agree with the premises of this argument, but disagree with the conclusion, because you believe that mathematics results from a deterministic process of logical deduction anchored in real-world experiences such as counting. So aliens will necessarily arrive at the same mathematics, given the same ancestral environment.
- You are asking for reasons that this argument should be interpreted to support Platonism given that there is an alternative explanation.
I am going to offer a frame challenge: This is not a good argument, and you should not concern yourself with refuting it. It's an intuition pump which presupposes that the aliens will think very similarly to us. There is no reason to assume that their mathematics will actually look similar to ours in the first place.
Humans are a heavily visual species. We (or at least, most of us) define our experiences in terms of the things we see. When humans first developed mathematics, it was heavily focused on geometry. Positive integers were a neat tool for counting, and the Pythagoreans got surprisingly far using integral ratios to describe geometric figures, but the idea of numbers existing "between the integers" took a very long time to catch on in the west. The same was true of zero, negatives, complex numbers, and many more abstract concepts. Even after these things were discovered, time and again western mathematicians stubbornly insisted that they were only meaningful insofar as they could be tied back to geometry. Western mathematics only began to shake this habit in the past couple of centuries or so (around the time of Cantor and Hilbert), which is extraordinarily recent compared to the history of mathematics as a whole.
Now I want to do a thought experiment. Imagine you have an intelligent alien species that hardly uses vision at all. Perhaps they spend most of their time underground, or perhaps they have a heavier reliance on other senses like touch or smell. Either way, they will not have the same relentless focus on geometry that early human mathematicians did.
In fact, let's go further. Imagine these aliens not only don't put geometry first, they take a long time to even invent it. Instead, their early mathematics focuses on algebra, number theory, and eventually modular arithmetic, because these fields are useful for describing (say) the complex network of tunnels that this species likes to build underground. By the time they get around to inventing geometry, they're pretty happy with the rational numbers, and perhaps the Gaussian integers, but they have not really encountered any irrational numbers yet. Geometry is seen as an esoteric, "weird" field, far away from what most of these aliens think of as "math." And, right out of the gate, geometry produces a number (sqrt 2) that does not exist. Obviously, the field must be nonsense. The species rejects geometry as a form of mathematics.
As time goes on, they eventually get around to inventing the p-adic numbers, and proceed to develop calculus in that setting. When talking about "a number," they intuitively think of a p-adic number for some suitable value of p, whereas most humans think of "a number" as meaning a real number. Perhaps, one day, the aliens return to geometry, and investigate it further, eventually developing a theory of the real numbers. But they still don't think of them as "the real numbers." They think of them as "those weird things that occasionally pop up when we play with the zeta function," in much the same way as we think of p-adics as "those weird things that occasionally pop up when we play with modular arithmetic."
If a human mathematician and one of these alien mathematicians were able to converse, they would both learn a lot of things from each other, which seems very counterintuitive if you insist that they're both working in the same overarching Platonic framework which provides the same basic set of truths and falsehoods to each side. It is almost inevitable that our chosen mathematical axioms would be incompatible with theirs, given the profoundly different starting points each of us had. So we would not be able to directly translate our theorems to their mathematics and vice-versa, without extensively tracing each theorem's underlying axioms. In other words, our mathematics and their mathematics would not just be dressed up differently, but they would entail different sets of theorems.
You might argue that these aliens are unrealistic, to which I agree. The true aliens will likely be even stranger and less comprehensible to our patterns of thought. Probably, their mathematics would not hit quite so many human-comprehensible milestones. The point is that we have no reason to believe that their mathematics would look any more similar to our mathematics than what I have described above, and in fact it would likely be substantially more different.
The moral of this story? Mathematics is not and has never been an absolute fixed point in "thought-space." This does not refute Platonism, of course, because we might suppose that the aliens have their own Platonic forms for their own mathematics, but it certainly puts a dent in the "alien mathematicians" argument for Platonism.
