Thanks for your reply. If I comment on such discussions in the future, I will try to phrase any possible disagreements in such a way as to not discourage discussion. Let me know if you think it's best if I delete the original comment and I will.
Let me just clarify some claims, just for the sake of having communicated some main points clearly.
My initial contention may be worded as follows: that mathematical and physical reality occupy separate worlds within the same universe. The existence of objects in one world is not predicated on the existence of objects in the other.
I called the mathematical phenomena 'coincidences' to emphasize just how surprising it is that they even exist. Why should the mathematical world have any such features at all? Let me give some brief examples. Forgive me if you already know this, and please bear with me if it's unfamiliar: there's no technicality involved. Groups are mathematical objects that encode symmetric patterns in some sense. Simple groups are groups that, roughly speaking, can not be described in terms of smaller constituent groups. They are like prime numbers. Finite simple groups have all been classified. They come in a few regular families, plus 26 specific sporadic simple groups. These latter ones don't belong to any family, and don't have any regular pattern among them. They simply exist, and are considered exceptional. The number of elements in the largest one among them is:
808017424794512875886459904961710757005754368000000000
It can be constructed as a part of the symmetries of some space with dimension at least 196883.
So there is a distinct symmetric pattern, one that can not be described in terms of simpler patterns, that can occur among the elements of a set of exactly that huge size, not one more or less. If you have one fewer element, it's not possible to almost construct this symmetric pattern. The pattern simply occurs at exactly that number. Furthermore, in sets with a higher number of elements there are no more exceptional symmetric patterns. That is the last one.
That is an example of a mathematical phenomenon. It's hard not to ask: why should this object exist? I mean, we know it exists because we have proof, but that doesn't really explain its exceptional existence. What compels it to exist? That number looks so arbitrary, and yet there is an extremely intricate symmetry that can only exist among exactly that many elements in a set.
In a completely different mathematical field, there is a famous "j-function", known since the 1800s at least, with very interesting properties having to do with solutions of certain algebraic equations. It can be expanded as an infinite sum, and its first few terms are:
j(q) = 1/q + 744 + 196884 q + 21493760 q2 + 864299970 q3 + 20245856256 q4 + ...
That second coefficient is 196884. It's one more than 196883, the number of dimensions needed to construct that monster group. These two objects, the group and the function, have completely different origins. And yet the fact that 196883+1=196884 turns out not to be coincidence. There happens to be a link between the two, a longe route through yet other mathematical fields, that can explain why these two specific numbers must have this relationship.
This is one of those phenomena I called 'conspiracies'. It's not an accident by any means, because we have proof of a logical link. But for such a link to exist, so many facts have to line up exactly right. it's not clear why this sort of thing happens in math. Why should the mathematical world have any such peculiar exceptional objects? What is compelling these and other apparent mathematical coincidences to occur?
There is clearly something in this mathematical world that "exists" independently from us. I mean that we may have put together the mathematical apparatus that lets us see the patterns, but we clearly did not put together that massively intricate monster group by gluing some rules together with logic. There is nothing in the axioms of ZFC that makes it at all obvious that if you continue to explore the facts it reveals, you will meet this monster.
ALL of this aside. The fact that the mathematical world is separate and yet analogous to the physical world, so as to be "unreasonably effective" in understanding it, is a completely different mystery in another direction.
Sorry for abusing your patience.