You should be asking this question of a physicist, not a mathematician — mathematical Platonism is just another religion.
Physics is clear on the question: there is a limit of entropy/information density in any finitly-bounded region of space. Initially this was demonstrated for flat spacetime in a result known as the Bekenstein bound, and was later extended to de Sitter spacetimes (and we're in an asymptotically de Sitter spacetime according to accepted cosmology). This means that physical quantities cannot be arbitrary precision (real-valued), because you can encode infinite information in a real number and that contravenes the aforementioned bound. Thus, real numbers are not real, and uncountable infinities do not exist in the physical universe. This severely limits the mathematics that actually applies to reality at a fundamental level.
Combining the above together with the fact that any causally connected system in the universe is finite in size (the limitations being accelerating expansion and the speed of light resulting in a cosmological horizon), any physical entity can be fully described by a non-deterministic linear bound automaton, which is a class of mechanistic information processing entities, even less powerful than Turing machines. That includes the human brain, and also the system comprised of the sum total of all human brains and any intelligent artifacts we ever create interacting together. The class of problems a non-deterministic LBA can solve is pretty limited. So how can mathematicians think and talk about concepts like uncountable infinities and everything in mathematics that depends on them, if their brains are based on physics in which these concepts play no part?
Let's separate the existence of thoughts on such concepts from the concepts themselves having any reality. The former are obviously connected to the physical universe via their neural correlates. As for the latter, they're easily explained as an extension of the sort of heuristics the brain uses in virtually all aspects of its functionality, as is well-known from cognitive psychology. As an example, concepts like the number pi (to a given number of digits) are just shorthands for their generative processes (to a given number of iterations or recursions). Even while mathematicians think about problems that are outside the class of those which are computable, their brains are not applying any magical non-computable processes to solve them. It's a combination of not really solving them (which would be impossible as they're not real) but processing them in other ways based on the assumption they're real, the luck and lack thereof of stochastic search that cognition oft relies on, and, without a doubt in some cases, accepting "solutions" which are wrong but unknowably so.