Relative plausability of platonism compared to standard hypotheses

Science shows that reality is so rich and complex that explanations will inevitably lead to scenarios which from our every day perspective seem fantastic, counter intuitive, and beyond common sense. Philosophy on the other hand, uses as one of its most widely applied tools of argumentation, the principle that a statement, in order to be believable, must be readily understandable, at hand, and well-grounded in the prevailing ontology (nowadays most often materialism). Science provides many examples that this principle is inadequate in our pursuit of understanding of the world. Indeed, there exist examples where scientific reasoning leads us to quite fantastic scenarios as the best explanations of the world.

Modern theoretical physics, in considering the anthropic principle, has brought us to basically two plausible scenarios: Either there is a Creator who has fine tuned the physics in our universe so that to make life like ours possible, or there is an incredibly high number of universes besides ours, where all logically possible physical theories are realized.
Now, which of the following would you say requires the greatest leap of faith, mathematical platonism, which posits a realm of mathematical objects and relations existing independently of our own minds, or the existence of God or infinitely many universes outside of our own?
If string theory with its ten spacial dimensions or the multiverse with billions (or even infinitely many) different universes outside our own are the best explanations for how physical reality is, and if it is reasonable for a rational person to believe in such theories, then what is so strange about the belief that mathematical objects exist independently of our minds?

It is a simple task to realize that both God and the structure of masses and masses of different universes are more complicated structures than the realm of mathematics. Still it is rational to believe in one of these extremely complicated transcendental existences. In comparison, a platonic realm of mathematics, seems like a small grain of sand.

The above examples show that the philosophical argument against mathematical platonism based on the difficulty to imagine where a transcendental world like the platonic realm could exist, is an argument which is not universally applicable. It appeals to down-to-earth bias (one could say geocentrism, to resurrect an old piece of terminology in a new context), but there exist situations where it is misguided, for example in the multiverse scenario mentioned above. To realize the shortcomings of the geocentric argument, one has to take a huge leap into abstraction. To cling to the argument is to effectively say "I can't make the required leap of abstraction, therefore I don't believe in the possibility of those things".

The most vocal current anti-platonists do not reject platonism mainly on account of the geocentric argument, but because they are in the business of promoting their own alternative views which underlie their research agendas. Did you expect anything else?