The Epistemological Argument Against Platonism
Over the years, antiplatonist philosophers have presented a number of arguments against platonism. One of these arguments stands out as the strongest, namely, the epistemological argument. In this section, we will discuss this argument and look at how platonists have tried to respond to it. The argument goes all the way back to Plato, but it has received renewed interest since 1973, when Paul Benacerraf presented a version of the argument. Most of the work on this problem has taken place in the philosophy of mathematics, in connection with the platonistic view of mathematical objects like numbers; thus, we will discuss the argument in this context, but all of the issues and arguments can be reproduced in connection with other kinds of abstract objects. The argument can be put in the following way (Balaguer, 1998):

Human beings exist entirely within spacetime.

If there exist any abstract mathematical objects, then they exist outside of spacetime. Therefore, it seems very plausible that:

If there exist any abstract mathematical objects, then human beings could not attain knowledge of them. Therefore,

If mathematical platonism is correct, then human beings could not attain mathematical knowledge.

Human beings have mathematical knowledge. Therefore,

Mathematical platonism is not correct.