Two characteristics of Platonism are paramount. The ontological ingredient is the claim that mathematical entities exist outside of space and time, independently from us, and mathematical statements are true (or false) in virtue of how things are. This is the realist part of Platonism. The epistemological ingredient is the claim that we can perceive or intuit these mathematical entities (at least some of them), and thereby come to know their properties. There are other sources of knowledge, as well, but intuition is the one that sets Platonism apart from other forms of realism. A semi-naturalist (as I shall use the term here) is someone who accepts Platonism's ontological ingredient, but rejects the epistemic claim. Typically, they are realists, though often reluctant, who resist anything beyond empirical sources of knowledge. The aversion to intuition is usually motivated by naturalistic reasons that are by now quite familiar. A typical argument directed against the epistemic aspects of Platonism runs:

Perception is a physical process involving sound waves, photons, or other natural entities; without such physical interaction perception of any sort is impossible. Sets, numbers, and functions don't emit “platons” or anything else of the sort that could make causal contact with us; hence, there is no way we can perceive them.