Instead, he took a philosophical position on the matter of whether Mathematics is discovered or invented: https://web.math.princeton.edu/~nelson/papers/rome.pdf
He sided with Computer Scientists.
> Plato was a dreadful fellow, the source of a persistent evil from which the world has not yet been liberated, [...]
Does anyone more familiar with Plato know what Nelson could have meant by that?
Edit:
Alright, I think I found the reason: "platonism" is a form of mathematical realism, while Nelson strongly favored mathematical formalism.
Instead, he took a philosophical position on the matter of whether Mathematics is discovered or invented: https://web.math.princeton.edu/~nelson/papers/rome.pdf
He sided with Computer Scientists.