We could go through the permutations of Platonism, nominalism, intuitionism, empiricism, and fictionalism. The guts of the question is whether, if Platonism is true, we can and do discover mathematical truths.
Platonism is very roughly the view that 'there is a realm of mind-independent mathematical objects (sets, numbers) whose properties mathematicians attempt to describe' ((P. Kitcher, 'The Nature of Mathematical Knowledge, Oxford, 1984, 58). In positing a mind-independent realm, Platonism is a form of realism. There are non-Platonic forms of mathematical realism, which is why 'realism' appears in the list, but I avoid them here since the question centres on Platonism, or Platonic realism, specifically.
Mathematical objects are abstract in the sense that they do not have spatio-temporal locations (Kitcher, 58) How we are to gain knowledge of them is not clear; causal knowledge is ruled out since abstract objects cannot enter into causal relations with our minds or anything else (Kitcher, 59). However, since mathematical objects belong to a mind-independent reality, any knowledge we can gain about them is discovery, not invention. If we could invent them, they would not be mind-independent.
Nominalism relies on convention, an agreement (tacit or explicit) to use mathematical notation in certain ways. There is no greater depth to mathematics than that. If convention involves invention, then nominalism involves mathematical invention. Empiricism and fictionism support invention in different ways from each other and from nominalism. Mozibur needs to get clearer about the particular view that he wants to oppose to, and contrast with, Platonism. This just needs time and inquiry.
Kitcher's book, cited above, is helpful as are P. Benacerraf & H. Putnam, eds, 'Philosophy of Mathematics', 2nd ed. (1984) and much more recently Mark Colyvan, 'An Introduction to the Philosophy of Mathematics' (2012) and S. Shapiro, 'Thinking about Mathematics' (2001).