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A perennial meta-mathematical question is whether mathematics is an invention or a discovery.

If mathematical platonism is true, it means that mathematical concepts exist as ideas, and therefore, or so it seems to me, that mathematics is a process of discovery of these mathematical platonic ideas. Is this right?

If on the other hand, nominalism is true, that is where mathematics describes objects of the world, is it then a process of invention?

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    Are you specifically asking about platonism and nominalism or are you asking about realism vs antirealism because platonism and nominalism are not the only views that fall under the realism/antirealism categories, e.g. naturalism is a nonplatonic realism and formalism is an antirealism. I also feel like your descriptions of platonism and nominalism are worded in a confusing way, specifically what you describe as platonism sounds a lot more like intuitionism (mathematical objects are mind dependent objects) and how you explained nominalism sounds like an empirical realism in line with Quine.
    – Not_Here
    Commented Dec 1, 2017 at 11:54
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    Platonism says that mathematical objects are abstract objects that exist, not that they are "concepts that exist as ideas", again that is intuitionism. The wording of "describes objects of the world" is mostly what I was referring to as being confusing though because again it sounds like Quine's empiricism that numbers are required to describe physical phenomena and therefore are real. I think that just using "realism" and "antirealism" as the views you're contrasting would lead to less ambiguity, and probably a better answer because it is more encompassing of the problem being discussed.
    – Not_Here
    Commented Dec 1, 2017 at 11:56
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    @not_here: I'm specifically asking about mathematical platonism & nominalism - thats why I referred to them; using the term 'idea' to describe platonism seemed fairly safe to me because thats how platonic ideas are referred to (or as forms), it would have been potentially confusing if I referenced Intuitionism in the question - but I didn't.
    – Mozibur Ullah
    Commented Dec 1, 2017 at 11:58
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    @not_here:personally, I think you're confusing the issue (rather than clarifying it) by bringing Intuitionism into this; the main reason why I asked the question is to disentangle that hoary chestnut about discovery/invention...
    – Mozibur Ullah
    Commented Dec 1, 2017 at 12:01
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    My entire point is that the explanations you used to describe platonism and nominalism are confusing because they do not sound like the actual views. I am not trying in anyway to bring intuitionism into your question, I am pointing out that you bringing in the statement "mathematical concepts exist as ideas" is the exact definition of intuitionism so it becomes confusing and you should either just go with the more general terms of realism and antirealism without trying to appeal to specific schools or change your definitions because right now they are clouded.
    – Not_Here
    Commented Dec 1, 2017 at 12:04

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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).

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    "causal knowledge is ruled out since abstract objects cannot enter into causal relations with our minds or anything else" Suppose I obtain causal knowledge that "A" and "A=>B" somehow. In principle then I should know B, but in practice I might not; it may involve computational work. We might call "A, A=>B means B" a Platonic object. So let's say I do the computation and arrive at B. Then, sure, the object doesn't cause me to know B; rather, the computation does. But is there reason to claim that the object cannot compel the result of the computation if performed to be B?
    – H Walters
    Commented Dec 2, 2017 at 17:42
  • &H. Walters. 'Then, sure, the object doesn't cause me to know B'. Isn't that my basic claim ? The computation may give you the relevant information about the abstract object but does it follow that the object fulfulled any causal role in the computation ? The computation may be determined by a computer programme.
    – Geoffrey Thomas ♦
    Commented Dec 2, 2017 at 19:11
  • I think we're crossing wires. To say this: "the computation may give you the relevant information about the abstract object" ...would be to grant that there is a way to access the object; namely, computation. This: "does it follow that the object fulfulled any causal role in the computation?" ...doesn't even really factor into my question at all. IOW, what I'm really asking about is this: "How we are to gain knowledge of them is not clear". (And here, I'm just focused on the Platonic view, per your interpretation).
    – H Walters
    Commented Dec 2, 2017 at 21:19
  • @H Walters. Thanks for taking my question seriously. I entirely agree that how we are to gain knowledge of them is not clear.
    – Geoffrey Thomas ♦
    Commented Dec 2, 2017 at 21:33
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The answer to the if-then question is "Yes". The text outlines two popular views which appear to present an either/or question. However this might not be the case because inventing/ discover are not a neat alternative in a neutral framework.

Richard Rorty has exposed at some length how vocabularies shape the creation and resolution of problems. Following him it seems reasonable to admit that 'discovering' is the adequate word within a platonist vocabulary, while 'inventing' pertains to some other. But it would be incoherent to say in a platonistic setting that mathematic objects are invented.

NB Rorty's view are in Phislosophy and the Mirror of Nature. Reading it should made obvious that there are no problems in Nature (pace Popper).

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