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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?

Mozibur Ullah
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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 Dec 01 '17 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 Dec 01 '17 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 Dec 01 '17 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 Dec 01 '17 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 Dec 01 '17 at 12:04
  • @not_here: before that phrase you've quoted, I say 'mathematical platonism'; so this phrase has/ought to be modified with that particular understanding ie 'mathematical concepts exist as platonic ideas'. – Mozibur Ullah Dec 01 '17 at 12:20
  • "if mathematics describes objects of the world, is it then a process of invention?" Are you sure about the reading od the discovery/invention dicotomy? Galileo's law describe the behaviour of falling bodies; is it an invention ? Has not it been "discovered" ? – Mauro ALLEGRANZA Dec 01 '17 at 12:55
  • @maura allegranza: whenever I've heard of the discovery/invention dichotomy its been in these fairly simplistic terms where no light has been thrown on the subject; bringing physics into it brings in another dimension altogether, but here I want to focus purely on mathematics, so for example the simplest notions - say the natural numbers; this is why I mentioned 'mathematical platonism', I don't know if there is such a notion of physical platonism. Is there? – Mozibur Ullah Dec 01 '17 at 13:06
  • Ate the level of "digging" allowed by a question-answer site (and allowed by my limited knowledge), the answer is simply: YES. See for a good review of math platonism: Marco Panza & Andrea Sereni, [Plato’s Problem: An Introduction to Mathematical Platonism](https://books.google.it/books?id=JCGCMAEACAAJ), Palgrave (2013). 1/2 – Mauro ALLEGRANZA Dec 01 '17 at 15:57
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    And see page 16: letter from Charles Hermite to Thomas Stieltjes, dated 13 May 1894, about the subject-matter of his own study: "I believe that numbers and functions of analysis are not the arbitrary product of our mind; I think that they exist outside of ourselves, with the same character of necessity as the things of objective reality, and that we encounter, or *discover* them, and that we study them like physicists, chemists, zoologists, etc." 2/2 – Mauro ALLEGRANZA Dec 01 '17 at 15:58
  • @Mauro Allegranza: thanks for the references; I wasn't asking whether mathematical platonism was true, note the important *if* beforehand - I was linking it to the question of discovery/invention. Ah, ok, now that I've read another time, I see that it does helpfully answer my question. – Mozibur Ullah Dec 01 '17 at 16:00
  • @Gordan: sure; but I'd point out that Euclidean geometry still plays an important role in non-euclidean geometry, its not simply a binary distinction which at times it seems to be often posited; I think the word sublated is useful here - Euclidean geometry is sublated into Non-Euclidean geometry. – Mozibur Ullah Dec 01 '17 at 17:04
  • @MoziburUllah sorry I erased that, but what if the first earthly project of say, the Egyptians, had been such that they had abstracted a nonEuclidean geometry first? I am just putting this as a thinking exercise. – Gordon Dec 01 '17 at 17:28
  • @gordan: Well, they would have discovered Euclidean geometry when they looked locally rather than globally; globally, they would have seen non-euclidean geometry and then some bright spark would have said, hang on, if we look locally we have another consistent geometry; the problem I have with that is that its more natural to look locally before looking globally - but its an interesting thinking exercise. – Mozibur Ullah Dec 01 '17 at 17:33
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    Survey: https://philpapers.org/surveys/results.pl – Gordon Dec 01 '17 at 17:35
  • @Gordan: this is kinda getting off topic, I was asking in the question about the discovery/invention dichotomy in relation to platonism/nominalism; thats why I put the 'if' in so as not to get entangled in the debate about the truth of mathematical platonism. – Mozibur Ullah Dec 01 '17 at 17:45
  • Yes sorry to go down that road. But it seems to me that inventions in mathematics come from deductions from "axioms". For discovery, we need a world. We need a problem posed to us by the world. – Gordon Dec 01 '17 at 17:54
  • Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/69573/discussion-between-mozibur-ullah-and-gordon). – Mozibur Ullah Dec 01 '17 at 17:54
  • You need to clarify several aspects of your question, along similar lines as suggested by @Not_Here. For one, there are realist versions of nominalism, in the sense that they are _truth value realists_ — mathematical theorems are truths — they are just not truths _about_ anything (they’re not _object realists_). See Burgess and Rosen’s _A Subject with No Object_. – Dennis Dec 01 '17 at 21:23
  • @Dennis: I'm happy with the question as it stands. There are always finer distinctions can be made, if an answer feels the need to disentangle them to anser this question then I'll read them. – Mozibur Ullah Dec 02 '17 at 03:38
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    @MoziburUllah the reason I said it needs clarification is because I think it is unanswerable in its current form. At least, any answer would be almost entirely concerned with the “disentangling”. I’m not sure if it’s close-worthy, but I do think a half-decent answer would require a _ton_ of work to disentangle. I think your chances of getting a good answer would increase substantially if you did a bit of that work. Obviously it’s ultimately up to you. – Dennis Dec 02 '17 at 03:43
  • @Dennis: I've done similar myself - there have been questions where I could have answered with a great deal of supplementary information; but I judge by the level the question its pitched at. Its written the way it is because I was prompted by an interview by three mathematicians who touched upon this topic in the simplest kind of way, had it been more in depth, then perhaps I would have asked a more in depth question. – Mozibur Ullah Dec 02 '17 at 03:56
  • @MoziburUllah perhaps you could then link to the interview that inspired it and excerpt some of the relevant portions. Maybe they’re confused about some philosophical distinctions, maybe they say something that clarifies the intended meaning. – Dennis Dec 02 '17 at 03:59
  • @Dennis: I think that the interview was pitched to a generally educated layman, rather than specialists in philosophy. – Mozibur Ullah Dec 02 '17 at 04:01
  • @dennis: here's the [radio interview](https://www.sciencefriday.com/segments/the-infinitely-surprising-career-of-a-mathematician/) - its half an hour long; have a listen and then ask whether I have pitched the question badly given what I was inspired by. – Mozibur Ullah Dec 03 '17 at 22:49

2 Answers2

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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).

Geoffrey Thomas
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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 Dec 02 '17 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 Dec 02 '17 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 Dec 02 '17 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 Dec 02 '17 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).

sand1
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