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I have read some of the philosophies on either side of the argument, but lack the historical overview to really get a picture of the main achievements chronologically. I'm taking nominalism to mean either rejection of universals or abstract objects, insight from either camp would be welcome.

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  • You can see : The Medieval Problem of Universals, Nominalism in Metaphysics, Nominalism in the Philosophy of Mathematics and Platonism in the Philosophy of Mathematics for historical overviews and recent debate.
    – Mauro ALLEGRANZA
    Commented Nov 2, 2016 at 7:56
  • @MauroALLEGRANZA Thanks, I had read the first two, but not the mathematics references so I will see if I can find any answers there. The problem I found with the first two is that whilst they gave an excellent historical account, I could not find in them an idea of what had been achieved in the field, as opposed to simply an account of what had been said.
    – Pseudonym
    Commented Nov 2, 2016 at 8:26
  • Achievements ? See Plato.
    – Mauro ALLEGRANZA
    Commented Nov 2, 2016 at 11:17
  • @MauroALLEGRANZA I'm sorry, I don't understand the reference, it's just taken me to a page about Plato, could you elaborate?
    – Pseudonym
    Commented Nov 2, 2016 at 13:35
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    @Conifold Thank you for your perseverance. I think I understand what you're saying. I'm not sure I like it as I think a lot of philosophers act as if it were something more than that, but your description explains the complication with my question perfectly.
    – Pseudonym
    Commented Nov 10, 2016 at 10:10

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Mark Balguer's Platonism and Anti-Platonism in the Philosophy of Mathematics is considered a critical text arguing that the ontology of mathematical objects is an open question such that there are perhaps equally good arguments on both sides for mutually exclusive positions.

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  • If I understand correctly, this would make the answer to my my question "there have been none whatsoever", at least with regards to mathematics. If in 2001 it can be argued still that both sides could be equally well argued for, then neither side has managed to produce a conclusive argument in two and a half thousand years. Has anyone considered it might be time to give up?
    – Pseudonym
    Commented Jan 3, 2017 at 11:12
  • @Pseudonym fictionalists take it that we should drop ontological commitment as a criterion of the utility of a theory. Basically we have the structure of realism, minus the realism
    – Lothrop Stoddard
    Commented Jan 3, 2017 at 13:34

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