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In the Stanford Encyclopedia of Philosophy the article about Realism has an argument in section 2 presented by a philosopher named Field regarding Platonism.

The argument goes this way.

  1. Platonic realism is committed to the existence of acausal objects and to the claim that these objects, and facts about them, are independent of anyone's beliefs, linguistic practices, conceptual schemes, and so on (in short to the claim that these objects, and facts about them, are language- and mind-independent).

  2. Any causal explanation of reliability is incompatible with the acausality of mathematical objects.

  3. Any non-causal explanation of reliability is incompatible with the language- and mind-independence of mathematical objects.

  4. Any explanation of reliability must be causal or non-causal.

  5. There is no explanation of reliability that is compatible with both the acausality and language- and mind-independence of mathematical objects. Therefore,

  6. There is no explanation of reliability that is compatible with platonic realism.

But I don't understand the 3rd one. How is that any non-causal explanation of reliability is incompatible with the language- and mind-independence of mathematical objects.

Can somebody please explain this in simple terms ?

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  • 1. numbers: they exist etc.
    – Mauro ALLEGRANZA
    Mar 8, 2018 at 10:12
  • 2. we agree on the reliability of our beliefs about the domain of e.g. numbers. But if numbers do not "intercat" with the material world, how we (that are part of it) can know them ?
    – Mauro ALLEGRANZA
    Mar 8, 2018 at 10:16
  • 3.but if mathematical objects are (as we suppose) mind- and language-independent and they have no spatiotemporal relations to anything, human being included, how we interact with them (how we know them) ? According to Field, the 2original" paltonic theory of some sort extra-corporeal world accessible to the mind is no longer teneable.
    – Mauro ALLEGRANZA
    Mar 8, 2018 at 10:19
  • This person seems very confused. If I keep rolling an ideal die until I get a six, the fact I get a six at the end is causally determined by my physical system. Each individual roll is not, being a (theoretically) random, independent value. Equally, mathematical objects can be outside of physical causality, but the fact we talk about those ones in particular is clearly not. We work on models with useful correspondences because those are useful, we don't work on ones that aren't because they are not. Note that brains are physical systems, which necessarily constrains logic too.
    – Veedrac
    Mar 8, 2018 at 10:45
  • Note also that though the thing being modelled doesn't necessarily interact causally, the models themselves do. We get power from number systems by manipulating symbols, not by manipulating the raw elements themselves (whatever that would mean).
    – Veedrac
    Mar 8, 2018 at 10:49

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See Hartry Field's Realism, Mathematics, and Modality, Blackwell (1989), page 230:

a realist view of mathematics involves the postulation of a large variety of aphysical entities - entities that exist outside of space-time and bear no causal relations to us or anything we can observe - and there just don't seem to be any mechanisms that could explain how the existence and properties of such entities could be known.

The problem arises in part from the fact that mathematical entities, as the platonist conceives them, do not causally interact with mathematicians, or indeed with anything else. This means that we cannot explain the mathematicians' beliefs and utterances on the basis of the mathematical facts being causally involved in the production of those beliefs and utterances [...]. Perhaps then some sort of non-causal explanation of the correlation is possible? Perhaps; but it is very hard to see what this supposed non-causal explanation could be.

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