Platonic Realism

A Platonic Argument

1. (Premise) We know about perfect circularity, equality, etc.
2. (Premise) If we know about something, then it exists.
3. Therefore, there is perfect circularity, equality, etc. (1 and 2)
4. (Premise) There is no perfect circularity, equality, etc. in, or even partly in, the physical world.
5. Therefore, there is perfect circularity, equality, etc. outside of the physical world. (3 and 4)
6. (Premise) What is dependent on the mind is subjective.
7. (Premise) Perfect circularity, equality, etc. are not subjective.
8. Therefore, perfect circularity, equality, etc., exist mind-independently outside of the physical world. (5-7)

What Platonic Realism Explains

• What objects that (univocally) are F all have in common.
• What imperfect versions of having F approximate. (Only Plato’s own version explains this.)
• What truths such as that all triangles have straight sides or that horses are mammals are about.
• How counting properties makes sense (“How many shades of green appear on this painting?” “George and Jennifer have more qualities in common than Patricia and Kenya.”)
• How relations between properties make sense (“Green is more like blue than red is like white”).
• How we can quantify over properties (“Two things are intrinsically indiscernible if and only if every non-relational, qualitative property had by one is had by the other.”)

The Third Man

1. (Premise) Whenever a bunch of beings are both human beings, there is a Form of Humanity apart from them such that these beings all participate in the Form.
2. (Premise) A Form of F is an F.
3. (Premise) Socrates and Plato are human beings.
4. Therefore, Socrates and Plato participate in a Form of Humanity which is distinct from Socrates and Plato. (3, 1)
5. Therefore, Socrates, Plato and the Form of Humanity participate in a Form of Humanity which is distinct from them. (4, 2, 1)
6. Therefore, Socrates, Plato, the first Form of Humanity and the second Form of Humanity participate in a Form of Humanity which is distinct from them. (5, 3, 2, 1)
7. And so on, viciously.

Russell’s Regress Argument Against Resemblance Nominalism

Not a valid argument as given below, but a regress generator:

1. Suppose that whenever we have a case a such that a is F, this is true because a resemble certain others things in an appropriate way.
2. Suppose a is green.
3. a is green because a resembles certain other things in color.
4. But then resemblance is to be explained in terms of resemblance, ad infinitum.

Regress of Universals Against Platonism

1. (Premise for reductio) Suppose that whenever we have a case a such that a is F, this is true because x instantiates the universal of Fness.
2. (Premise) Suppose a is green.
3. Claim (2) holds because a instantiates greenness. (2, 1)
4. a is something that instantiates greenness. (3)
5. Claim (4) holds because a instantiates instantiation of greenness. (4, 1)
6. Etc.

Pears’ Argument Against Platonism

1. (Premise) Platonism explains “green” being applicable to a by a’s instantiating greenness.
2. (Premise) To explain something about the applicability of “green” in terms of greenness is circular.
3. (Premise) To explain circularly is pointless.
4. Therefore, Platonism does something pointless.

Field’s Epistemological Argument Against Mathematical Platonism (from SEP, s.v. “realism”)

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.