MATH-lec 13

Question 1

The epistemological argument against Platonism

Answer 1

The epistemological argument against Platonism says that Platonism should be rejected as an
answer to the metaphysical question because it violates this constraint:
1 An account of the subject matter of mathematics must be consistent with the possibility of
mathematical knowledge.
2 Our knowledge across different subject matters is interdependent.

3 A right account of the conditions for knowledge with respect to one subject matter must apply
to knowledge with respect to all subject matters. (It cannot be that conditions that are necessary
and sufficient for knowledge with respect to one subject matter are not necessary and sufficient
for knowledge with respect to another.) [From 2]

4 The right account of knowledge in the realm of ordinary objects and their physical properties
will include among the necessary conditions for knowledge a causal relation between the
subject’s belief and the constituents of the state of affairs that makes the belief true. (For
example, if I am to know that my dog is asleep right now, there must be a causal relation between the constituents of the [dog-asleep] state of affairs and my belief.) [The causal theory of
knowledge for the everyday case]

5 Platonism is the view the subject matter of mathematics is a realm of abstract (non-material)
objects.
6 Abstract objects cannot enter into causal relations.

7 If the Platonist is right about the subject matter of mathematics, the account of knowledge that
is right in the realm of ordinary objects and their properties does not apply to the mathematical
case. [From 4, 5, 6]

8 If the Platonist is right about the subject matter of mathematics, there can be no mathematical
knowledge. [From 3, 7]

9 The Platonist is wrong about the subject matter of mathematics. [From 1, 8]