Date: Jan 22, 2013 3:31 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Matheology § 196
Matheology § 196

Platonism about mathematics (or mathematical platonism) is the

metaphysical view that there are abstract mathematical objects whose

existence is independent of us and our language, thought, and

practices. Just as electrons and planets exist independently of us, so

do numbers and sets. {{No there is a difference. All electrons and

planets exist, but ideas do not exist unless someone has them. If all

sets would exist as complete sets, then obviously they would exist in

the platonic shelter. But then this shelter would contain all sets -

and its cardinality would be greater than its cardinality. If,

however, the shelter would not exist as a complete set, but only as a

class or so, why then should any set be complete?}} And just as

statements about electrons and planets are made true or false by the

objects with which they are concerned and these objects' perfectly

objective properties, so are statements about numbers and sets.

Mathematical truths are therefore discovered, not invented. {{This is

a proof by naive belief.}}

The most important argument for the existence of abstract mathematical

objects derives from Gottlob Frege and goes as follows {{Gottlob

Frege: "Foundations of Arithmetic", Blackwell, Oxford, Translation by

J.L. Austin (1953)}} language of mathematics purports to refer to and

quantify over abstract mathematical objects. And a great number of

mathematical theorems are true. But a sentence cannot be true unless

its sub-expressions succeed in doing what they purport to do. So there

exist abstract mathematical objects that these expressions refer to

and quantify over. {{This argument is similar to Kant's ontological

proof of God (1763). Contrary to Frege Kant noticed his slip during

his lifetime (in 1781).}}

[Øystein Linnebo: "Platonism in the Philosophy of Mathematics",

Stanford Encyclopedia of Philosophy (2009)]

http://plato.stanford.edu/entries/platonism-mathematics/

Regards, WM