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Matheology § 196
Posted:
Jan 22, 2013 3:31 AM
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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
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