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But what are all these precise and beautiful mathematical statements?
(An aside to philosophers: It may appear that I am implicitly endorsing the Kantian idea that mathematical statements are synthetic a priori truths.
Mathematical statements are written in a defi- nite system of symbols, and one reduces all of the kinds of mathematical rea- soning one is interested in doing to a precise set of rules for manipulating those symbols.
In mathematics conceived of in this purely "formalistic" way, mathematical statements do not "mean" anything.
Statements about numbers and their properties we can call "mathematical statements," while statements about mathematical state- ments we can call "meta-mathematical statements."
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