Recommended reading
Section outline
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Please see articles and resources in this sections. They will be the primary focus of this class, but we will consider other articles as well.
The collection "Philosophy of mathematics" (Eds. Benacerraf, Putnam) contains "old" influential papers, some of them are certainly worth reading. We will read some of them.
The collection "The Oxford Handbook of Philosophy of Mathematics and Logic" (Ed. Shapiro) contains more recent papers, some of them again worth reading. We will read some of them.
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Students mustMark as done
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We have discussed Hilbert article on the infinite and Moore's historical context. We have illustrated the use of infinitary methods in analysis of arithmetics using Loeb measure on omega_1-saturated models. See Cutland's article below for more details on non-standard analysis and measures.
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In the next topic, we will extend our focus to type theory. Originally developed by Russell and Whitehead in response to set-theoretic paradoxes, it started to be influential due to connections with functional programming and constructive mathematics. Let us start by reading a paper by Wadler, "Propositions as types" and Zach "The significance of Curry-Howard isomorphism" to gain some initial understanding of the concepts.
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With the advent of forcing, concepts of infinity extended from "size", i.e., how hight a given infinity is in terms cardinals, to "width", i.e., to combinatorial properties making the universe wide in the low parts of the cumulative hierarchy. See the enclosed link for a review of "logical" and "mathematical" compactness principles (corresponding to infinity concepts).
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