Intuitionism and Constructive Mathematics 80-518/818 — Spring 2016
From the course description:
In this seminar we shall read primary and secondary sources on the origins and developments of intuitionism and constructive mathematics from Brouwer and the Russian constructivists, Bishop, Martin-Löf, up to and including modern developments such as homotopy type theory. We shall focus both on philosophical and metamathematical aspects. Topics could include the Brouwer-Heyting-Kolmogorov (BHK) interpretation, Kripke semantics, topological semantics, the Curry-Howard correspondence with constructive type theories, constructive set theory, realizability, relations to topos theory, formal topology, meaning explanations, homotopy type theory, and/or additional topics according to the interests of participants.
Texts
- Jean van Heijenoort (1967), From Frege to Gödel: A Source Book in Mathematical Logic 1879–1931, Cambridge, MA: Harvard University Press.
- Michael Dummett (1977/2000), Elements of Intuitionism (Oxford Logic Guides, 39), Oxford: Clarendon Press, 1977; 2nd edition, 2000.
- Michael Beeson (1985), Foundations of Constructive Mathematics, Heidelberg: Springer Verlag.
- Anne Sjerp Troelstra and Dirk van Dalen (1988), Constructivism in Mathematics: An Introduction (two volumes), Amsterdam: North Holland.
Additional resources
- Constructivists’ Hymn by S. Y. Maslov.
- Constructive Mathematics by Douglas Bridges and Erik Palmgren in the Stanford Encyclopedia of Philosophy (SEP).
- Intuitionism in the Philosophy of Mathematics by Rosalie Iemhoff (SEP).
- Intuitionistic Logic by Joan Moschovakis (SEP).
- The Development of Intuitionistic Logic by Mark van Atten (SEP).
- Luitzen Egbertus Jan Brouwer by Mark van Atten (SEP).
- The nLab: a wiki-lab on Mathematics, Physics and Philosophy from the point of view of (higher) category theory.
Not online but a Spring course at Carnegie Mellon with a reading list that should exercise your mental engines!
Any subject with a two volume “introduction” (Anne Sjerp Troelstra and Dirk van Dalen), is likely to be heavy sledding. 😉
But the immediate relevance to topic maps is evident by this statement from Rosalie Iemhoff:
Intuitionism is a philosophy of mathematics that was introduced by the Dutch mathematician L.E.J. Brouwer (1881–1966). Intuitionism is based on the idea that mathematics is a creation of the mind. The truth of a mathematical statement can only be conceived via a mental construction that proves it to be true, and the communication between mathematicians only serves as a means to create the same mental process in different minds.
I would recast that to say:
Language is a creation of the mind. The truth of a language statement can only be conceived via a mental construction that proves it to be true, and the communication between people only serves as a means to create the same mental process in different minds.
There are those who claim there is some correspondence between language and something they call “reality.” Since no one has experienced “reality” in the absence of language, I prefer to ask: Is X useful for purpose Y? rather than the doubtful metaphysics of “Is X true?”
Think of it as helping get down to what’s really important, what’s in this for you?
BTW, don’t be troubled by anyone who suggests this position removes all limits on discussion. What motivations do you think caused people to adopt the varying positions they have now?
It certainly wasn’t a detached and disinterested search for the truth, whatever people may pretend once they have found the “truth” they are presently defending. The same constraints will persist even if we are truthful with ourselves.