That was my (charitable) interpretation too, until, to my dismay, Eliezer confirmed (at a meetup) that he had "leanings" in the direction of constructivism/intuitionism -- apparently not quite aware of the discredited status of such views in mathematics.
And indeed, when I asked Eliezer where he thinks the standard proof of infinite sets goes wrong, he pointed to the law of the excluded middle.
His idol E.T. Jaynes may be to blame, who in PTLS explicitly allied himself with Kronecker, Brouwer, and Poincaré as opposed to Cantor, Hilbert, and Bourbaki -- once again apparently not understanding the settled status of that debate on the side of Cantor et al. One is inclined to suspect this is where Eliezer picked such attitudes up.
Eliezer confirmed (at a meetup) that he had "leanings" in the direction of constructivism/intuitionism -- apparently not quite aware of the discredited status of such views in mathematics.
Can you elaborate on constructivism, intuitionism, and their discrediting? And what that has to do with the law of the excluded middle? I thought constructivism and intuitionism were epistemological theories, and it isn't immediately obvious how they apply to mathematics. Does a constructivist mathematician not believe in proof by contradiction?
Also, I don't know what you mean by "the standard proof of infinite sets".
[edit: sorry, the formatting of links and italics in this is all screwy. I've tried editing both the rich-text and the HTML and either way it looks ok while i'm editing it but the formatted terms either come out with no surrounding spaces or two surrounding spaces]
In the latest Rationality Quotes thread, CronoDAS quoted Paul Graham: