Apr 05, 2017

Thanks, will look at both! (I've been interested in HoTT but never looked closer.) Paper looks good, too (for future reference / interested folks, full version available at http://www.ams.org/journals/bull/0000-000-00/S0273-0979-2016... )

Feb 24, 2017

It's not like this issue hasn't come up, you know. That was at the core of the controversy between Hilbert and Brouwer. But if you think that math must be constructive you must realize that the ramifications are significant and the resulting math is very different and has its own counter-intuitive results. For one, analysis becomes much harder (at least so far) and very different; for example, in intuitionistic math, every function on the reals is continuous. Some counter-intuitive results are that a countable set can have an uncountable subset; in fact, not all subsets of a finite set are even finite. Andrej Bauer wrote a nice, mostly readable introduction to constructive math, which you may find interesting: http://www.ams.org/journals/bull/0000-000-00/S0273-0979-2016...

There are numerous philosophical justifications for non-constructive math, but the pragmatic one is that it is -- as far as we know -- consistent, and coincides with constructive math in all physically realizable instances. Hilbert, for example, said that math has "real propositions" (with physically observable consequences), as well as formalistic propositions, which we can accept as they agree with the real propositions in all instances.