Foundations of Constructive Analysis

What Hacker News says

1. black_knightApr 2020
   If you want to work with the real numbers intuitionistically (or constructively), you quickly find out that infinite decimal expansions are not what you want.

   In classical mathematics all the usual definitions of real numbers (decimal, Cauchy sequences and Dedekin cuts) are equivalent. If you overthrow the Law of Excluded middle, these are all different.

   Infinite decimal expansions are bad intuitionistilcally for several reasons. The first one which come to mind is that you cannot add numbers together. Imagine your numbers started 0.33333 and 0.66666. OK, so far it would seem that the sum would start 0.99999, but somewhere down the line one the firs number could contain two 4s, making a 1 carry all the way up and leaving one behind, so that it should in reality be 1.00000000001…

   On the other hand there could also show up a 2 later, making it 0.999999998. Thus, you cannot decide weather the first decimals should be 1.00 or 0.99 without looking at infinitely many decimals. And the fact that 0.999… = 1.000 will not help you out, since 1.00000000001 ≠ 0.999999998.

   Being able to define addition on decimal expansions is equivalent for constructivists to solving the halting problem. It cannot be done.

   It turns out Cauchy sequences are better behaved, and (with a bit computational improvement) you can make a lot of things work out. See Bihshop's book, Foundations of Constructive Analysis, for details.

   Show morethread
2. bigbillheckMay 2021
   As a mathematician, I have yet to encounter a reason as to why I should care about 'constructive mathematics' or 'neutral mathematics'.

   Bauer brings up non-Euclidean geometry but the thing there is about 40-ish years into it Beltrami and then Klein started showing why hyperbolic geometry is interesting and important (per Milnor[0]). Constructive mathematics is older than that by now ('Foundations of Constructive Analysis' came out in 1967, and the field was old even then), where are its applications to other fields of math? By 'application' I'm meaning stuff like, for example, how certain isometries of the plane (PSL2(R)) are connected to the Poincare disk.

   I'm even less impressed by 'neutral mathematics'. OK, you proved a well-known result without using excluded middle or choice, that's cool, I beat Doom with only the pistol. Tell me why that's not just stunts and gimmickry.

   [0]: https://projecteuclid.org/journals/bulletin-of-the-american-...

   Also of note by the same author: https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016...

   Show morethread