Five Stages of Accepting Constructive Mathematics
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users cite it when asking about or explaining constructive mathematics, especially proof by contradiction, the law of excluded middle and the behaviour of the reals.
Recommended by 2 people across 2 different months, 2014 to 2019.
What it is
A talk by Andrej Bauer at the Institute for Advanced Study, uploaded by Mike Minestrone, on the stages of coming to terms with constructive mathematics. It covers proof by contradiction versus proving a negation, what changes without the law of excluded middle, and surprising consequences such as reals behaving differently.
What people said
Recommends Nikolaj-K · 2014-12 · Mathematics Stack Exchange · on “"Novel" proofs of "old" calculus theorems”
Constructive math is cautious about it's tools and reproves many theorems. Also, some theorems don't go through (as viewed form the constructive perspective) and are substituted by variations, e.g. this [LINK] . The are improvement in that they actually contain algorithms for forming this and that object. See also this talk [this video] . It also reminds me of Reverse mathematics [LINK] .
Recommends Thomas Davis · 2019-02 · Mathematics Stack Exchange · on “Proof by contradiction in Constructive Mathematics”
I’m watching this video on Constructive Mathematics Five Stages of Accepting Constructive Mathematics [this video] , and Andrej Bauer makes the following claim: Mathematicians call two different things “Proof by Contradiction”. One is assume $\neg p$ , blah blah blah contradiction. Therefore, $p$ . The other is assume $p$ , blah blah blah contradiction. Therefore, $\neg p$ . The first is not constructively valid, but the second is. The second is how you prove negation. I’m confused why the latter is constructively valid. Aren’t the two statements dual to each other in some sense, even constructively speaking? Perhaps I’m confused about proving negation.
Cites NotSure · 2017-01 · Mathematics Stack Exchange · on “Bijection between natural numbers and the real numbers in constructive mathematics”
In this video: [this video] (called Five Stages of Accepting Constructive Mathematics) at exactly 24:09 [this video] the lecturer says that it is possible to find a bijection between natural numbers and the real numbers in constructive mathematics. I wonder how one might do so. Thanks.
Cites Maxime Ramzi · 2018-12 · Mathematics Stack Exchange · on “Nondecreasing converging sequences without the LEM”
@Potato44 : in synthetic differential geometry, every function is differentiable but it's probably not the same model for the reals ( [this video] right after 26:35)