
The death of Classical logic and the (re?)birth of Constructive Mathematics - ukj
https://forum.philosophynow.org/viewtopic.php?f=26&t=26183&start=105
======
ukj
The law of identity is the cornerstone of Arostotelian/Classical logic. A = A
is True.

In the 2nd half of the 20th century the American mathematician Haskell Curry
and logician William Alvin Howard discovered an analogy between logical proofs
and working computer programs. This is known as the Curry-Howard
correspondence. Mathematical proofs are working computer programs.
[https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...](https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence)

Therefore, if we can write a working computer program which asserts that A = A
is false without producing an error then we have living proof contradicting
the founding axiom of Classic/Aristotelian logic.

I hereby reject the law of identity, and give you the law of humanity: A = A
is False.

A thing needs not be the same as itself!

Version 1:
[https://repl.it/repls/SuperficialShimmeringAnimatronics](https://repl.it/repls/SuperficialShimmeringAnimatronics)

Turing-completeness/equivalence is the bar for "reason": λ-calculus ⇔
λ-calculus ⇔ λ-calculus ⇔ λ-calculus ⇔ λ-calculus ⊇ Type theory ⊇ Mathematics

I will spell this out in English: Turing-completeness guarantees GLOBAL
consistency. Type theory allows for the containment of LOCALIZED
contradictions thus preventing explosions. This is why intuitionistic logic is
vastly superior to any "complete" logic that is NOT Turing-complete.
Consistency paralyzes human thought! We are wildly inconsistent!

Being able to contain LOCAL inconsistencies actually allows for the GLOBAL
system to become more and more consistent. This is completely and utterly
counter-intuitive to most logicians!

