3 ms·

The underlying problem is that infinity doesn't exist. It's a convenient illusion to make special cases go away. It's possible to have entirely constructive mat

The underlying problem is that infinity doesn't exist. It's a convenient illusion to make special cases go away. It's possible to have entirely constructive mathematics. In a true constructive model, everything can be constructed in a finite number of steps. There are only integers, no reals.

pjbeam 2y ago
I like to think of continuity as always having more "resolution" available if a sharper "picture" is required. It is _weird_ though.
- Animats 2y ago
  Constructive mathematics can handle rational numbers. Rational numbers are continuous, in the sense of being infinitely sub dividable. That is, between N/M and (N+1)/M lies (2N+1) / (2M).
dilyevsky 2y ago
Integers are already infinite and you need infinity to express things like convergent sequences. Real numbers are a whole new level of wtf altogether
- erehweb 2y ago
  See https://math.stackexchange.com/questions/4216831/what-does-constructive-mathematics-say-about-limits https://math.stackexchange.com/questions/4216831/what-does-c... for some discussion on convergence using constructive mathematics.
- kreetx 2y ago
  It's the definition of "exists" that is contested. To me, the infinity of anything is a generator (a program that continues to print digits). Does some infinite number exists "now"? Not really (it's being printed). But is the number being printed larger than any given integer? Yes. So, does infinity exists? ¯\_(ツ)_/¯ "The Emperor's New Mind" is a great book on this and related topics.
throwaway81523 2y ago
Well sure, there is a constructive subset of the topic we call "mathematics", just like there is a subset that admits only numbers less than or equal to 5. But if you want the whole topic of mathematics, it's going to include nonconstructive theorems like the Banach-Tarski paradox. One could take a philosophical view for or against the idea that the non-measurable set in the paradox platonically "exists", but either way, those theorems are a legitimate part of mathematics. At best you can say the theorems are about mythological entities rather than "real" ones.

pjbeam 2y ago

I like to think of continuity as always having more "resolution" available if a sharper "picture" is required. It is _weird_ though.

Animats 2y ago
Constructive mathematics can handle rational numbers. Rational numbers are continuous, in the sense of being infinitely sub dividable. That is, between N/M and (N+1)/M lies (2N+1) / (2M).

dilyevsky 2y ago

Integers are already infinite and you need infinity to express things like convergent sequences. Real numbers are a whole new level of wtf altogether

erehweb 2y ago
See https://math.stackexchange.com/questions/4216831/what-does-constructive-mathematics-say-about-limits https://math.stackexchange.com/questions/4216831/what-does-c... for some discussion on convergence using constructive mathematics.
kreetx 2y ago
It's the definition of "exists" that is contested. To me, the infinity of anything is a generator (a program that continues to print digits). Does some infinite number exists "now"? Not really (it's being printed). But is the number being printed larger than any given integer? Yes. So, does infinity exists? ¯\_(ツ)_/¯ "The Emperor's New Mind" is a great book on this and related topics.

throwaway81523 2y ago

Well sure, there is a constructive subset of the topic we call "mathematics", just like there is a subset that admits only numbers less than or equal to 5. But if you want the whole topic of mathematics, it's going to include nonconstructive theorems like the Banach-Tarski paradox. One could take a philosophical view for or against the idea that the non-measurable set in the paradox platonically "exists", but either way, those theorems are a legitimate part of mathematics. At best you can say the theorems are about mythological entities rather than "real" ones.