[xscott]

No matter how small you go, between every two real numbers is a computable number, and between every two computable numbers is a real number that's not computable. If you restricted yourself to computable numbers, are you sure there are any holes in your graphs and functions? Can you point to or name one?

:-)

[seanhunter]

I understand the point about density and I'm fine with the concept of functions being continuous over a restricted domain (eg the rationals even) but if I draw a line and label one end 0 and one end 1 then I have plotted the set of all numbers in that interval, not just the ones we find computationally convenient.

The historical context about the constructivist movement is that it was a religiously-inspired objection to the work of Cantor, who some random bishop said was challenging God with his work on transfinite numbers because God owned infinity. I just find it weird that now people try to pretend that it's somehow more rigorous when really it's just an alternative axiomatic perspective that started in this shonky way and has grown to a point where it's just about respectable.

[xscott]

I can't speak to the religious bits or history. That's certainly not my motivation for thinking about this stuff. It's not about computational convenience either. Both of those seem like strawmen, but maybe they're relevant to other people.

The problem to me is that the Reals which aren't computable are absurd. We've never used any of them in all history. We can only put names on a very special few, those are countable, and we don't even know their value very well.

And the main way we prove that the Reals are uncountable is to use a proof by contradiction. It would take too long to spell it out, but they aren't really just contradicting "Reals are Countable". It's "All that other stuff we think is true AND Reals are Countable" that gets contradicted.

Once you accept the Reals, the Axiom of Choice is not simply obvious any more. And if you go down that path you get things like the Banach-Tarski paradox. To me THAT ought to be a proof by contradiction that we've made a mistake somewhere.

More interesting than that though: If the universe we live in requires non-computable Reals to describe it carefully, then it says something very weird about determinism. In order to compute a future state of a system, we need to use numbers we can't compute?!?

[seanhunter]

> We've never used any of them in all history.

You just used them yourself in your previous post to make your argument that computable numbers are dense in incomputable numbers.[1] So presumably that makes you the first person in all of maths history to use them. Congratulations I guess? The other possibility is they get used a lot and we just don’t make a fuss about it because most of the time it’s exactly like you used them - to express an argument where it doesn’t matter whether they are computable or not.

I have no idea why you think they are absurd but as I say it’s an alternative axiomatic perspective. It just makes a huge amount of maths very inconvenient without really yielding (as far as I can see) much of anything.

I don’t have a perspective on your questions about the universe. Non-computable numbers don’t really trouble my world view. We invented the calculus and the language of continuous functions which gave rise to the rigorous construction of the real numbers precisely to better describe the universe, so it seems that they are pretty useful in that context.

[1] Which is interesting right, because this is a trivial argument in the sense that in standard first year analysis you learn that the rationals are dense in the irrationals and rationals are obviously computable so it must be that rationals are dense in the non-computable numbers. But this is a very basic argument that we can only make because we permit the Reals to be complete. It’s not possible to express the argument in the restriction to computable numbers because there is no ambient space outside the computable numbers for them to be dense into.

[seanhunter]

> And the main way we prove that the Reals are uncountable is to use a proof by contradiction. It would take too long to spell it out, but they aren't really just contradicting "Reals are Countable". It's "All that other stuff we think is true AND Reals are Countable" that gets contradicted.

Btw that's not true. Cantor's diagonal argument isn't a proof by contradiction, it's a purely constructive argument. The way he made it in his original paper is a bit more technical than this but this is the "modern" version that's a bit easier to put into layman's terms.

Say you say you can construct a (countably finite) list containing all the real numbers. I say I don't care how you made your list I can give you a number that's not on it, and in fact cut your list down to just numbers between 0 and 1. If you have all the real numbers you must have all the numbers between 0 and 1, but even if you make a countably infinite list of numbers between 0 and 1 I'll give you a procedure that will construct a number that's not on your list no matter how you made it.

1) Read the first number on your list. If it has 1 in the first decimal place, make the first decimal place of my number a two otherwise make it a 1. 2) Read the second number on your list. If it has 1 in the second decimal place, make the second decimal place of my number a two otherwise make it a 1. ....

Proceed in that manner.

At the n-th step I read the n-th number on your list. If it has 1 in the n-th decimal place make the n-th decimal place of my number a 2 otherwise make it a 1.

Now: My number is clearly nowhere on your list as it differs at in least one decimal place from every number on your list.

Therefore it is not possible to construct a countably infinite list of real numbers.