[PaulHoule]

My grad school mate Ron Maimon one day told me in a bar about the problem of computable numbers in a way that made him sound like a serious crackpot. I thought about it enough to conclude that the “real” numbers were “phony” numbers because unlike the integers or rationals most of them don’t have a name and can’t be referred to specifically.

I found out later that Turing had introduced the computable numbers idea and that was the work he had really done as opposed to the modern formulation of the halting problem.

As for Ron he really descended into conspiracy theory insanity and got kicked off Quora because he was saying the Boston bombing was an inside job. I still wish Steve Wolfram would grow some balls, take his constructivist program seriously, and reject the axiom of choice.

[kbelder]

I found his Physics Stack Exchange answers to be immensely valuable. Although, I did have to read everything he wrote with a little bit of a critical eye, due to some of the rants he went on.

I'm glad to see he bought into the abiotic generation of oil; it's my favorite fringe theory I just can't shake.

[anon291]

I think it's one of the more unfortunate thing in mathematics that the real numbers are as popular as they are. I'm with your roommate. I don't believe they exist. I don't think every set of rational numbers has a least upper bound

[auggierose]

I don't believe that either. But every set of rational numbers bounded from above has a least upper bound in the reals.

[anon291]

That's my point. I don't believe they do. I don't believe the reals are well defined since no one can name them. In general, I lean towards mathematical constructivism: https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_...

I agree that computable real numbers exist, even if they're intractable to compute.

[auggierose]

If only those things existed that have a name, the world would be a pretty small and boring place. Also, that would be a world entirely defined by humans, and that just doesn't make sense.

[anon291]

Sorry, to be clear, it's irrelevant whether humans are doing it or someone else. They're not identifiable, and there's no example of such a thing. There are noncomputable reals which we can identify. I'm willing to say those exist. There are non-computable reals that we cannot identify (basic countability argument), that must exist as part of the reals if defined as they are.

But we've no example of them. They're just kind of there. They don't exist.

Either way, you don't need to argue with me. Much smarter thinkers than me have written extensively on it, and it's a widely held view. It's an indictment of one's own curiosity when the response to a new idea is to suggest the introducer is somehow lacking.

[auggierose]

It's not really a new idea though. I just don't think it is a particularly good one.

The cool thing about mathematics is that we can use finite reasoning to talk about things that are otherwise hard to grasp, and that are yet part of our shared reality. Saying that the reals don't exist is pretty much the same as saying that the natural numbers don't exist. After all, you are going to have a hard time to name each and every one of them. Oh, you say that countability makes a difference here? Well, that's a purely mathematical concept, and if you don't believe that mathematics is real, how are you going to convince me that this concept makes sense?