[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?

[anon291]

> Saying that the reals don't exist is pretty much the same as saying that the natural numbers don't exist

I mean... it's really not. The natural numbers have a straightforwards definition, where one can theorize the 'existence' of them pretty straightforwardly.

Are you familiar with the definition of the reals? The reals don't appear anywhere in nature, and are only useful as modeling really. You can't 'measure' a real-valued quantity and get a result that is not rational or at least computable.

The complex numbers over the rationals are more 'real' in that you can measure complex numbers with rational coeficients.

> 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?

I'm not sure you understand exactly what you're talking about. Mathematical constructivism doesn't claim that mathematics is not real. In fact, quite the opposite. It's typically a view held by very experienced mathematicians.

Well regardless countability is easily shown to be constructivist by first constructing a finite description of the natural numbers (and showing that the description of any natural number is itself finite and decidable).

Then you show that you can create decidable functions mapping each element of particular sets onto the natural numbers. It is trivial to construct these from any alphabet, including our own. Same with the rationals, etc.

Then you get to the reals and you ask questions like 'what does it mean to have a set of rationals' [1], 'what does it mean to have a least upper bound', etc.

[1] a constructivist would say a decision procedure to determine if a number is in the set. I.e., a set is real if you can decide whether a thing is in it or not. From this it follows that significant portions of the real numbers do not exist because there is no decision procedure that can produce a set of rationals for which they are the least upper bound.