This began as a comment on Russell’s post, “Do Real Numbers Exist?” over at Social Mode, and got a bit lengthy, so I decided to post it here.

Russ’s main question, in my estimation, is that if you cannot ever accurately figure out pi to the last digit, yet you can loop a six-inch string and measure its radius to obtain pi, does pi exist though you can write it down symbolically?

I don’t know if this is the correct parallel, but it reminds me of an argument in the NKS Forum about language, mathematical statements, and physical truth.

I think it’s important to keep in mind that constructs like the real number system and pi are mathematical language elements which help to describe what we see in the real world. Puzzling over whether the real numbers exist is like puzzling over whether the statement “This statement is true” connotes the existence of truth in that statement (you can see what I’m trying to get at here). Just as logic represents a model of what is real, you can screw with it to get things that aren’t real, using a certain transformation scheme (in this case, words with a certain logical meaning).

And so we come to — the infinity problem.

If something takes infinitely long to compute, how can it be a thing in itself, since there is no “end”? Do only things in themselves exist? Do ideas — like the infinitely small and infinitely large, which can be approached using limits of simply-constructed series — exist as symbols for physical reality first and foremost, and have no real existential meaning in and of themselves? Does it make sense to ask whether a model and a symbol, no matter how computable, “exists” in the real world?

Perhaps what we can agree upon is that what exists is knowledge. Then the problem boils down to predictability. Are things which are determinate yet unpredictable knowable? If not, do they “exist”? Do they only exist when they happen, i.e., pi only exists as long as you keep computing its digits, and only to the accuracy you’ve so far computed?

At this point, I think that the question of whether the real numbers or pi “exists” doesn’t make any sense. Mathematics is a symbolic language — you can argue that none of its elements “exist” in physical reality, yet they can be used to communicate information about things which are real. I think the question of whether, conceptually, the real numbers are relevant, is more apt than whether, conceptually, they exist. You can argue that while the counting numbers are an infinitely large set their discreteness makes them more computably relevant, while the real numbers being uncountably infinitely large makes them not computably relevant. And if everything is computable — everything knowable, that is — the real numbers aren’t useful in computation.

(oh yes, and I know how to construct the real numbers. So they are also computable, but that’s not the same in my meaning as computably relevant!)

You’re just reframing the problem and avoiding an answer.

“At this point, I think that the question of whether the real numbers or pi “exists” doesn’t make any sense. Mathematics is a symbolic language — you can argue that none of its elements “exist” in physical reality, yet they can be used to communicate information about things which are real. I think the question of whether, conceptually, the real numbers are relevant, is more apt than whether, conceptually, they exist. You can argue that while the counting numbers are an infinitely large set their discreteness makes them more computably relevant, while the real numbers being uncountably infinitely large makes them not computably relevant. And if everything is computable — everything knowable, that is — the real numbers aren’t useful in computation.”

Yes, I suppose we can kill of the line of questioning by saying it no longer makes sense.

I remain unconvinced that we’ve added much to the discussion.

My question is more about this:

We have a physical reality – a circle, a string pulled into a circle, whatever geometric object you want to consider.

Its measure is something we’re not sure is a reality (pi).

At somepoint something that isn’t discrete (pi) manifests itself physical. (perhaps that’s not the best way to say it)

I think I’m getting at more of a discussion of is the universe discrete rather than the infinity problem or truth in mathematics.

so… add to this with that context in mind.

Ah, okie dokie. That was a point I was thinking of addressing, but I abandoned it in favor of being a bit more philosophical. But I get where you’re coming from and it’s relevant.

The question is — if I understand it correctly — if something like pi exists, does that mean the universe can’t possibly be discrete, because pi is a real number, and it is a property of something in the real world (the circle, or string pulled into a circle)?

Not sure if I have any answers, but I’ll throw my two cents in…

Certainly the *idea* of real numbers is very useful for computation (at least in traditional science). It’s just that what we’re actually computing with are not real numbers but just approximations. So Pi is a very useful idea, and an approximation to Pi is very useful, even if Pi in its entirety is not.

Similar to measuring string… What about the speed of light? When we measure the speed of light, we only get an approximation… but an approximation to what? Is the speed of light a genuine real number, or is it a fuzzy thing that only has finitely many digits we’ll ever be able to compute? Whereas Pi comes out of your axioms of geometry, the answer to this question will come out of your axioms of the world: If the world is fundamentally discrete, then perhaps we’ll be able to write the speed of light in terms of purely mathematical constants, like Pi and E (once we’ve found the right units, of course). On the other hand, if the rules of physics take an infinite amount of information to write down, then we’ll never know the exact value of the speed of light.

As a physicist I would say that real

numbers are useful approximations to

rational numbers. After all, the only

numbers from a measurement are rational.

As a mathematician I see it the other

way around.

in terms of the physical world, and maybe even at some level our brains themselves, the real numbers do NOT exist. for example, we can CONSTRUCT a length of √2, but we cannot measure it EXACTLY. in practical terms, this doesn’t matter much, because we can get “close enough” (for government work, at least :P).

at the abstract level, it is a different story. it IS possible, but rather cumbersome, to do without the irrational numbers. you cannot define limits properly, and most functions are not continuous, anymore. if one restricts oneself to rational approximations, you wind up with a lot of “almost continuous” functions, and “nearly true” theorems. and what we mean to get at, with the notion of convergence (which IS what irrational numbers are, after all) is that by considering limits of rationals as algebraic entities that obey the computational restrictions of a complete ordered field, we can perform computations with a lot less qualification that we would have to add if only rational numbers were allowed.

consider the standard epsilon-delta notions of limits and continuity. these make sense when you only consider rational numbers as possible values, and have an easily realized real-world application as bounds for error tolerenaces in measurement and predicted results. but to solve even the simplest of expressions like x^2 – 2 = 0, we can easily get lost in a sea of rectifications of errors. it gets messy.

the notion of continuity, and the algebraic completion (under the ordering of “<") of te rational numbers is USEFUL. it allows us to use irrational numbers in computations involving rational numbers with a high degree of confidence. the system is directly analogous to using complex numbers as an aid to solving problems with only real variables.

insofar as our finite, and halting (a turing pun, here) bodies are concerned, irrational numbers, and that bastard love-child of philosophy and mathematics, infinity, have no tangible existence, per se. but as ideas, they exist just as much as the notions of "red" or "existentialism" do.