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!)