I think something could be two units long. Two is discrete, and at a microscopic level, things can be exactly discrete values (i.e. two angstroms...). I think asking if something could be exactly pi long is a different question.
Actually, no they can't. At a quantum level things do not have definitive sizes. They have sort of "clouds", where the center of the cloud is more likely to be their size, and the edges are less likely - but still possible.
Yeah I gotta think about that one. If we assume that there is some smallest discrete base unit of space (which I think is plausible), then all string lengths could only be some integer multiple of that base unit. Therefore length pi is out of the question.
Now presumably 1 meter will be equivalent to some integer multiple of base units, though that may turn out not to be the case. Perhaps the current definition of a meter in terms of certain wavelengths of light will turn out to have a remainder of 1/7 of a base unit. ;) In that case you'd just define a slightly adjusted meter as a whole number of base units. So for simplicity let's assume that a meter is a whole number of base units.
Then it's clear that a string of any integer meter length would be possible. A string of length 1/3 meters might be possible, assuming that the number of base units in a meter was divisible by 3. But a string of length pi meters would be impossible.
As a computer programmer I tend to think of transcendental numbers in terms of processes which relentlessly converge, and that of course is the theory of limits. But I also recognize the financial constraints on running processes. So there will always be a "good enough" aspect to any physical measurement.
Right, and I'm sure you're aware of the Planck constant for energy quanta as well. That's the kind of thing that made me say that a base unit of length is "plausible".
So although it's a fun thought experiment to think about strings that that are precisely 4998997308233 base units long, actually measuring such a thing is too expensive or even impossible due to Heisenberg.
Nevertheless, mathematics has demonstrated that it's useful to think of "real numbers" as infinitely precise things, because that way your number system doesn't impose any preordained limits on your measurements -- even though nature itself does.
By far most real numbers are actually non-computable, meaning that there is no finitely expressible procedure for listing their digits to any desired length.
Now it's always seemed clear to me that if a thing is fundamentally unobservable and unidentifiable in any way, you might as well say that thing does not exist at all. Nevertheless, the theory of real numbers implies that the uncomputable numbers "exist" in some sense.
That actually simplifies the theory. Otherwise you'd have to confine yourself to the computable numbers, namely all strings of binary digits that can be produced by some Fexl function (see http://fexl.com/). For example, the number .1010... could be expressed as:
\number == (1; 0; number)
(I use Fexl because it's based on combinatorics, which behave according to very simple rules. Ultimately any Fexl function can be expressed as a binary tree with only "S" and "C" at the leaves.)
That might make the strict constructivists happy, but it might also hamper the free reigning thought processes of mathematicians.
