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I could make the opposite argument. In nature/reality there are no irrational numbers. Say you want to measure the circumference of the visible universe. This

I could make the opposite argument. In nature/reality there are no irrational numbers. Say you want to measure the circumference of the visible universe. This is all the space we have, the rest is outside our light cone and we can't interact with it to the point that we can only infer it might exist. You need about 55 decimals of pi to get this circumference to the accuracy of a Planck length. Or about that many, give or take. So that's measuring the very largest real thing that could possibly matter, down to the accuracy of the very smallest thing we can conceive of. So that's it for pi. Any further decimals are strictly theoretical. We can prove they must be these decimals and not others, using math, but it's only theoretical knowledge, these additional decimals serve absolutely zero purpose in nature or reality. You cannot get to them by measuring reality, you can only theorize about what these numbers would be if you could measure to infinite precision, which we can't, because there are limits. The "Real" numbers is really a misnomer. Even if you don't buy the above accuracy argument, and want to describe nature as something infinite (even though we're strictly limited to interacting with a finite subset of it), then at least agree that it's countable. The real numbers are way too stretchy and insane (see the Banach-Tarski paradox). In nature you can't stretch things infinitely far, nor can you cut up things to arbitrary precision. And yes occasionally we discover new smaller particles, or sub-particles, but what we don't discover is a continuum. And it would be really weird if we did, because you can do crazy tricks to the Real numbers.

geebee 7y ago
Very interesting response. The thing that keeps me from entirely agreeing (with an admission that I don't have a background in the science you mentioned[1], so I can't really understand it) is that the number is pi - the 55 decimals is just an approximation that can be expressed as a ratio of two integers. The number isn't the rational that can be made arbitrarily close to a limit, the number is the limit. That number is pi. pi is every bit as much a number as 1, 2, 3. So is sqrt(2). So is e. My understanding is that almost all numbers are real, but not rational. They can be expressed as the limit of a sequence that can be made arbitrarily close to the limit. But the number is not the sequence, the number is the limit of the sequence. [1] = I had to look up what a Planck is. Should have taken more physics along with the math.

geebee 7y ago

Very interesting response. The thing that keeps me from entirely agreeing (with an admission that I don't have a background in the science you mentioned[1], so I can't really understand it) is that the number is pi - the 55 decimals is just an approximation that can be expressed as a ratio of two integers. The number isn't the rational that can be made arbitrarily close to a limit, the number is the limit. That number is pi. pi is every bit as much a number as 1, 2, 3. So is sqrt(2). So is e. My understanding is that almost all numbers are real, but not rational. They can be expressed as the limit of a sequence that can be made arbitrarily close to the limit. But the number is not the sequence, the number is the limit of the sequence. [1] = I had to look up what a Planck is. Should have taken more physics along with the math.