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Well, suppose the following: Given a universe finite in extent, with a finite lifetime and a finite mass-energy content, then for all possible practical purpos

Well, suppose the following: Given a universe finite in extent, with a finite lifetime and a finite mass-energy content, then for all possible practical purposes only reals up to a finite precision are needed to calculate and describe observable properties of the universe. With that constraint, there is e.g. an isomorphism between integers and all needed reals and as such every observable property in the universe can be quantified using the integers. It may seem that that would 'invalidate' a lot of mathematical machinery, which uses concepts that can not be enumerated by the integers (i.e. the set of reals), but that need not be the case. The mathematical machinery we have developed may partly be a very effective 'thought experiment': what if the reals were uncountable? It turns out that makes things easier to calculate than to take their countability into account, while giving results indistinguishable from taking their countability into account. I lack the mathematical sophistication to prove this is possible. I'm just proposing a view of mathematics that would make it obvious why it is effective. I don't believe in miracles or coincidence and I'm not a Platonist or other sort of idealist, so that doesn't leave me much choice, does it? :)

yequalsx 12y ago
Suppose you get into your car and start driving. You drive for 1 hour and travel 70 miles. At the end of 1 hour your car is at rest since you've reached your destination. Now let x be a number less than or equal to 70. Was there a point in time you were traveling x miles per hour? Is this true for all x in [0, 70]? If so then there is a need for an uncountable number of reals. If not then which values of x did you skip? Do these values of x become part of the needed numbers since they are an excluded set of a numbers of an experiment?
- judk 12y ago
  In the first branch, you are assuming the existence of uncountably continuous time and space with perfect precision. Scientific observation is not on your side. (Plancks constant, QM, for example) For the second, if the universe were somehow rational, there is no need to drag in irrational numbers to describe it.
  - yequalsx 12y ago
    I don't believe I'm assuming anything. I asked specific questions. The answers to those questions are relevant to what the OP wrote. Your last sentence does not make sense to me. The universe is not a number and therefore is not a rational number. It is not rational in the sense of thought either since it does not think (as far as I can tell). I have not idea what is meant by the statement, "if the universe were somehow rational...".
- Confusion 12y ago
  So I would indeed say that x cannot be any number less than, or equal to, 70. In this universe, it probably is not possible to drive exactly 70 miles per hour, because mile or hour may not be an integer multiple of a fundamental unit. Everything would be discretized, including possible speeds. Note that actually only speed differences need to be discrete, as those are the only thing actually observable. You cannot measure any absolute speed in this universe.
judk 12y ago
You are slightly confused about the difference between irrationality (the universe certainly has irrationals) and uncountability (the observable Universe almost certainly does not have this). That is, irrational real numbers do exist in nature (diagonal of a square, to start), but only countably many of them exist. The rest are just an intellectual completion to create an elegant-looking symmetry.
- yequalsx 12y ago
  I'm not confused about the difference between what an irrational number is and the cardinality of the real numbers and it's various subsets. It all depends on one's philosophy and existence. When you say, "only countably many of them exists....The rest are just..." you are admitting that at least in some sense they all exist. Indeed, one must be careful when making a statement, "irrational real numbers exist in nature". You can't point to anything and say, "There is the number square root of 2." It does not exist as a physical object. Even if one were to make a giant philosophical leap and believe that one can make right triangle whose legs have precisely the same length (down to the last particle!) you don't have the existence of an irrational number. You have the hypotenuse of a triangle. The number itself does not exist in the physical sense. Numbers are concepts and to say that one such concept exists but not another because there is some physical quantity is "equal" to it in former case but not the latter is an absurdity in my opinion. There are errors in all physical measurements. How is one going to distinguish between which numbers exist in your sense and which ones don't when the numbers are close enough together that no measurement can possibly distinguish between them?
- Confusion 12y ago
  The perfect diagonal of a square is an ideal. Actually measurable diagonals of squares may only be able to have a finite set of lengths. Note that this may imply (perhaps: does imply) that ideally square angles do not exist. I'm not bothered by those things :)

yequalsx 12y ago

Suppose you get into your car and start driving. You drive for 1 hour and travel 70 miles. At the end of 1 hour your car is at rest since you've reached your destination. Now let x be a number less than or equal to 70. Was there a point in time you were traveling x miles per hour? Is this true for all x in [0, 70]? If so then there is a need for an uncountable number of reals. If not then which values of x did you skip? Do these values of x become part of the needed numbers since they are an excluded set of a numbers of an experiment?

judk 12y ago
In the first branch, you are assuming the existence of uncountably continuous time and space with perfect precision. Scientific observation is not on your side. (Plancks constant, QM, for example) For the second, if the universe were somehow rational, there is no need to drag in irrational numbers to describe it.
- yequalsx 12y ago
  I don't believe I'm assuming anything. I asked specific questions. The answers to those questions are relevant to what the OP wrote. Your last sentence does not make sense to me. The universe is not a number and therefore is not a rational number. It is not rational in the sense of thought either since it does not think (as far as I can tell). I have not idea what is meant by the statement, "if the universe were somehow rational...".
Confusion 12y ago
So I would indeed say that x cannot be any number less than, or equal to, 70. In this universe, it probably is not possible to drive exactly 70 miles per hour, because mile or hour may not be an integer multiple of a fundamental unit. Everything would be discretized, including possible speeds. Note that actually only speed differences need to be discrete, as those are the only thing actually observable. You cannot measure any absolute speed in this universe.

judk 12y ago

You are slightly confused about the difference between irrationality (the universe certainly has irrationals) and uncountability (the observable Universe almost certainly does not have this). That is, irrational real numbers do exist in nature (diagonal of a square, to start), but only countably many of them exist. The rest are just an intellectual completion to create an elegant-looking symmetry.

yequalsx 12y ago
I'm not confused about the difference between what an irrational number is and the cardinality of the real numbers and it's various subsets. It all depends on one's philosophy and existence. When you say, "only countably many of them exists....The rest are just..." you are admitting that at least in some sense they all exist. Indeed, one must be careful when making a statement, "irrational real numbers exist in nature". You can't point to anything and say, "There is the number square root of 2." It does not exist as a physical object. Even if one were to make a giant philosophical leap and believe that one can make right triangle whose legs have precisely the same length (down to the last particle!) you don't have the existence of an irrational number. You have the hypotenuse of a triangle. The number itself does not exist in the physical sense. Numbers are concepts and to say that one such concept exists but not another because there is some physical quantity is "equal" to it in former case but not the latter is an absurdity in my opinion. There are errors in all physical measurements. How is one going to distinguish between which numbers exist in your sense and which ones don't when the numbers are close enough together that no measurement can possibly distinguish between them?
Confusion 12y ago
The perfect diagonal of a square is an ideal. Actually measurable diagonals of squares may only be able to have a finite set of lengths. Note that this may imply (perhaps: does imply) that ideally square angles do not exist. I'm not bothered by those things :)