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By overpowered, I mean that a model that allows the idea of something having infinite precision leads to apparent paradoxes. For instance you get constructs li

By overpowered, I mean that a model that allows the idea of something having infinite precision leads to apparent paradoxes. For instance you get constructs like Gabriel's Horn that have greater surface area than volume, or the case of choosing a random real number between 0 and 1: the probability of picking any particular real number is (approaches?) zero, yet some number will be picked. But in the case of the Cantor argument specifically he seems to be encoding an infinite amount of information in one number simply by extending the digits after the decimal point. It's almost like - cheating, or more specifically an "exploit" allowed by what I call reals being overpowered. (Here I have to make an aside: the value Pi, which is actually a computable number, can also encode any amount of information because any number will appear at some point in the expansion. But you'd have to indicate where in Pi your number starts, and your index values would (on avergage) be so large as to consume at least as many bits of information as it would to simply state the number you want.) In the physical world, when you do something like measure the voltage on a capacitor you can only measure it to within a certain accuracy & precision. If you wanted to make an analog computer that uses different voltages to represent different values, you'd be limited by the accuracy and precision which you could measure it. Note that, even if the voltage is infinitely variable, your ability to differentiate those values is still limited, so you still do not have infinite precision. (In Turing's paper on computable numbers, he makes the case that even if a human brain can have infinite states, there would at some point be a threshold of similarity at which the brain couldn't distinguish two similar states from each other.) So my claim is that real numbers imply the ability to measure something to infinite precision, and that that is contradictory to physical experience.

lmkg 17y ago
Reality and measurable reality were separate concepts, until quantum mechanics came along. If the intricacies of the Reals demonstrate the philosophy they come out of, it has as much to do with predating the atom in chemistry and the Heisenberg Principle in quantum mechanics as it does with predating float-point numbers. I feel that math is intended to appeal to logical intuition rather than physical intuition, so my belief is that adding limits to precision is an unnecessary complication to the number system. By extension, I don't believe that math or numbers reveal 'fundamental truths' of the physical universe, nor are they intended to. I view math as a disembodied exercise in logic, and if we happen to find a physical system that satisfies the hypotheses of a mathematical system, then the results will apply as well. The result of this is that I view the reals as 'purer' math because they have less hypotheses, and numerical approximations as a set of additional hypotheses that are more commonly seen in practice. The natural result is that there's a lot of mathematical ideals with cool properties that are ruled out in practical terms. I have a few questions for you and your viewpoint (for my own curiosity, not antagonistic). 1. Do you believe that Pi actually does have an infinite number of decimal places, or that it's limited by the precision with which we can measure a circle? I think I read somewhere that 50 decimal places is enough to calculate the radius of the visible universe from the radius to within an atom's width of accuracy. 2. What do you think of the Random Oracle Model in cryptography[1]? I think there's an anology between the ROM vs real hashes, and how I conceive of the reals vs finite-precision mathematics. In particular, simple and (logically) intuitive idealized models have properties that are not held by any possible real approximation. Feel free to ask me more if you're totally confused =P. 3. Do you accept infinite-precision rational numbers? Why or why not? My motivation in asking is that there's a difference between computability and measurability, and you appeal to both as a source of intuition. [1] http://en.wikipedia.org/wiki/Random_oracle_model http://en.wikipedia.org/wiki/Random_oracle_model

lmkg 17y ago

Reality and measurable reality were separate concepts, until quantum mechanics came along. If the intricacies of the Reals demonstrate the philosophy they come out of, it has as much to do with predating the atom in chemistry and the Heisenberg Principle in quantum mechanics as it does with predating float-point numbers. I feel that math is intended to appeal to logical intuition rather than physical intuition, so my belief is that adding limits to precision is an unnecessary complication to the number system. By extension, I don't believe that math or numbers reveal 'fundamental truths' of the physical universe, nor are they intended to. I view math as a disembodied exercise in logic, and if we happen to find a physical system that satisfies the hypotheses of a mathematical system, then the results will apply as well. The result of this is that I view the reals as 'purer' math because they have less hypotheses, and numerical approximations as a set of additional hypotheses that are more commonly seen in practice. The natural result is that there's a lot of mathematical ideals with cool properties that are ruled out in practical terms. I have a few questions for you and your viewpoint (for my own curiosity, not antagonistic). 1. Do you believe that Pi actually does have an infinite number of decimal places, or that it's limited by the precision with which we can measure a circle? I think I read somewhere that 50 decimal places is enough to calculate the radius of the visible universe from the radius to within an atom's width of accuracy. 2. What do you think of the Random Oracle Model in cryptography[1]? I think there's an anology between the ROM vs real hashes, and how I conceive of the reals vs finite-precision mathematics. In particular, simple and (logically) intuitive idealized models have properties that are not held by any possible real approximation. Feel free to ask me more if you're totally confused =P. 3. Do you accept infinite-precision rational numbers? Why or why not? My motivation in asking is that there's a difference between computability and measurability, and you appeal to both as a source of intuition. [1] http://en.wikipedia.org/wiki/Random_oracle_model http://en.wikipedia.org/wiki/Random_oracle_model