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.<p>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.<p>I have a few questions for you and your viewpoint (for my own curiosity, not antagonistic).<p>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.<p>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.<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.<p>[1] <a href="http://en.wikipedia.org/wiki/Random_oracle_model" rel="nofollow">http://en.wikipedia.org/wiki/Random_oracle_model</a>