We need a new number system, for it is obvious that our current "reel numbers" are flawed in multiple ways.
1. The measure theoretic notion of probability is inconsistent with reality. "Sets of measure zero" should be assigned a non-zero value if an event within them is actually physically possible. It is erroneous to say for example "Under a normal distribution the probability of drawing the number 100 is 0" even though it is very well possible that this number can be drawn.
2. The notion of convergence to infinity is incompatible with analytic continuation, leading to paradoxes like . The paradox itself would be okay as long as you distinguish between analytic continuation and convergence of sums. However replacing the sums with analytic continuation turns out to be consistent with physical reality as shown in renormalization of QFT, e.g. in the Casimir effect. Therefore, apparently a deeper connection between sums diverging in the reels but converging in analytic continuation needs to be made explicit in a new number system replacing the reels.
3. The usual paradoxes regarding infintesimals, e.g. 0.999... = 1 (I don't need to go into depth here, we already have other threads for that), Zeno's paradox, or the fact that for all effective purposes you can do calculus by treating dy/dx as a fraction of infinitesimals (highschool calculus fags will call it handwavy physicist math but it does actually work out, in particular there where you can't work with "rigoros" limits anymore).
Any suggestions how to fix math?
1. The measure theoretic notion of probability is inconsistent with reality. "Sets of measure zero" should be assigned a non-zero value if an event within them is actually physically possible. It is erroneous to say for example "Under a normal distribution the probability of drawing the number 100 is 0" even though it is very well possible that this number can be drawn.
2. The notion of convergence to infinity is incompatible with analytic continuation, leading to paradoxes like . The paradox itself would be okay as long as you distinguish between analytic continuation and convergence of sums. However replacing the sums with analytic continuation turns out to be consistent with physical reality as shown in renormalization of QFT, e.g. in the Casimir effect. Therefore, apparently a deeper connection between sums diverging in the reels but converging in analytic continuation needs to be made explicit in a new number system replacing the reels.
3. The usual paradoxes regarding infintesimals, e.g. 0.999... = 1 (I don't need to go into depth here, we already have other threads for that), Zeno's paradox, or the fact that for all effective purposes you can do calculus by treating dy/dx as a fraction of infinitesimals (highschool calculus fags will call it handwavy physicist math but it does actually work out, in particular there where you can't work with "rigoros" limits anymore).
Any suggestions how to fix math?
