| > An interesting idea that follows from this: what other kinds of "numbers" might we come up with if we relax our logical blinders? It depends on exactly what you mean by this, but I would argue that we do this absolutely everywhere. For example, matrices have similar properties to numbers. You can add/subtract them, you can multiply them, you can (sometimes) divide them. Depending upon how you restrict your set of matrices, ab might equal ba or it might be the case that ab makes sense while ba does not (i.e. not only is it not commutative, but it doesn't even make sense to multiply the other way). > In the same way, what if we invent a new class of real numbers that are quantized (such that there exists a smallest quantity x)? Here's an example of something similar: https://en.wikipedia.org/wiki/Dual_number Basically you're adding in a number smaller than everything else that squares to 0. > What if we propose the existence an "imaginary" algorithm (call it omega, if you like) that can decide the halting problem? I'm no logician, but I believe that they would refer to this sort of thing as "model theory". I.e. you're basically taking proofs (sequences of logical statements) and studying different logical systems in which these statements make sense. For example, you could extend your theory/model to take as an axiom that there exists an algorithm to solve the halting problem (though this would be stupid since you can already prove within most theorems that the halting problem is impossible...i.e. you would have a self-contradictory axiomatic system). > These mathematical object do not have to "exist" for them to be useful. I think there is a whole lot of new and interesting math that would be unlocked by dispensing with our logical blinders. I 100% agree. Math is a tool that we invent to better understand the world and our thoughts (as well as do more). However, you'll probably have to track down a true logician if you were to go down that rabbit hole... |