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> ...anyway, this is quite a convoluted proof. Agreed :) > Theorem: Given a rectangle R with non-zero side-lengths... And this is the key problem with the "p

> ...anyway, this is quite a convoluted proof. Agreed :) > Theorem: Given a rectangle R with non-zero side-lengths... And this is the key problem with the "proof" I laid out. My theorem does not specify non-zero dimensions. Therefore standard arithmetic division by a dimension is not well-defined. Since we're pondering intuition, it's interesting to consider how fixing the "proof" was approached. On the one hand, you fixed it by in effect back-progating invariants to the problem definition. Another approach would be to use a modified definition of "division" ("d = 0 if a or b are zero"--would still require the constraint "c !=0 unless a*b=0"). Mathematically, the former appoach is almost always what you're going to want, since a modified definition will be pretty useless to build upon consistently. From an application standpoint, the latter would be much more robust. This is kind of one of the points made in the Sci Am article--pure vs. applied proofs are different.

Twisol 8y ago
Yes, this is a fair point! Interestingly though, projective geometry already provides us with a space potentially suited to this question — I find https://en.m.wikipedia.org/wiki/Projectively_extended_real_line https://en.m.wikipedia.org/wiki/Projectively_extended_real_l... to be a good introduction, although I’m not terribly familiar with the field myself. The intuition behind this choice is, if we’re squishing the rectangle so hard that one side approaches 0 length, the constant area has to go somewhere. If the other side length approaches infinity, why not literally let its measure be infinity? One would argue that the real numbers are too impoverished to give a satisfactorily general answer to our original question. Of course, we still obtain a zero area, so we’re no longer really solving the same problem as we started with. If we want a zero side length and a non-zero area, we may need to move into non-Euclidean geometries. It really depends on what we’re doing. My point is that when an application needs to make changes to the foundation, usually there’s going to be a richer pure setting we can move to. And if there isn’t yet, that’s a great opportunity to find out how robust of a new foundation we can create. Hiding edge cases away in an otherwise unobtrusive definition bothers me as both a mathematician and as a software developer. In all cases, I want to highlight the assumptions we’re making and find the right abstraction to tame them.

Twisol 8y ago

Yes, this is a fair point! Interestingly though, projective geometry already provides us with a space potentially suited to this question — I find https://en.m.wikipedia.org/wiki/Projectively_extended_real_line https://en.m.wikipedia.org/wiki/Projectively_extended_real_l... to be a good introduction, although I’m not terribly familiar with the field myself. The intuition behind this choice is, if we’re squishing the rectangle so hard that one side approaches 0 length, the constant area has to go somewhere. If the other side length approaches infinity, why not literally let its measure be infinity? One would argue that the real numbers are too impoverished to give a satisfactorily general answer to our original question. Of course, we still obtain a zero area, so we’re no longer really solving the same problem as we started with. If we want a zero side length and a non-zero area, we may need to move into non-Euclidean geometries. It really depends on what we’re doing. My point is that when an application needs to make changes to the foundation, usually there’s going to be a richer pure setting we can move to. And if there isn’t yet, that’s a great opportunity to find out how robust of a new foundation we can create. Hiding edge cases away in an otherwise unobtrusive definition bothers me as both a mathematician and as a software developer. In all cases, I want to highlight the assumptions we’re making and find the right abstraction to tame them.