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I copy an interesting paragraph: "A notable result is that integration (in the sense of the Riemann integral) is computable. This might be considered surprisin

by singularity2001 4y ago

I copy an interesting paragraph: "A notable result is that integration (in the sense of the Riemann integral) is computable. This might be considered surprising as an integral is (loosely speaking) an infinite sum. While this result could be explained by the fact that every computable function from {[0,1]} to R is uniformly continuous, the notable thing is that the modulus of continuity can always be computed without being explicitly given. A similarly surprising fact is that differentiation of complex functions is also computable, while the same result is false for real functions." Are real functions not a subset of complex functions? Oh, if you redirect the output to R it gets harder, just like with finding zero roots.

VyseofArcadia 4y ago
I think you could get an idea some of the subtleties of this if we try to make what you mean be "subset" explicit. Starting off on a set theoretic foot, what even is a function f:A->B? Well, it's a set of pairs (a, b) with a in A, b in B, such that (a, b) in f and (a', b) in f implies a = a'. In this sense, you could say a function r:R->R is a subset of some c:C->C by defining r = {(x, x') in c : x in R} On the other hand, the set of functions C->C is a different beast entirely, because that consists of all functions whose domain is C, whereas the set of functions R->R is all functions whose domain is R. While each individual function in R->R is a subset of some function in C->C, R->R itself is not a subset of C->C. I'm no analyst, complex or real, but my intuition is that C simultaneously has more wiggle room in the topological sense (you have a whole extra dimension) but less wiggle room in an algebraic sense (deg n polynomials are guaranteed to have n roots in C), it's pretty easy to just stumble upon surprising and unintuitive results. Thinking back to my grad school complex analysis course, limits are fundamentally different in C. In R, you have two directions from which you can approach a point, but in C, any sequence converging on the point is a valid perspective from which the limit must make sense. You can approach the point from a straight line in any direction, or a squiggly line, or a spiral, etc etc. So this extra dimension of wiggle room puts a hell of a strong condition on what functions can even be differentiable in the first place, and this restriction on differentiability often translates to stronger results.

VyseofArcadia 4y ago

I think you could get an idea some of the subtleties of this if we try to make what you mean be "subset" explicit. Starting off on a set theoretic foot, what even is a function f:A->B? Well, it's a set of pairs (a, b) with a in A, b in B, such that (a, b) in f and (a', b) in f implies a = a'. In this sense, you could say a function r:R->R is a subset of some c:C->C by defining r = {(x, x') in c : x in R} On the other hand, the set of functions C->C is a different beast entirely, because that consists of all functions whose domain is C, whereas the set of functions R->R is all functions whose domain is R. While each individual function in R->R is a subset of some function in C->C, R->R itself is not a subset of C->C. I'm no analyst, complex or real, but my intuition is that C simultaneously has more wiggle room in the topological sense (you have a whole extra dimension) but less wiggle room in an algebraic sense (deg n polynomials are guaranteed to have n roots in C), it's pretty easy to just stumble upon surprising and unintuitive results. Thinking back to my grad school complex analysis course, limits are fundamentally different in C. In R, you have two directions from which you can approach a point, but in C, any sequence converging on the point is a valid perspective from which the limit must make sense. You can approach the point from a straight line in any direction, or a squiggly line, or a spiral, etc etc. So this extra dimension of wiggle room puts a hell of a strong condition on what functions can even be differentiable in the first place, and this restriction on differentiability often translates to stronger results.