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As a programmer I find constructivism is my default position if I think of math intuitively; perhaps that's why I butted heads so much with many of my math teac

As a programmer I find constructivism is my default position if I think of math intuitively; perhaps that's why I butted heads so much with many of my math teachers. But, your argument actually makes me less sure of my agreement with it: it seems clear that at bottom constructivism must lead to the rejection of the concept of continuously variable quantities. It is the horns of a dilemma: if you reject Cantor you reject the existence of "analog" as even an admissible concept, on the other hand if you accept continuous variability then you must also accept the shocking conclusion that the overwhelming majority of the set you assert you believe exists (the reals) cannot even be described or calculated.

btilly 14y ago
But constructivism does not give up continuity. All of the traditional definitions carry over. What you do have to give up, though, are discontinuous functions on continuous sets. They are not well defined for numbers that are the result of an infinite sequence converging near the border of the discontinuity. You can still give them a formal treatment, of course, but they aren't really functions. However if you're attached to your step functions, never fear. It would not be the first function-like thing that isn't a function. For another example consider the well-known derivative of the step function, the Dirac delta. Not actually a function in classical mathematics, though it is certainly useful.