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Line, Surface and Contour Integration “Find the integral of the function ” is a typical core thing one wants to do in calculus. But particularly in applications of calculus, it’s common to want to ask slightly more elaborate questions, like “What’s the integral of over the region ?”, or “What’s the integral of along the line ?”
eventually finding generalizations of things like differential geometry and algebraic topology that answer questions like what 3 -dimensional curvature tensors are like, or how we might distinguish local gauge degrees of freedom from spatial ones in a limiting hypergraph. OK, so that’s a lot of projects.
But with the multicomputational paradigm there’s now the remarkable possibility that this feature of physics could be transported to many other fields—and could deliver there what’s in many cases been seen as a “holy grail” of finding “physics-like” laws. I know of a few perhaps-closer approaches to our conception of multiway systems.
But with the multicomputational paradigm there’s now the remarkable possibility that this feature of physics could be transported to many other fields—and could deliver there what’s in many cases been seen as a “holy grail” of finding “physics-like” laws. I know of a few perhaps-closer approaches to our conception of multiway systems.
For example, we know (as I discovered in 2000) that (( b · c ) · a ) · ( b · (( b · a ) · b )) = a is the minimal axiom system for Boolean algebra , because FindEquationalProof finds a path that proves it. But what about other models of computation—like cellular automata or register machines or lambda calculus?
It didn’t help that his knowledge of physics was at best spotty (and, for example, I don’t think he ever really learned calculus). Then McCarthy started to explain ways a computer could do algebra. It was all algebra. Gliders are usually transported with their wings removed, with the wings attached in order to fly.
Any integral of an algebraic function can in principle be done in terms of our general DifferentialRoot objects. When you do operations on Around numbers the “errors” are combined using a certain calculus of errors that’s effectively based on Gaussian distributions—and the results you get are always in some sense statistical.
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