basic train of thoughts: Gödel's incompleteness theorem (and Turing's subsequent adaptation to computational machinery) shows that mathematics itself is incomplete. that was a shock: mathematics is the domain of all formal languages, but mathematics cannot be generated using them
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there is a small branch of mathematics that only accepts statements as true that have actually been constructed. constructive mathematics turns out to be identical to computation. Church and Turing demonstrated that computation contains itself, i.e. we can compute all computers
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constructive mathematics is not time-less like classical math. a function that has not been computed yet does not have a defined value. pi is a function, not a number. the axiom of choice does not hold. true infinities and continua cannot be constructed
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construction means that there has to be a machine that can perform the actual computations. in the strong, intuitionist form, we must actually run the proof, i.e. we need to compress it into a form that actually runs on our machine (which leads to a small foundational crack)
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since we cannot know that our interface to the machine is entirely coherent, we may always have to bracket our theories into a residual probability that a bit flipped somewhere in the proof making machine or its verification machine or the interpretation of the verification, etc
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the main claim of AI and cognitive science however is that the brain is a computer, and because all the mathematics we need to account for run on our brains, we can build something that can satisfy all desires a classical mathematician can hope to fulfill with a finite automaton
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this is where Penrose disagrees: Penrose believes that the mind and by extension the universe are implemented in non constructive mathematics. (that should imply that he thinks that QM is the wrong foundation for physics, since it is a [slightly hyper]computational model)
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there is good hope though; the program of constructive analysis in math seems to be successful, and there does not seem to be any obvious limit to the capacities of computer algebra systems
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so, basically, the solution is: while we cannot build math in math, and computation is much weaker than math, every bit of math that we will ever come across will have been built by computation
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I sometimes like to use the idea of a derivative here. A line cannot be reified, but it’s derivative can. Pi cannot be reified, but a algorithm to compute it can. An infinite set cannot be reified, but can be specified. Etc. The difference between classic and constructive math?
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It would seem that continuous integration is a bit fishy, yes… I think about it as difference between specification and implementation.
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