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Raises a tough question about mathematics generally. Is math anything other than memorizing a set of rules (and being able to regurgitate them when shown a pro

Raises a tough question about mathematics generally. Is math anything other than memorizing a set of rules (and being able to regurgitate them when shown a problem)? Obviously for anything like arithmetic, multiplication, etc. the answer is "yes this is just memorized". Then for more complex math, can't it be argued that this is just recombination of memorized theories and algos, but using some kind of random search? I'm not sure math involves as much symbolic reasoning as we like to think, but I could be wrong. The same goes for programming - mostly just regurgitating known algos (which break into smaller memorized sub-algos). When presented with something novel, generally just use random approaches until one fits. A few people in history who had to come up with "the first" such programs had some novelty, but even those were mostly just applications of known / memorized solutions in a random way (e.g. quicksort).

whatshisface 7y ago
For at least a hundred years (as long as symbolic logic has existed, however long that really is) it has been widely known that you can iterate through every single theorem without any special creativity or thinking. The problem with that strategy is that you will spend most of your time on theorems like (not not X => X), (not not not not X => X), (not not not not not not X => X) and so on. What mathematicians actually do is use their human intuition and aesthetic sense to pilot the mechanical formalism towards goals that are worth achieving.
- vladf 7y ago
  That’s not a really fair way of putting it. Unless you’re suggesting that there’s something deeply magical about our human “aesthetic” intuition why can’t that itself be coded into a set of syntactic indicators (e.g., favor short equations, don’t repeat yourself, connecting theorems otherwise far from each other in some distance metric is valuable) to be used as a search heuristic?
  - whatshisface 7y ago
    The human aesthetic can't practically be programmed as a classical algorithm. The human ability to identify cats can't even be programmed as a classical algorithm, that's why neural networks were invented. Maybe a neural network could predict how interesting mathematicians would find a theorem but by that point there's no point for anybody to do anything, even artists.
throwlaplace 7y ago
differential equations at this level (ie odes that have tabulated solutions or are a transformation away from such an ode) are definitely a set of rules that you memorize. differential equations as an active area of research is much more abstract, dealing with things like existence/uniqueness of solutions (when picard-lindelof doesn't apply), or stability when a solution is known to exist but cannot be analytically written down (and so integration has to be performed). hell one of the clay problems is about a solution to a pde. so no it's not this straightforward except in basically toy examples.

whatshisface 7y ago

For at least a hundred years (as long as symbolic logic has existed, however long that really is) it has been widely known that you can iterate through every single theorem without any special creativity or thinking. The problem with that strategy is that you will spend most of your time on theorems like (not not X => X), (not not not not X => X), (not not not not not not X => X) and so on. What mathematicians actually do is use their human intuition and aesthetic sense to pilot the mechanical formalism towards goals that are worth achieving.

vladf 7y ago
That’s not a really fair way of putting it. Unless you’re suggesting that there’s something deeply magical about our human “aesthetic” intuition why can’t that itself be coded into a set of syntactic indicators (e.g., favor short equations, don’t repeat yourself, connecting theorems otherwise far from each other in some distance metric is valuable) to be used as a search heuristic?
- whatshisface 7y ago
  The human aesthetic can't practically be programmed as a classical algorithm. The human ability to identify cats can't even be programmed as a classical algorithm, that's why neural networks were invented. Maybe a neural network could predict how interesting mathematicians would find a theorem but by that point there's no point for anybody to do anything, even artists.

throwlaplace 7y ago

differential equations at this level (ie odes that have tabulated solutions or are a transformation away from such an ode) are definitely a set of rules that you memorize. differential equations as an active area of research is much more abstract, dealing with things like existence/uniqueness of solutions (when picard-lindelof doesn't apply), or stability when a solution is known to exist but cannot be analytically written down (and so integration has to be performed). hell one of the clay problems is about a solution to a pde. so no it's not this straightforward except in basically toy examples.