Oh, I think they are equivalent. In both mathematics and musical professional training, there is “perfect practice makes perfect”. Professional training with musicians you must stop right when you make an error, back track, go slowly and repeat until you get it right before continuing with the piece. I would place discovery and use of heuristics as equivalent to improvisational practice. Mathematics is an art and there are many equivalent ways to express similar things as with music. With music, you can “hear” the calculations as they happen but with mathematics you would only hear the calculations if you can read the steps and understand the placement of heuristics in surprising places. There is a difference in quality of process and results between a student using heuristics and a professional using heuristics just as there is a difference between the improvisation of a professional musician vs the improvisation of a student. I can’t read mathematical formulas but I can read code and I can “feel” what it’s doing as I look it, which I suspect people who can read mathematical formulas can do, which is why I believe it is equivalent to musical improvisation: the same musical idea can be present but be brought forth in numerous ways.

Oh, I think they are equivalent. In both mathematics and musical professional training, there is “perfect practice makes perfect”. Professional training with musicians you must stop right when you make an error, back track, go slowly and repeat until you get it right before continuing with the piece.

I would place discovery and use of heuristics as equivalent to improvisational practice. Mathematics is an art and there are many equivalent ways to express similar things as with music. With music, you can “hear” the calculations as they happen but with mathematics you would only hear the calculations if you can read the steps and understand the placement of heuristics in surprising places.

There is a difference in quality of process and results between a student using heuristics and a professional using heuristics just as there is a difference between the improvisation of a professional musician vs the improvisation of a student.

I can’t read mathematical formulas but I can read code and I can “feel” what it’s doing as I look it, which I suspect people who can read mathematical formulas can do, which is why I believe it is equivalent to musical improvisation: the same musical idea can be present but be brought forth in numerous ways.

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What I wonder is:
Is what is discovered, whether a previously unheard musical improvisation or a novel conceptual space in mathematics a region that did not exist before, or is it a permutation / rotation that was always possible but never attempted?
 
 Math is in a strange place too. It is not science but it is treated as such. It supports science but it is a creative act. Programming finds itself in a similar bind and I don’t know the answer honestly.

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