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Note by Juan Otalora
1 year, 1 month ago

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This diagram--this is invention, motivated by some non-rigorous truths? By the way, does "rigorous truths" have a place in this scheme, and if so, where would they be? Michael Mendrin · 1 year, 1 month ago

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@Michael Mendrin This is an approximation of the debate about the discovery or invention of mathematics. Maybe no one will ever be right. the fact that rigorous truths appear in this diagram just explains the ramification of mathematics; when some advance is made, there are plenty of new areas discovered or invented (as you prefer). Juan Otalora · 1 year, 1 month ago

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@Juan Otalora Well, it's been long debated whether mathematics is about invention or discovery. A native wood carver, for example, may speak of "discovering" the animal inside the log, and seeks to "free" it. Others will say he is merely creating animal art from an otherwise featureless log.

I am, and many others, are of the strong opinion that mathematics is not all arbitrary invention. For example, given the axioms and definitions of Euclidean geometry, there are only 5 regular polyhedra, no matter how some may wish to creatively make happen many more. As another example, after the discovery of complex numbers and their extremely useful geometrical properties, it was quite natural for mathematicians to wonder and hope for a 3 dimensional analog of it. Sir William Rowan Hamiltonian sought to generalize it, and ended up with quaternions, which have very interesting properties of their own, but nevertheless not quite like complex numbers. Generalization of such algebras is possible through what's called Cayley-Dickson construction, proceeding from complex to quaternions, octonions, sedenions, etc., each with new properties and very little or no leeway for any "creative license in inventing novel algebras." Another famous example is the classification of finite simple groups--which fall into exactly four categories, which are cyclic groups, alternating groups, Lie groups, and "sporadic" groups. And there is only a finite number of such categories, and mathematicians, after more than a century of work, are finally getting around to completing the proof of this, which includes finding all 26 sporadic groups. Again, no creative invention here--that's all there is, folks! However tremendously complicated it may be.

Adam Strange of "Mythbusters" is famous for his line, "I reject your reality and substitute my own!" In mathematics, you really don't have that choice. Creative invention takes you only so far, the rest is discovery of unalterable reality. Michael Mendrin · 1 year, 1 month ago

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