Piecemeal Discovery in Mathematics

31 03 2007

I recently saw an interesting post pointing out that the invention/discovery of calculus didn’t take place totally out of the blue (as we’re used to thinking) but was really the culmination of a lot of historical developments that happened to be coming together at the right time. (Possibly including the development two centuries earlier, in southern India, of many particular cases of Taylor series!)

Anyway, this seems to relate to an earlier discussion that Brian Weatherson and I had about the notion of discovery in mathematics. On some pictures of discovery (such as where a proposition is discovered to be true iff someone brings it from being not believed to being known), if someone earlier had a justified true belief in a proposition, then it turns out that no one really discovered it. But in many instances of the real, complicated mathematical and scientific history, we don’t really want to attribute a particular discovery to anyone because particular cases and broad patterns had been noticed by many people independently, and the person to whom the discovery is usually attributed was just the first to codify it in some useful and very general way. Of course, there are also many cases (like the one of Cavalieri’s principle mentioned in the post referred to above, and the case of Lagrange’s theorem in group theory, which came almost a century before the modern definition of a group) in which the standard attribution pushes the discovery anachronistically early, because the development of the special case was thought of as the key to the discovery, rather than the generalization that came later on.

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