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I think, as with many paradigm changing and major ideas, its novelty required a change in the prevailing thinking of the epoch, which always makes people uncomf
by voidhorse 9y ago
I think, as with many paradigm changing and major ideas, its novelty required a change in the prevailing thinking of the epoch, which always makes people uncomfortable--according to wikipedia most objections were to Cantor's notion of transfinite numbers: https://en.wikipedia.org/wiki/Georg_Cantor https://en.wikipedia.org/wiki/Georg_Cantor
There's even a separate article on objections to his theory: https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_th...
Seems like the notion of "completed infinity" which sort of gives infinity ontological status (? I'm no mathematician) did not sit well with some--I guess I could understand why--it definitely isn't an intuitive notion:
"Before Cantor, the notion of infinity was often taken as a useful abstraction which helped mathematicians reason about the finite world; for example the use of infinite limit cases in calculus. The infinite was deemed to have at most a potential existence, rather than an actual existence.[18] "Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already".[19] Carl Friedrich Gauss's views on the subject can be paraphrased as: 'Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics'.[20] In other words, the only access we have to the infinite is through the notion of limits, and hence, we must not treat infinite sets as if they have an existence exactly comparable to the existence of finite sets."
I guess this shows that the overall pragmatic value of your work may eventually win out against formal refutations.
- chriswarbo 9y agoIt may be that a more deep-seated change has taken place within mathematics: the rise of formalism, with Frege, Russell, etc. separated mathematics from intuitions about 'reality'; a formal theorem can't be refuted by saying it's 'unrealistic', since it's a watertight consequence of the chosen axioms and deduction rules. We can attack those axioms and deduction rules as being 'unrealistic', but since Goedel shot down the attempts to define "one true" formalism, it seems that mathematicians have come to accept axioms and logics based on their own merits (e.g. whether they lead to interesting, non-trivial consequences) rather than whether they're accurate models of 'reality'. In this relaxed view, Cantor's arguments make sense from his premises, and it's interesting to entertain the idea of his premises because they lead to such interesting results. Whether or not "completed infinities" are "real" is delegated to the philosophers, while mathematicians 'play the proof game', and change the rules if they make it more fun.