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completely-ineffable44 karma

Wittgenstein, in his 1939 Lectures on the Foundations of Mathematics is critical of mathematics done to produce a "pleasant feeling of paradox":

There is a kind of misunderstanding which has a kind of charm:

[picture of a circle and a line not intersecting]

"The line cuts the circle but in imaginary points." This has a certain charm, now only for schoolboys and not for those whose whole work is mathematical...

The kind of misunderstanding arising from this assimilation is not important. The proof has a certain charm if you like that kind of thing; but that is irrelevant. The fact that it has this charm is a very minor point and is not the reason why those calculations were made.—That is is colossally important. The calculations here have their use not in charm but in their practical consequences.

It is quite different if the main or sole interest is this charm—if the whole interest is showing that a line does cut when it doesn't, which sets the whole mind in a whirl, and gives the pleasant feeling of paradox. If you can show there are numbers bigger than the infinite, your head whirls. This may be the chief reason this was invented. (page 16, emphasis mine)

Now it's quite certainly false that Cantor's work on 'multiple infinities' was done with this sort of motive. His work has application to real, serious mathematics. (Indeed, in his 1874 paper where he introduced his famous theorem that the reals and the integers are not equipotent, he used this fact to prove a new proof of Liouville's theorem that every interval contains transcendental numbers.) However, what is true is that his work is often presented as having as its main interest this charm of a pleasant feeling of paradox. The incompleteness theorems are often subjected to a similar presentation.

In the short description of the Paradox and Infinity course on the edX page, it seems like it might be presenting these results in this way.

Learn about how some infinities are bigger than others, and explore the mind-boggling hierarchy of bigger and bigger infinities.

My question to Prof. Rayo is: Do you think that we should shy away from presenting work on the higher infinite and on incompleteness as a sort of mind-boggling thing, invoking a 'pleasant feeling of paradox'? If so, how do you aim to avoid students in the edX course getting that impression about these results?

completely-ineffable3 karma

Thanks for the response!

completely-ineffable2 karma

Hi Prof. Grayling!

What is your opinion on groups like Atheism+, formed in response to a perception of sexism and other bigotry within the secular community. Do you think they are correct that such problems exist in the secular community? Are they going about addressing these problems in a productive way? If not, how should such issues be addressed?