Since fidotron has been piled on, let me defend the point in his/her post. Good mathematics has come out of being worried that what other mathematicians have done isn't quite right, and I think the perspective of the article doesn't acknowledge that.
For example: Cantor's theorem is quite true, only cranks doubt it [1]. But many mathematicians take it to have the corollary that cardinalities greater than that of the natural numbers exist, which does not follow: it is perfectly coherent to say that constructions such as the power set of natural numbers do not exist as a definite whole, and so do not have a cardinality. These kinds of doubt have driven constructivism which has led to interesting work in topology, measure theory, and type theory, and led to such useful applications as calculators for exact real arithmetic.
Cantor's paradise seems to be coherent (likewise I would be deeply surprised if large parts of mathematics turned out to be misconstrued) but the assumptions of large cardinal set theory are grandiose and poorly justified, and yet for a long time those people who wondered if it was wise to embrace the whole edifice were marginalised. It seems that now there are many more mathematicians who are interested in revisiting this perspective [2].
To put Wiles' metaphor in perspective, it is good if some mathematicians step outside the mansion from time to time, to see if the superstructure is up to all the crashing about that happens in the dark rooms.
For example: Cantor's theorem is quite true, only cranks doubt it [1]. But many mathematicians take it to have the corollary that cardinalities greater than that of the natural numbers exist, which does not follow: it is perfectly coherent to say that constructions such as the power set of natural numbers do not exist as a definite whole, and so do not have a cardinality. These kinds of doubt have driven constructivism which has led to interesting work in topology, measure theory, and type theory, and led to such useful applications as calculators for exact real arithmetic.
Cantor's paradise seems to be coherent (likewise I would be deeply surprised if large parts of mathematics turned out to be misconstrued) but the assumptions of large cardinal set theory are grandiose and poorly justified, and yet for a long time those people who wondered if it was wise to embrace the whole edifice were marginalised. It seems that now there are many more mathematicians who are interested in revisiting this perspective [2].
To put Wiles' metaphor in perspective, it is good if some mathematicians step outside the mansion from time to time, to see if the superstructure is up to all the crashing about that happens in the dark rooms.
[1]: https://www.math.ucla.edu/~asl/bsl/0401/0401-001.ps (Postscript file)
[2]: http://homotopytypetheory.org/book/ has been very successful