Not knowing what you are sure of: e.g. “π + e is irrational.”

It is as unexceptionable as philosophical principles ever are that we can claim to have knowledge if, in addition to belief, we have the proper sort of justification. It seems we can have excellent justification for believing that that π + e  is irrational, leading to high confidence that it is. Yet no mathematician would say that we know the sum is irrational. Why is that?

Nothing is Certain – Cromwell's Rule

I here defend the proposition that absolutely nothing is absolutely certain, where certainty is understood as a flat 0 or 1 as a Bayes prior.

The Continuum Hypothesis and Mathematical Platonism

Does the fact that an apparently straightforward mathematical proposition can be neither proved nor disproved show something deep about the nature of mathematics?

To be a little superficial, the Platonist position in philosophy of mathematics is that the number 17, and all of its friends and relations down to distant cousins, exist quite independently of us. Mathematical truth is discovered in something closely analogous to the way rare jungle fauna are discovered by exploring zoologists. The truths of mathematics are not invented, stipulated, or constructed, not directly, not indirectly, not even very indirectly.