Mathematics, beyond doubt?
On the axis: Relation to certainty · an axiomatic system
Long the very model of certainty, do mathematics escape all doubt, or do the choice of axioms and the limits revealed by the incompleteness theorems show that even here certainty has a horizon?
Justification
Against , Descartes turns doubt against itself: he drives it to the point of , supposing an evil genius who deceives him about everything, to see whether anything withstands it. Yet the very act of doubting presupposes a subject who doubts: I can call everything into doubt except that I am thinking, and therefore that I am. The cogito is no dogma laid down in advance but a certainty won through doubt itself, an indubitable foothold from which the whole of knowledge can be rebuilt. The conclusion is thus firmly : full certainty is attainable, and skepticism refutes itself when pushed to its limit.
I think, therefore I am.Discourse on Method, IV
Meditations on First Philosophy, I and II · Discourse on Method, IV
Anchors: Descartes · Gödel · Hilbert