At 12:41 AM, November 09, 2006, Bruce said...
Are you implying that religious belief is as solid as fundamental mathematical concepts? They are not even close. You are comparing apples and oranges. Your analogy doesn't work...Yeah, I'm being a smart ass tonight. But seriously, to imply that the existence of God is as certain as 2 + 2 = 4 is an insult to mathematicians everywhere.
As paradoxical as it may sound, even our most rigorous reasoning rests at bottom upon human intuitions. Formal reasoning (mathematics, formal logic, etc.) cannot proceed without at least some basic axioms, derivation procedures, formation and transformation rules, and other inferential resources. Those in their turn can be justified only as basic givens which have the property of just seeming right (or necessarily true, self-evident, incorrigible, etc.), or which when employed in ways sanctioned by the system itself generate results which exhibit some required virtue (consistency, etc.). But either way, there will be an ultimate dependence upon some human capacity for registering or recognizing the special character involved. That capacity might be some judgment concerning consistency or coherence, or concerning the rational unacceptability of contradictions. Or it might be an unshakable sense that the foundational logic operations that seem absolutely right to us, really are absolutely right - that our inability to even imagine how denials of such intuitions could even be thinkable, testify to their absolute legitimacy. Or it might be something else entirely.
Mathematician Keith Devlin notes (more or less apologetically) that:
"if you push me to say how I know [that Hilbert's proofs are correct], I will end up mumbling that his arguments convince me and have convinced all the other mathematicians I know."