Dear Country Joe:
I am still wipin' the tears from my eyes after reading your post. Its nice you took the time to look up information on logical positivism and Kripke.
:I don't get the impression here that old Kurt [Godel] would say, "WELLLL! Not necessarily cuz we ain't all that sure that the plus sign is really a plus sign even though it looks to be a plus sign, and who is to say what plus really means anyway in the grand scheme of things. A plus sign is nothing more than 2 minus signs at right angles to one another, so really, a plus could actually be a double minus sign which would indicate that 2 minus minus 2 is not a 4 at all but a -2."Yeah! That works for me. How about you?:
I have also been heavily influenced by Paul Davies' _The Mind of God_. If you will recall, he notes that Godel demonstrated the veracity of his incompleteness theorem in 1931. This theorem tells us that there are certain mathematical statements "for which no systematic procedure could determine whether they are either true or false" (Davies 101). In other words, it now seems that undecidable propositions obtain in mathematics. This problem, says Davies, occurs when a mathematician engages in self-referentiality. There are thus paradoxes in math akin to the medieval conundrum:
Socrates: "What Plato is about to say is false."
Plato: "Socrates has just spoken truly."
"Tom cannot prove this statement to be true."
It is no wonder that Davies relates this experience:
"John Barrow has remarked wryly that, if a religion is defined to be a system of thought which requires belief in unprovable truths, then mathematics is the only religion that can prove it is a religion."
To sum things up, I am not saying that 2 + 2 does not equal 4. My point is that one cannot PROVE 2 + 2 = 4 apodictically. Davies suggests that we KNOW such propositions as 2 + 2 = 4 on the basis of a prioricity. Here he takes a page from Immanuel Kant's epistemology (theory of knowledge), which makes a distinction between a priori and a posteriori ideas. Kant also differentiated between analytic, synthetic, and a priori synthetic propositions. Davies' thoughts are thus similar to phenomenological realism, which generally posits the notion of eideational intuition. That is, we KNOW that 2 + 2 = 4 since we eidetically grasp the solution of a certain proposition.
But I enjoyed your post.
Take care!
Dan
Duns the Scot
"Nobody is taller than himself or herself."