"But science, spurred by its powerful illusion, speeds irresistably toward its limits where its optimism, concealed in the essence of logic, suffers shipwreck. For the periphery of the circle of science has an infinite number of points; and while there is no telling how this circle can ever be portrayed completely, noble and gifted men nevertheless reach, e'er half their time and inevitably, such boundary points on the periphery from which one gazes into what defies illumination. When they see to their horror how logic coils up at these boundaries and finally bites its own tail -- suddenly the new form of insight breaks through, tragic insight which, merely to be endured, needs art as a protection and remedy."
-- Friedrich Nietzsche
The Birth of Tragedy
When I was 12 I got a pair of flippers and goggles. One day I got into the water of this dock where I could stand. I decided to do the backstroke out about 50 feet. I then made the mistake of looking down. Looking through the clear water with my goggles I saw a beam of light enter the water, and go deeper ... and deeper .. and deeper, and fade ever so slowly until it was finally completely absorbed by the water; I could tell this was nowhere near the bottom. Immediately my stomach practically dropped right out of me. I panicked, flailing my legs as fast as I could to get back to the shore. I will never forget that experience as long as I live.
I think about that experience often; I use it as a metaphor for intellectual experiences I sometimes have -- especially in mathematics. When I come to understand something surprising in math that violates my intuition, or seems mysterious to me, and of course the experience can be really cool. But sometimes, when I've had to solve some problem or other, and there is no one I can turn to for the answer, the experience takes on a sinister, confidence undermining aspect as well.
I wonder how Bertrand Russell felt when he realized his now famous paradox: "The set of all sets that do not contain themselves." Call this set S. Now, is S a member of itself? Well, if S is a member of itself, then it must satisfy the definition of S, which is not being a member of itself. Thus, if S is a member of itself, then it is not a member of itself. What? Has logic coiled back on you yet? No? OK, suppose S is not a member of itself, then it satisfies its own definition, thus it IS a member of itself. So, if it isn't a member of itself, then it is a member of itself. Down the runny toilet(see chapter 4) goes grounding all arithmetic in set theory!
Hofstadter says that Russell was too timid when confronted with this problem, actually Hofstadter is right here, by eliminating all self-reference:
"This trauma instilled in him a terror of theories that permitted loops of self-containment or of self-reference, since he attributed the intellectual devastation he had experienced to loopiness and to loopiness alone."(Hofstadter, pg.88 Nookbook)
Remember that Russell and Whitehead worked for years on this project. I think Hofstadter's a little hard on him when he says: "...I was disappointed for a lifetime with the oncebitten twice-shy mind of Bertrand Russell."(Hofstadter pg. 89 Nookbook).
It is this type of paradox that Goedel later exploits in his famous incompleteness theorem. I will be going into this stuff when I get to that chapter. I'm taking this opportunity to read the original paper closely. But I'll say here that the goal reducing all of mathematics to formal systems -- perhaps a hubristic goal -- was shown impossible. Perhaps then there is indeed something tragic here. NO, we cannot take all knowledge and completely represent it in a single formal system because we can't even do it with arithmetic!