Before ending Godel week I want to go over a few other thoughts on both him and his theorem.

Point 1

I am struck how similar Einstein and Godel were. I don’t think I would have said that before. It seems odd they were close friends in New Jersey. But there area number of commonalities. First, they both thought big. While at the Institute, Noam Chomsky crossed paths with Godel and asked him what he was working on. Godel replied, “I’m trying to prove the laws of nature are a priori”. Godel ultimately in his last years even attempted a proof, or at least a line of reasoning, for the existence of God or some uber power. In both cases Einstein and Godel wanted to do something that transcended their fields and in both cases they succeeded.

Second, both were Platonist’s philosophically. While Godel’s theorem has been co-opted by Positivists they were both firmly entrenched in the belief that there was a true reality. Godel firmly believed in the objective and independent existence of mathematical reality. Einstein held a similar worldview with physics. It explains Einstein's reticence for things like the Copenhagen interpretation of quantum mechanics. In short, man is not the measure of all things. Rather, things are what they are and they are real.

Point 2

An awful lot of famous and smart physicists and mathematicians committed suicide. Does their brain state lead to this or is it more due to the fact that creating world class work declines to almost nothing after reaching the age of 40. And this fall from grace seems unbearable. In short this book touched a few of them,

- Boltzmann, the famous thermodynamicist, who committed suicide after his advisor pooh poohed his work
- GH Hardy, a famous mathematician, attempted suicide, failed, wrote a book on a mathematicians life, and then tried again and succeeded
- Wittgenstein, Godel’s arch-nemesis who never believed Godel’s work was right, attempted suicide until he moved to Cambridge and Bertrand Russell (Principia Mathematica) pronounced him a genius
- Turing, mathematician who would learn of Godel at the Institute and in talks with Von Neumann & Wittgenstein and go on to prove a sister theorem to Godel’s, also committed suicide

The seeds of problems that Godel confirmed were evident long before Godel completed his work. After Hilbert rallied the troops in 1900 to find a way to prove the arithmetical formal system was complete and consistent, paradoxes started cropping up all over the place. Bertrand Russell, the Welsh mathematician, who was constructing Principia Mathematica to lay down the formal systems for all math, found a horrendous paradox in set theory (the set of all sets that do not contain themselves) and Cantor did the same (the set of all sets). Hilbert himself even voiced some concern that all was not well. It really took the fresh face and the Platonist mind of Godel to avoid this bias.

Point 4

Godel formulated another theorem. It follows from the first and Von Neumann spotted it right away after Godel spoke his short sentence at the conference. If a formal system (S) is consistent, it’s possible to form a proposition G that is true but unprovable (1st theorem). Trivially if S is consistent then G is true (from the previous sentence). The proof Godel made is carried out in a formal system devised for arithmetic. If the consistency of S can be proved in S, then G would have been proved in S, since it follows from the consistency of S that G is true. This contradicts that G is unprovable. The only way out of this contradiction is to say that S cannot be proved consistent within the context of the formal system S. It is thus impossible to formally prove the consistency of arithmetic within an arithmetic formal system (2nd theorem).

It is probably this second theorem that caused more of a stir. I would argue this was because Von Neumann was already considered a mathematics titan at this time when Godel was unknown outside the Vienna circle. He was thus able to popularize it at the Institute more effectively than Godel in Austria.

Let me rephrase a little. To show a system is consistent, you need a 'higher order' system. The original system can't do it by itself.

Here’s where people have gone with this. A computer runs on a formal system. In other words it follows hard wired rules. It is mechanistic. If any given formal system cannot be proved consistent within itself (2nd theorem), then consistent systems can only be proved by a larger formal system. So within a single computer you have one formal system. Within itself it cannot prove its own consistency and therefore cannot have intelligence. That last bit is a bit of a stretch. But people are arguing that we believe we are self-consistent and therefore computers can never mimic our intelligence. Or another way to think of it is that mathematics didn't take a nose dive after Godel's theorem. We don't have issues in math with proving contradictions or not having formal systems to guide our way. Math does not suffer because of Godel. We still understand true mathematical statements. We, as humans, go beyond pure formal systems. It is therefore impossible to build a computer with artificial intelligence. Minds cannot be explained as machines.

Another way to think of this is that sitting here right now your brain constitutes a formal system. If you should happen to need to determine if something you are deciding is consistent you can rewire brain as it were to create a higher order system to look at that old one. We can keep stepping outside of ourselves.

No matter how complex a computer, it is run by hard wired rules. If we ask the computer what the true propositions are there will be some that it cannot grasp by definition of an incomplete formal system. Some truths about arithmetic that it cannot understand are true. Yet we ultimately don’t have this problem. We may not be able to prove it within a formal system yet we know some mathematics are true that fall outside the purview of formal system proofs. Roger Penrose thought similarly. He said, one cannot use mechanistic descriptions and formalisms to define the mind. He argued quantum mechanics in its non-mechanistic nature might prove a valuable area of examination.

Godel was far more reserved on this topic. He understood the line of reasoning above but thought there were outs for a mechanistic basis for our thoughts. Namely that we are deluded in thinking we can create mathematical proofs outside of a formal system. This seems very possible to me. But if we aren’t deluded, he does agree that we are not machines.

Update

My friend docrpm has posted a nice review of the book Incompleteness here. Rather than covering the mathematics he does a nice job of talking about some of the philosophical stances during Godel's time as well as some historical aspects.

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