Friday, January 14, 2005

Reflections on Godel's Theorem

by Asim Jalis

I am struck by the fact that Godel's proof is really simple and elegant. It has the same flavor as Russell's paradox (consider the set that contains all the sets that don't contain themselves -- does this set contain itself?). The trick of expressing mathematical statements as numbers seems obvious now given that we do this in computers all the time, but was probably a little less accessible in 1931. Now the interesting this about this is that in my model theory class at Wisconsin we spent several weeks on forcings, which are a generalization of Godel's idea. And then finally covered Godel's proof as a minor example of the powerful concept of forcings. In retrospect this was precisely the wrong way to do this. The course should have talked about Godel first. Everyone is motivated to learn about Godel's theorem. That can easily be the highlight of the course. And then once people became familiar with Godel's ideas, then we could have generalized it. But this was not a one-off incident. For some reason the general->specific route was taken in the majority of the courses that I took, when the specific->general would have been a much more entertaining and engaging journey. In Utah the program was much easier. In some ways I was educating myself, and was free to go from specific->general, if that was what I wanted. So I am trying to think why the professors would go from general->specific? General->specific makes more sense if people are already familiar with the material -- if they are already familiar with the specific cases. So the general case will immediately appeal to them. It would also make sense if they are familiar with both the general and the specific, and if the course is meant to be a quick summary of the field. It is much faster to move through general ideas than to move through specific examples. So maybe the professors were interested in catching up with the cutting edge of the field quickly, so that they could introduce students to current research. Another explanation might be that the professors actually found it easier (in the way they thought) to go from general->specific. This was how they organized the information in their heads. It was like there were two kinds of people in math. The first kind can take a system in complete abstraction, without needing any kind of grounding to reality, and can play with it and extract all kinds of results related to it. The upside of this approach (if your mind works like this) is that you can get up to speed quickly. You can crank out a lot of results. You can get published. In fact you can get published regularly. Math turns into work. The downside to this approach might be that it is harder to evaluate which results are more interesting or relevant. The second approach is to start with concrete grounded examples, such as Godel's proof, and then generalize the ideas up. The downside of this approach might be that it takes longer to get up to speed. The upside might be that you always have several alternate models to fall back on. If you hit a problem in the general case, you can fall back on the specific case and see how you would solve the problem there. Plus, this might not even be a matter of choice. Maybe you go with whatever approach fits your brain naturally. Perhaps this is the INTP-INTJ difference. INTJ's are much better at taking an abstract ungrounded system and carrying it to its logical conclusion, while INTPs need the grounding to make the system appear meaningful and to make real progress in it. At different phases in the evolution of a science or a field of mathematics, different kinds of skills might be necessary. In choosing graduate schools it seems to me that there are two ways to go about it: (a) Either one should find a school where the INTP approach is dominant. In this case the faculty will automatically be interested in the kinds of problems that follow this grounding pattern. It will be easy to fit into this ecology. (b) The other approach is to go to a school which is not extremely competitive, where there is a lot of slack, and free time to pursue one's own interests.