Thursday, August 20, 2009


It occurs to me, I haven't talked much about my progress with the logic paper I have mostly written. Today I had an interesting dream that I think was about it, so I'll say something about it. I finished a first draft of it last year. But there are errors. The few very significant errors I have corrected very easily. (Since I am not much of an example-oriented person, I have a tendency not to notice much when I prove something ridiculous if error causes me to do that.) That's not the problem, though—these were easily corrected. What I most hope to do in the current draft (which I hope to make public) is give the paper polish. In particular, the part of logic that most bothers me is model theory. Mostly, I think the subject should be done away with. But there are necessary bits and pieces there that I need. How to present these bits and pieces in the right way has proved more difficult than I had anticipated.

Everything came to an impasse in early May, I think it was. I was sputtering along steadily typesetting the paper and polishing it up, but then a little less than half way through, I realized that there must be some general result that allows one in certain circumstances to replace entailment with implication (in the context of the silly logic). One of my professors at North Carolina had taught me that entailment ought to be distinct from implication, and indeed I believe it. The formula A implies B should be thought of as the most general formula whose conjunction with A entails B, and the entailment relation should be thought of as the same thing as the "less general than" relation (where "less" is not interpreted strictly, in fact any formula should be considered less general than itself). Annnywayyy, though it may be true that entailments and implications tend to hold in similar circumstances, they really are different. For instance, in my silly logic, A entails B precisely if not B entails not A; but it turns out that if A implies B is silly, then not B implies not A is just plain false. Still, though, it was intuitively clear to me that there must be a metatheorem allowing one in a certain natural subset of inference rules to replace entailment with implication (replacing metalogical connectives with ordinary logical connectives) to yield theorems. In other words, I could sense the existence of a (for me) advanced applied logic theorem that would enable me in one giant swoop to prove about, idk, maybe 20% of the basic logic theorems that I otherwise would have to prove separately. In the course of about a week or two in March or April I guess it was, I really went at it, (metaphorically) engaging in much banging my forehead against the wall, and came up with two cute similar little results that gave what I was looking for. The problem is that the whole structure of the results was very demanding of the little bits and pieces of what is called model theory. And when thinking about it, I couldn't at all easily separate all the nonsense I had learned about model theory from the essence of the matter. Somehow where my brain had previously been fairly clear-headed, I now became confused; even though I felt the problem was just making a few very simple definitions, my mind became so much like a tornado full of irrelevant flying debris, logic became hard. I guess I felt I had to just get away from logic for a while to let things settle or I would be unable to sense what these definitions were. And clearly it was very important for elegance (and even usefulness) that the right definitions be made. At any rate, the clarity was just not there. I haven't thought about logic since May.

Mostly, I have been blaming myself for having been too applied. But the dream I had this morning makes me think I'm taking the wrong outlook about it. I dreamed I had just showed up at Michigan (the University in Ann Arbor) again, and I was just about to move into a dorm room or efficiency or something when there outside on a sidewalk someone was selling a sturdy utilitarian and attractive antique dresser. Wow! I thought. I bought it right away, good deal. Then it occurred to me: I will need to put this in my room. It might be a tight fit and I have no truck (or even car) to get it there. Somehow this dresser represented my advanced applied logic theorem. It first appeared very useful to me, but then on second thought, it wasn't as useful as it seemed, making life difficult for me. The wood was interesting. It seemed some kind of very high quality wood, but wasn't quite as dark I think as black walnut, which made me afeared 'twas the dreaded white walnut, but mostly I didn't think it was that either. It looked kind of like Oak but without the open grain. And then a kindly-looking black (he wouldn't have been black if it was made of white walnut) man saw me in my quandary standing in front of the dresser I had bought, and offered to help me put it beneath an awning while I had to go away to prepare to put it in its proper place. And as he was doing that it turned out (it became revealed) that there were two little diorama cabinets that you had to get (free) with the dresser. The sorts of things which if presentable you might expect to find ensconcing a very small nativity scene or surrounding a show-and-tell presentation if kids could afford that sort of thing and the teacher wanted to keep it on display. But the thing is, they were covered in peeley green paint. Perhaps some would have claimed the effect antiquesy looking, but it was hideous. These hideous things, I figure, do represent the necessary parts akin to model theory that I need to find or keep. But first I have to remove the awful paint, which represents all the rubbish in that subject. The best I can figure, the moral of the dream is that I should appreciate the theorem for what it is made of, which is a fine wood easy to underappreciate from its ad hoc quality. The insights the theorem may cast on what is called model theory may be more important than the applications. In fact, probably, I have been thinking, I should prove its applications separately anyway, only introducing the theorem after I have proven the results in the normal way. It is too big and particular and advanced a result to present early. But instead of looking at it as something that introduces ugliness I should more see it as something that, if enough light is cast upon it, can be stripped of its ugliness, revealing fine wood perfect for displaying beauty and truth.

I dream quite a bit about going back to university, and mostly it is always wrong. When I have dreams about going back to Michigan, I typically end up having to study a bunch of stuff too fast I am not interested in, which I probably can't or won't do anymore, and I always do things in more insane ways not good for efficiency, ruining everything (like buying the dresser). And I'm similar when I dream of going to UNC, except it's worse inasmuch as also I frequently am just wandering about hiding from my math professors because I don't want to have to tell them I didn't get my Ph.D. from Michigan like I had hoped. So, typically (in my dreams), I end up going there with enthusiasm but never go to a class, and each day I'll think to myself, I should really show up to class, but I don't, and then the tuition bill comes and of course I've missed the drop deadline because I was obsessed with understanding why I'm not going to class, and everything (my scholastic record and savings) gets ruined. And with my roommates, in my dreams I never fit in with them.

I used to dream a lot about going back to high school. Similarly, that would always turn out wrong. I'd dream that I'd catch the bus and show up, but then think, now wait a second, I forgot, I'm 30 or 40 or whatever, and people that old are not allowed to go to school--I wouldn't be on the homeroom roster. And then I'd realize I have to go, and so I'd sneak out (without trying to be too obvious that I am an older person a sneaking out), and then after leaving the building I'd have to walk home because I'd remember I was too old to take the bus, but Needwood Road somehow in my dream becomes like some sort of transcontinental highway full of drawbridges, fords, lakes that have to be rowed across (after finding the hidden rowboat), etc., with distracting museums along it. But now I more often instead dream of college, and that goes wrong too, but not so much because of my age (except when it comes to relating to roommates).

I need to be intellectual and practical, but not probably in a university setting. That's what I mostly feel, and my dreams seem to support that. I guess the internet offfers opportunities, and if I can become wise enough, I figure I won't have to make much enlightened effort at selling myself. I can't really imagine how it will work, but eventually, if I keep becoming wiser (what I'm most convinced I am skilled at doing), I'll become so preposterously wise eventually it will become obvious to people (if I maintain openness and don't get considered an enemy by the search engines) that I am extraordinarilly wise, whereupon, something good might happen. That's about the extent of my plan as to how to be practical in life.

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