10 March, 2005

Sometimes, thinking is a bad thing

I was working on my thesis tonight; this is the third draft, and I was making corrections to the sixth chapter (which contains the most important result, insofar as its being the one that, when my advisor saw it, he told me that I could graduate). I get to one point, and I realize, Maybe I should explain in more detail what I'm doing here.

The smart thing to do would be to think about why I'm doing what I'm doing, and in fact that's what I started doing. Fortunately, I found a major mistake in a definition — thank God I found it before my advisor did. Since he hasn't emailed me in days, I imagine that trip to Japan must have cut him off from his email, so I can get him a revised copy.

For those of you who understand logic sufficiently well, I can describe the problem thus: I had defined

A = [ B OR C ]
I went about proving A by proving B. This is logically valid.

I also went about "disproving" A by disproving C. Of course, that's logically invalid: if A = [ B OR C ] then by our friend De Morgan

This is what had me in such a tizzy. To disprove A by my definition, I had to disprove BOTH B AND C. I hadn't done that, so I hadn't really disproved A. This was A Very Bad ThingTM, and what made it worse is that I had no idea how to disprove B. The entire reason that I had been disproving C, is that disproving C is easy.

Making it worse was the embarasmment that I had made a very basic logical error that should not appear in any mathematical writing, least of all a thesis. We're talking page 29 of my Symbolic Logic textbook here. (Wow, that's an expensive book.)

I knew that by disproving C, I had disproved what I wanted to disprove. I also knew that by proving B, I had proved what I wanted to prove.

The contradiction is plain as day. I knew that C implies B, but not the converse: that is,
[ C => B ] AND [ B =/=> C ]

It should have been a stroke of luck that I realized quickly that I had to change the definition of A. It should also have been a stroke of luck that I realized immediately after that I wanted
A <=> B
Unfortunately, I knew that in some problems I needed to disprove C, and I thought that meant that I was disproving A. Logically speaking:

The logically savvy among you have seen the problem: if NOT C implies NOT A, then A implies C (A => C). By definition, A and B are equivalent (A <=> B), so B implies C (B => C). This contradicts the observation above that B does NOT imply C, and I'm in danger of being laughed out of a career if I submit such an assertion to my PhD committee and they notice it before I do.

A clever fellow would have chuckled here and realized that what B proved is not the same as what [ NOT C ] disproved. Okay, I did think of that, too, but — I didn't chuckle, maybe because I'm not all that clever at 10pm. For some reason, I thought the paradox invalidated my work, rather than proving I had simply written a definition wrong.

After thinking about this a looooooong while (too long), I realized what the problem was: I was confusing matters. It was correct to define A <=> B. The trouble was that I did not want to disprove A by disproving C; I wanted to disprove a different predicate D, which is related conceptually to A, but not the same. That conceptual relationship is what had me in a tizzy.

You'd think this wouldn't be so hard, since I've been working on this problem for some years. However, while writing my thesis, I realized that no one had ever defined rigorously (as far as I could tell) this concept A that we have lots of results on. So, I defined it, only I defined it wrong. (It seemed reasonable at the time, and for good reason. In fact, that old definition isn't really wrong, but inelegant: if C implies B, then [ B OR C ] is equivalent to B.) All the thinking on the thesis has got me stressed and worked up & confused! I worked on this one thing for at least two hours, probably three, which set back the rest of the corrections that I was working on. Whatever the case, it's now 2.45am; and I finished chapter 6 about half an hour ago. (Now, in revising this post, it's 3.30am. Ugh.)

I've titled this post "Sometimes, thinking is a bad thing," and that's true. It's not the thinking that's bad, but the time at which it's done. :-)

In any case, all that confusion had two good results: first, I now have a working definition with no inherent paradoxes; second (and more important), I have a much better understanding of what's going on. This will help me enormously on my defense next week if someone should ask the same question I asked tonight.


Alessandra said...

Hi Jack,

I don't know if what you described is just an oversight or if it is the problem with having to revise something over and over again, and then getting to that point where we stop seeing in a thinking way. It's a very interesting cognitive phenomenom (don't know if cognitive here would be the best label). This happens to me a lot with text. I am so flabbergasted to see how my mind completely blanks out at registering certain spelling or sequence of words when I revise a text (specially if over 2-3 times) because there is an overriding text already in my brain, so I am not reading my text as if for the first time where everything is new and highly *visible*.

jack perry said...

You have a point. I've noticed the phenomenon you describe. I'm not sure that's it, but I'm not sure that I can exclude it entirely, either.