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

NOT A = [ NOT B AND NOT C ]

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 Thing^TM, 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:

NOT C => NOT A

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.