When I was in high school taking math, the thing I hated more than anything else was doing geometric proofs. Not because I didn’t understand them or couldn’t get them right (which was, in fact, my problem when we moved on to trig...) but because they were so bloody meticulous. I could usually see the solution pretty early on, but writing out every step to get there was such a drag and a bore and I hated it.

In college I was introduced to symbolic logic, which is effectively doing “proofs” with a sentence by replacing its clauses with symbols. My professor, who remains one of my favorite teachers ever, was convinced solving logical proofs was the greatest thing any human being could aspire to do. I had to disagree (actually, even though I respect this man tremendously, we didn’t agree on much of anything...). Learning the rules of logic was extraordinarily useful, but it was not what got me going in the morning.

Logic was touted to me by my very analytic professor and a few others as a way of thinking objectively. It was as if logic put limits on reality. Which was an interesting notion that I never really dwelt on until very recently. For whatever reason, I was thinking about the kinds of proofs we did in that logic class or in the geometry class I took in High School. Some of those proofs were designed to “prove” the very rules we used to solve other problems. In thinking about that process, it occurred to me that you can only “prove” one rule at a time. Further, when you do that you assume other rules are true so that you can solve the proof. It is as if the rules of logic are the supporting beams of a structure. You can perhaps take one beam out at a time to inspect it, but if you do more than that the structure becomes unstable. There is no such thing as starting a logic problem from complete scratch with no rules whatsoever. It just wouldn’t work. To me at least it seems that the implication of this is that the system of logic is dependent on itself, that the full system in some sense assumes itself. There is, in other words, a subtle circularity to any system of logic.

Now by invoking the term “circular,” which is a term generally associated with fallacious reasoning, I seem to have just thrown logic out. That is not actually the move that I would like to make, but I want this realization to be the kind of starting point for the next move I do intend to make. So even though this post is insanely short compared to how much I normally write, here is where we end for now. Later on I will consider how we might “redeem” logic as it were from the apparent problem we have uncovered.

The "circularity" that you are identifying are known more generally, and less ominously, as "first principles." Yet first principles are hardly endemic to logic; they are found in every academic discipline, and they are foundational to, if not the goal of, many of the experimental sciences.

ReplyDeleteSo, in one sense, your post is too narrow. You claim that you retrieve a "subtle circularity" to "any system of logic," yet you have found something much greater. What you have found is the origin and goal of knowledge itself--whether that knowledge be theology, philosophy, english, math, whatever. Any structure of knowledge that even purports to be systematic will be one in which first principles will be operational. I could only see a nihilist viewpoint as evading this charge.

On the other hand, your post is way too broad, since if you are going to "redeem" logic, you may as well be "redeeming" anything and everything a student (at least in the Western tradition of schooling) has ever learned.

Thanks for the comment! I think you are getting at a foundationalist approach in your answer, and to tip my cards a bit I'll go ahead and say that I am probably not a foundationalist. While I'm not getting into the technicalities of this discussion in either this post or the follow-up post that I am working on, I think what I'm getting at could be seen as a kind of critique of foundationalism by implying that the "first principles" are in fact circular. But I don't think this is particularly endemic either, I think it just means we need to redefine what our objective is in using anything like a "first principle," which I will explain in a bit more detail in the upcoming post.

ReplyDeleteWell from a mathematical standpoint, you're right and you're wrong. Yes, the bases of proofs are often other proofs, and those starting proofs are founded on some assumptions. And yes, by altering those assumptions you can dramatically alter the mathematical "system." However, that doesn't make them in any way invalid, or even circular per say. It just means that this is math/logic according to this particular set of assumptions. There are in fact systems of mathematics that do not share the same basic assumptions that the math that we know of hold to; we just never talk about them since they're not of much use to anyone other than theoretical mathematicians (at least as far as I know). And those systems are completely logically and mathematically valid (as a side note, apparently it's surprising how much stays the same even in those systems). So really, by framing logic as something other than a consistent set of rules, you're ascribing properties to logic that shouldn't be there (if that makes any sense).

ReplyDeleteAgain, strictly from a math perspective.

The point you are making is very close to where I want to go with this. Next post should be up soon!

ReplyDeleteSo I definitely can't comment on your most recent post.

ReplyDeleteDisregard that the comment was just too long

ReplyDelete