In my high school mathematics courses, 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. And that 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 circularity, which is a term generally associated with fallacious reasoning, I seem to have just thrown logic out with the bathwater.
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.