It is literally my job to prove mathematical theorems. Well, it's part of my job; another (around this time of year, much bigger and more taedious) part of my job is examining other persons' attempts at proving particular theorems and judging their success or failure in so doing. In either case, I had better be fairly sure of my ability to distinguish a valid proof from a piece of B.S.
Now, let me distinguish a theorem from a theory. A theorem is a statement in a formal language that is necessarily true, while a theory is a well-supported explanation for observed events. A theorem, once proven to satisfaction, can only be falsified by showing that the language in which it is written is inconsistent; whereas a theory, by its very nature, is falsifiable -- if in some given situation a theory predicts one event but a completely different event is observed to occur, then the theory is invalid.
Here is a theorem, known as the "triangle inequality":
If a and b are non-negative real numbers, then a + b >= sqrt(a^2 + b^2).
Proof. (a + b)^2 = a^2 + a*b + b^2 >= a^2 + b^2 = (sqrt(a^2 + b^2))^2, because as both a and b are non-negative, a*b >= 0. Thus a + b = sqrt((a + b)^2) >= sqrt(a^2 + b^2).
Here is a theory, known as the "ideal gas law":
If P represents the pressure upon a mass in gaseous state, V represents the volume of that gas, n represents the number of molecules in that gas, and T is the absolute temperature of the gas, then P*V = R*n*T for some constant R irrespective of all the other values P, V, n, and T.
The theory is falsifiable by inspection, for if P is held constant and T goes to zero, then V goes to zero as well -- which is simply impossible, because all mass has volume. Nevertheless, it is a good
approximation to the real behaviour of gases for values of T significantly greater than zero.
Now, there are many ways to establish the truth of a theorem. Here are a few:
- Direct proof, as I used above. That is, take the statement of the theorem and show by deductive steps how the premises require the conclusion. In short, if I have a statement like "P implies Q" then I assume P holds and by calculation, show that Q holds as well.
- Proof by reductio ad absurdum. In this one, if given a statement like "P implies Q", I make the initial assumption of the opposite, that P implies not Q -- but then show that that cannot be the case because some contradiction must then hold if it does.
- Proof by mathematical induction. Given a statement like "For all n, P implies Q", I first show by some other method that for some n=k P does in fact imply Q, and then using that fact and some way of expanding n from k to other values, I show that P implies Q for all possible n.
- Proof by counterexample. (This is the one MiB hated so much.) Given a statement like "All P imply Q", if I can find some P for which Q does not hold, the initial statement is disproven. This is important! Really!
- Proof by case analysis. Given a statement like "P implies Q", I divide P into possible initial sub-cases P_1, P_2, ... P_n, and then show by differing methods that for each sub-case P_i, Q must hold nevertheless.
- &c.
The common underlying structure of all these proof methods (and all others, indeed) is the use of the strictures of the predicate calculus or some other equivalent formal system as rules for steps in the proof. One identifies the premises and the conclusion of the statement with atoms of the predicate calculus, and by means of mathematical calculation and by logical deduction, the proof is established. If a calculation is done erroneously or a deduction is invalid then the proof as a whole is invalid.
So much for math, though. Math is simple. Even physics is pretty simple. People and their activities are complicated. Let me take for example something said in another place:
Angel on Crack wrote:
There is no good and no evil. If there were, God would have to exist, and absolute morals would have to exist. Without proof of these two requisites, there can be no concept of good and evil except those that are individual.
By the exacting standards I would apply to marking a math exam, this attempt at proof is a total washout. To be fair, though, it might not really be an attempt a proof at all, but rather a sketch of how a proof might go. If the terms were clarified, the statement were made precise, and the logical steps between deductions were explicit (and in the style of the predicate calculus!), it might become a proof that I could accept. I admit now my doubt that such a proof could ever be constructed, but who knows for sure. Besides, whether or not AoC's contention is true is a topic for another discussion entirely.
Now, I fall back on boring old logic as a means for deducing things about the world perhaps merely out of ingrained habit, perhaps indeed instinctually. My brain is my primary organ for processing information, and whatever else it is, it can act like a computer so in a very fundamental way it
is a computer.
But, when I step away from the realm of analytic sentences into the much larger realm of observations, I lose the certainty of the a theorem and have to live with the well-foundedness of a theory. Theories are made to unify observations into predictive laws. However, if a theory is truly well-founded, then its predictions will be accurate and treatable as premises in a logical process. Human experience is that well-founded theories do combine with each other logically to make new theories that are also well-founded, though if we are honest with ourselves we test those new theories with new observations just to make sure. In this way, the fact that my brain is a computer (and perhaps more than that, but again, who knows) is useful to help me cope with the outside world.
Many philosophical statements are analytical statements: e.g. "Act as though the maxim of your action should become the maxim of all action." Many are not, but are nevertheless rather similar in form to analytic statements: e.g. "It is erroneous to fail to distinguish the representation of a thing from the thing itself." We are accustomed to combining all these kinds of statements
logically in order to deduce new statements, and I believe we are doing things right thereby.
Intuitively -- don't you have trouble understanding how the Christian God can be both three persons and one person at the same time? It's a mystery of the Christian faith, but it's also a logical impossibility. We frown and say, "I believe it is true, though I do not understand it" or "that's utter balderdash" according to our preferences, but we do not say "Sure, I saw Josh the other day. He was there by himself all alone, and his two other selves were there standing beside him at arm's-length from each other." That latter pseudo-sentence is in fact not even valid English: it is semantically ill-constructed and thus at best a meaningless noise and at worst ... pernicious.
Besides -- do you have a
better way of making sense of things?