A couple of people have weighed in with mathematics, and that's always a good choice. If we're just going to go charging ahead discussing "absolutes", we aren't going to get past our navels until we give the question some context. The three areas where I think the question will be most interesting are: 1) Are there absolutes in non-empirical/quasi-empirical inquiry? 2) Are there absolutes in empirical inquiry? 3) Are there moral absolutes?
I stay away from ethics, so I'll avoid (3). It is perhaps best to discuss (2) first. Empirical investigation is primarily concerned with the examination of information derived from sensory experience, the a posteriori. It's been beat to death, so I won't mention it in more than passing, but empirical investigation suffers from a problem of induction that prevents us from making any claim of an empirical nature with absolute certainty. There are several proposed remedies to this, and the most commonly accepted one is the pragmatic approach offered by Popper with his criterion of falsifiability. If we cannot positively verify something in the course of empirical investigation, we should at least be able to prove it wrong through experiment. Then, by rigging experiments so that they deliberately target any part of the theory which we are skeptical of, we can at least gain evidence for the accuracy of a theory by failing to disprove it despite several attempts to do so. We kick the theory's tires, so to speak.
Too much is made of the need for certainty in empirical investigation. Hard-line supporters of philosophical skepticism and solipsists create a false dichotomy in which we must either be entirely certain or abstain from making claims altogether. That's rediculous, and actually an informal fallacy. In any case, I don't know for certain that when I let go of a pen it will fall to the ground. But it would be foolish to claim otherwise, or to plead agnosticism, since a lifetime of experience suggests it will fall to the floor. What's more, stating that there is a problem of induction is something different from saying just how much of a problem of induction there is. We've developed language and intelligence. The existence of cognitive systemizability necessarily implies a certain degree of ontological systemicity. The world may not be perfectly stable, but it is at least stable enough to make thought possible, and thought is induction. So, even though we can't be entirely sure that our inferences are utterly inerrant, we are still at liberty to make them, and that is evidenced by the fact that you are now reading words on a screen.
Mathematics doesn't suffer from these kinds of problems. For starters, mathematics doesn't have the problem of induction of empirical inquiry. It's proofs are positive verifications, something that we cannot have in empirical investigations, and once a proof is given, the result is known to be true. There is, however, a price to pay for that. The thing that lends mathematics this certainty is its formal structure of axiomatics. Formal axiom systems aren't real in any way. They're created, in our heads, and accepted by convention. It doesn't even make sense to speak of the truth or falsehood of a mathematical axiom -- only relative consistency and independence from other axioms we might posit. We simply take the axioms for what they are, and see what kinds of results we can obtain from them. The uncertainty will creep in when we make the claim that the mathematics are in any way useful. When we apply a mathematical result to a physical problem, the hope is to give the physical process a quantitative description. This, as Einstein said, only gives us a certain measure of security, however. For mathematical theorems to be completely true, we must mean true with respect to the relevant axioms. Once we apply the mathematics, we are making a claim that the axioms model the real world somehow, and that the hypotheses of our theorems have been satisfied by the physical objects in question. To know that absolutely, we'd have to know the properties of those objects absolutely. So we can't gain absolute certainty there, but can gain more certainty than by experiment alone.
Now:
Huehuecoyotl said:
Then tell me....can one divide by zero and why not?
Multiplication and division are essentially the same process. In fact, in abstract mathematics we don't recognize the distinction and tend to think of everything as something analogous to multiplicaion by fractions. This happens in a study we call "Ring Theory". It's one of the structural theories of algebra and arithmetic, and one of the most widely pursued areas of mathematics. Algebra, being a formal manipulation of symbols, comes from structure. That's what makes the formal manipulation meaningful. That structure comes from the Ring Theory Axioms. Two important rules are the Left and Right Cancellation Laws for multiplication which, as a theorem, only hold in what we know as division rings (rings lacking zero divisors). If zero divisors are present, the structure of the ring is such that division cannot happen in a well-defined manner, if they are not present, then division can happen as a consequence of the left-right cancellation laws. I won't prove it here, but a rigorous proof is given in any book with a title that sounds like "An Introduction to Abstract Algebra".
In any case, the real numbers are a division ring. Actually, they're a very special kind of division ring called a field.
Edited by AlephOne (05/20/07 11:24 PM)
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