Saturday, September 03, 2005

On Meaning

I've been debating philosophy on other blogs lately. Occasionally, I come across a debating colleague of good temperament, who challenges me to think carefully and refine my ideas. The following is an edited version of one of my recent posts about meaning.

leibniz: What reason do we have to think that 'the meaning of a proposition is its method of verification or falsification' can be empirically verified or falsified?

Clearly, we cannot answer this question without some definition of meaning.

My claim is that the definition of meaning is such that, for a meaningful proposition, P, one can determine one or more distinct propositions that are consistent with P and one or more propositions that are not consistent with P (excluding ~P). This determination need not be complete. For example, I may determine that test proposition P1 is consistent with P, and that test proposition P2 is inconsistent with P, but not know whether test proposition P3 is consistent with P. In other words, P must have logical consequences to be meaningful.

Indeed, this is the principle of verifiability, i.e., the claim that any meaningful proposition is ultimately equivalent to some set of empirical test propositions or else it is part of some self-consistent mathematical system (i.e., it is equivalent to mathematical test propositions). Again, in my view, mathematics is empirical. Let's refer to propositions that meet my definition of meaning as verifiable propositions.

Now, you can challenge my claim with the counterclaim that my definition of meaning is wrong or incomplete.

Let's suppose that meaning is deeper than I have suggested and that not all meaningful propositions are verifiable propositions. In that case, we might be able to construct two propositions Q1 and Q2 that have different meanings, but which are compatible with exactly the same set of logical or empirical test propositions. This hardly seems reasonable, since we could not then distinguish the meanings of the two propositions except by looking at the symbols used to express the two propositions. For example, we would have to admit the possibility that given that the two propositions, "the cat is in the hat" and "the tac is in the hat", can be substituted for one another within any set of logical and empirical test propositions, they don't mean the same thing. In other words, we would have to claim that raw symbol substitution changes meaning. This isn't supported by the evidence that meaning can be preserved under language translation or changes in typeface.

Suppose the challenge to my definition of meaning is dropped, but instead the claim is made that we can still form a mutually consistent set of metaphysical statements that is meaningful. Let's look at what distinguishes a mathematical system from a metaphysical one.

We can create a large number of independent mathematical systems by devising axioms and deriving theorems within each system using the rules of that system. Each mathematical system is independent of the others when its axioms differ. If I start with an algebraic system, A1, and add the axioms x = 5, x + y = 10, I am not precluded from creating a new system, A2, in which x = 7, x + y = 8. The two systems are never in contradiction because their contexts are different. The axioms of A1 may contradict those of A2, but this is unimportant because the systems exist in isolation. Every proposition of A1 coexists with A2 and vice versa.

Similarly, all of these mathematical systems are independent of the physical world. However, we may be able to construct a new system that incorporates an association between a chosen mathematical system and physical phenomena (i.e., a scientific theory).

For example, noncommutative algebras (NCAs) may form a consistent mathematical system, independent of whether or not free neutrons decay in 15 minutes. However, we could form a scientific theory about neutrons that relied on NCA, and which incorporates connected empirical and mathematical propositions from NCA. However, if the theory were falsified by some experiment, the mathematics of NCA would not be falsified, because the mathematics is subject only to its own axioms. Only the new system incorporating the association between the mathematics and the empirical propositions would be falsified.

Now, to metaphysics.

Metaphysical systems also consist of a set of propositions {M} = M1, M2,...,MN which are inter-related by logical operations. For example, we can say that God is supreme and therefore Satan is not supreme and so on. In that sense, {M} is as meaningful as any mathematical system because, for any Mi with can find a distinct Mj that is consistent with Mi or a distinct Mk that is inconsistent with Mi.

However, metaphysical systems need more than just internal mathematical relations. Metaphysical systems have to mean something about the world. That means that we must be able to locate a set of empirical propositions about the world that contains propositions consistent with and inconsistent with {M}. If we fail to do this, and every proposition in {M} is consistent with every empirical proposition, then how can we distinguish {M} from a mathematical system? We cannot. Metaphysics just pushes symbols around on paper, and, at best, creates some mathematically consistent system. It never has anything to say about the world because it never makes a falsifiable prediction.

In summary, I have tried to show that

1) meaning is defined as a logical property in the context of a set of propositions.

2) that if meaning were not so defined, then one would have to conclude that meaning is dependent on the specific symbols used to express it. This is not supported by empirical evidence of meaning.

3) under my definition of meaning, mathematical systems are meaningful in their own context.

4) that scientific theories associate mathematical systems and physical empirical propositions in a falsifiable way.

5) that metaphysical systems are indistinguishable from isolated mathematical systems, and have no meaning relative to the world, only to themselves.

Imagine taking a test in algebra class, and solving problem #1 that begins "x = 5, ...". Then imagine being profoundly confused when problem #2 on your exam begins "x = 3,...". How can x be both 3 and 5?

Metaphysicians suffer from this very same delusion. They confuse truths within one isolated mathematical system with truths within other, independent systems.

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