Sunday, December 05, 2004

Language Induction

This weekend, I realized something interesting about natural language and its relation to logical positivism. What seems deep to me now might actually be trivial to the professional philosopher. Still, I haven't heard it explained before, so I'm going to try and explain it here.

Quine writes about the indeterminacy of translation. His classic example is that of a field linguist learning a new/unknown language from a group of native speakers. The linguist and his party encounter a rabbit, to which a native speaker points says "Gavagai!" This could mean "rabbit" or "rodent" or "gray rabbit" or "he's eating my carrots," etc. Of course, the linguist creates theories about the word's meaning, and does experiments to determine which theory is correct. The more subtle the meaning of the word "gavagai", the longer it will take the linguist to determine the meaning of the word. An infinitely subtle language might take forever to learn.

This analysis applies to our own native language, too. As children we learn our native tongue in the same way. Quine's work is often interpreted as placing limits on the formality of natural languages, and therefore limits on how any formal theory of verification can be applied.

Along related lines, Wittgenstein's Philosophical Investigations is often abridged to a single sentence: "meaning is use." What Wittgenstein is saying is that, in natural language (e.g., in English), the meaning of words is dicovered by induction. We look at how a word is used in social context, and infer the word's meaning. This makes it difficult to isolate individual propositions for logical analysis. Can we rigorously analyze the proposition "all men are mortal" when the meaning of its constituent terms (e.g., the word mortal) are defined by their use in our society?

I claim that we can indeed do so, and quite easily.

When we look carefully, we see that the analysis of natural language is a microcosm of the scientific method. We construct theories and confirm them using roughly Bayesian methods. Each new observation is taken in the context of what we already know, and the confidence levels for our theories of meaning are updated accordingly.

Formal science uses a perfect language that has total precision. It's called mathematics, and it enumerates all logically-consistent structures. A theory is a model of empirical observation that is described in mathematical language. Theories are scientific when they are testable, i.e., when experiments can be performed that will alter our confidence level in the theory using Bayesian methods. In science, there are always an infinite number of theories consistent with a finite set of observations. This isn't a problem, it just means that any effective theory we have developed might one day be replaced by one that is more effective.

Natural language is part of the empirical world, so it is not surprising that natural language can be probed with science. Correspondingly, there are many theories of meaning consistent with a finite number of linguistic observations, but, in principle, this need not cripple linguistics any more than non-unique theories cripple particle physics (i.e., not at all).

Back to our original question.

Each theory of meaning we construct for the proposition "all men are mortal," is implicitly a theory about all of the words in the proposition. However, this is no different from the task at hand when we tackle a scientific proposition like "the mass of the electron is 0.511MeV/c2." To understand this scientific proposition we need to know about the theory of electrons, the special theory of relativity and so on. Scientific propositions build theories upon theories, yet the result is always a single overall theory. In the case of Quantum Electrodynamics, the theory makes predictions accurate to ten decimal places. In principle then, theories of meaning might be no less imprecise than theories about particle physics.

Previously, I had only grasped this concept at an intuitive level: if natural language is a scientific phenomenon, the indeterminacy can always be overcome by studying its underlying physical mechanisms, e.g., psychology and neuroscience.

Why is any of this a big deal? One of the attacks on logical positivism is that indeterminacy and "meaning as use" add so much fuzziness to natural language propositions that it is futile to to speak of semantic meaning at high precision. This analysis makes a compelling case that natural language is as deeply analytical as science because it is science.

There's one more part to this story. I have claimed that a child's natural language comprehension is an informal kind of science. Yet, such science - the kind that we do intuitively - makes no explicit use of formal mathematics. What is the equivalence of formal and informal science?

A good example might be our comprehension of the behavior of falling objects. For example, the speed of a falling ball is v(t) = gt2/2 where t is the time since release and g is the acceleration due to gravity. A nine-year old child can't do algebra, so how does she play catch?

If we graph the curve of v(t) we get a parabola. This geometric shape is an alternate representation of the theory of motion for a dropped ball. Further, any physical system (electronic, optical, thermal, nuclear) that provides a parabolic response curve can serve as a representation of the laws of motion. Hence, the child does not need algebra, she only needs some electro-biochemical representation of a parabola to model the motion of the ball.

The neural networks in our brains have more than enough computing power to apply the methods of informal science:

1. Observe patterns.
2. Generate a collection of structures that are consistent with the observed patterns.
3. Create theories by mapping structures to the observations.
4. Test the theories with further observations. Add the new observations to the master set of patterns.
5. Add successful theories to the set of observed patterns.
6. Repeat

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