Saturday, January 14, 2006

Science Series: Math



Is math a science? Does math exist? Is math created or discovered by humans?

These are old and played-out questions for sure, but I'm finding them interesting right now for some reason. And I did promise some math during our science series this month. It does seem pretty clear to me that a number, say "4", does not exist in the same sense that mass and gravity exist. The number is an abstract and linguistic concept, just like the letter "E" or the word "shower". Of course, this gets you into the dizzying quandary of whether a "shower" exists even if we don't have a word for it. Of course it does, but in what sense? How is it -- if it cannot be verbalized -- differentiated from the flow and flux of the rest of the world?

Anyway, numbers and math seem to present even more problems than words: numbers are eerily useful for describing the physical universe and the laws of physics. A famous 1960 paper about the ontology of mathematics called this the "Unreasonable Effectiveness of Mathematics in the Natural Sciences". The theory that mathematics's uncanny usefulness in predicting natural phenomena means that it must "exist" is generally known as "mathematical realism":
One of the most intriguing features of mathematics is its applicability to empirical science. Every branch of science draws upon large and often diverse portions of mathematics, from the use of Hilbert spaces in quantum mechanics to the use of differential geometry in general relativity. It's not just the physical sciences that avail themselves of the services of mathematics either. Biology, for instance, makes extensive use of difference equations and statistics. The roles mathematics plays in these theories is also varied. Not only does mathematics help with empirical predictions, it allows elegant and economical statement of many theories. Indeed, so important is the language of mathematics to science, that it is hard to imagine how theories such as quantum mechanics and general relativity could even be stated without employing a substantial amount of mathematics.

From the rather remarkable but seemingly uncontroversial fact that mathematics is indispensable to science, some philosophers have drawn serious metaphysical conclusions. In particular, Quine (1976; 1980a; 1980b; 1981a; 1981c) and Putnam (1979a; 1979b) have argued that the indispensability of mathematics to empirical science gives us good reason to believe in the existence of mathematical entities. According to this line of argument, reference to (or quantification over) mathematical entities such as sets, numbers, functions and such is indispensable to our best scientific theories, and so we ought to be committed to the existence of these mathematical entities. To do otherwise is to be guilty of what Putnam has called "intellectual dishonesty" (Putnam 1979b, p. 347). Moreover, mathematical entities are seen to be on an epistemic par with the other theoretical entities of science, since belief in the existence of the former is justified by the same evidence that confirms the theory as a whole (and hence belief in the latter). This argument is known as the Quine-Putnam indispensability argument for mathematical realism.
From Indispensability Arguments in the Philosophy of Mathematics

Mathematical formalism, on the opposite end of the spectrum, is the theory that mathematics, like verbal language, is a game with its own internal rules, which inexorably produce "correct" answers. See Wikipedia.



The tantalizing question is whether in developing mathematics and applying it to the world (e.g., the trajectory of comets, the acceleration of falling objects, the melting point of metals, etc.) we are discovering -- innately -- universal laws. Could it be that mathematics, which started for us with counting "separate" units, and which would appear to be as human a product as language, could correspond or be useful in measuring and predicting the phenomena of the universe? This appears to be the case.


ancient tokens from the Near East, precursors of writing and math


I feel that I'm out on the edge of my ability to discuss this meaningfully at this point. First, I don't know a tremendous amount about math, or its specific applications to the universe. (For example, don't we have to create new rules of math to describe quantum mechanics?) Second, there's something here that is a deep logical puzzle, which my relatively feeble mind cannot process: what does it mean that humans created math if math corresponds to the world? Does that make math a science of sorts? But math is not developed by observation and use of the scientific method of experimentation and observation; rather, it is developed -- as far as I know -- largely through the application of logic. The formalism argument does not appear to account for math's eerie applicability to natural phenomena. So if math is developed through logic, the implication is that logic is connected in some rather profound way to universal laws of the universe. As you might expect, this possibility is hotly debated. (Quantum mechanics again throwing in the monkey wrench. I wish I understood what the hell quantum mechanics was.)

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