Sunday, September 7, 2008

truesdell on theory, nature, and mathematics

The revolutions of world war and socialism and quantum mechanics, however right and necessary and fruitful, have clouded the massive solidity, the serene confidence of classical mechanics. Classical mechanics has weathered through, standing fast behind the smoky putrid mists. . . . While "imagination, fancy, and invention" are the soul of mathematical research, in mathematics there has never yet been a revolution.
The ancient Greek philosophers speculated whether matter were an assemply of tiny, invisible, and immutable particles, or a continuous expanse. As the quantitative, mathematical science of the West developed, the debate continued but became more and more definite and detailed. The great theorists proposed specific mathematical theories, restricted to certain specific kinds and circumstances of bodies, for example, to "aeriform fluids" subject to moderate pressures.

Until the first decades of this century it seemed possible that one or another theory would turn out to be the final one, the one that would explain everything about matter and thus be universally accepted as "correct", while all competitors would be defeated. Far from being borne out, this hope now seems childish. Our picture of nature has become less naive. While in the nineteenth century more and more aspects of the sensible world were shown to be mere appearances, mere "applications" of a few fundamental "laws" of physics or biology, the recent enormous production of experimental data has undeceived us of our former simplisms. The line between the living and the inanimate has been blurred if not erased. Within the once indivisible atoms has been found an ever growing host of mysterious "elementary particles" whose nature and function are scarcely clearer than those of dryads and familiar spirits.
Of course these discoveries have brought with them different attitudes toward theories of nature. Those who push forward the frontiers of experiment cannot wait for the thoughtful, critical, and hence cautious and slow analysis that mathematics has always demanded. Mathematicians, for their part, cannot afford to waste their time on physical theories of passing interest.

These contrasting standpoints are reconciled by a keener appraisal of the role a theory is to play. A theory is not a gospel to be believed and sworn upon as an article of faith, nor must two different and seemingly contradictory theories battle each other to the death. A theory is a mathematical model for an aspect of nature. One good theory extracts and exaggerates some facets of the truth. Another good theory may idealize other facets. A theory cannot duplicate nature, for if it did so in all respects, it would be isomorphic to nature itself and hence useless, a mere repetition of all the complexity which nature presents to us, that very complexity we frame theories to penetrate and set aside.
If a theory were not simpler than the phenomena it was designed to model, it would serve no purpose. Like a portrait, it can represent only a part of the subject it pictures. This part it exaggerates, if only because it leaves out the rest. Its simplicity is its virtue, provided the aspect it portrays be that which we wish to study. If, on the other hand, our concern is an aspect of nature which a particular theory leaves out of account, then that theory is for us not wrong but simply irrelevant. . . . With this sober and critical understanding of what a theory is, we need not see any philosophical conflict between two theories, one of which represents a gas as a plenum, the other as a numerous assembly of punctual masses.

Truesdell, Clifford, from An Idiot's Fugitive Essays on Science, largely "Statistical Mechanics and Continuum Mechanics," 1979

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