Abstract
Mathematics has been the language of the core of theoretical physics for three centuries or more, at least since the times of Newton and Leibnitz. Tat core, particle physics, explained the movements of bodies here on Earth and in the starry heavens, using essentially the same approach involving the calculus those two worthies invented and Newton’s famous three laws of motion.1 Over time the mathematics was deepened and other branches of physics brought into that core, such as thermodynamics and electrodynamics during the nineteenth century. In the twentieth century new fields emerged, especially quantum mechanics, relativity, and in the past couple of decades a newly respectable cosmology. Over the centuries mathematics proved itself as the only language in which theoretical physicists could express themselves.2
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Notes
A short account of the rise in the use of mathematics in physics can be found in Bochner’s The Role of Mathematics in the Rise of Science (Princeton: Princeton University Press, 1966), especially chs. 6 and 7. Bochner makes much of the “complexifcation,” as he calls it, of physics over the past century and a half. Here too economic theory has followed suit with a long lag time, but higher math is now common in the upper reaches of mathematical economics. Complex variables appear intermittently in Simon and Blume’s Mathematics for Economists (New York: Norton, 1994), a standard text for graduate students, but the calculus of variations does not.
Blaug’s Economic Theory in Retrospect, 3rd ed. (Cambridge: Cambridge University Press, 1978)
and Napoleoni’s Economic Thought in the Twentieth Century (New York: Wiley, 1972) tell much of this story. I began graduate work in economics in 1953 and followed this development with enthusiastic approval, for a while.
Surveyed in Napoleoni, 1972, and Nasar’s Grand Pursuit (New York: Simon & Schuster, 2011).
The later book is Mas-Colell, Whinston & Green’s Microeconomic Theory (New York: Oxford University Press, 1995). The Nobel Committee supplies potted biographies of its awardees. Varian’s third edition (1992) testifies to its longevity; it’s still in classroom use.
The similarities between Simon and Blume’s book, noted above, and AA Boas’s Mathematical Methods in the Physical Sciences, 3rd ed. (Hoboken: Wiley, 2006) are truly striking. Because much of basic microeconomics is static, the budding economist is likely to first encounter the calculus of variations in an advanced macroeconomic growth course. It is fundamental to particle physics and would show up in a good undergraduate course in classical mechanics.
The principle of least action, as it’s sometimes called, is briefly described in Bochner, 203–8, and more extensively discussed in Taylor’s Classical Mechanics (Sausalito, CA: University Science Books, 2005), ch. 6,
and Byron and Fuller’s Mathematics of Classical and Quantum Physics (New York: Dover Publications, 1992), ch. 2.
The early chapters of Wilczek’s The Lightness of Being (New York: Basic Books, 2008)
describe this expanding connectedness, as does Segre’s A Matter of Degrees (New York: Penguin Books, 2002), for quantum mechanics. Isaacson’s biography of Einstein, Einstein (New York: Simon & Schuster, 2007), discusses this issue of integrative general theory in chs. 15 and 23. See note 12 below.
For Ehrenfest’s theorem and complementarity, see AA Griffiths, Introduction to Electrodynamics, 3rd ed. (Upper Saddle River, NJ: Prentice-Hall, 1999).
See AA Tannor’s Introduction to Quantum Mechanics (Sausalito, CA: University Science Books, 2007), ch. 4, sec. 1, for an account of classical-quantum wave function similarities based on the theorem.
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© 2016 Benjamin Ward
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Ward, B. (2016). Physics and Economics—A Theory Comparo. In: Dionysian Economics. Palgrave Macmillan, New York. https://doi.org/10.1057/9781137597366_2
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