[Archives] Emanuel Derman: Metaphors, Models & TheoriesVW Staff
Metaphors, Models & Theories
November 22, 2010
Theories deal with the world on its own terms, absolutely. Models are metaphors, relative descriptions of the object of their attention that compare it to something similar already better understood via theories. Models are reductions in dimensionality that always simplify and sweep dirt under the rug. Theories tell you what something is. Models tell you merely what something is partially like.
Metaphors, Models & Theories
Sleep is the interest we have to pay on the capital which is called in at death; and the higher the rate of interest and the more regularly it is paid, the further the date of redemption is postponed.
So wrote Arthur Schopenhauer, comparing life to finance in a universe that must keep its books balanced. At birth you receive a loan – consciousness and light borrowed from the void. leaving an absence in the emptiness. Nightly. by yielding temporarily to the darkness of sleep, you restore some of the emptiness and keep the absence from growing limitlessly. Finally you must pay back the principal. make the void complete again. and return the life originally lent you.
By focusing on the common periodic nature of both sleep and interest payments. Schopenhauer extends the metaphor of a loan to life itself. The principal is life and consciousness. and death is the final repayment. Along the way. sleep is the periodic little death that keeps the borrower solvent.
Good metaphors are expansive; they compare something we don’t understand (sleep), to something we think we do (finance). They let you see in a new light both the object of interest and the substrate you rest it on. and enlighten upwards and downwards. The common basis of Schopenhauer’s metaphoric extension is periodicity. Taking an analogy based on matching regularities and then extending it into distant regions is a time-honored trick of mathematicians. You can see it at work in the extension of the definition of the factorial function n!=n x (n-1) x (n-2)…1.
Using the exclamation point is traditional but clumsy. Since n! is a function of n , it’s more elegant to express it via the function F (I!) defined by F (n) = (n – l)! . which satisfies the recursive property Fo! + l) = “F (n). You can regard this property as almost a definition of the factorial function. If you define F(l) = l . then F (n) for all integers greater than 1 can be found from the recursive definition.
The definition n! = n xo1 – l)x (n – 2) 1 works only for positive integers n. The definition F(n + l) = nF(n) seems more malleable. Why shouldn’t there be some a frmction F(x) that satisfies the relation F(x +1): 117(1) where x is not necessarily a positive integer? Why shouldn’t the factorial function exist both for x = 3 and, say, X = 3.2731
The Swiss mathematician Leonhard Euler discovered (invented?) the gamma function I ‘(x) that does indeed satisfy I'(x + I): xl'(x) for all x . For integer values of x. it agrees with the traditional factorial function. For non-integer or even complex values of x. I’(x) serves as a smooth extrapolation or interpolation of the factorial from integer to non-integer arguments. It’s smooth because it coincides with the factorial function for positive integer arguments, but maintains the crucial recursive property for non-integers. Mathematicians call this kind of extension called analytic continuation.
The gamma function is a metaphorical extension of the factorial, in which one property. its recursion. becomes its most important feature and serves as the basis for extending it. It’s a bit like calling an automobile a horseless carriage. preserving the essence of carrying and removing the unnecessary horsefulness. or like calling a railroad ferrovia in Italian or Eisenbahn in German, focusing on the fact that it’s still a road, but one made instead of iron. Analytic continuation is a method of modeling a function. But whereas most models are restrictive – a model train is less than a real train – in mathematics, a new model can be something greater rather than diminished. That’s because mathematics deals entirely with its own world. and everything you do extends it rather than confines it.
Most of the words we use to describe our feelings are metaphors or models. To say you are elated is to say you feel as though have been lifted to a high place. But why is there something good about height? Because in the gravitational field of the earth all non-floating animals recognize the physical struggle necessary to rise. and experience wonder when they see the world spread out beneath them. Being elated is feeling as if you’d overcome gravity. People dream of flying.
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