Posted on June 19, 2007 by Peter Turney
Consider an analogy of the form A:B::C:D, “A is to B as C is to
D”; for example, “mason is to stone as carpenter is to wood”. This kind
of analogy is often called a
- A/B = C/D → A/C = B/D
- 1/2 = 3/6 → 1/3 = 2/6
- mason:stone::carpenter:wood → mason:carpenter::stone:wood
Aristotle talks about this in Nichomachean Ethics, “As the term A, then, is to B, so will C be to D, and therefore, alternando, as A is to C, B will be to D.” Proportional analogy is central in Aristotle’s theory of justice (Book 5 of Nichomachean Ethics). Consider an analogy in which two of the four terms are equal: - A:B::B:C
- “my arm is to me as I am to my country”
Aristotle mentions this kind of analogy in Nichomachean Ethics, “as the line A is to the line B, so is the line B to the line C.” Now, suppose that C is the whole formed by joining A and B: - A:B::B:(A+B)
- “the citizen is to the emperor as the emperor is to the country”
- “the lesser is to the greater as the greater is to the whole”
If we look at this analogy numerically, we get the Golden Ratio: - A/B = B/(A+B) → B/A = (A+B)/B
- Golden Ratio = phi = (sqrt(5) + 1) / 2 = 1.618
- B/A = (A+B)/B = phi
- phi/1 = (1+phi)/phi
Thus the Golden Ratio seems to say something about ethics and politics, or at least the Greeks seemed to think so. Hofstadter and French also make a connection between analogy and ethics, in Fluid Concepts and Creative Analogies (in Chapter 8). It is well-known that “tit for tat” is a good strategy for playing the Iterated Prisoners Dilemma. This observation seems to have some lessons for ethics and cooperation in human societies. The problem with “tit for tat” in the real world (as opposed to the abstract world of game theory) is that two situations (two opportunities for cooperation) are never exactly the same (just as “tit” and “tat” are similar words, but not identical). To play “tit for tat” in the real world, two agents need to perform analogical reasoning. If you lent me a book last week, then I should be willing to lend you a video this week. The two lendings are “proportional”, to use the terminology of Aristotle’s Nichomachean Ethics. Hofstadter and French are concerned with the cognitive mechanisms by which we recognize this “proportionality”. This line of thought suggests that evolutionary pressure (natural selection) for cooperation may have been a large factor in driving the evolution of the human capacity for analogical reasoning. |