[2 April 2009]

Why does Nelson Goodman want to work so hard just to avoid talking about classes or sets?

Earlier this year I spent some time reading the section on the calculus of individuals in Nelson Goodman’s The structure of appearance (3d ed. Boston: Reidel, 1977) and the paper Goodman wrote on the subject with Henry S. Leonard (Henry S. Leonard and Nelson Goodman, “The calculus of individuals and its uses” The journal of symbolic logic 5.2 (1940): 45-55).

I was struck by the lengths Goodman goes to in order to avoid talking about sets, although his compound individuals which contain other individuals seem to be doing very much the same work as sets. Indeed, the 1940 paper makes a selling point of this fact. On page 46, Leonard and Goodman write “To any analytic proposition of the Boolean algebra will correspond a postulate or theorem of this calculus provided that …” (In other words, with some few provisos, if you can make a true statement about sets, you can make a corresponding true statement about individuals in the calculus of individuals. The provisos aren’t even statements you can’t make, just restrictions on the form you make them in. Instead of saying “the intersection of x and y is the empty set” you have to say they are discrete. And so on.) And the concluding sentence of the paper (p. 55) is: “The dispute between nominalist and realist as to what actual entities are individuals and what are classes is recognized as devolving upon matters of interpretative convenience rather than upon metaphysical necessity.“

In other words, Goodman seems at first glance to be simplifying the world by eliminating the notion of sets and classes, and then to be complicating it again in precisely similar ways by taking all of the fundamental ideas we have about sets or classes, and reconstructing them as funny ways of talking about individuals. Cui bono?

This afternoon I saw a review by Anthony Gottlieb, in the New Yorker, of a recent book about the Wittgenstein family (Alexander Waugh, The House of Wittgenstein: A family at war), which seems to suggest a solution. Gottlieb quotes a suggestion from the physicist Heinrich Hertz:

Hertz had suggested a novel way to deal with the puzzling concept of force in Newtonian physics: the best approach was not to try to define it but to restate Newton’s theory in a way that eliminates any reference to force. Once this was done, according to Hertz, “the question as to the nature of force will not have been answered; but our minds, no longer vexed, will cease to ask illegitimate questions.”

(Throws a new light on Wittgenstein’s remark about not wanting to solve problems but to dissolve them, doesn’t it?)

It’s true that once you rebuild the ideas of set union, intersection, difference, etc. as ideas about individuals which can overlap or contain other individuals, and eliminate the word ‘set’, it becomes a lot harder to describe a set which contains as members all sets which are members of themselves, or a set which contains as members all sets which are not members of themselves. The closest you can conveniently get are statements about individuals which overlap themselves (they all do) or which do not overlap themselves (no such individual). Good-bye, Russell’s Paradox!

And consider the surrealist joke I ran into the other day:

Q. What is red and invisible?
A. No tomatoes.

A user of the calculus of individuals can enjoy this on its own terms, without having to worry about whether it’s a veiled reference to the fact that some typed logics end up with multiple forms of empty set, one for each type in the system. One for integers, if you’re going to reason about integers. One for customer records, if you’re going to reason about customers. And … one for tomatoes?

Q. What is red and invisible?
A. The empty set of tomatoes.