Minor Note: Equations

640px-codomain2-svgOne peculiarity of the way I think about quantitative relationships that took me years to figure out seems to stem from different ways of treating functions in economics (and the social sciences generally) and mathematics. My economics professors would write down equations, like π=pY-wL-rK, where the equation as a whole represented an accounting identity and the variables were symbolized with the appropriate abbreviation: w for wages, L for labor, and so on. Then if you wanted to consider a causal hypothesis about the determinants of one of the variables, you simply stipulate that the variable is in fact a function of those variables, without choosing a functional form: thus w(x, y) >>> π=pY-w(x, y)L-rK, and so on.

I assume there are two reasons why social scientists converged on this approach to model-building — well, three. The first and most obvious is that there is no hard and fast line in the social sciences between the various uses of formulae as metaphors, as shorthand for rough hypotheses, as descriptions of ideal types, and as fully-specified empirical models. Thus it is useful to have an amphibious formal convention which can do double duty as a pseudo-accounting identity and as an actual mathematical notation. The second reason may perhaps be a special case of the first: if you move fluidly between rough hypotheses and fully-specified models, you are going to want to use an elastic notation that can easily be adjusted back and forth between an explicit functional form and one or two weak first-order conditions. But the third reason to use them, at least in lecture courses and textbooks, is that it acclimates people to a certain way of thinking about (a) causation and (b) the relationship between causation and rates of change. Whatever phenomenon you are trying to explain can be interpreted as a function of its causes, both in a logical sense (reducing the explanandum to an interaction between the explanantia) and in the analytical sense (refining and constraining the explanation with the vocabulary of monotonicity, continuity, and differentiation).

But mathematicians think about functions in way that seems different to me. At least since Bourbaki, the standard convention has been f:xy, “f is a function from domain x to range y.” (Right? Sometimes I’m surprised by the conventions other people learn. Reading the TeX symbol list gives me the impression that God is babelling us at this very moment…) The range and the domain determine what sorts of arguments the function can take and what values the function can have. In pure mathematics the main questions about these spaces are their algebraic properties; i.e. are the values of the function sets? vectors? ordered triplets? elements of a ring?

If you typically think about functions this way (which is salutary!) it promotes the instinct that when a function is modeling some causal process, the types of phenomena you are giving an explanatory role correspond to the range of the function, and the type of phenomenon you are taking as the outcome to be explained corresponds to the domain of the function. So to return to our earlier example, where variations in x and y cause variations in wages, we would instead think of the function as f:(x, y)w, “f is a function from x and y to wages.”

This may seem like a pedantic difference in notation. But it’s really a quite important difference in approach to abstract thought. Once you’ve explicitly identified wages as the space of outcomes under investigation (the range of variation), it would be senseless to describe the function itself (the mapping between the domain and the range) as “wages”. So if the function isn’t wages, what is it? Well, when I start thinking about relationships in this manner, I start to identify it with the specific causal process I have about what hypothetical interaction between x and y that causes the prevailing wage to rise and fall. And, having identified it with a particular causal process that maps x and y onto w, I am usually no longer thinking of the function as an elastic catch-all which could, at least in principle, take all of the causal influences on w as its arguments, if it were necessary to stretch it to do so.

I don’t know whether the two approaches to formalism matter much, in the end. I am a dilettante in both realms, alas, and the main reason I noticed the conflict is that I will unconsciously switch from one style to the other while I am thinking about a problem, generating tangles of inconsistencies and category errors. But I suspect that the two formalisms correspond to two different ways of looking at the world. One treats functions as analytic statements (in the philosophical sense: not synthetic; tautologous) about formal identities. The other treats functions as processes, as transformations of inputs into outputs, or as sequential relationships between abstract states.


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