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Monday, May 23, 2005

LI read a fascinating article by Mary Morgan in the Winter Philosophy of Science journal, and we want to write about it. Mary Morgan, Margaret Morrison, Nancy Cartwright, and Ronald Giere form, in LI’s mind, a sort of collective that bridges the distance between the Latourian Science in Action school and the last gasp tradition of the Popperians. Significantly, many come out of the London School of Economics, long a stronghold of the late Popperian school.

As philosophers of science know, the first thing that scientists will mention when asked for a philosophy of science is falsification. This is less a thoughtful judgment of the practice of science as they have observed it than boilerplate. As is well known, the falsification criteria comes from the Logical Positivist school in the twenties. More specifically, it comes from Karl Popper, as a leading thesis in the investigation of the “logic of discovery.” What those scientists usually don’t know is that the leading thesis was part of a program that claimed to offer a devastating and final refutation of induction in science.

The idea that science could be captured in a logic is an essential move in the logical positivist program of reducing all salient questions of knowledge to questions of formal language. This inflation of the notion of language signals the lineage of the philosophy: language functions, here, much as Kant’s reason functions in the Critiques. We won’t go over the adventures and impasses of this program. Suffice it to say that Popper’s picture of science was confessedly abbreviated. It didn’t tell us much about how statistics decisively changed the practice and meaning of experimentation in science. It didn’t tell us much about models. It made assumptions about hypothesis building that isolated that activity from science practice. But, mainly, it was aimed at telling us about the truth – with the idea that science is ultimately constructed around the truth. Other Popperians – Lakatos, Feyerabend, Kuhn, etc – extended to Popperian impulse to larger views of research programs, and in the process destroyed Popperian rationality. It was self mined from the very beginning. But the essential idea – that at the heart of science there is a wholly deductive program that is theoretically capturable in a formal language – still remained, yearly becoming much worse for wear. As, indeed, the original and simple thesis of falsification proved itself unable to account for large swatches of science, and fell into logical difficulties of its own (Hempel’s White Raven paradox).

Actually, in LI’s eyes, the logical positivists simply encoded, in a new form, the reaction of philosophy to science that arose during the early modern period – notably, the Cartesian idea that science advances by the hypothetico-deductive method. It was that idea which Newton fought against from the correspondence around his first paper, the great 1676 letter on light and color, to the Opticks which he published after the death of his inveterate enemy, Robert Hooke. Newton’s entire seriousness in not “framing hypotheses” was a great step towards separating, utterly, physics from metaphysics. It is a step the philosophers have never wholly forgiven him – or even wholly understood in him. Hence, the perennial urge to annex the natural sciences as a branch of logic.

Where does this leave philosophy of science, then? PoS has an uneasy relationship with Sociology of Science, insofar as it gives up the pretence of deducing the principles of science and applies itself to the observation of scientific practice. In one way, this makes PoS a very exciting field. Where other branches of philosophy amuse themselves with dubious thought experiments, PoS observes real ones.

Morgan’s paper takes a case from Economics – a model called the Edgeworth Box – and shows how it permutated over the course of a century, as economists mathematized their discipline. The Edgeworth Box (see this history by Humphrey ) was invented by Francis Edgeworth in “his now famous Mathematical Psychics ([1881] 2003), a book of almost impenetrable erudition from this Irish economist. For Edgeworth, mathematics was a form of expression, a language, and because of its special qualities it was a tool or instrument both for expression of economic ideas and for reasoning about them. But in Edgeworth's mind it was also an instrument of imagination.”

Edgeworth imagines two individuals with two goods to exchange. The world of these individuals is closed, to an extent: the traders do not have competitors. But they are free to contract or not. In other words, the Robinson Crusoe story so savaged by Marx. As Morgan puts it, the Edgeworth box “defines the locus of points at which exchange might be contracted as those where, whichever direction a move is made away from that set of points, one trader gets more and the other less utility. This set of points is termed the "contract curve.’” From Morgan: “Edgeworth's diagram refers to individual traders alongside their goods, and provides an indifference curve for each individual and their contract curve. And while it seems initially that the whole space is open for trade as in Marshall, the argument defining the contract curve in conjunction with the indifference curves through the origin (i.e., points at which utility is equivalent to that obtained from zero exchange) rules out some areas of the ninety-degree total space. Edgeworth is so impressed by his own diagram and the way that it allows him to work out some results which had previously failed to yield to general analysis, that he writes that his figure "is proved to be a correct representation" and that the diagram provides "an abstract typical representation" of a process (Edgeworth [1881] 2003, 36; my underlining).”

Now, the interesting thing about this abstract typical representation is that it represents a dynamic – although Morgan doesn’t mention it, surely there is some slight reference, here, to Maxwell’s fields, which are also constructed to capture trajectories. Morgan, instead, references Marshall’s theory of trade between two countries as the template for Edgeworth. LI notes this as a limit to exploring model building with an exclusive endogenous focus.

Morgan points out that Edgeworth’s original representation is not a box: “What might now be taken as the irreducible shape of the Box--namely, a closed set of two amounts of exchangeable items represented by the sides of the box, and two traders at opposite corners, each with two axes of potential commodities to trade with--are not there from the beginning.” Yet by 1950, the standard form of the Edgeworth diagram was a box. LI won’t reproduce Morgan’s history. But we are interested in the conclusion of that history: “For the economists in my case, learning to represent the economy in new ways was drawing new things. The mathematically expressed economic elements inside the Edgeworth Box--the indifference curves, the contract curve, the points of tangency and equilibrium, etc.--are new, mind's eye, conceptual elements, not old, body's eye, perceptual elements. Scitovsky's 1941 use of the diagram provides an excellent example of this point. The critical point of his article is the difference between allocative efficiency in which the total resources in the economy are fixed (denoted by a fixed size box) and those in which the resources change (denoted by a change in box size). The representation of the effect of this change proves to be quite difficult to understand for the modern user of such boxes. It is tempting for the reader of the diagram to suppose that, by expanding the box, there are just longer axes, more goods (for example, cheese and wine) to be exchanged for given indifference maps (representing tastes, which have no reason to alter). But of course these indifference lines represent contours in conceptual space, and increasing the total resources effectively expands the box from the middle. As the axes are lengthened, perceptual space expands, but so does the conceptual space, so that the original contract curve opens out to provide a region in the middle through which the new contract curve runs. This distinction between conceptual space and perceptual space also helps us to distinguish when a diagram is doing any work in the argument. If the diagram is about perceptual space but the argument about conceptual space, the reasoning will take place, as Mahoney describes it, "off the diagram" and the diagram will be, at best, an illustration, rather than a tool for experimentation and demonstration. (9)
Yet, as we know from Humphrey's 1996 history, during the early-twentieth-century period, the Edgeworth Box diagram was a creative tool used to derive propositions and prove theorems in economics. It was indeed a tool for reasoning about the economic world using the conceptual resources of the diagram.”

To evoke an entirely different philosophical tradition – the notions at play here, in Derridian terms, subsist in the gap between language and text. Those who read Derrida as collapsing text into language – as a run of the mill social constructionist, with the usual language idealism -- don’t understand him at all. The Edgeworth box is an excellent example of the trajectory of signs that constitutes a “text”, in Derrida’s terms. And what Morgan says, finally, about the ontological status of the Box is exactly what deconstruction would predict:

“I should be careful here to point out that when the Edgeworth Box is described as a mathematical model, it is not only made of mathematics. We can illustrate this best by considering the allowable movements or manipulations which can be made in the model. The notion that the two traders will be at some kind of optimum when their indifference curves meet at a tangency makes use of mathematical concepts and logic. But the apparatus of offer curves, indifference curves, and so, for example, the spaces in which trade is ruled out, depends on understanding the conceptual content of the elements in the model. Thus, Scitovsky's diagram showing the implications of increasing the resources requires manipulations of the diagram which are determined by the economic meaning of these curves, not by the logic of geometry. Both mathematical and subject-matter conceptual knowledge constrain the details of the representation and define the allowable manipulations. This is surely not particular to models in the form of diagrams, and indeed it seems likely that most if not all "mathematical" models in economics depend on economic subject information to constrain or define their rules of manipulation. From this point of view, there would be as much difficulty in "translating" the Edgeworth Box into "just mathematics" with no subject content as into "just words" with no mathematical content. The Edgeworth Box diagram carries an independent representational function: (10) it contains conceptual apparatus which could not be represented, or manipulated, in verbal form and indeed cannot be entirely expressed in purely mathematical terms.”

Which last sentence opens up a few too many vistas.

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