June 18, 2011
Keynes is rather dismissive of what he calls ‘“mathematical” economics’. The following passage is from chapter 21 of The General Theory:
The object of our analysis is, not to provide a machine, or method of blind manipulation, which will furnish an infallible answer, but to provide ourselves with an organised and orderly method of thinking out particular problems; and, after we have reached a provisional conclusion by isolating the complicating factors one by one, we then have to go back on ourselves and allow, as well as we can, for the probable interactions of the factors amongst themselves. This is the nature of economic thinking. Any other way of applying our formal principles of thought (without which, however, we shall be lost in the wood) will lead us into error. It is a great fault of symbolic pseudo-mathematical methods of formalising a system of economic analysis, such as we shall set down in section vi of this chapter, that they expressly assume strict independence between the factors involved and lose all their cogency and authority if this hypothesis is disallowed; whereas, in ordinary discourse, where we are not blindly manipulating but know all the time what we are doing and what the words mean, we can keep “at the back of our heads” the necessary reserves and qualifications and the adjustments which we shall have to make later on, in a way in which we cannot keep complicated partial differentials “at the back” of several pages of algebra which assume that they all vanish. Too large a proportion of recent “mathematical” economics are mere concoctions, as imprecise as the initial assumptions they rest on, which allow the author to lose sight of the complexities and interdependencies of the real world in a maze of pretentious and unhelpful symbols.
There is much truth in the above, I think, and it is truth that applies not only to “economic thinking” but to any kind of thinking that relies on formalization. Statistical analysis is plagued with this kind of problems. Keynes does lay too much stress on the matter of interaction between factors. The problem with formal methods is not particularly with neglecting various effects – it is that they simply are false in various ways (neglecting various effects is only one of the sources of falsehoods). Informal methods have the same problem, of course – and in addition have problems associated with informality.
The special problem with formal methods is their privileged status in discourse. Formal methods are usually difficult for the uninitiated to analyze – pushing them out of the realm of contestability for a large section of the audience. The initiated, on the other hand, are often also indoctrinated into accepting certain assumptions that are convenient for their formal analysis. In addition, the initiated often have a vested interest in maintaining the privileged status of the formal analysis. Thus those who can contest often don’t. Another important source of privilege is that mathematical models enjoy a reputation for scientific rigor and practical power which makes out-of-hand dismissal a difficult position.
Thus the issue is not inherent in formalization. There is no particular problem in keeping things at ‘the back of our heads’ when manipulating formal symbols. People who carry out such manipulation don’t do so in a blind manner (I certainly don’t and I would be surprised to learn others do). Each manipulation step is motivated by an intuitive understanding of the meaning of the step and the implication of the result. The implications of assumptions on the analysis can be examined and alternatives can be considered. When this is not done, it is not done due to practical considerations, not because the method does not allow it. Such practical considerations are present when carrying out a word-argument as well.
The strength of informal methods, then, is that they are usually obviously contestable. People are used to examining informal arguments and contesting them, opening the possibility of rational critique. Of course, not too infrequently informal argumentation becomes hard to examine. A non-trivial book length argument can easily be outside the realm of examinability for most people. This is the situation with, say, Marxist theory, various philosophical schools, or Keynes’s own work. In such situations, there seems to be little advantage to the word-arguments over formalized arguments. (Contrary to the situation with mathematics, a blanket statement dismissing such work is socially acceptable, due to the commonplace and undeniable fact that word-arguments are not made truer by being longer or more complicated. But such out-of-hand dismissal is not any more of a rational decision than the uncritical acceptance of an argument.)
Formalization does have its uses. Its main advantage over words is that it eliminates ambiguity. Associated advantages are often brevity and facility in ascertaining consistency and identifying missing details, and similar effects. That is, it does a good job at providing “an organised and orderly method of thinking out particular problems.”
All this goes to explain why I think that having a mathematical model of Keynes’s General Theory is useful despite contravening Keynes’s own strictures in that same work. I have spent considerable time and effort understanding that work. I find Keynes’s thinking compelling in its method – it is careful, self-reflective and wide-ranging; it is philosophical in the most positive sense of the word. The book is well worth reading. Still, I believe the presentation suffers due to a lack of a general formal framework. Oddly, Keynes does not avoid mathematics altogether (as he mentions the passage above) – but his few mathematical sections focus largely (as far as I can tell) on minor or tangential points. They do not seem to illuminate the main argument but to obscure it.
The model I presented attempts to correct this situation. Unlike Hicks’s oft-referred-to paper, Mr. Keynes and the Classics (1937), this model describes Keynes’s main lines of thinking as they appear in the book (as I understand them, of course), rather than by trying to reconcile them with contemporary convention.