Suppose you’re a social scientist who wants to explain why something occurs. You have a model in your head: *x* causes *y*. How do you go about making your case?

There are basically two approaches you can choose. In the first, you try to be as airtight as possible. You develop your theory in terms of concepts that people can understand, note the caveats, and in general try to capture as much insight as possible. You end up with a very long treatise, and if anyone argues with you, you can say “Yes, I accounted for that on page 354.”

Alternatively, you can try to ignore all the caveats, claim *ceteris paribus*, and draw up a highly stylized proof of concept that shows the mechanism you have in mind. You end up with a dense but elegant academic paper full of Greek letters.

The conventional wisdom in economic theory is that the latter is preferable to the former. It’s better to have a precise mathematical model that bears a fuzzy relationship to the real world than a fuzzy verbal model that bears a direct relation to the real world. Mathematical models are *precise*, and make their assumptions explicit. By contrast, verbal models afford too many degrees of freedom and run the risk of being unfalsifiable. Without mathematical precision, the logic goes, it’s too easy to be drawn into the illusion of insight without actually having gained any falsifiable knowledge.

But it’s unclear that the interpretive fuzziness of a mathematical model – when you try to map model concepts to real world objects – affords fewer degrees of freedom than the “internal” fuzziness of a verbal model. You haven’t eliminated your degrees of freedom; you’ve just moved them from the model stage to the mapping stage.

It might *look* like you’ve eliminated degrees of freedom if you assume the mapping is straightforward. In the physical and experimental sciences this is usually an ok assumption. The mathematical description of “electron” is for all intents and purposes a good representation of actual electrons, and we have a pretty good idea about what “actual electron” means in terms of observations. But what about concepts like capital? Or savings? Or even money? What about things like utility, payoffs, or welfare? A highly abstracted measure aggregated over social facts is not nearly so straightforward. With complex and meaning-laden (as opposed to objective) concepts like these, eliminating degrees of freedom in the model itself *necessarily* introduces degrees of freedom into the mapping. The more straightforwardly “capital” figures into your model, the less straightforwardly your model applies to the world.

Mathematized models, therefore, face a dilemma. If they naïvely regard the mapping process as straightforward, theories will be *spuriously falsified* if the mapping process was in error. I’ve argued that a naïve mapping of the concept of “money” onto M2, for example, led to the spurious rejection of the monetarist theory of the business cycle in the early 1990s, a move which had (and continues to have) grave consequences following 2008. Falsifiability, when achieved by a rigid commitment to a faulty concept map, is not a virtue but a vice.

On the other hand, if mathematized models are honest about the ambiguities in the mapping process, a mathematized model has no fewer degrees of freedom than a (good) verbal model.^{1} Did the test of your model come back negative because your causal process is totally wrong? Or just slightly wrong? Or entirely correct but you operationalized one of your concepts wrong? These are exactly the same kinds of ambiguities that a verbal model faces.

I’m *not* arguing here that verbal models are always preferable, or that mathematization is illegitimate, or even that the critique of verbal models is invalid. But on the margin, we should prefer mathematical models when concept mapping is relatively unambiguous, and verbal models (in order to provide legitimate room for complexity) when it is not.^{2} Mapping can, of course, become less ambiguous over time. It makes sense that the collection of national income statistics led to an efflorescence of mathematical economics. Theory can reduce ambiguity too; returning to the example of models of money in the business cycle, one could argue that the development of Divisia indices allows the concept of “money” to be applied much more straightforwardly than before. Nevertheless, there are strong reasons to believe that, in social systems, the ambiguity of concept mapping is often a feature, not a bug. Where this is true, verbal models have the potential to add to our legitimate social-scientific knowledge in ways that precise and self-contained mathematized models cannot in principle do.

- For fairness’ sake, I’ll restrict us to comparing the best verbal models to the best mathematical models.
- Koppl’s (2002) argument that mathematization is legitimate where a “system constraint” reduces the significance of individual choice is a subset of this argument: a tight system constraint reduces mapping ambiguity, hence the fact that many neoclassical constructs only have unambiguous meaning “in equilibrium”.

## James

Jul 29, 2019 at 17:17 |https://www.jstor.org/stable/23026500

## Richard

Jul 29, 2019 at 17:41 |When one works better.

## Roger

Jul 30, 2019 at 3:10 |See also

The Problem of Economic Assumptions in Mathematical Economics Author(s): Diran Bodenhorn

Source: Journal of Political Economy, Vol. 64, No. 1 (Feb., 1956), pp. 25-32