Use of mathematics and imprecise definition of terms

Question

As a postgraduate student of economics I've been trying to expand my mathematical "toolset". While doing so I've talked to engineers, physicists and mathematicians, many of which have disdained the use of mathematics in economics. Their arguments vary, but one common theme is summed up by mathematician Michael Edesess' critique:

Economics pretends to be mathematics, but it is not mathematics. There is a major difference. No mathematician uses a term in a formula, or a statement of a theorem, unless that term has first been defined with excruciating precision.

And while economists may think they’ve defined terms like "aggregate demand" or "economic growth", they should try reading some real mathematics to see what a precise definition truly is. The economists, I think, leave the work of definition to be inferred from the way the terms are used in the formulas.

I believe I know the precise definition of (quite a few) economic terms, but maybe Edesess is pointing out to some more profound mathematical foundations which I may not be familiar. Could someone expand on his argument and maybe even counter back?

Answer 1

Edesess is attacking what is really just a straw man of economics. I'm not sure he really understands the field. To start, economics is not math. We're not claiming that it is. It's more of an "applied" science. Economists have never claimed that these definitions are precise in the way that mathematics is. These definitions are modeling constructs---they're for applied work. They're use is temporary in a way. The point is to try to convey an idea in a more precise way than just in words---but everyone knows that they're not a precise as we would like and not as precise as they ultimately should be. They're meant to be debated and later refined. But, as all applied scientists know, you've got to start somewhere and sometimes ideas are best conveyed through simpler---if less detailed mean.

Coming up with better definitions is a huge part of the economic science. Consider these examples. When the Cowles Foundation was founded in 1932 its motto was "Theory and Measurement" (the motto was first adopted in 1952). Measurement is not an easy thing to do. As another example, a lot of Larry Kotlikoff's work has dealt with how a lot of fiscal measures are not economically well-defined concepts.

Einstein taught us that neither time, nor distance are well-defined physical concepts. Instead, their measurement is relative to our frame of reference – how fast we were traveling in the universe and in what direction. Our physical frame of reference can be viewed as our language or labeling convention. ... Kotlikoff, along with Harvard's Jerry Green, offered a general proof of the proposition that deficits and a number of other conventional fiscal measures are, economically speaking, content-free, concluding that the deficit is simply an arbitrary figment of language in all economic models involving rational agents.

Also, take another example of current interest. Lars Hansen's recent work (winner of the 2013 Economics "Nobel" prize) has focused on the difficulty and ongoing failure to define certain economic concepts, including "bubbles" and systemic risk. See his essay "Challenges in Identifying and Measuring Systemic Risk". I'm a fan of the dictum he relays, attributed to Lord Kelvin,

I often say that when you can measure something that you are speaking about, express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of the meagre and unsatisfactory kind: it may be the beginning of knowledge, but you have scarcely, in your thoughts advanced to the stage of science, whatever the matter might be.

He notes that "an abbreviated version appears on the Social Science Research building at the University of Chicago." So, yeah, economists (as social scientists) definitely take this seriously.

So, the point is that economists are well aware of the problems in these "definitions." They are a part of ongoing research in the field; sometimes they're ignored if people don't think they're first-order to the problem; etc...

Answer 2

Economics pretends to be mathematics, but it is not mathematics.

God forbid, if you excuse my language. As many other scientific disciplines, Economics uses Mathematics, it most certainly is not Mathematics, and it can never become Mathematics.

Mathematics may get inspiration from the real world, but it then defines and works with its concepts without regard to whether they remain connected to the source of inspiration.
Economics on the other hand, is obliged to define its concepts in a way that preserves some degree of relevance to the real-world aspects which it attempts to study. And since the "real-world" that preoccupies Economics is the social world, full of uncertainties and laws nobody has discovered yet, it follows that Economics can never achieve "excruciating precision", and remain relevant. So what? Economics is not Mathematics, we already said that. Economics is harder than Mathematics, exactly because it cannot impose such precision to itself and remain useful. But it abides by the scientific method, and so instead of being confined to verbal arguments, it attempts to "mathematize" them (use symbolic language, that is) so that they can be more transparent and focused as regards their conclusions and their internal consistency.

It would be so much more easy to produce verbal treatises, that first would require a round of semantics analysis, and then, if this round eventually concludes somewhere, to discuss the argument per se. But once we put it in symbolic language, we clear the fog and we let our premises (and so our limitations and imperfections) shine for anyone interested to see. That's what I call scientific integrity in Social Sciences, and this is why I consider Economics to be the Avant-garde of Social Sciences.

Answer 3

The critique of Edesess, misses the point. The reality is much more profound than mere misuse of mathematical definitions.

The truth of a mathematical statement depends strongly upon the ability to retrace any definitions used to axiomatic logical level. In this sense, one would not find any mathematician wildly using definitions that cannot be reduced to the mathematically/logically true body of knowledge that already exists. But this is stating the obvious.

In the applied science fields (biology, medicine, engineering, etc, etc), one starts with a real problem (problem domain), or phenomenon, and models the problem in the language of mathematics. The aim is to solve/study/simulate a mathematical problem, to be able to say something about the real problem.

The critique is actually about the mathematical-ization of economics (which started with Samuelson in 50's 60's). The claim is that some economists make the transformation to the mathematical domain and lose sight of the original problem and never transform back to the problem domain (i.e. the interaction of people, firms, resources, etc). These economists appear content with formulating linear algebraic relations, or solving vector auto-regressive equations, with no empirical justification - or worse - claiming that such economics is above short run considerations (i.e. my theory cannot be falsified in any of our lifetimes).

There are many examples of this. One obvious one is the the so-called theory of general equilibrium - which not only was shown to be mathematically flawed (via multiple equilibria (see Sonnenschein, Mantel, Debreu theorem) in the 1970s), but is hypothesised to lack any empirical content. As a result, some economists prefer to stay in the mathematical-ized domain - perhaps chasing a more accurate model (computable GE, dynamic GE, Stochastic GE, Dynamic stochastic GE, etc, etc) - hence the misunderstood critique that economists masquerade as mathematicians. One could make the case such people are more accurately described as pseudo-mathematicians, masquerading as economists (in the problem domain sense).

Answer 4

Definitions in Mathematics

The field of mathematics is much more than just the applications. In fact, the applications are a result of actual mathematics which comes in the form of proofs and theorems. For example, in ring theory, mathematicians needed to prove that a * 0 = 0 for all values of a. Below is the proof: Observe a * 0 = a(0 * 0) = a * 0 + a * 0. (1) Then we add -(a * 0) to both sides to get (a * 0) + -(a * 0) = (a * 0 + a * 0) + -(a * 0) (2) This gives us 0 = a * 0. (3)

The applications of this benefit many individuals when used to show 5 * 0 = 0, but this is merely the result of a more broad result that has been proven.

How are these proofs constructed? Through definitions. To prove the above result, we could not assume that a(0 * 0) = a * 0 + a * 0; instead, we needed to use the definition of a "ring" which, by definition, allows for line (1). Similarly, we needed to use the definition of a "ring" in order to know that we were allowed to use -(a * 0) in line (2).

Definitions in Economics

Economics, however, does not use definitions in the same capacity. Here, definitions are used strictly for the "defining of terms" rather than the "relation of terms." In Economics, one cannot prove that, in the short-run, expanding the money supply (which causes inflation) will decrease unemployment. The definitions in economics are not set up to do that; even more so, they cannot do that.
Part of the reason that the definitions in economics cannot do this is because of the definitions. Think of the terms "good," "market," and "demand." All of these terms have sloppy definitions. They do not really relate to anything else. On the other hand, we have terms such as "currency" and "GDP" that have extensive, precise definitions. These definitions have been chosen purposefully, and measurements of "currency" and "GDP" are precise because of this.
Another part of why Economics has "poor" definitions is due to the study of economics itself. Economics relies heavily on the demand of individuals. This demand cannot be quantified nor is there any guarantee that it will remain the same from one moment to the next. Thus, no real way to construct a proof that will be true beyond any particular moment. Because of this, economics does not need rigorous definitions. In Mathematics, however, we can construct proofs regardless of the numbers we use and, thus, transcend the limitations up to a very broad context. In the proof above, we used a instead of a number so that we did not have to rely on using that number and only that number. By using a, we know that multiplying any number by 0 will give us 0.

Response to Edesess

Edesess is mostly (probably 95%) correct. In truth, most of the definitions of economics are not "precisely defined" to the same level that mathematical definitions are required to be. In Mathematics, the definitions are carefully considered and decided upon by the Mathematical Community as a whole (not to say that Economic definitions aren't, but that's outside my knowledge). Also, by the nature of Economics, the use of the definitions cannot be used to prove anything.
In response to Edesess, however, economics should not be treated as Mathematics because of the fundamental differences in how they make discoveries. Economics is advanced through polls, market data, supply and demand graphs; Mathematics is furthered by research, proofs, and theorems.