# Use of mathematics and imprecise definition of terms

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?

• Many mathematicians that have become economists have defined appropriately aggregate demand, economic growth is a loosely defined term but true economists not use growth loosely, rather they refer to the growth of some economic variable and growth is a simple notion. Also, economists as physicists, biologists, and others do not do math for the sake of math, so econ is not math and we do not pretend it to be, we use it as a way to understand some real phenomenon. So what we do instead of defining arbitrary mathematical objects we care about how using this definitions and relations for science. – user157623 Dec 6 '14 at 16:41
• This seems to me to be subjective, argumentative, too broad, and therefore off topic in multiple ways. Here's an example of a good question: "In such and such a paper, by such and such an author, the term "foo" is used repeatedly but I am not able to locate a precise definition. Does this term have a standard definition that I'm supposed to know about before I read the paper?" – Steven Landsburg Dec 6 '14 at 20:06
• The critique is misunderstood. It is not about the mere misuse of mathematical definitions. It is about mathematics masquerading as economics. See my answer below. – Rusan Kax Dec 14 '14 at 15:43

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...

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.

• There is nothing wrong with economics being different from mathematics, but the pretending is a problem. I notice the practice in economics papers of defining terms implicitly during an argument quite often; sometimes this is innocuous, but sometimes it is used - intentionally or not - to sneak in hidden assumptions or hide the underlying model, and that is bad science. I don't insist that economists account for every last variable, but they should be explicit about the assumptions they are making, and this is the function of a proper definition. – Paul Siegel Dec 6 '14 at 18:24
• @PaulSiegel "Bad science" exists everywhere (in Math, the last I remember is the storm of frivolous papers around Invexity), and it certainly exists in Economics. But the issue is not whether it exists -but how much widespread it is. And the impression of "quite often" is no evidence, although this is how scientific research indeed starts. It should be quantified, at least statistically. Because, for example, my impression is that "rarely" have I read an Economics paper and had doubts about the assumptions made. – Alecos Papadopoulos Dec 6 '14 at 18:34
• Regarding the Invexity example : see math.uaic.ro/~zalinesc/papers3.php?file=invexity.pdf – Martin Van der Linden Dec 6 '14 at 20:40
• I agree that there may be some work that tries hard to be mathematics, but the key distinction is falsifiability. Are you able to generate predictions about the real system that is being represented that can be proved wrong or false in finite time. Then this is a scientific or economic model. – user157623 Dec 6 '14 at 21:00
• Whatever your reservations about invexity, the phrase "invex function" has a precise definition which can easily be looked up on, say, Wikipedia. It is, in my experience, accepted practice in economics to define terms using what you called "verbal arguments" and then skip straight to detailed calculations without taking the time to lay out the mathematical assumptions required for those calculations to make sense. Sometimes this is done responsibly and sometimes not; in either case the practice is quite contrary to your claim that "we clear the fog and let our premises shine". – Paul Siegel Dec 6 '14 at 22:49

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.

• You write In Economics, one cannot prove that, in the short-run, expanding the money supply will decrease unemployment. But of course one can. One can prove exactly this in some models and exactly the opposite in some others, just as, in mathematics, one can prove that some rings are commutative and others are anticommutative. Yes, it is perfectly possible to be sloppy when you're talking about economics, just as it's perfectly possible to be sloppy when you're talking about math. The future of this site will largely depend on how much that sort of sloppiness is tolerated. – Steven Landsburg Jan 7 '15 at 5:15
• @StevenLandsburg Ah, but how did they prove that? It was not proven via definitions and logical reasoning in the same way that mathematical proofs are constructed. If you continue to the next sentence and following of the one that you quoted, I explain further. – Mathematician Jan 7 '15 at 14:44

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).

• First, there is nothing flawed about having multiple equilibria. Physical systems can have multiple equilibria and that is not a problem for physics either. Second, general equilibrium theory does have testable implications, as the work of Brown-Matzkin shows. – Michael Greinecker Dec 15 '14 at 0:48
• Thanks. I was not implying multiple equilibria was the actual, and only, problem. Yes, Brown and Matzkin demonstrated the existence of testable "restrictive" cases on the equilibrium manifold. They gave exact solutions for some "special cases": a two-agent economy, and a Robinson Crusoe production economy. As extensions, in particular in the presence of externalities - there were negative results (i.e. non-falsifiable) obtained by Carvajal in early 2000s. Sounds like pure maths to me. – Rusan Kax Dec 15 '14 at 1:27
• There is a gap between "which not only was shown to be mathematically flawed" and "Sounds like pure maths to me." – Michael Greinecker Dec 15 '14 at 1:29
• Sounds like pure maths in the sense of: " 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)". – Rusan Kax Dec 15 '14 at 1:33