How Many Traders Can You Fit into a Model?
Most people have at least once in their lives heard a how-many-can-you-fit joke, for example:
Q: How many contrabass clarinetists can you fit into a phone booth?
A: Both of them.
The how-many-can-you-fit jokes are rarely laugh-out-loud funny (although fans of classical music will probably agree that the one above is actually quite witty), but what is important here is that, ironically enough, they bear a certain resemblance to most mainstream economic models whose concern is how many traders can fit, not into a phone booth, but into a market.
Obviously, we are not talking here about finite quantities — that would be child's play. Mainstream economists, in their search for generality and a somewhat irritating "pretence of knowledge," deal almost exclusively with infinities.
Infinities? Does that word even have a plural?
As a matter of fact it does, which should already hint how dangerous and counterproductive might be the use of mathematics in economics.
As unlikely as it may seem, mathematicians have a way of distinguishing the "sizes" of infinities. In mundane speech, "infinite" simply means not finite or as Webster puts it, "without limits, boundless." This seems to be in accordance with the mathematical concept of infinity as long as we think of, say, the set of all positive integers: it, too, is boundless, since if any number were to be its upper bound, it would suffice to add 1 to it, thus obtaining a number that is even greater than the postulated bound.
Our intuition fails to square with mathematical theory, however, once we start to think about the unit interval (0,1); it too is infinite but at the same time bounded from below by 0 and from above by 1!
Now, here comes the big surprise: although both the unit interval and the whole set of integers are infinite, the former can be meaningfully said to be "larger" than the latter. But how can one even compare the two sets if they both have infinitely many elements? Well, imagine a huge and crowded ballroom during a prom night, and say you want to check whether there are more girls than boys at the ball. Obviously there's no use in trying to count all of them — it's dark, people move around and you'll probably make a mistake sooner or later.
What you could do, however, is have all the seniors pair off and start dancing — then if you discover that, say, a couple of boys end up sitting by themselves instead of dancing, it means that boys are more numerous than the girls! It follows from this simple example that one need not count the elements of two sets (which is clearly impossible if they are infinite) to know which of them is "greater."
Now, using a little bit of a mathematical apparatus, it can be similarly proven that any unit interval — say (0,1) — has necessarily the same number of elements as the whole set of real numbers (-∞,∞) and at the same time is greater than the set of positive integers: 1, 2, 3…
Finally, there is one terminological point that should be mentioned before turning to economic models: the infinity of real numbers is often called uncountable infinity or the continuum, which — being fancy terminology — has become unavoidable in mainstream journals.
Take for example the prestigious American Economic Review. A simple JSTOR search indicates that, starting in the mid-1960s, the word continuum (used to denote a "bigger" infinity) was used in roughly 400 articles, which, if one takes into account that AER publishes about 40 articles per year, certainly indicates the birth of a new, blatantly absurd fashion in economics. Nevertheless, not many economists seem to be bothered by this rather distressing turn of events — the notable exception being Paul Ormerod, who in his book The Death of Economics writes:
"To take just one example, the phrase 'assume a continuum of traders' will be encountered in many theoretical papers on the idealised market economy… But what does this phrase 'continuum' actually mean? It sounds quite innocuous, yet spelt out in words it might lead people to query the realism of any academic paper based on this assumption, or even begin to doubt whether the article was worth writing in the first place. For the phrase means that the number of people, whether as individuals or as firms, carrying out trade in this theoretical economy is not just large but quite literally infinite. In fact, to be strictly accurate, it even means rather more than this. If one were to start to count the whole numbers — one, two, three and so on — one could go on for ever. There is an infinite number of them…. But mathematicians have the apparently bizarre but nevertheless logical concept of infinities which are even bigger than this infinity! A continuum is exactly one of these. In other words, it is assumed that there is not just an infinite number of traders, in the sense that the set of whole numbers is infinite, but there is an even bigger number of them than this."
The invasion of infinities into economic theory started with Robert J. Aumann, the 2005 Nobel laureate in economics, who has argued in his influential Econometrica article of 1964 that the most natural (sic!) model for the purpose of developing the notion of perfect competition "contains a continuum of participants, similar to the continuum of points on a line or the continuum of particles in a fluid."
It has, however, long been recognized by physicists specializing in atomic and molecular theory that everything in the material world is of finite character (see, e.g., Ronald E. Mickens, ed. Mathematics and Science, World Scientific, Teaneck NJ, 1990). The mathematician Eric Schechter in an illuminating essay "Why Do We Study Calculus?" explains:
"[U]se a pencil to draw a line segment on a piece of paper, perhaps an inch long. Label one end of it '0' and the other end of it '1,' and label a few more points in between. The line segment represents the interval [0,1], which (at least, in our minds) has uncountably many members. But in what sense does that uncountable set exist? There are only finitely many graphite molecules marking the paper, and there are only finitely many (or perhaps countably many) atoms in the entire physical universe in which we live. An uncountable set of points is easy to imagine mathematically, but it does not exist anywhere in the physical universe."
Uncountable infinity is therefore not "real" in the sense in which physical objects are real — or even in the sense that arithmetic is "real." The latter, as pointed out by Paul Lorenzen, is an a priori–synthetic discipline, rooted in our conception of reality and our understanding of the repeatability of action. The former, on the other hand, is pure mental gymnastics, a mere façon de parler. The concept of multiple infinities lacks any constructive foundation, and, in fact, leads to inconsistencies and contradictions in mathematical theory itself. As Hans-Hermann Hoppe notes, it should therefore be recognized as "epistemologically worthless symbolic games."
It is indeed puzzling that a concept that causes problems in pure mathematics itself is employed in economics, whose purpose, after all, should be the study of real things and real-life phenomena. Aumann is certainly much too good a mathematician not to have realized that. And, indeed, a careful reading of his paper reveals that he was fully aware of what he was doing, for he explicitly admits that "the idea of a continuum of traders might seem outlandish to the reader" and that everywhere around us there is always a "large but finite number of particles (or traders or strategies or possible prices)," but — and here comes the rationalization for the use of "outlandish" concepts — "the chief result [of the developed theory] holds only for a continuum of traders — it is false for any finite number."
Furthermore, and perhaps more importantly, while Aumann is perfectly aware that the structure of, say, a shopping mall looks completely discontinuous to its clients, nevertheless "the economic policymaker in Washington … [who] works with figures that are summarized for geographic regions, different industries, and so on" doesn't care about the individual consumer (or merchant), and would much rather treat them in a continuous manner just as the physicist treats individual molecules. The real motivation behind introducing "bad metaphysics" into economics was certainly not to help identify the mechanisms of a market economy, but rather to provide a useful tool for policymakers.
However, in order to properly assess Aumann's contribution in economic terms, it is crucial to ask what exactly he means when he postulates that the number of market agents should be uncountably infinite. We already know that nowhere in the real, physical world does there exist anything even remotely resembling the continuum.
"An approximation" of the way things really are on a market where individual agents have little influence on price — an approximation still exaggerated but at least intuitively imaginable — would perhaps be to assume that traders are like integers, i.e., that there are so many of them that any given quantity can be shown to be insufficient to include them all, but nevertheless that one can list or number them, e.g., by saying that Mr. X is the first, Mr. Y second, Mr. Z third, and so on. But no such thing can be done with an uncountable set!
Moreover, if agents were to have three dimensions and some minimum "size," then it follows from elementary set theory that there could only be countably many of them in a three-dimensional space. Now, since Aumann wants his traders to be located on the real line, then his assumption essentially boils down to thinking of people as points with no bodies, mass, or any physical characteristics.
Unsurprisingly, though, the flirtation with badly conceived abstraction goes even further, as pointed out by eminent mathematical economist Nicholas Georgescu-Roegen: Donald Brown and Abraham Robinson, in a paper presented for publication in the official periodical of the US National Academy of Sciences, assume that there are more traders even than the elements of the continuum, which implies that there are so many of them that even Euclidean space is too small to accommodate them all!
Following Aumann's example, Rudiger Dornbusch, Stanley Fischer, and Paul A. Samuelson set out to develop a generalization of Ricardo's classical international trade analysis by assuming that the number of goods is uncountably infinite. Obviously, as Ludwig von Mises observes, not every generalization is a sensible one.
We could build a whole theoretical economic edifice encompassing all thinkable conditions (e.g., labor not causing disutility), and then logically deduce from them all the possible implications, but this (to use Mises's accurate expression) would be merely mental gymnastics, since true science, as long as its purpose is to discern the mechanisms of the real world, need only take into account those conditions and circumstances that are given in reality, and not in some imaginary world.
But there is more to it than that. For again, as in Aumann's model, we should ask what is the significance and real meaning of the assumption that there is a continuum of goods in the economy. Mathematically speaking, if a set is uncountable, then uncountably many elements can be subtracted from it leaving still no less than uncountably many! What's more, this procedure can be repeated over and over again without significantly decreasing the "size" of the original stock of goods. (Think, for example, of removing "odd" unit intervals — (0,1), (2,3), (4,5), etc. — from the real line; there still remain the "even" ones, each being uncountably infinite.)
What does that mean in economic terms? It can mean only one thing: that the fact that a single individual consumes a unit of a good does not reduce the total available supply of goods, which implies in turn that the postulated model assumes away the most fundamental fact of economic inquiry, namely, scarcity! Samuelson in his own legendary Principles of Economics admits that "economics is the study of how people choose to use scarce or limited productive resources." Mainstream economists do not always practice what they preach.
Mathematical economists would probably object at this point that the aforementioned "unfortunate" examples are merely a corollary, a necessary price one has to pay for the scientific mathematization and the use of abstraction. As Roderick Long pointed out, however, abstraction doesn't have to be unrealistic.
Mark Blaug, himself obviously a mainstream economist, could not have been more accurate, when in an interview he described present-day economics as a kind of "social mathematics," which, to be sure, uses words such as "price," "market," or "commodity," but gives them purely mathematical meaning with no likeness to real world observations.
What, then, can be done with economic theory whose models resemble jokes about how many elephants can be fit into a refrigerator? And how, given the overrepresentation of often indecipherable mathematical symbolism, is one to distinguish good economics from bad?
Diran Bodenhorn writes,
"Many mathematical economists have suggested that communication between literary and mathematical economists would be greatly facilitated if literary economists would learn more mathematics…. However, it is also possible that communication might be improved if mathematical economists would learn more economics … Moreover, communication might be improved if mathematical economists would state their economic assumptions clearly in literary form and discuss fully the economic implications of the mathematical model which they are employing."
A good acid test of the usefulness of an economic theory consists in a careful examination of its assumptions. If they are manifestly absurd, unrealistic, or even unrealizable — like the assumption of a continuum of traders or goods in the economy — then such a theory has nothing important to say about the way things really are, and should be treated as the joke it actually is.
 Strictly speaking, infinity has been used before in economics, since the continuum manifests itself in the assumption of marginal analysis which assumes infinite divisibility of units of goods. It seems, however, that Austrian critics of mathematical method have typically understand by it — much like Leibnitz or even Newton — merely a countably infinite process — i.e., such a division of a given unit of good which at each step has finitely many predecessors — yielding arbitrarily small parts.
 See Paul Lorenzen, Constructive Philosophy, The University of Massachusetts Press, Amherst, 1987, pp. 195–248.
 See Ludwig von Mises, Human Action. Scholars Edition. Ludwig von Mises Institute, Auburn Al., 1998, p. 65.
 Roderick Long, "Realism and Abstraction in Economics: Aristotle and Mises versus Friedman." The Quarterly Journal of Austrian Economics, Vol. 9, No. 3, 2006, pp. 3–25.
 Diran Bodenhorn, "The Problem of Economic Assumptions in Mathematical Economics," The Journal of Political Economy, Vol. 64, No. 1. (Feb., 1956), p. 32.