A previous entry quotes a foot-note from Austrian Marginalism and Mathematical Economics by Karl Menger; that foot-note is tied to a sentence that I found particularly striking.

Menger has already explained that there is nothing intrinsically more rigorous about reasoning with formal symbols that now characterize most discussions of mathematical subjects as such than reasoning verbally (though he also cites Irving Fisher's helpful analogy between taking a train for a trans-continental trip, as opposed to walking the distance). And he further argues that the Austrian School marginalism was more general in conception than was that of the overtly mathematical marginalists of the late Nineteenth and early Twentieth Century. None-the-less, he raises a question of why the early Austrian School economists didn't master the calculus. (At the least, it would have allowed them to better follow the work of the other marginalists.)

Menger's explanation is that texts on the differential calculus were (and in 1971, when the essay was first presented, generally continued to be) written in a manner that used terms ambiguously and equivocally, so that learning typically relied upon inter-action with an instructor, including working through examples with that instructor. (Menger compares this process with how children learn language, as opposed to how languages are learned by adults using grammars and lexica.) In the absence of an instructor, the exposition would tend to frustrate, rather than to enlighten, a careful thinker.

The sentence that really struck me, though, did so by way of an incidental reference.

While a logical and analytic mind is perfectly sufficient for the self-study of any of the numerous good books on the theory of real functions or on abstract algebra (fields as yet of little use for economics) such a disposition hinders more than helps a mature man, with limited time at his disposal, in a self-study of calculus.14

(Underscores mine.)

Since my readers come from various backgrounds, let me indicate the nature of abstract algebra. The algebra to which one is first introduced — the only algebra to which most people are ever introduced if they are introduced to any at all — is a matter of going from the arithmetic of identified numbers to that of variable numbers. It is, then, in an important sense, already abstract. However, it's really just a particular case of a more general class of structures. For example, there's the Boolean algebra, with the operations of negation, disjunction, and conjunction. Its variables can take one of two values (true or false), and the result of any operation is always one of these two values. Within this set of values, there is an identity element for the operation of disjunction (false) and an identity element for conjunction (true), much as addition and multiplication have identity elements (0 and 1, respectively) in arithmetic algebra. The operations have properties of associativity, commutativity, and distributivity. Boolean algebra is not just like arithmetic algebra, but there are enough similarities to see these two are variations on some more general theme. When a mathematician refers to abstract algebra, she's referring to the generalized study of such themes.

(Sometimes, a mathematician will in fact just use the bald word algebra for this scope. While a book entitled College Algebra is likely to have nothing but math that one ought to have learned in-or-before high school, a college text entitled just Algebra may be something rather more challenging. However, with an article a[n] or the, algebra will refer to a structure in a narrower class of systems. An algebraic system may have no more than some set with a single operation, but an algebra must have significantly more.)

Traditionally, mathematics was defined as the concern with quantity as such, in which case mathematics really always is about arithmetic. But, especially as attempts were made to place mathematics on a firm foundation, the relationship of mathematics to deeper things, especially to logic and to order, began to be reässessed, and the scope of mathematics came to include formal structure more generally.

Mainstream economics entails a lot of overt mathematics, but this mathematics is primarily that of the calculus. Other sorts of mathematics may be used, but there is typically an attempt to introduce a sufficient set of assumptions to allow the calculus to be employed in place of those mathematics, as when preferences are proxied by differentiable utility functions. And even when the calculus is avoided, the mathematics used is that of arithmetic — abstracted beyond specific numbers perhaps, but still a matter of addition and of its derivative operations (subtraction, multiplication, &c).

Mainstream economists imagine themselves themselves not only as more rigorous but also as more scientific than non-mathematical economists, who rely upon verbal reasoning. However, although the physical sciences use a considerable amount of overt mathematics, science itself is not about using mathematics as such; to use mathematics because one sees the physical scientists doing so, and to use a sort of mathematics because that is what one has seen them use, is to behave like a cargo cult, expecting desired results will follow from uncomprehending mimickry of surface forms. And, although the non-mathematical economists are relying upon verbal reasoning, they are doing so where logic as such is to be brought into play; which, as it turns-out, is true of the mainstream economists as well. One doesn't much see mainstream economists employing symbolic logic; and, when they do, they often do so sloppily; because they're really just substituting single-characters for words, which words they don't quite understand. (I once saw an economics professor complete an ostensible proof with the line ⇒ ∃x.) Meanwhile, as Menger noted, it is also quite possible to be every bit as careful with words as one is with mathematical symbols.

Some of the non-mathematical economists have no particular objection to the overt use of mathematics; it's just not what they do. Others feel that formal mathematics more often obscures than clarifies. Some, especially the more faithful followers of Ludwig Heinrich Edler von Mises, insist that mathematics is actively inapplicable except in very limited circumstances. I have no argument with the first group, and perhaps none with the second. (Even economists who think that the effects of formal mathematics are generally salutary will often agree that too many alleged economists are too often truly lost in formulæ decoupled from economic reälity.) But when I attend to the claims of the third I find that they seem always to equate mathematics with arithmetic and with that which follows upon it.

In the case of v. Mises himself, he dated to an era when the definition of mathematics in terms of quantity would have been almost universal (though the aforementioned reässessment began before his birth) and when few mathematicians would have been doing much of any sort of overt mathematics except such stuff. Later anti-mathematical economists have less of an excuse, though it certainly hasn't helped that their mainstream opponents often have no better real grasp of the scope of mathematics, nor of the possibilities of formal structures. But a point to take away here is that these anti-mathematical economists are, for the most part, really somewhat confused anti-arithmetical economists. That these radically non-mathematical economists misunderstand the basic nature of mathematics might initially have seem utterly fatal to their case, but their mistake is one of confusing a part with the whole. If one wished to salvage their case, a first step would be to recast it as an attack on the part and not on the whole. If they've made any case against the use in economics of mathematics more generally, it's just the practical claim that formal mathematics more often obscures than clarifies.

While there are doubtless a few howling loons who call themselves economists and propose to disregard logic, it is happily embraced (or at least courted) by the vast majority of the anti-arithmetical economists. Well, the form of logic (or at least of logic as most of us understand it) is captured by an algebra, the Boolean algebra mentioned above. Further, they attempt to reason in terms of order without quantification; again, the forms of orderings (complete or incomplete) and of the processes for manipulating orderings fall within the scope of abstract algebra. If one were to practice both an overtly mathematical economics and yet an anti-arithmetical economics, the work would be one of abstract algebra.

In fact, one way or another, it's largely in terms of abstract algebra that the disagreement between mainstream and anti-arithmetic economists would be clearly and definitively resolved. If the present approach of mainstream economists is as general as they assert it to be, then a proof for that generality could be presented in terms of abstract algebra. And if it is as wrong-headed as the anti-arithmetical economists assert it to be, then a proof of the wrong-headedness could be presented in terms of abstract algebra.

But, forty years after Menger's parenthetical remark, there's still just not much explicit use being put to abstract algebra. It appears here-and-there, but almost always only until the author has, again, introduced enough assumptions to start employing arithmetical algebra and all that; often the gain is not in tractability but only in familiarity.

But when I was exposed to the possibility of applying to economics a mode of reasoning that was overtly mathematical yet did not presume any sort of quantification, my thought was to achieve greater generality by pushing ahead without those assumptions.

For good or for ill, the sorts of mathematics that I've been explicitly applying to economics and plan to continue to apply are primarily algebras other than that of arithmetic. My paper on incomplete preferences is relatively heavy on set algebra and on Boolean algebra, and relatively light on arithmetical algebra, the last playing a rôle only amongst the probabilities describing lotteries. And the paper upon which I'm presently working discards quantification (and even completeness of order) of the plausibilities associated with outcomes, so that the arithmetic remaining in the earlier model is abandoned. (The earlier model was just to prove a specific point — that indecision could operationalize differently from indifference, and thus that it would make sense to invest in construction of models of incomplete preferences.)

Where I find myself at present can be viewed as a frontier or as a wilderness. Indeed, the apparent problem had by Theory and Decision in finding a reviewer for the paper on incomplete preferences was almost certainly an artefact of that paper being unquestionably very mathematical, but employing a sort of mathematics with which most economists are distinctly uncomfortable. (I would caution against any conclusion that its sort of mathematics is instrinsically more difficult.)

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14 What? No! Foot-note 14 is quoted in that previous entry! It's not a foot-note to this entry! Get out of this foot!

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