Who owns mathematics?

Do you know what "math envy" is? While mathematics is not particularly popular, many people are eager to press it into service. You can see this in such trends as the "mathematization" of the social sciences, where something stated in terms of an equation apparently carries more weight than the same thing stated in mere prose. The inherent problem in such mathematical approaches is that not everything is amenable to expression as a numerical or symbolic model. Even in highly mathematical undertakings as physics, the solutions of the modeling equations are only as good as their congruence with the observed phenomena. Many a lovely mathematical model has been sent back to the drawing boards when refuted by experimental observation.

We who are exponents of mathematics have a responsibility to cast a jaundiced eye at the misappropriation of mathematical tools. I think, however, we are too often flattered by the "math envy" exhibited by those who write fanciful equations to make their work look more algebraic and less prosaic. Let us be on guard.

My thoughts were turned in this direction by some coincidences: The death of Serge Lang, the appearance of William Dembski on The Daily Show, and a recollection of a passage from Claude Lévi-Strauss.

Lang vs. Huntington

Lang was a tireless combatant against anything he perceived as inaccurate or sloppily reasoned. When political scientist Samuel Huntington was nominated for membership in the National Academy of Sciences (of which Lang was already a member), Lang launched a vigorous and successful campaign to defeat Huntington's candidacy for the honor. What mathematical sin had Huntington committed? Lang was adamant that anyone who appropriated mathematical language was obligated to respect its consequences. Huntington's use of quasi-algebraic reasoning in a 1971 paper (see Sullivan, 1998) resulted in such statements as

political instability = (political participation)/(political institutionalization).

What are we to make of this? Apparently Huntington posits that political instability will grow as political participation increases. Inversely, instability will decrease as political institutionalization increases. It is difficult to know what to make of this, given that Huntington did not bother to define his terms specifically enough to allow anyone to check out his equations. (Nor, for that matter, did he define in what units the quantities were to be given.)

While this simple example does not on its own refute Huntington's argument (which I lack the background to address anyway), it nevertheless shows a mathematical formalism being pressed into duty where it adds nothing to (and, indeed, is likely to detract from) any accompanying narrative discussion.

Claude Lévi-Strauss

Lévi-Strauss is the grand old man of structuralism, a school of thought no longer popular in an era of deconstruction. Still, his name is well-known in anthropology and it would be unkind to dismiss him because of his current unfashionability. However, when I saw the news of Serge Lang's death and browsed several articles about Lang's career, I recalled having seen something similar to Huntington's expropriation of mathematics, but in a more literary context. I couldn't remember who had written it. Thanks to a friend with a better recollection than mine, I soon had from him the exact quote and the name of the author of same. Brace yourselves. Here is the quote, right out of Lévi-Strauss's The Structural Study of Myth:

Finally, when we have succeeded in organizing a whole series of variants into a kind of permutation group, we are in a position to formulate the law of that group. Although it is not possible at the present stage to come closer than an approximate formulation which will certainly need to be refined in the future, it seems that every myth (considered as the aggregate of all its variants) corresponds to a formula of the following type:

Here, with two terms, a and b, being given as well as two functions, x and y, of these terms, it is assumed that a relation of equivalence exists between two situations defined respectively by an inversion of terms and relations, under two conditions: (1) that one term be replaced by its opposite (in the above formula, a and a − 1); (2) that an inversion be made between the function value and the term value of two elements (above, y and a).

For whom do you think Lévi-Strauss wrote this? Professional mathematicians, who are unlikely to be reading about the structure of myth? Fellow anthropologists, who would not know the difference between a group and a tribe? While I am hesitant to make pronouncements out of hand about anthropology, a discipline in which I have had but one elementary college course, I do know just a little bit more about mathematics. For example, Lévi-Strauss says he can organize a series of myth variants into a "kind of permutation group". Okay, I know what a permutation group is. But then he purports to "formulate the law of that group". Law? Does he mean to characterize the group operation? If it's a permutation group, then the operation involves rearrangement of the components of its elements. I do see some items, variously called terms and functions, being rearranged, but what the heck is F? And how did a and a − 1 become inverses of each other? (As elements from [0, 1], where 1/2 is its own opposite?)

You know what? I think this is gibberish. The mathematical "formula" will be impenetrable to those who know no math and highly suspect to those who do. Perhaps I need to read the entire book. Perhaps I am simply not sophisticated enough to grasp the keen group-theoretical insight represented by Lévi-Strauss's formulation, but until someone enlightens me, I will tend to believe that the meaningfulness was in the author's head alone. Sorry, Claude.

William Dembski

Dr. Dembski is the current Wunderkind of Intelligent Design and is, surprisingly enough, an actual mathematician with an earned doctorate. Good for him! Dembski represents the other side of the "math envy" coin. Instead of applying mathematics where its applicability is doubtful, Dembski wields math as a bludgeon to hide the doubtfulness of his conclusions. My examples from Huntington and Lévi-Strauss showed mathematics going astray into foreign fields. Dembski, however, uses mathematics in logical chains of symbolic reasoning. What could be more suitable?

Mark Perakh provides the definitive take-down of Dembski's mathematism, the deliberate use of mathematics to obscure the paucity of one's arguments. In his on-line paper, A Consistent Inconsistency, Perakh homes in on an example in which Dembski just about gives the show away. Examine the following logical statements in ordinary prose:

Premise 1: E has occurred.
Premise 2: E is specified.
Premise 3: If E is due to chance, then E has small probability.
Premise 4: Specified events of small probability do not occur by chance.
Premise 5: E is not due to regularity.
Premise 6: E is due either to a regularity, chance or design.
Conclusion: E is due to design.

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Supposing that a satisfactory definition of "specified" was previously provided, the chain of reasoning is reasonably clear, is it not? However, Dembski is not content to explain with words what he can tart up in mathematical language to impress the bourgeoisie:

Premise 1: oc(E)
Premise 2: sp(E)
Premise 3: ch(E) → SP(E)
Premise 4: ∀X[oc(X) & sp(X) & SP(X) → ch(X)]
Premise 5: ∼reg(E)
Premise 6: reg(E) ∨ ch(E) ∨ des(E)
Conclusion: des(E)

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See how much better this is? While the general reader might peruse Dembski's prose argument and follow the steps with a modicum of confidence, who but a logician would be more comfortable with the abbreviated and symbol-laden alternative? Dembski makes the mistake here (taken from his book The Design Inference and quoted in Perakh's review) of revealing how unnecessary is his resort to mathematical formalism. While mathematics can shed light on complex problems, it can also be used to confuse and confound. With his many articles and books, Dembski has raised "mathematism" to a fine art, so that all who do not understand his symbols but agree with his objectives (replacing evolution with creationism) can praise his intellectual attainments and the supposed rigor of his arguments.

In Summation

Mathematics is powerful, so people seek to enlist its support for their positions. This is completely understandable. But mathematics is also demanding. If you cannot meet its demands for consistency and rigor, then back away slowly from it. You'll be safer that way. If, on the other hand, you are mathematically competent, then the question becomes whether you use math to inform (as with fruitful mathematical models in physics) or distract (à la Dembski and his creationist tracts). Let's be honest about math.