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.

---

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)

---

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.

Meme of the day

I picked this up from Pharyngula, but it's been popping up in a number of different locations. Most of what follows is quoted text, with my responses inserted:

Overview:
The following survey is for bloggers who are actual or aspiring academics (thus including students). It takes the form of a go-meme to provide bloggers a strong incentive to join in: the 'Link List' means that you will receive links from all those who pick up the survey 'downstream' from you. The aim is to create open-source data about academic blogs that is publicly available for further analysis. Analysts can find the data by searching for the tracking identifier-code: "acb109m3m3". Further details, and eventual updates with results, can be found on the original posting:

http://pixnaps.blogspot.com/2005/09/academic-blog-survey.html

Instructions:
Simply copy and paste this post to your own blog, replacing my survey answers with your own, as appropriate, and adding your blog to the Link List.

Important (1) Your post must include the four sections: Overview, Instructions, Link List, and Survey. (2) Remember to link to every blog in the Link List. (3) For tracking purposes, your post must include the following code: acb109m3m3

Link List (or 'extended hat-tip'):
1. Philosophy, et cetera
2. Pharyngula
3. Halfway There
4. Add a link to your blog here

Survey:

Demographics
Age - 54
Gender - Male
Location - Northern California
Religion - Thoroughly lapsed Roman Catholic
Began blogging - August 2005
Academic field - Mathematics
Academic position [tenured?] - Professor [yes]

Approximate blog stats
Rate of posting - weekly
Average no. hits - beats me
Average no. comments - 0.5/day
Blog content - Miscellaneous, including math, teaching, and politics

Other Questions
1) Do you blog under your real name? Why / why not?
- No. If I talk about colleagues or students, I'd like to maintain a smidgen of pseudonymous discretion.
2) Do colleagues or others in your department know that you blog? If so, has anyone reacted positively or negatively?
- A few do. Their reactions are mostly positive. So far, anyway.
3) Are you on the job market?
- No.
4) Do you mention your blog on your CV or other job application material?
- No. What's a CV? ;-)
5) Has your blog been mentioned at all in interviews, tenure reviews, etc.? If so, provide details.
- Halfway There is too darned new for anyone to be talking about it anywhere. I am, however, in the blogroll at Tall, Dark, & Mysterious. Are you?
6) Why do you blog?
- Beats me. Perhaps I think everyone is entitled to my opinion. One of my friends (yes, I can honestly use the plural) says I'm blogging to fill the gap left by the completion of my graduate program. Interesting theory, but I hope it's not true. I spend so little time on blogging that so far it's a bad reflection on my recent studies.

The counting numbers

When the power goes out, all stoplights are supposed to convert to four-way stop signs -- but in Hollywood, nobody can count that high.

Los Angeles blacked out this afternoon for reasons to be determined (a cut cable?). Xeni Jardin on BoingBoing predicted traffic problems in Hollywood because no one can count very high (unless, she said, the numeral is preceded by the words "Star Wars Episode"). I have another suggestion. People in Hollywood can count pretty high in Roman numerals: I, II, III, IV, V, VI, VII. (Star Trek anyone?)

The Sign of the Fraud

[W]hen you have eliminated the impossible whatever remains, however improbable, must be the truth...

The Sign of the Four, Sir Arthur Conan Doyle

It doesn't matter that they're calling it "Intelligent Design" now. Creationism is the same thing it was in the beginning, is now, and ever shall be: a hollow hypothesis striving frantically to stave off its own extinction. Although creationism responds to environmental pressures by adapting in hopes of survival -- adding "science" in its guise as "creation science" or "scientific creationism", or even dropping "creation" entirely along with the identity of the creator in "intelligent design" -- the evolution of creationism cannot disguise that it is a wholly negative enterprise.

The idea is simple enough. First, establish creationism (in whatever form) as the only alternative to evolution. Second, destroy evolution. Third, proclaim the sole survivor as victor. Unfortunately for the creationists, proving a negative is as difficult as they have always said it is (as when they criticize those who believe God does not exist). William Dembski has his explanatory filter, which purports to detect the presence of purposeful design. The only problem with Dembski's EF is that it doesn't work, since he can't give any examples of its use (although it does seem to suggest that God was intelligently designed). Not to worry, Dembski's still working on it. At least he's a trained mathematician, which means he can throw in fancy symbols whenever he needs to distract the unwary.

Michael Behe has his irreducible complexity, which means that some biological mechanisms cannot be functional if even a single component is missing. This is difficult to prove, as you might imagine, but assume Behe is correct for a second. What he misses, however, is that while the reduced mechanism might no longer be able to perform the task it was executing, it might be perfectly capable of performing some other useful task. Behe has failed to come to grips with natural selection's unapologetic opportunism, taking whatever is close at hand and shaping it in response to environmental influences. Remember that this was a gambit, by the way. We assumed Behe had demonstrated an irreducibly complex biological mechanism, but he really hasn't. His showcase arguments about bacterial flagella and the blood-clotting sequence have been picked apart by biologists with more imagination that he has.

While nature supposedly abhors a vacuum, creationism is desperate to create one that will suck it into the mainstream. Since creationism has no explanatory power ("God did it" is not an explanation), its proponents must carry the battle to the enemy in a war against science and the scientific method. While scientific controversies are settled in a contest of facts and observations, the "controversy" over evolution versus creation is a political dispute between creationist propaganda and scientific research. In the long term, science has a good record of ousting superstition. I just wish I were more confident about the near term.

You are the world

A game the whole family can play

This is a mind game: Imagine, if you will, the state of the world if you were the ultimate arbiter of good taste and success. It's a simple game, inspired by the gobsmacking stupidity of who is famous and what is popular these days. Ready? I'll go first:

Reality television programs? Off the air, each and every one, including Survivor (never watched it) and Nanny 911 (every program is exactly the same; I suspect drugs are used to tame the monster children quickly enough).

Organized sports? Bankrupt (I mean even more than they already are). While this may horrify people, remember that I am the one playing the game this time. You can let the football teams prosper when it's your turn. I, on the other hand, have never in my life been to a professional football game or watched one on television (and I've been to only one pro baseball game, when my father dragged the whole family to the Oakland Coliseum; actually, I was the only one who had to be dragged). ESPN could be used for chess tournaments.

Paris Hilton? Gone already. Remember, I ditched the "reality" stuff first.

Health clubs? Abandoned. I'm not opposed to health, but facilities devoted to sweating and straining are noisome. I'll go for a walk instead. Maybe I'll get a bike, but not a stationary one.

Math and science would become much more popular majors in college. Astrology, creationism, homeopathy, and other frauds would vanish for lack of practitioners and clients. Tarot cards might survive as a party game (although there will probably be a lot fewer parties). I'm a little worried about medicine, since I'm too squeamish to be a doctor yet would sure like to have a few good ones around.

Cultural events would do okay, and movie theaters would survive. However, there'd be dismal sales at the concession stands (I don't like to eat at the movies) and Mel Gibson would probably be out of work (although I did see his Hamlet, so there's hope for him). Peak hours would shift, too, since I usually prefer to see movies in the early afternoon on weekdays because it's not as crowded. (See the flaw here? If I set the standard for public taste, then everyone else will want to crowd into my favorite time slots with me. Damn.)

Alcoholic beverages? Gone. Churches? Shuttered or converted into museums. The Republican Party? Stuffed and mounted for display in those ex-churches. The various Bushes? In the rogues' gallery.

Are we having fun yet?

Yes, it's a game everyone can play, but it has only one winner at a time.