The Unbearable Objectivity of the Positive Integers

June 23, 1999

Book Review
Philip J. Davis

1,000 Years, 1,000 People: Ranking the Men and Women Who Shaped the Millenium. By Agnes Hooper Gottlieb, Henry Gottlieb, Barbara Bowers, and Brent Bowers, Kodansha International, New York, 1998, 332 + xix pages, $17.00.

What we have here is an amusing and occasionally informative piece of lightness that the authors themselves seem not to have taken seriously. Contained herein are the names of the thousand most important movers and shapers of the last thousand years, listed in order of significance: 1, 2, 3, . . . , 1000. After checking to see whether I had made the list---and I had not---I checked to see whether Leonhard Euler had made it. He had not; while I was comforted in that regard, I concluded that the list was just another piece of Y2K-iana, designed largely to sell on the basis of its implicit sensationalism. I was about to dismiss the book when I had another reaction. It struck me that it can be interpreted as a satire on today's tendency to rate practically everything, from colleges to cucumbers, employing mathematics (often very elementary) to do so.

But first, the book under review. The four authors---two couples, all media people---selected a few thousand candidates for greatness or rememberability from the several billion people who have lived during the last thousand years. Then, using their rating system, they pared the list down to the published thousand. Each entry in the list is accompanied by a serial number, the name of the selectee, dates, a cute epithet, a short biography, and the number of points received. Beside the outstanding deeds, the biography often contains a few wisecracks or cliches associated with the particular notable.

As an example: Number 294 in the list (notability goes down as the serial number goes up) is Jean-Baptiste Colbert (1619-1683), "architect of French economic grandeur," the man who, among other things, ordered that acorns be planted in central France to provide timber for France's future navy. Colbert received 14,412 points. Notice the five-figure accuracy---well, how else could you separate 1000 items?

How did the authors arrive at their ratings? They identified five dimensions of significance: lasting influence, effect in the world, influence on contemporaries, singularity of contribution, and charisma. Each dimension was assigned a maximum number of points; the total (24,000) is the highest score possible. The reader is not told how the points were assigned in the individual categories. I suspect that, having checked out some biographies or encyclopedia entries, the group just sat around drinking beer and having a whale of a good time assigning points according to their personal knowledge and feelings. There is also evidence of some list massaging in the service of political, geographical, or other types of inclusiveness. An admissions officer in any respectable college would know all about such things.

Read page by page, the book becomes tedious quite rapidly. But there are uses to which it might be put, other than seeing whether, for example, Florence Nightingale beats out Emily Dickinson. In a parlor game of the Trivial Pursuit type, one might flip through the book at random and ask participants whether they can identify the notable selected. Readers can judge the level of difficulty of this game by trying their hand on a few examples: Toyotomi Hideyoshi (No. 57), Andreas Marggraf (No. 621), Sciopone Riva-Rocci (No. 969). Curious to know who these people are? Buy the book and find out!

Well, there's not much harm and perhaps a bit of fun in this kind of thing, but let me move on to cases of list making or index making, or, more generally, mathematizations of various kinds whose effects are more questionable. This cannot be done intelligently without talking about the distinctions between objectivity and subjectivity, the quantitative and the qualitative, and how these problematic categories that give philosophers of science much heartburn overlap one another or change with time. In the space allotted me, I can only throw out a few remarks.

Quantification in Today's World
Our age is awash in rank-orderings and quantifications of all sorts. In recent months, I have seen quantified the quality of a restaurant, the aesthetic content of a work of art, the consonances and dissonances in a musical composition, the degree of compatibility of two young lovers, the prestige value of checking into the Plaza Hotel, the degree of belief in a statement, the degree of risk in an enterprise, the competitive utility of a product.

Let me go on. I have seen quantified the performance of an Olympic skater, the productivity of a nation, the appropriateness of job J for individual I, the caliber of a university department of mathematics, the tendency of Nation A to go to war with Nation B, the degree of acceptability of a given prose style, the condition of the world's poor, the excellence of a movie: one to five stars.

Readers can easily extend these lists. While the results often embody a consensus, reached by averaging the results of a salad bar of responses or the opinions of a panel of experts, they do not as yet command firm respect or agreement as to their appropriateness or utility.

The questions of what can be quantified usefully, what can be rank-ordered, and how one goes about doing it, can be answered only historically, and not in a general way. The equations of physics are phrased in terms of quantification. The relation between physics and the quantifiable is an intimate one, and its necessity is now considered absolute:

"The demand for quantitativeness in physics seems to mean that every specific distinction, characterization, or determination of a state of a physical object and the transmission of specific knowledge, must ultimately be expressible in terms of real numbers, either single numbers or groupings of numbers, whether such numbers be given 'intensively' through the medium of formulae or 'extensively' through the medium of tabulation, graphs or charts" (Salomon Bochner, "The Role of Mathematics in the Rise of Science," in The Dictionary of the History of Ideas).

As the great physicist James Clerk Maxwell said, if something is expressible in numbers, then it is understandable and potentially scientific. The psychologist E.L. Thorndike made a more sweeping assertion: "What exists, exists in some amount." Hence, all becomes grist for the mathematical mills, which can grind quantities with ease and then have them baked into "scientific" bread.

We are used to this. But all is not that easy in physics. As the mathematician Andr´┐Ż Weil remarked, mathematics provides a palette of objects and operations from which we must pick and choose with care.

How can musical consonances and dissonances be described quantitatively? The reader might believe that since knowledge of a simple relation between the length of a vibrating string and harmony dates back to the ancients, this problem was solved long ago. Not so. Would you believe that the famous astronomer Johannes Kepler (1571-1630) had a theory of musical consonances involving the particular polygons that can be constructed by rule and compass means? R.B. Cohen (The Quantization of Music, 1984) says that the consonance problem is not yet solved. Theories are still being advanced.

Move away from physics and consider concepts like happiness and pleasure. F.Y. Edgeworth (1845-1926), the famous statistician, believed that pleasure could be quantified, and that, for example, my pleasure could be added to your pleasure to arrive at something significant. In this way, a "hedonic calculus" could be set up. He asserted later that utility and degrees of belief could be measured, and in this way came to econometrics and rediscovered one of the standard interpretations of probability. All this can be found in his book Mathematical Psychics (1881).

Consider, for example, the question of the IQ (intelligence quotient). For many years, intelligence has been measured or described by a single number. Psychologists now tell us that an individual may possess many "intelligences," perhaps as many as eight, so that quantification of intelligence, if possible at all, must be done vectorially. Let's assume that this has been done. Since an eight-dimensional number loses the relationships of greater than and less than, and since the space of eight dimensions is far richer than the one-dimensional space of traditional IQ, we can rightfully ask what conceivable use we could make of eight-dimensional intelligence quotients, and how we could go about validating such usages.

Another example, not necessarily more dubious than IQ, is the notion of freedom. In 1990-1991, the UN promulgated a mathematized notion of freedom, and on this basis ranked countries as to their levels of freedom. Entering into the "freedom quotient" were such considerations as freedom of women to have abortions and freedom to engage in homosexual practices. Later, due to adverse criticism, the UN discontinued this index. We could compile, I feel sure, an Index of Prohibited Indices.

Quality vs. Quantity
According to some philosophers, quality and quantity are independent attributes. Once perceived, moreover, a quality cannot be reduced to more primitive concepts. Scientific and mathematical minds tend to see the matter differently. Quality is seen as depending on and subordinate to quantity. As Lord Rutherford, Nobelist in physics, once put it: "The qualitative is nothing but poor quantitative." What do we mean when we speak of the quality of a piece of cloth? We could be referring to its thickness, tightness of weave, stretchability, durability, shrinkage after washing, all of these characteristics being measurable. Mathematics is employed to convert questions of quality to measurable and easily processed quantities.

In some mathematical situations, particularly in the pre-electronic computer days, qualitative statements were more easily arrived at than quantitative statements. The result has been, in the last century, an emphasis on qualitative theories and a denigration of the quantitative. (I'm thinking here of qualitative differential equations.) More generally, some believe that the Bourbaki school of mathematicians (c. 1950) was based on structures and on qualities rather than on quantities.

The Subjective Becomes Quantifiable
Whatever is quantitative carries with it the cachet of the scientific, the objective. What is subjective is merely your experience, your perception, your opinion, and is not necessarily mine and may not be expressive of the "true" situation.

Now it is the easiest thing in the world to pass, or to claim that one has passed, from the subjective to the quantitative. How? One of the ways, surely the crudest, but one that is extremely popular, is merely to ask people to rate their perceptions on a scale:

On a scale from 0 to 10, how much do you like broccoli?
On a scale from 0 to 10, how well do you think the President is doing his job?
On a scale from 0 to 10, indicate the degree of social alienation caused by the suburbanization of Crescendo, Michigan.

There are no difficulties in eliciting answers to such questions: The public is delighted to answer. Once we have numbers, we can operate on them mathematically and automatically, and derive consequences or cause certain actions to be initiated.

Much thought has gone into constructing scales and into considering the invariants deemed necessary. There is a branch of mathematics known as abstract measurement theory; begun in 1887 with Hermann v. Helmholtz ("Counting and Measuring"), it is pursued largely by mathematicians and psychologists. The theory is both highly abstract and philosophically moot.

If I have come down heavily against the transition from the subjective to the quantifiable and thence to the objective, let me point out that in some "hard" sciences, the preference for the objective over the subjective has had its ups and downs. What, for example, could be considered more objective than a photograph? Yet in astronomy and in anatomy, reliance on the "evidence" of drawings, often done by skilled artists, has given way to reliance on photographs, and thence back to drawings. (See "The Birth and Death of Mechanical Objectivity," Peter Galison's paper in Picturing Science, Producing Art, Routledge, 1997.)

Returning, finally, to the book under review, how then should one go about making a list of the best shapers of the millennium? Why do it, say I, even while admitting that the temptation to do it is irresistible. G.H. Hardy, after all, made his list of the best mathematicians of his day, pairing them with the best cricketers.

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island.

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