Mathematics, Substance and Surmise: Views on the Meaning and Ontology of Mathematics (2015)
Philip J. Davis1
Department of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA
Philip J. Davis
A prominent mathematician recently sent me an article he had written and asked me for my reaction. After studying it, I said that he was proposing a mathematical “product” and that as such it stood in the scientific marketplace in competition with nearby products. He bridled and was incensed by my use of the word “product” to describe his work. Our correspondence terminated. What follows is an elaboration of what I mean by mathematical products and how I situate them within the mathematical enterprise.
Civilization has always had a mathematical underlay, often informal, and not always overt. I would say that mathematics often lies deep in formulaic material, procedures, conceptualizations, attitudes, and now in chips and accompanying hardware. In recent years the mathematization of our lives has grown by leaps and bounds. A useful point of view is to think of this growth in terms of products. Mathematical products serve a purpose; they can be targeted to define, facilitate, enhance, supply, explain, interpret, invade, complicate, confuse, and create new requirements or environments for life.
What? Mathematical “products”? Products in an intellectual area that is reputed to contain the finest result of pure reason and logic: a body of material that in its early years was in the classical quadrivium along with astronomy and music? How gross of me to bring in the language of materialistic commerce and in this way sully or besmirch the reputation of what are clean, crisp idealistic constructions! Products are the routine output of factories, not of skilled craftsmen whose sharp minds frequently reside far above the usual rewards of life. The notion that mathematics has products or that its content is merchandise, might tarnish both its image and the self image of the creators of this noble material.
And yet … The world is full of mathematical products—mathematics produces knowledge hence we have mathematical products—many of them. As O’Halloran [1, 2] claims, mathematics is functional; it permits us to construe and reason about the world in new ways that extend beyond our linguistic formulations. The world of today embraces the product of that knowledge.
1 Examples of Mathematical Products
Yes, the world is full of mathematical products of all sorts. I will name a few. A slide rule is a product. A French curve is a product. An algorithm (recipe) for solving linear equations is a product. A theorem is a product and stands among hundreds of thousands of theorems, ready to be interpreted, appreciated, used, updated, reworked or neglected. A text book on linear algebra is a product. A polling system is a product. The statutory rule for allocating representatives after a new census is a product. A tax or a lottery or an insurance scheme or even a Ponzi scheme is a product. Telephone numbers are a product. A professional mathematical society is a product. A medical decision that depends in a routine manner on some sort of quantification is a product. A computer language is a product. A supermarket cash register and the Julian calendar are products. The act of taking a number at a delicatessen or a bakeshop counter to facilitate ones’ “next” is a product. Matlab is a product. Google is a product. Encryption schemes are products. Sometimes a mathematical product is designed for very specialized usage; it may then be called a package or a tool box.
Admittedly these examples might seem to indicate that in my mind anything at all that has to do with mathematics can be considered a product. Is Cantor’s diagonalization process a product? Is a T-shirt imprinted with the face of Kurt GödeI a mathematical product? Well, I would find it exceedingly difficult to propose a formal definition. In any case, let us see the extent to which one might describe and discuss the mathematical enterprise from the point of view of its products that I have or will cite.
What is the clientele for mathematical products? While mathematical products are the brain children of inspired individuals or groups, the targeted users of the products may vary from a few individuals to entire populations. Those targeted may be aware of the availability of a product that has been claimed to be of use; they may either use it or reject it. In many cases the product is built into a whole social system and one cannot easily opt out of its use. Examples: phone numbers and area codes; the US Census; more locally, passwords at the ATM around the corner.
1.1 Scientific/Technological aspects of Mathematical Products
Mathematics was called by Gauss “the Queen of the Sciences” and a good fraction of its products relate to science/technology: e.g., packages for the factorization of large integers; for the analysis of architectural structures or packages marketed; for on-site DNA analysis. A scheme for constructing and interpreting a horoscope can be a mathematical product of considerable sophistication and complexity. The “wise” may reject its conclusions, yet the product flourishes.
Without in any way dethroning the Queen, it should be pointed out that the employment of mathematics has always gone far beyond what are now called “the sciences.” Mathematics has made an impact on commerce, trade, medicine, biology, mysticism, theology, entertainment, warfare, etc.
1.2 The Transmission or Communication of Mathematical Products
Transmission is done by a wide variety of “signs” or “semiotic products.” Short texts, books, pictures, programs, flash drives, chips, formal classroom teaching or the informal master/apprentice relationship, word of mouth, etc. The international or intercultural transmission and absorption of mathematical products (e.g., the adoption by the West of Arabic numerals) has been and still is the object of scholarly studies.
1.3 Commercial aspects of Mathematical Products
The commercialization of mathematical products has grown by leaps and bounds since World War II. A mathematical product can be promoted in many of the same ways that a brand of breakfast food is promoted: by ads, by the praises of well-known personalities or groups, etc, (plugs). On MATLAB’s website, you can find a list of MATLAB’s available products, listed openly and labeled clearly as “products.” Investment and insurance schemes are called “products.”
A product can be sold, e.g., a hand held computer or the Handbook of Mathematical Functions. A product can be licensed for usage, or it can be made available as a freebie. In the case of taxes (qua mathematical product), it is “promoted” by laws and threats of punishment. Rubik’s Cube, a mathematical product, caught the imagination and challenged the wits of millions of people and has earned fortunes. Sudoku, a mathematical puzzle, is sold in numerous formats. If a product is income producing, its sellers can be taxed. A product can be copyrighted or patented; the owners of such can be contested, sued for infringement.
1.4 Competitive Aspects of Mathematical Products
A mathematical product is often subject to competition from nearby products. Think of the innumerable ways of solving a set of linear equations. Textbooks, a source of considerable income, compete in a mathematical market place that involves educationists, testing theorists and outfits, unions, publishers, parents’ groups, local state and national governments.
1.5 Social aspects of Mathematical Products
If a mathematical product finds widespread usage, it may have social, economic, ethical, legal or political implications or consequences. The repugnant Nuremberg Racial Laws (Germany, 1935) with their numerical criteria caused incredible suffering. DNA sequencing and its interpretations is a relatively new branch of applied mathematics, resulting in a host of new products. In a number of States, the level of mathematical tests for the lower school grades has been questioned. The social consequences of mathematical products, benign or otherwise, may not emerge for many years.
1.6 Legal aspects of Mathematical Products
There are innumerable examples of this. The US Constitution is full of number processes. Consider
“Representatives and direct Taxes shall be apportioned among the several States which may be included within this Union, according to their respective Numbers, which shall be determined by adding to the whole Number of free Persons, including those bound to Service for a Term of Years, and excluding Indians not taxed, three fifths of all other Persons.”
Some mathematical products have been subject to judicial review. As an example, the mathematical scheme for the 2010 Census was vetted and restricted by the US Supreme Court.
An example of a statutory product is the method of least proportions used to allocate representatives in Congress. It was approved by the Supreme Court in Dept. of Commerce v. Montana, 503 U.S. 442 (1992). Another example: A multiple regression model used in an employment discrimination class action is another such example; it was approved by the Supreme Court in Bazemore v. Friday, 478 U.S. 385 (1986).
1.7 Logical or Philosophical aspects of Mathematical Products
A mathematical product, considered as such, is neither true nor false. Of course, it may embody certain principles of deductive logic, but these do not automatically make the employment of the product plausible or advisable. A product can be made plausible, moot or useless on the basis of certain internal or external considerations. An interesting historical example of this is the dethroning of Euclidean geometry as the unique geometry by the discovery of non-Euclidean geometries.
A product may raise or imply philosophical questions such as the distinction between the subjective and the objective or between the qualitative and the quantitative, between the deterministic and the probabilistic, the tangible and the intangible, the hidden and the overt.
Numerical indexes of this thing and that thing abound. Cases of subjectivity occur when a product asks a person or a group of people to pass judgment on some issue: “On a scale of zero to ten, how much do you like tofu?” The well-known Index of Economic Freedom embodies a number of items, expressed numerically:
“We measure ten components of economic freedom, assigning a grade in each using a scale from 0 to 100, where 100 represents the maximum freedom. The ten component scores are then averaged to give an overall economic freedom score for each country. The ten components of economic freedom are: Business Freedom | Trade Freedom | Fiscal Freedom | Government Size | Monetary Freedom | Investment Freedom | Financial Freedom | Property rights | Freedom from Corruption | Labor Freedom”
1.8 Moral aspects of Mathematical Products
Society asks many questions. Does the manner of taking the US Census account properly for the homeless? Are tests in Algebra slanted towards certain subcultures? Does the tremendous role that mathematics plays in war raise questions or angst in the minds of those who are responsible for its application? Are results of IQ testing being misused?
2 Judgments of Mathematical Products
As mentioned, mathematical products serve a purpose; they can be targeted to define, facilitate, enhance, invade, any of the requirements or aspects of life. Ultimately, a product can be judged in the same way that any product can be judged: by the response of its targeted users or purchasers. In the case of a mathematical product what criteria are in play? The cheapest? The most convenient? The most useful? The most comprehensive? The most accurate? The most original? The most seminal? The most reassuring? The safest or least vulnerable? The most esthetic? The most moral? Is the product unique? Are there pressures from investors or the various foundations that support their production?
Is “survival of the fittest” a good description of the judgment process? Probably not. There are fashions in the product world attracting both excited consumers and producers. Time, chance and what the larger world requires, appreciates, or suffers from mathematizations that are always in play.
O’Halloran, K. L. (2005). Mathematical Discourse: Language, Symbolism and Visual Images. London and New York: Continuum.
O’Halloran, K. L. (2015). The Language of Learning Mathematics: A Multimodal Perspective. The Journal of Mathematical Behaviour. http://dx.doi.org/10.1016/j.jmathb.2014.09.002