# Should mathematics be pursued for its own sake, not for its social utility

# Should mathematics be pursued for its own sake, not for its social utility?

**Viewpoint:** Yes, mathematics should be pursued for its own sake, as doing so will eventually present practical benefits to society.

**Viewpoint:** No, mathematics should first be pursued by everyone for its societal utility, with the mathematically gifted being encouraged to continue the pursuit of math for math's sake at advanced levels.

One of the great problems in mathematics is not a "math problem" at all; rather, it is a question of the means by which the study of mathematics should be pursued. This may be characterized as a debate between "pure" mathematics, or mathematics for its own sake, and applied mathematics, the use of mathematics for a specific purpose, as in business or engineering. This is a concern that goes back to the historical beginnings of mathematical inquiry.

The earliest mathematicians, those of Mesopotamia, were concerned almost entirely with what would today be called applied mathematics. The Sumerians, for instance, developed the first mathematical systems not for the purposes of abstract inquiry, but rather to serve extremely practical needs. They were businesspeople, and their need for mathematical techniques was much the same as that of a modern-day corporation: merchants needed to keep track of their funds, their inventory, their debts, and the monies owed to them.

Today, accountants and mathematicians employed in business use concepts unknown to the Sumerians, utilizing mathematics for such esoteric pursuits as calculating risks and benefits or optimizing delivery schedules. The techniques involved originated from abstract studies in mathematics, meaning that the thinkers who developed them usually were not driven by immediate needs, but today the application of those methods is decidedly down-to-earth.

The other great mathematicians of ancient Mesopotamia were the Babylonians, who likewise used numbers for highly practical purposes: the study of the heavens and the development of calendars. Astronomy arose from ancient astrology, which first came into mature form in Babylonia during the second and first millennia before Christ. Today, astrology hardly seems like a "practical pursuit," but the situation was quite different in ancient times, when rulers relied on astrological charts the way modern politicians consult polls and focus-group results.

To a high-school student confronted with the vagaries of trigonometry, the discipline might seem quite abstract indeed. Trigonometry analyzes the relationships between the various sides of a right triangle, as well as the properties of points on a circle, and requires the student to understand concepts and values such as sine, cosine, tangent, and so forth. Yet trigonometry did not originate from the imagination of some abstracted mathematical genius; rather, it had its origins in the astrological and astronomical studies of the ancient Egyptians and Babylonians.

For centuries, in fact, trigonometry had the status not of a mathematical discipline, but of an aid to astronomy. Only in the late Middle Ages, on the brink of the voyages of discovery that would put trigonometry to use in navigation, would it be fully separated, as a mathematical discipline, from its many applications. This separation, however, began nearly 2,000 years earlier, and is one of many legacies for which the world of mathematics is indebted to the ancient Greeks.

Despite the advances made previously by the Babylonians, and to a lesser extent the Sumerians and Egyptians, the Greeks are almost universally recognized as the originators of mathematics as a formal discipline. This is true even when one takes into account the advances made by the Chinese, of which the Greeks were of course unaware, because Chinese mathematicians were likewise primarily concerned with the application of mathematics to practical, everyday problems. Only the Hindu mathematicians of early medieval India, who developed the number system in use today (including zero, a number unknown to the Greeks and Romans), may be regarded as mathematical innovators on anything like a par with the Greeks.

What was different about the Greek mathematicians? Their interest in mathematicians for its own sake. Whereas the Sumerians had envisioned numbers in purely concrete terms, so much so that they had no concept of 5, for instance (only "five goats" or "five fingers" and so on), the Greeks recognized the abstract properties of numbers. Likewise they pursued geometry, not for the many practical applications it would eventually yield, but rather as a great puzzle or game in which the attempt to find solutions was its own reward.

Greek mathematicians became fascinated with problems such as the squaring of the circle (finding a circle with an area equal to that of a given square using only a compass and straight edge) or determining prime numbers. Many of these pursuits might have seemed, to an outside observer, like so much useless theoretical speculation, but in the centuries to come, they would yield a vast wealth of mathematical knowledge that find application in everything from architecture and engineering to computers and rocket science.

The two essays that follow examine the question of applied vs. pure mathematics, not as a dilemma that has only one possible answer, but as a question that offers several answers of differing relative worth. Thus, for the defender of pure mathematics, it is implicit that the pursuit of math for math's sake will eventually present practical benefits to society. Likewise, for the partisans of applied mathematics, there is no censure attached to pure mathematics; rather, it is a question of which pursuit is most appropriate for the largest number of students.

—JUDSON KNIGHT

## Viewpoint: Yes, mathematics should be pursued for its own sake, as doing so will eventually present practical benefits to society.

Social utility may be defined as whatever contributes toward making life easier, safer, longer, more comfortable, harmonious, invigorating, fulfilling, and enjoyable. Every human endeavor ought to promote social utility somehow or to some degree, but not necessarily in a direct or primary way.

### The Nature and Role of Mathematics

Mathematics promotes social utility by enabling scientific advances that may ultimately result in new products, services, systems, or insights that make life better. The old slogan of E. I. du Pont de Nemours was "Better Things for Better Living Through Chemistry." The company that created nylon, Teflon, Lucite, and Freon could easily say that, but it must also be remembered that chemistry depends on mathematics. So does physics, biology, technology, and even medicine. The "better things" constitute scientific utility, or the products of applied science, and the "better living" is the social utility that emerges from applied science and ultimately from pure science, including mathematics.

Since the late seventeenth century, statistics, algebra, and calculus have all been components of epidemiology, especially since the pioneering theory of deterministic epidemiological models by William H. Hamer in 1906 and Sir Ronald Ross in 1911 began to make biometrics a rigorous mathematical science. The subsequent mathematical understanding of epidemics has saved countless lives. Hamer and Ross were deliberately working toward practical medical goals, but Sir Isaac Newton (1642-1727) and Gottfried Wilhelm Freiherr von Leibniz (1646-1716), who invented the calculus that biometricians now use, were each working on purely theoretical levels. The practical applications of their mathematical discoveries came later.

Mathematics is the epitome of theoretical thought, even more so than philosophy, which is grounded in some kind of perceived reality. Pure mathematics is the epitome of apparently impractical thought, even more so than speculative philosophy, which typically has an object in view. Mathematicians are interested in determining what fits, or what coheres to make each whole theoretical system work. Other thinkers may discover how to apply these coherent deductive systems, such as calculus, trigonometry, Lobachevskian geometry, set theory, etc., to everyday problems in the real world. If mathematicians had to consider the practical ends of their thought as well as its theoretical coherence, very few of them would have the intellectual talent, resources, or stamina to achieve any worthwhile results, either practical or theoretical. Society is better served by dividing the intellectual labor between the theoreticians and the practical thinkers.

The ancient Babylonians and Egyptians developed the theoretical science of trigonometry out of astronomy, to explain quantitatively their observations of the movements of stars and planets. Many centuries later the ancient Greeks found practical applications for trigonometry in surveying, military tactics, and navigation. Similarly, elaborate theoretical developments regarding Hilbert space, topology, probability, statistics, game theory, model theory, computer science, and many other areas of mathematical inquiry may not yet have realized their potential in the practical business of improving society, but that does not make them any less valuable, either as mental exercises or as cultural products.

Paradoxically, pursuing mathematics as a whole for its own sake promotes social utility more reliably than pursuing it specifically for the sake of either social utility or any other ulterior purpose. However, since utility is more likely to be served by unrestricted mathematical investigations than by mathematical projects designed with specific, practical ends in view, even the most theoretically inclined mathematicians should remain aware that their results may ultimately have practical uses. By thinking "outside the box," as it were, these mathematicians actually aim at such practical uses, but not directly.

Ancient Greek philosophy was not distinct from the natural sciences and included mathematics. The philosopher Thales (ca. 624-ca. 547 b.c.) studied mostly geometry, astronomy, navigation, meteorology, and ontology. Once he inferred from his meteorological calculations that the next year's olive crop was likely to be abundant. Therefore he bought as many shares as he could afford in all the olive presses in the area. When his prediction proved correct, he made a fortune by selling his shares at whatever price he named. The point is that Thales did not study mathematics and science in order to become rich, but when a practical application of his self-acquired knowledge presented itself, he saw his opportunity and took it. This story, whether true or not, illustrates that to pursue mathematics and science only for their utility limits their scope and hinders mathematicians and scientists, but to pursue them for their own sake does not preclude finding practical uses for them.

Theoretical results usually have broader and more varied application than practical results. Practical applications of theoretical results are sometimes recognized only much later, and sometimes even by accident or serendipity. Results in pure mathematics may sit idle long after their discovery until someone finds practical uses for them. Such was the case with the equations for determining plane functions from line integrals. In 1917 Austrian mathematician Johann Radon discovered these equations, which turned out to be necessary for computerized axial tomography (CAT) scans, but South African physicist Allan Macleod Cormack only learned of them in 1972, nine years after he had independently solved that problem and successfully applied it to computerized composite imaging. Cormack's work at first almost completely escaped the notice of physicians. Not until another physicist, Godfrey Newbold Hounsfield, unaware of both Radon and Cormack, led a team of British radiologists in the construction and installation of the first clinical CAT scanner in 1971 did the medical community show interest. CAT scans have proved to be the greatest single advance in radiology since x rays. Thus, not from ingratitude but from ignorance, Hounsfield failed to acknowledge either Radon or Cormack in the Nobel Prize-winning article which introduced CAT scans to the world. The Nobel Committee, however, allowed Cormack to share the prize with him.

Theoretical researchers sometimes serve practical or social purposes despite themselves. In 1994, mathematicians Julius L. Shaneson of the University of Pennsylvania and Sylvain E. Cappell of New York University, building on the earlier work of University of Cambridge mathematician Louis Joel Mordell, published a result in theoretical algebra that they believed had no practical value. They were not seeking any practical value when they were working on the problem. Nevertheless, economists recognized almost instantly the importance of the Mordell-Shaneson-Cappell method for certain kinds of business problems and began using it successfully.

John Forbes Nash's results in game theory, especially his discovery of mutual gain for all players in non-cooperative games, first published as "Equilibrium Points in N-Person Games" in the 1950 *Proceedings of the National Academy of Sciences*, were later applied to practical economics in ways that he had not foreseen, yet for this work in pure mathematics he shared the 1994 Nobel Prize in economic sciences.

### Philosophical Underpinnings

The question of the relative importance of theoretical interest versus practical value is at least as old as ancient Greek philosophy and can be applied to any area of knowledge or research. Aristotle (384-322 b.c.) distinguished between pure and applied mathematics, claiming that the practical was justified by the theoretical, not by its relevance to the physical world. He also systematically discussed *theôria* ("human thought"), *praxis* ("human action"), and *poiêsis* ("human creativity or production"). Logically, but not always chronologically, *theôria* precedes *praxis*. *Poiêsis* emerges from the cooperation of the two.

For Aristotle, *poiêsis* not only opposes *praxis*, but also enlists *theôria* to supersede *praxis* and create new products. *Praxis* and *theôria* are each primary and irreducible, and although *poiêsis* is secondary to both *praxis* and *theôria*, ultimately both *poiêsis* and *praxis* depend on *theôria* for their impetus and advancement.

Karl Marx (1818-1883) had little use for pure theory, declaring in his eleventh thesis on Feuerbach that the point of philosophy is not to understand the world, but to change it, and implying in the eighth of these theses that *theôria* is misled by the kind of thinking that has no practical aim or design, and can be rescued from its airy wandering only by *praxis*, which essentially is the same as social utility or common life. For Marx, the purpose of *theôria* is to understand and assist *praxis*, not to ruminate on vague, impractical mysteries such as God, pure science, or *theôria* itself. His attitude directly contrasts Aristotle's, for whom the highest form of thinking is contemplative thought about nothing but itself.

Martin Heidegger (1889-1976) distinguished between "meditative thinking" (*besinnliches Denken*) and "calculative thinking" (*rechnendes Denken*). This distinction is similar to Aristotle's *theôria* and *praxis*, except that, for Heidegger, the gulf is wider and the extremes are more sharply defined. While Aristotelian *theôria* may eventually have a practical end, Heideggerian meditative thinking has none, and would be compromised and denigrated if any practical use were ever found for it. In this regard, Heidegger resembles the Taoists.

The subtle power of pure thinking has long been known in Chinese wisdom. Lao-Tse (604?-531? b.c.), the legendary founder of Taoism, called it *wu wei*, "non-action." He wrote in the *Tao-te Ching*, Chapter 37, "The Tao never takes any action, yet nothing is ever left undone." The relation between Taoism and the other great Chinese religious system, Confucianism, is analogous to that between *theôria* and *praxis*, the former centered in individual contemplation, intellectual advancement, and mental tranquillity, the latter on social action, political organization, and worldly progress.

Heidegger and Taoism (pro- *theôria*) and Marxism and Confucianism (pro- *praxis*) represent the two extremes. Aristotelianism takes a middle ground by regarding *theôria* and *praxis* as opposite and equally essential elements of human life, which together find their greatest manifestation as *poiêsis*, the practical yet creative production of the substance of human culture and civilization.

### Serving Practical Needs

The Radon/Cormack/Hounsfield case illustrates the need for better communication among pure mathematicians and those scientists who may be seeking practical applications of mathematical discoveries. Corollary to that would be a subsequent need for better cooperation between academic and vocational educators, to emphasize the *poiêsis*, or productive power, generated by reciprocity between *theôria* and *praxis*. But this is not to say that pure mathematicians should not remain free to let their highest thoughts proceed wherever they will, without the impediment of practical considerations.

Applied mathematics is taught not only in secondary schools on non-college-preparatory tracks, but also at the Massachusetts Institute of Technology (MIT), where it constitutes a separate department. In 1959 at the University of Cambridge, England, George Batchelor founded the Department of Applied Mathematics and Theoretical Physics. At first blush, such a name might seem incongruous, but not when we consider that it does not refer to "applied mathematics" in the usual sense. Rather, it means mathematics applied to theoretical physics, that is, mathematics as the basis of theoretical physics.

Immanuel Kant (1724-1804) famously wrote in the *Critique of Pure Reason*, "Percepts without concepts are empty; concepts without percepts are blind." By this he meant that merely being in the world, experiencing, perceiving, working, and living day-to-day is meaningless if we do not think about, reflect upon, or criticize what we experience; and by the same token that merely thinking, ruminating, and contemplating is misguided if we do not also somehow act in the world, or put our thoughts to use.

Drawing the lines between concepts and percepts, theory and practice, *theôria* and *praxis*, theoretical and practical science, pure and applied mathematics, or academic and vocational education, is difficult. The lines are gray. In general, though, we can say that concepts, theory, *theôria*, theoretical science, pure mathematics, and academic education are characterized by being open-ended and making progress in the sense of achieving a Kantian overarching reason (*Vernunft*), while, on the other hand, percepts, practice, *praxis*, practical science, applied mathematics, and vocational education are characterized by not progressing beyond the narrow focus of the solution to each particular defined problem, and thus achieving only a Kantian understanding (*Verstand*) of circumstances. But the union or harmony of all these dichotomous factors would be *poiêsis*, which is indeed a social good.

To concentrate primarily on present particular problems instead of the broader theoretical elements of mathematics prevents the development and accumulation of the very knowledge that will be needed to solve future particular problems. The attitude of those who prefer to focus narrowly on *ad hoc* puzzles rather than ponder the general truths or overarching unity of mathematics recalls the closing lines of "Ode on a Distant Prospect of Eton College" by Thomas Gray: "Thought would destroy their paradise. / No more;—where ignorance is bliss / 'Tis folly to be wise."

—ERIC V.D. LUFT

## Viewpoint: No, mathematics should first be pursued by everyone for its societal utility, with the mathematically gifted being encouraged to continue the pursuit of math for math's sake at advanced levels.

There is widespread agreement that the pursuit of mathematics starts at an early age. Mathematics is a part of the curriculum from kindergarten through grade 12 in essentially all public education today. The Executive Summary in *Before It's Too Late: A Report to the Nation from the National Commission on Mathematics and Science Teaching for the 21st Century* identifies "enduring reasons" for children to achieve competency in mathematics. Among them are the widespread workplace demands for mathematics-related knowledge and abilities; the need for mathematics in everyday decision-making; and for what the authors describe as "the deeper, intrinsic value of mathematical knowledge that shapes and defines our common life, history, and culture." Except for the last and quite esoteric reason, the Commission is squarely on the side of studying mathematics for its practical utility in society.

Disputes abound among educators when it comes to mathematics. Although educators generally agree on the importance of pursuing mathematics, and that it should be promoted for its utility, there are heated differences on how it should be presented, and how the results should be measured. The extent of the use of computers and calculators is another point of contention among some educators.

There are many theories on how humans learn. One of the most influential in education, Jean Piaget's, provides some clues on how students process abstract concepts at various stages of development. According to students of Piaget's theory, the percentage of the population that will ever reach the level of learning needed to pursue math for math's sake is not large. Academic enrollments at high-tech universities such as MIT support that theory.

### Why Teach/Learn Math?

Before going further in this discussion, it must be pointed out that teaching anything does not guarantee that learning will result. That can be a problem. There is no doubt that both teaching and learning mathematics is important to the future of our children, our economy, our progress in technology, and even our success on the global scene. Just how to accomplish this is the big question.

In the forward to the *Before It's too Late* report, John Glenn, Commission Chairman, says, "at the daybreak of this new century and millennium, the Commission is convinced that the future well-being of our nation and people depends on just how well we educate our children generally, but on how well we educate them in mathematics and science specifically." Since each and every science has quantitative components, that makes the teaching, and learning, of mathematics even more important.

Glenn continues, "From mathematics and the sciences will come the products, services, standards of living, and economic and military security that will sustain us at home and around the world. From them will come the technological creativity American companies need to compete effectively in the global marketplace."

The Commission report expresses strong concern that in the United States, the education systems as a whole are not teaching children to understand and use the ideas of science and mathematics, with the least success coming at the higher grades. Studying mathematics for math's sake is most likely to begin at the higher grades when students are beginning to follow their interests and aptitudes. Unfortunately, the number of students going on to major in math make up a very small percentage of those pursuing higher education.

### How to Teach Math: The Math "Wars"

The debate concerning how to teach mathematics has been described as a "war" by some, among them David Ross, a mathematician at Kodak Research Labs. The National Council of Teachers of Mathematics (NCTM), a nonprofit, non-partisan education association with 100,000 members dedicated to improving mathematics teaching and learning, takes the position that every student will enter a world that is so different from the one that existed a few years ago that modifications in school mathematics programs are essential. They point out that every student will assume the responsibilities of citizenship in a much more quantitatively driven society, and that every student has the right to be mathematically prepared. They believe that quality mathematics education is not just for those who want to study mathematics and science in college—it is for everyone.

NCTM states that students must learn basic computation facts and know how to compute, and that practice is important, but practice without understanding is a waste of time. Although the opponents of the teaching techniques promoted by NCTM do not see much merit in what they are doing, NCTM does include basic skill mastery in their goals. NCTM just takes a more flexible approach to teaching mathematical concepts. NCTM reasons that the old method did not work very well and did little to promote students' problem-solving skills. Citing from the NCTM *Standards 2000* :

"Many adults are quick to admit that they are unable to remember much of the mathematics they learned in school. In their schooling, mathematics was presented primarily as a collection of facts and skills to be memorized. The fact that a student was able to provide correct answers shortly after a topic was generally taken as evidence of understanding. Students' ability to provide correct answers is not always an indicator of a high level of conceptual understanding."

The NCTM emphasizes giving the teacher freedom to determine how topics should be approached in any given lesson with an emphasis on problem-solving skills and real-life applications. This all sounds very idealistic. Opponents call it "fuzzy math" and take a firm stand that students have to be well grounded in fundamentals. They do not see all the old rote learning as bad. Opponents to the NCTM approach definitely stand by the old saying that practice makes perfect.

### The Place for Calculators

The dispute on how best to achieve math literacy includes debates on how, when, and where calculators and computers should be used. The NCTM points out that students live in a fast-paced world of TV and video games. They describe today's students as "Internet savvy, smoothly juggling multiple images and inputs" in one of their "Setting the Record Straight about Changes in Mathematics Education" bulletins. They describe calculators as one of many tools to enhance students' learning. NCTM believes that calculators support the learning of mathematics and knowing how to use calculators intelligently is part of being prepared for an increasingly technological world. NCTM recommends calculators be made available for use by students at every grade level from kindergarten through college. Various other organizations agree, according to research compiled at Rice University for the Urban Systemic Initiative/Comprehensive Partnership for Mathematics and Science Achievement (1997).

The Rice University survey suggests that by reducing the time spent on tedious paper-and-pencil arithmetic and algebraic algorithms, calculators allow students to spend more time developing mathematical understanding, reasoning, number sense, and applications. Quite understandably, those on the other side of the math "wars" do not see the time spent on paper-and-pencil calculations as time wasted. They see calculators as useful tools, but feel it is important to learn the fundamentals before relying on them.

The use of "tools" in the teaching, learning, and application of mathematics has been going on since the first person looked at his or her fingers to count. The Chinese have been using an abacus for at least 700 years. The National Museum of American History at the Smithsonian has a Web site titled "Slates, Slide Rules, and Software," where one can learn about the various devices students have used to master abstract mathematical concepts.

The debates and disputes on how to get students interested in learning mathematics, for either its utility or for its own sake, are ongoing. According to a report to the President of MIT, only 3.3% of the students enrolled at the school during the 1999-2000 academic year were math majors. Since so few students pursue mathematics as a discipline, it make sense to encourage the study of math for its social utility.

—M. C. NAGEL

## Further Reading

Aristotle. *Basic Works*, edited by Richard McKeon. New York: Random House, 1941.

Bailey, Norman T. J. *The Mathematical Theory of Infectious Diseases and Its Applications*. London: Charles Griffin, 1975.

Ball, W.W. Rouse. *A Short Account of the History of Mathematics*. New York: Dover, 1960.

*Before It's Too Late: A Report to the Nation from the National Commission on Mathematics and Science Teaching for the 21st Century* [cited July 29, 2002]. <http://www.ed.gov/americounts/glenn/toolateexecsum.html>.

Calinger, Ronald. *A Contextual History of Mathematics: To Euler*. Upper Saddle River, NJ: Prentice Hall, 1999.

Cappell, Sylvain E., and Julius L. Shaneson."Genera of Algebraic Varieties and Counting Lattice Points." *Bulletin of the American Mathematical Society* 30, no. 1 (January 1994): 62-69.

"Du Pont Heritage" [cited July 25, 2002].<http://heritage.dupont.com/>.

"The Du Pont Story" [cited July 25, 2002].<http://www.dupont.com/corp/overview/anniversary/story.html>.

Grattan-Guinness, Ivor. *The Norton History of the Mathematical Sciences: The Rainbow of Mathematics*. New York: Norton, 1998.

Hamer, William H. "The Milroy Lectures on Epidemic Disease in England: The Evidence of Variability and Persistence of Type." *Lancet* 1 (1906): 733-39.

Haven, Kendall F. *Marvels of Math: Fascinating Reads and Awesome Activities*. Englewood, CO: Teacher Ideas Press, 1998.

Heidegger, Martin. *Discourse on Thinking*, translated by John M. Anderson and E. Hans Freund. New York: Harper and Row, 1969.

Hogben, Lancelot. *Mathematics in the Making*. London: Macdonald, 1960.

Katz, Victor J., ed. *Using History to Teach Mathematics: An International Perspective*. Washington, DC: Mathematical Association of America, 2000.

Kincheloe, Joe L. *Toil and Trouble: Good Work, Smart Workers, and the Integration of Academic and Vocational Education*. New York: Peter Lang, 1995.

Lazarsfeld, Paul F., ed. *Mathematical Thinking in the Social Sciences*. Glencoe, IL: Free Press, 1954.

Marx, Karl, and Friedrich Engels. *The Marx-Engels Reader*, edited by Robert C. Tucker. New York: Norton, 1978.

*An Overview of Principles and Standards for School Mathematics*. Reston, VA: NCTM, 2000.

Paulos, John Allen. *A Mathematician Reads the Newspaper*. New York: Basic Books, 1995.

Penn, Alexandra, and Dennis Williams. *Integrating Academic and Vocational Education: A Model for Secondary Schools*. Alexandria, VA: Association for Supervision and Curriculum Development, 1996.

Phillips, George McArtney. *Two Millennia of Mathematics: From Archimedes to Gauss*. New York: Springer, 2000.

Sarton, George. *The Study of the History of Mathematics and the Study of the History of Science*. New York: Dover, 1957.

Stasz, Cathleen, et al. *Classrooms That Work: Teaching Generic Skills in Academic and Vocational Settings*. Santa Monica, CA: RAND, 1993.

"Slates, Slide Rules, and Software." Smithsonian Institution [cited July 25, 2002]. <http://americanhistory.si.edu/teachingmath/>.

Stein, Mary Kay. *Implementation Standards-Based Mathematics Instruction: a Casebook for Professional Development*. New York: Teachers College Press, 2000.

Stillwell, John. *Mathematics and Its History*. New York: Springer, 2001.

Sylla, Edith Dudley. "The Emergence of Mathematical Probability from the Perspective of the Leibniz-Jacob Bernoulli Correspondence." *Perspectives on Science* 6, nos.1-2 (1998): 41-76.

## KEY TERMS

### ABACUS:

An ancient counting instrument characterized by sliding counting beads on rods.

### ACADEMIC EDUCATION:

Study and teaching intended to widen and deepen the students' grasp of concepts, without regard for the practical value of the curriculum. *Library of Congress Subject Headings* calls it "critical thinking."

### APPLIED (OR PRACTICAL) MATHEMATICS:

Mathematics as put to practical use in the physical world, to solve either specific problems *ad hoc* or classes of recurring problems. These problems may be very simple, such as measuring household gardens, or very complex, such as programming missile guidance systems.

### POIÊSIS :

Aristotle's technical philosophical term meaning"production" or "creativity," i.e., theory as applied to practice, or the union of *theôria* and *praxis*. The English word "poetry" derives from it.

### PRACTICAL (OR APPLIED) SCIENCE:

Any area of knowledge or research pursued specifically and solely for the sake of its immediate or obvious practical applications.

### PRAXIS :

Standard philosophical term from the ancient Greek meaning "practice" or "human action." It exploits *theôria* and its adherents live the "active life." Marx valued it above *theôria*.

### PURE (OR THEORETICAL) MATHEMATICS:

Mathematics with no apparent practical purpose.

### SLIDE RULE:

A calculating instrument consisting of a ruler with an inserted sliding rule in the middle, and the rules marked with numbers in a logarithmic scale.

### THEORETICAL (OR PURE) SCIENCE:

Any area of knowledge or research pursued "for its own sake," or without any immediate practical application in view.

### THEÔRIA :

Standard philosophical term from the ancient Greek meaning "theory" or "human thought." It scrutinizes *praxis* and its adherents live the "contemplative life." Most non-Marxist and non-Utilitarian philosophers value it above *praxis*.

### VOCATIONAL EDUCATION:

Study and teaching intended to give students a "trade," without regard for the students' wider or deeper grasp of concepts outside the necessary skills of that trade. *Library of Congress Subject Headings* calls it "occupational training."

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