The ancient Greeks, building upon earlier work by the Egyptians and Babylonians, transformed mathematics into an integral part of liberal education during the fourth century B.C.E. The academic disciplines (mathemata) of arithmetic and geometry were then sharply distinguished from the menial rules of practical calculation (logistica) necessary for the everyday work of artisans, tradesmen, and money changers. Arithmetic studies the properties of whole numbers such as divisibility and factorization by primes, while geometry studies properties of magnitudes such as congruence, similarity, and proportion. Both are concerned with aspects of measurement, understood in a broad sense, but arithmetic deals with discrete quantities (multitudes of a unit) while geometry considers continuous magnitudes (line segments, planar areas, and solids).

The notion of a ratio (logos)—the size of one thing relative to another—plays a major unifying role, yet many advances in both classical and modern mathematics have sprung from the inherent tension between the continuous and the discrete. The tension we may sense today between our flowing, or continuous, temporal existence and the discrete digital world of the modern computer reveals the distinction between these cooperating opposites and suggests the possibility of a powerful interaction.

Pythagorean and Platonic connections

Measurement is made by expressing a ratio of the thing to be measured to a second thing, usually to a standard unit that is more familiar—nowadays taken to be a meter, second, liter, or the like. In the fifth century B.C.E. the Pythagoreans made much of the fact, said to have been well known already in China, that ratios of small whole numbers in arithmetic are related to harmonious musical intervals. Thus, to speak in modern terms, the easily recognizable octave is produced by two pitches in the ratio 2:1, while the ratio 3:2 yields a musical fifth, and 5:4 determines a third. Our ability to sense the ratios between pitches in music and their identification with ratios between numbers may have helped inspire the Pythagorean dictum, "All is number." By this is meant, presumably, that integers and their ratios (logoi) have the power to express underlying harmonies in nature that will be hidden from those ignorant of mathematics. Perhaps the most familiar modern (nineteenth-century) example of this power is the order induced in the periodic table by the assignment of an appropriate atomic number—an integer—to each basic chemical element.

Pythagoras (c. 560–c. 495 B.C.E.) is traditionally credited with putting together two Greek words to coin the word philosophy ("love of wisdom") and with objectifying the notion of order by taking the Greek word for it, cosmos, and giving this name to the universe. Despite his mystical leanings, Pythagoras is sometimes seen as the founder of Western science because his followers continually promoted mathematics as a means of finding order and harmony in the natural world. The Pythagoreans used the connection between arithmetic and the science of music to develop a musical scale based upon just intonation (and they appreciated the difficulties that were finally ameliorated in the eighteenth century by well-tempering). They also noted the more obvious connection between geometry and astronomy. Stars are like points and the constellations are formed by line segments joining pairs of stars—so that problems in navigation may become problems in geometry.

Aspects of astronomy are thus naturally modeled by geometry, just as some properties of music are modeled by arithmetic. But these sciences deal with things in motion—the rotating celestial sphere, the vibrating strings of a lyre—whereas the mathematics of arithmetic and geometry deal with idealized static objects such as whole numbers and stationary line segments. A striking analogy is due to Archytas (fifth century B.C.E.), a latter-day Pythagorean: Arithmetic is to Music as Geometry is to Astronomy. Almost a thousand years later these four mathemata became collectively known as the quadrivium, a name given them by the Roman philosopher Boethius (c. 480–c. 524), although his practical countrymen prized logistica more highly. Eventually, the quadrivium became an integral part of the classical liberal arts in medieval European universities.

The word ratio has long been associated with measured study and hence with reason itself, while logos, the Greek word for ratio, has taken on a wide-ranging religious significance as well. The unit generates all numbers, whose logoi, according to the Pythagorean faith, have the power to measure (know) everything in the cosmos. Thus, for the Pythagoreans, the logos is a mathematical means of expressing cosmic harmony. The variety of basic roles that the logos plays in mathematics, science, philosophy, liberal education, and religion is suggested by the wide usage of such cognate terms as logic and analogy, and the host of academic words with the suffix -logy. Pythagoras seems to have been drawn toward a holistic view encompassing all these spheres, but their explosive growth would make this view ever more difficult to sustain.

Plato (c. 427–347 B.C.E.) became familiar with Pythagorean doctrines through Archytas and endorsed their emphasis upon mathematics and their insistence upon the same basic education for men and women. Plato thought that our power of direct apprehension of idealized mathematical forms like the circle might be refined to help us apprehend such things as truth, beauty, and goodness—Platonic forms whose properties, moreover, might also be studied by deductive reason. If, as Plato insisted, mathematics helps train the mind to rise from the apparent and ephemeral to the true and permanent, then its study should promote both science and religion. Indeed, when Jewish and early Christian thinkers began to view Platonic forms as ideas in the mind of God, an important link was established between Platonism and Judeo-Christian thought.

Plato even suggested that the immortality of the soul is intimated by geometry, especially when learned by the Socratic method, where it may appear that we are remembering—rather than learning anew—connections between geometric forms that we had somehow forgotten. To Plato this implies the existence of some earlier state of fuller communion with the forms. We must therefore (re)search in order to remember where we came from. In the midst of this perhaps fanciful argument, however, is Plato's admonition with which all modern scientists would agree, that in research we must look beyond mere sensory impressions. The laws governing the stars are fairer than the stars.

Plato comes close to espousing a religious motivation for scientific inquiry by taking the position, ardently embraced much later by Johannes Kepler (1571–1630), that the universe is, in some sense, an expression of the nature of its creator. Many researchers in mathematics and science, including some to whom Plato's views might appear naïve, have occasionally expressed a belief that they are, so to speak, reading the mind of God. "We cannot read [the great book of Nature]," wrote Galileo Galilei (1564–1642), "unless we have first learned the language and the characters in which it is written .… It is written in mathematical language."

Mathematics as a human endeavor

A quick excursion sketching the rise of seventeenth-century calculus may help to put a human face upon the making of mathematics. In the early Middle Ages a slowly growing quantitative sense began to evolve, later bolstered by the convenience of working with numerals developed in India that would eventually be used in Indo-Arabic decimal fractions. The preservation, refinement, and advancement of Greek and Indian ideas during the rising tide of the Islamic movement led to the development of algebra—the very word for which comes from Arabic (al-jabr) and has somewhat the sense of "rearrangement." Mohammed ibn Musa al-Khwarizmi (c. 780–c. 850 C.E.) began his influential algebra book of the ninth century by praising God for bestowing upon man the power to discover the significance of numbers. The word algorism (later, and more commonly, algorithm) derives from the author's patronymic.

Calculus may be seen as a post-Renaissance blending of these developments with a new propensity to think in terms of the intuitive notions of variable, function, and limit, coupled with the development of analytic geometry, which unites large parts of algebra and geometry through the use of Cartesian coordinates. The joining together of such diverse ideas gave mathematics (and physical science) an astounding vitality in the seventeenth century. Isaac Newton (1642–1727) and Gottfried Wilhelm Leibniz (1646–1716) were the first to see the calculus as a unified whole that studies the interplay between functions and derivatives. This interplay casts light upon previously perplexing philosophical and scientific problems concerning the notions of instantaneous velocity and acceleration, gives new and efficient ways to find optimal solutions to many types of problems, and provides natural and effective methods for solving equations and for finding lengths of curves and sizes of areas and volumes. Newton used the calculus, together with his physical laws (axioms) of motion, to show how Kepler's observations about planetary motion follow from the law of gravity.

The scientific successes of "reason" inspired attempts to extend its methods beyond science. The philosophy of René Descartes (1596–1650), who developed analytic geometry, drew a clear distinction between reason and ecclesiastical authority. Descartes—and, later, both Newton and Leibniz—made serious, rational contributions to theology.

The early reaction to such efforts by Blaise Pascal (1623–1662), who had helped develop several nascent branches of mathematics (probability, projective geometry, and calculus), would be telling. Repelled by the idea of a god "of philosophers and scholars," Pascal abandoned everything for theology, returning to mathematics only once, in 1658, when he published some pretty results about the cycloid that calculus students still study. Pascal's writings exalting heart over mind ("Humble thyself, impotent reason!") would be seen to help inspire romanticism during a much later period, which left in its wake a great gap between the sciences and the humanities. Mathematics would find itself stretched ever more tenuously across this gap.

Ironically, the great mathematical advances of the so-called Age of Reason owe more to the imagination and intuition of mathematicians than to their logic and reason. The development of calculus was facilitated, as its developers were well aware, by a relaxation of the strictures of rigorous geometrical methods that proceed from precise definitions and clear first principles. Instead, mathematicians embraced loose numerical methods allowing unending decimal expansions and other infinite sums—thus going far beyond the finite arithmetic of the Greeks. This attitude led both to unprecedented progress in research and to occasional confusion and contradiction. The logical difficulties encountered were principally due to the suggestive, but slippery, notion of an infinitesimal, which was supposed to be a discrete entity that retained qualities of the continuous. Not until the precise formulation of the notion of a limit by Augustin-Louis Cauchy (1789–1857) and others were these difficulties decisively overcome.

In the meantime the shaky foundations of the calculus were exposed by the philosopher George Berkeley (1685–1753), an Anglican bishop, who published in 1734 a witty and acerbic essay called The Analyst, where he famously (and justly) ridiculed infinitesimals as "ghosts of departed quantities." His subtitle—To an Infidel Mathematician—reflects his purpose, to rebuke mathematicians of his day by showing that their discipline contains mysteries no less subtle than those of theology. Perhaps the best eighteenth-century advice to those who would learn the calculus was given by the French mathematician Jean le Rond d'Alembert (1717–1783): "Go forward, and faith will follow."

The search for coherence: Euclid's legacy

The axiomatic method consists in somehow intuiting basic accepted facts (axioms) about a discipline and logically deducing all else. Axiomatization of the real number system in order to derive rigorously the results of calculus—and thereby answer criticisms of Berkeley and others—did not occur until the late nineteenth century, when finally rational sense was made out of the huge mass of calculus-inspired research largely due to, but overly dependent upon, an unbridled trust in mathematical intuition. Pressure to provide such coherence to a discipline usually comes only when its elements have been basically established and it is time to synthesize a great web of connections into a consistent body of work.

The most celebrated example of such a synthesis is Euclid's Elements, which appeared in Alexandria around 300 B.C.E. Here, the towering edifice of geometry appears to be solidly built up by logic, unerringly applied to a small number of "self-evident" facts that we are willing to accept at the outset. The Elements is doubly valuable, however, because its study—with the help of a skilled tutor—will also impart the dual thinking techniques of analysis and synthesis that are indispensable in achieving rational coherence in any discipline. Analysis, as Plato used the term, refers to the testing of the truth of a proposition by deducing implications from it. If one of these implications is false, then the proposition must of course be false (reductio ad absurdum); otherwise, one hopes to deduce a consequence that is self-evidently true, and a synthetic proof is said to be obtained if the steps in this deduction can be reversed so as to obtain the given proposition as a logical consequence of self-evident truths.

The power of such analysis had been strikingly felt when the central tenet of the Pythagorean faith—the proposition that every ratio can be expressed as a ratio of whole numbers—was tested and proved false by reductio ad absurdum: If the proposition were true, then the ratio of the diagonal of a square to its side would be expressible as a ratio of integers. But this implies (to use modern terminology) that the square root of two is rational, which leads to contradiction, as first noted by the Pythagoreans about 430 B.C.E. Perhaps partly as a consequence of the limitations of arithmetic revealed by this shock, the Greeks came to look more favorably upon geometry, which Euclid attempted to put on a firm, rational foundation. It was not, however, until the nineteenth century that the foundations of mathematics were seen to require substantially more careful attention than Euclid had provided.

Archimedes (287–212 B.C.E.) effectively invented mathematical physics by giving an axiomatic development to hydrostatics, beginning by deriving from simple axioms the fundamental law of the lever. He then went on to discuss rigorously how to find centers of gravity of complicated solids, solving problems that are routinely handled today, but only by using calculus in a fairly sophisticated way. Mathematical physics came of age with Newton in the seventeenth century, and physicists today who seek an axiomatic basis for quantum mechanics follow in this tradition.

Western civilization has absorbed over a thousand editions of the Elements, whose influence is sometimes subtly felt. As noted by Bertrand Russell in Wisdom of the West (1959), a revealing moment in the Enlightenment occurred in 1776 when Benjamin Franklin spotted the phrase "sacred and undeniable" in the penultimate draft of the American Declaration of Independence and suggested that "self-evident" be substituted. A revolutionary list of moral and political rights of individuals was thus introduced to the world not with a religious invocation, but with an implicit salute to Euclid: "We hold these truths to be self-evident."

In contrast to Euclid, who presumably thought that his basic axioms about geometry were obviously true, both Nicolaus Copernicus (1473–1543) and Kepler on occasion spoke of an "axiom" of astronomy as a provisional truth that one might someday hope to establish. Axioms of empirical disciplines may alternatively be viewed simply as facts to be tested by analyzing their implications to see how well they model reality. The scope of axiomatics was decisively extended beyond the sciences when Baruch Spinoza (1632–1677) set down philosophical axioms and deduced the consequences in his Ethics. Systematic theology embraces a similar method of exposition when it exhibits the collective implications of basic religious tenets as a rationally coherent system.

In light of these modern points of view, the existence of non-Euclidean geometry—a startling development when Euclid was thought to represent "absolute truth"—is now seen as unsurprising. If "light rays" of physics are to be modeled by "lines" from geometry, why should the lines satisfy Euclid's axioms, now that we know of consistent mathematical structures developed by N.I. Lobachevsky (1792–1856) and G.F.B. Riemann (1826–1866) in which "points" and "lines" can be defined in such a way that Euclid's parallel postulate fails while the other axioms hold? Modern physicists routinely use non-Euclidean geometry to model the cosmos.

Faith in Euclid's absolute truth is thus clearly unfounded. In fact, modern mathematicians, when presented with axioms defining a vector space or some other mathematical structure, typically do not ask whether the axioms are "true," but instead set about deducing theorems that must hold for every structure satisfying the given axioms. The existence of foundational mathematical structures such as the real number system, out of which vastly complicated, useful, and interesting structures can be constructed, is generally regarded as unproblematic by working mathematicians. Mathematical logicians, on the other hand, study foundational questions intensely, usually basing their work upon the theory of sets. The surprising "incompleteness" theorem proved in 1931 by Kurt Gödel (1906–1978) demonstrated unforeseen limitations in the power of the axiomatic method and has sparked further study.

Conclusion

Modern mathematics has expanded far beyond the study of calculus and differential equations that has helped scientists to cope with continuous processes and, as well, beyond the developments in probability and statistics that have advanced the mathematical treatment of discrete processes. Carl Friedrich Gauss (1777–1855), perhaps the greatest modern mathematician, made deep contributions to almost all areas of the subject. By the early twentieth century, however, the scope of mathematics had grown so large that no single mathematician could claim to have mastered more than a small portion of the field.

The attraction of mathematics as a worthy human interest lies in discovering and establishing surprising and interesting connections between apparently disparate mathematical ideas that have not yet been fully comprehended. Mathematicians pursue useful goals, but while attaining them they often meet new ideas without immediate practical value that are appealing in their own right. Sometimes, intriguingly, these ideas prove to be surprisingly useful, whereas their initial appeal is only aesthetic in the sense that they seem to call for an imaginative synthesis expressed with clarity and style. "The love of a subject in itself and for itself, where it is not the sleepy pleasure of pacing a mental quarter-deck, is the love of style as manifested in that study," said the mathematician and philosopher Alfred North Whitehead (1861-1947).

Whitehead contended that pure mathematics, in its modern developments, may claim to be the most original creation of the human spirit. A similar claim might be made in connection with an often overlooked feature of its ancient development. Howard deLong perceptively observes in A Profile of Mathematical Logic (1970) that early Greek interest in abstract thought owes much to the expansion, from the physical to the mental arena, of the familiar spirit of competition and play. The sportive aspect of the play of the mind, which animates mathematics in its purest form, is bound up with this remarkable growth of the human spirit so long ago.

In A Mathematician's Apology (1940), G. H. Hardy (1877-1947) bases his defense upon aesthetic grounds and confesses a genuine passion for his calling. Something akin to Hardy's passion is known to all who have experienced the revelation that follows a spell of total concentration and have found themselves echoing in their own tongue Archimedes's famous cry of eureka ("I have found it"). Mathematicians count heavily upon the spirit that compels such engagements and articulates such an involuntary cry of delight. What transpires under its spell may even seem like something done to—rather than by—a mathematician. No one seems ever to have argued, however, that a calling to an Archimedean engagement implies the existence of a "caller." Attitudes of mathematicians toward religion range from Whitehead's well-known sympathy for the religious experience to Hardy's strongly opposing view.

See also ALGORITHM; GALILEO GALILEI; PLATO

W. M. PRIESTLEY