Reasoning and proof in society

Summary: Many aspects of society have inherited from mathematics the desire for a method of proof that is demonstrable and irrefutable.

Reasoning and proof are fundamental components of human existence. Children begin applying reasoning as soon as they can make connections between actions and consequences. They then go on to explore more formal methods of reasoning and proof throughout their educational careers, not just in mathematics. Although people often associate mathematics solely with deductive proofs, many other types of reasoning are important to mathematics, including inductive logic, evidence-based reasoning, and computer-assisted arguments. Furthermore, the concept of truth being produced by reasoning and proof also pervades other fields, including philosophy, the natural and social sciences, and political and legal discourse.

Origins of Mathematical Proof

What proves a statement? Generally, it is believed that statements are proved by deducing the statement as a logical consequence of something already believed to be true. One might think that proofs are necessary only when what is being proved is not apparent. The Greeks, however, did not limit proving to non-obvious statements; they gave a logical structure to all of geometry, assuming as its basis the smallest possible number of “already believed” statements. They also employed a method called “proof by contradiction” in which a truth is not demonstrated directly, but rather by showing that its opposite cannot be maintained.

Why did Greek culture give geometry this kind of logical structure, and why did the Greeks think that doing so was significant? The question is important because the causes that produced mathematical proof still exist in the twenty-first century, where they continue to operate and promote the use of proof.

First, proofs give a way to reconcile discordant opinions. Greek mathematics was heir to two earlier traditions, Egyptian and Babylonian mathematics, whose results did not always agree. For instance, in studying circles, the Babylonians approximated π first as 3, and later as 3.125. Egyptian computations give a value for π of about 3.16. The Greeks wanted to know π’s true value. One way to avoid having multiple answers to the same question is to make no assumptions other than those with which nobody could disagree, like “all right angles are equal,” and then deduce other facts solely from those un-doubtable assumptions. What is amazing is how many results this approach produced.

Second, proofs are a natural outcome of the search for basic principles. The pioneering Greek philosophers of nature of the fifth and sixth centuries b.c.e. sought simple explanatory principles that could make sense out of the entire universe. Thales, for instance, said that “everything is water,” and Anaximenes claimed that “everything is air.” The Pythagoreans asserted that “all is number,” while Democritus said that “everything is made of atoms.” As in nature, so in mathematics, the Greeks wanted to develop explanations based on simple first principles, on the so-called elements.

Third, the logic of proofs can arise from the process of discovery. One effective way to solve a problem is to reduce it to a simpler problem whose solution is already known. For instance, Hippocrates of Chios in the fifth century b.c.e. reduced finding the area of some lunes (areas bounded by two circular arcs) to finding the area of triangles. In reducing complicated problems to simpler problems, and then reducing these to yet simpler problems, the Greek mathematicians were creating sets of logically linked ideas. If such a set of linked ideas is run in reverse order, a proof structure emerges—simple statements on which rest more complex statements on which rest yet more complex statements. The simplest statements at the beginning are called the “elements”; the intermediate ones are the fruitful results that are now called “lemmas”; and these in turn demonstrate the final and most important results.

Fourth, logical reasoning played essential roles in classical Greek society. In the sixth and fifth centuries b.c.e., Greece was largely made up of small city-states run by their citizens. Discourse between disputing parties, from the law courts to the public assemblies, required and helped advance logical skills. A good way, then and now, to persuade people is to understand their premises, and then construct one’s own argument by reasoning from their premises. A good way to disprove someone’s views is to find some logical consequence of those views that appears absurd. These techniques are beautifully illustrated in Greek legal proceedings and political discourse, as well as in the dialogues of Plato.

Finally, Greek mathematics developed hand in hand with philosophy. Greek philosophers began by trying to logically refute their predecessors. Zeno, for instance, presented his paradoxical arguments not to prove that motion is impossible but to challenge others’ intuition and common-sense assumptions. That Plato wrote in dialogue form both illustrates and demonstrates that Greek philosophy was as much about the method of logical argument as it was about conclusions. Aristotle wanted every science to start, like geometry, with explicitly stated elementary first principles, and then to logically deduce the key truths of the subject. Greek philosophy issued marching orders to mathematicians, and men like Euclid followed these orders.

Philosophy returned the favor. Plato made mathematics the center of the education of the rulers of his ideal Republic and mathematics has remained at the heart of Western education. Plato championed mathematics because it exemplified how, by reasoning alone, one could transcend individual experience. Such transcendence is most striking in the case of proof by contradiction. The argument form, “If you accept A, then you must also accept B, but B contradicts C,” was part and parcel of the educated Greek’s weapons of refutation. But proof by contradiction is not merely destructive, it also allows people to rigorously test conjectures that cannot be tested directly and, if they are true, to demonstrate them.

For example, Euclid defined parallel lines as lines in the same plane that never meet. But it can never be shown directly that two lines can never meet. However, it can be assumed that the two lines do, in fact, meet and then prove that this assumption leads to a contradiction. This process made Euclid’s theory of parallels possible.

As another example, consider the Greek proof that √2 cannot be rational (it cannot be the ratio of two whole numbers). Because the Pythagorean theorem holds for isosceles right triangles, √2 must exist.

But no picture of an isosceles right triangle can allow one to distinguish a side of rational length from one of irrational length.

Nor can one hope to prove the irrationality of √2 by squaring every single one of the infinitely many rational numbers to see if its square equals 2. However, if one assumes that there is a rational number whose square is two, logic then leads to a contradiction, so it is proved that √2 cannot be rational.

By such means the Greeks proved not only that √2 was irrational but also that a whole new set of mathematical objects existed: “irrational numbers.”

Proof in general, and proof by contradiction in particular, transformed the nature of mathematics. Logic lets people reason about concepts that are beyond experience and intuition—about ideas that cannot be observed. Mathematics had become the study of objects transcending material reality, objects visible only to the eye of the intellect. There could be truths about such objects and such truths could be proved. These developments had profound consequences far beyond mathematics.

Beyond Mathematics

The ideal of logical proof in mathematics took on a life of its own. Since mathematicians apparently had achieved truth by means of proof, practitioners of other areas of Western thought wanted to do the same. So in theology, politics, philosophy, and science people tried to imitate the mathematicians’ method.

In 1637, Rene Descartes wrote in his Discourse on Method, “Those long chains of reasoning… which enabled geometers to reach the most difficult demonstrations, made me wonder whether all things knowable to men might not fall into a similar logical sequence.” If so, he continued, there cannot be any propositions that cannot be eventually discovered and proven.

Building on Descartes’s ideas, Baruch Spinoza in 1675 wrote a book called Ethics Demonstrated in Geometrical Order. Like Euclid, Spinoza first explicitly defined his terms, including “God” and “eternity.” He then stated axioms about existence and causality. On the basis of his list of definitions and axioms, Spinoza logically demonstrated his philosophical conclusions, including the existence of God.

Isaac Newton wrote his great Principia in 1687. This work includes Newton’s laws of motion and theory of gravity. He did not structure the Principia like a modern physics book; he gave it the same definition-axiom-theorem structure that Euclid had given the Elements. Newton expressly called his famous three laws “Axioms, or Laws of Motion.” From these axioms, Newton logically deduced the laws of the universe, including universal gravitation, just as Euclid had deduced his own theorems.

The American Declaration of Independence of 1776 also pays homage to the ideal of Euclidean proof. The principal author, Thomas Jefferson, was well versed in the mathematics of his time. Jefferson began with axioms, saying, “We hold these truths to be self-evident,” including the axioms “that all men are created equal” and that, if a government does not preserve human rights, “it is the right of the people to alter or abolish it, and set up new government.” The declaration then says that it will “prove” that King George III’s government had not protected human rights. Once Jefferson proved this, the Declaration of Independence concludes: “We therefore… publish and declare that these United Colonies are and of right ought to be free and independent states.” Indeed, Jefferson could have ended his argument, as had Spinoza and Newton, with the geometer’s “QED.”

Jefferson’s argument exemplifies the characteristic program of Enlightenment philosophy—using reason to reach conclusions on which everyone will agree. This program is epitomized in the words of Voltaire in his Philosophical Dictionary: “There is but one morality, as there is but one geometry.”

Abstraction, Symbolism, and their Power

Logical proof in mathematics and the use of mathematical models of reasoning in the larger intellectual world were not limited to geometry. In mathematics in the seventeenth and eighteenth centuries, proof methods moved beyond the geometric to include the algebraic. This shift began when Françis Viète, in 1591, first introduced general symbolic notation in algebra, an idea with incredible power.

School children learn that for every pair of distinct numbers, not only does 9+7=16, so does 7+9. Viète’s general symbolic notation allows one to write down the infinite number of such facts all at once: B+C=C+B.

A century later, Isaac Newton summed up the power and generality of Viète’s idea by calling algebra “universal arithmetic.” Newton meant that one could prove algebraic truths from the universal validity of the symbolic manipulations that obey the laws of ordinary arithmetic. For instance, consider the quadratic equation 2x²-11x+15=0. Simply stating, “3 and 2 1/2 are the solutions” gives no information about how those answers were obtained. But every quadratic equation has the general form of ax²+bx+c=0. Solving that general equation by the algebraic technique of completing the square gives the well-known quadratic formula for the general solution:

This general solution contains the record of every operation performed in getting it. The original example had a=2, b=-11, c=15. As such, it is known exactly how the answers, 3 and 2 1/2, are obtained from the coefficients in the equation. More important, this process proves that these and only these must be the answers.

In the seventeenth century, Gottfried Wilhelm Leibniz was so inspired by the power of algebraic notation to simultaneously make and prove mathematical discoveries that he invented an analogous notation for his new differential calculus. Furthermore, he envisioned an even more general symbolic language that would, once perfected, find the indisputable truth in all areas of human thought. Once such a language existed, Leibniz said, if two people were to disagree, one could say to the other, “let us calculate, sir!” and the disagreement would be resolved. This idea made Leibniz the prophet of modern symbolic logic.

By the eighteenth century, many mathematicians thought discovery and proof should be based on abstract symbolic reasoning. Imitating mathematics, scientists introduced analogous notations in other fields. For instance, Antoine Lavoisier and Claude-Louis Berthollet developed a new chemical notation that they called “chemical algebra,” which is used when balancing a chemical equation.

These ideas, both within and beyond mathematics, led the Marquis de Condorcet to write in 1793 that algebra contains within it the principles of a universal instrument, applicable to all combinations of ideas. Such an instrument, he said, would eventually make the progress of every subject embraced by human intelligence as sure as the progress of mathematics.

In the nineteenth century, George Boole produced the first modern system of symbolic logic and used it to analyze a wide variety of complicated arguments. His system, developed further, underlies the logic used by digital computers in the twenty-first century, including applications embodying Condorcet’s dream, from automated theorem-proving to translators, grammar checkers, and search engines.

Non-Euclidean Geometry: The Triumph of Euclidean Logic

Unthinkable as it may have been to Enlightenment philosophers like Voltaire, there are alternatives to Euclid’s geometry. But non-Euclidean geometry was not invented by imaginative artists or by critics of mathematics speculating about alternative realities. Like irrational numbers, non-Euclidean geometry was discovered by mathematicians. Its discovery provides another example of human reason and logic trumping intuition and experience and it—like Euclid’s geometry—has had a profound effect on other areas of thought.

Non-Euclidean geometry grew out of attempts to prove Euclid’s parallel postulate:

If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, then the two straight lines, if produced indefinitely, meet on that side where the angles are less than two right angles.

Such attempts were made because the postulate seemed considerably less self-evident than his other postulates. From antiquity onward, mathematicians felt that it ought to be a theorem rather than an assumption, and many eminent mathematicians tried to prove it from the other postulates. Some attempted to prove it indirectly; assuming it to be false, they deduced what appeared to be absurd consequences from that assumption. For instance, that parallel lines are not everywhere equidistant, and that there is more than one line parallel to a given line through a point in the same plane. These results contradict our deep intuitive sense of symmetry.

But in the nineteenth century, three mathematicians independently realized that these conclusions were not absurd at all, but were perfectly valid theorems in an alternative geometry. Nicolai Ivanovich Lobachevsky, by analogy with imaginary numbers, called his new subject “imaginary geometry.” Janos Bolyai more theologically called it “a new world created out of nothing.” But Carl Friedrich Gauss, acknowledging the logical move that made it possible, called the new subject “non-Euclidean geometry.”

The historical commitment of mathematicians to the autonomy of logic and to logical proof enabled them to overcome their scientific, psychological, and philosophical commitments to Euclidean symmetry to create this new subject. Logical argument once again let mathematicians find and demonstrate the properties of something neither visual nor tangible—something counter-intuitive. Non-Euclidean geometry is the ultimate triumph of the Euclidean method of proof. But there are wider implications.

From this discovery, nineteenth-century philosophers concluded that the essence of mathematics (as opposed to the natural sciences) is its freedom to choose any consistent set of axioms that meets the mathematician’s sense of what is important, beautiful, and fruitful—just as long as the logic is right. There could even be real-world applications of systems that contradict all past mathematical orthodoxies. In physics, for instance, the type of non-Euclidean geometry studied by Bernhard Riemann in the 1850s turned out to be exactly what Albert Einstein needed for his general theory of relativity; the new mathematics can explain gravitation, describe the curvature of space, and account for black holes.

Knowing that alternative systems of mathematical thought are logically possible has also had philosophical and social implications. José Ortega y Gasset, for instance, contrasted the view of the old geometry (interpreted as saying that nations may perish but principles will be kept) to the new perspective, which he interpreted as saying that people must look for such principles as will preserve nations, because that is what principles are for.

Proof and the Citizen

Citizens of democracies need to be able to evaluate arguments presented to them, whether by friends, adversaries, politicians, or advertisers. In the words of Jacques Barzun, “The ability to feel the force of an argument apart from the substance it deals with is the strongest possible weapon against prejudice.”

Citizens also need to be free to work out the logical implications of the principles they treasure. In the words of Winston Smith, a character in George Orwell’s novel 1984, “Freedom is the freedom to say that two plus two make four. If that is granted, all else follows.” This kind of “proving” has driven the progress of the idea of universal human rights. For instance, building on the Declaration of Independence, Elizabeth Cady Stanton, a pioneer in fighting for women’s rights in America, wrote in the Seneca Falls Declaration of 1848, “We hold these truths to be self-evident; that all men and women are created equal.” Similarly, Martin Luther King, Jr., in his “I Have a Dream” speech, spoke of “the promise that all men, yes, black men as well as white men, would be guaranteed the unalienable rights of life, liberty, and the pursuit of happiness.”

Now, just as in ancient Greece, the ability to reason and prove and the liberty of expressing and acting upon the results of proofs are essential to a free and democratic society. The historical function of proof in mathematics has not been just to prove theorems but also to exemplify and teach logical argument in areas such as philosophy, law, politics, religion, and every area of modern life.

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