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Why Mathematical Reasoning Should Be a Part of Civic Education

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We now live in a world where reason and truth are under siege on a daily basis. The Economist declares that we have entered an era of “post-truth politics.” Falsehoods are called “alternative facts.” Science is subject to ideological manipulation. On both sides of the aisle, moral relativism and partisan politics have largely replaced commitment to constitutional principles and our inalienable rights. For many conservatives, religious freedom is important only if the religion is Christianity. For many liberals, the core liberal tenet of free speech is now somehow contingent on the identity of the speaker. While people everywhere used to struggle for our indivisible and inalienable human rights, we now like to manipulate these principles to fit our own narratives. I think part of the problem is lack of appreciation; learning about history or politics in oppressive regimes can help. The other part of the problem is inability or unwillingness to reason on both sides. Mathematical reasoning, I argue, provides an antidote to this culture of relative truth.

First, some clarifications on terminology. By relativism, I mean the belief that moral truths are relative to individual perspective, and that universal moral truths do not exist. By mathematical reasoning, I mean the ability to use proofs and rigorous arguments to generate indisputable mathematical knowledge. Mathematical reasoning is different from numeracy in that the latter denotes only the ability to work with numbers and calculate them.

The culture of truth and certainty that is generated by the method of proof in the mathematical world is sorely needed in the political and social sphere. In mathematics, once someone has proven that there are infinitely many prime numbers, no distortion or reframing or political maneuver or “alternative facts” can change that. This does not mean that skepticism has no place in mathematics, or, as political scientist Andrew Hacker apparently believes, that mathematics thrives under oppressive regimes. Basic assumptions in mathematics are often revised in light of evidence that shows a system can generate paradoxical results. For example, our intuitive understanding of sets as collections of objects was discarded after the discovery of the famous Russell’s paradox: the set of all sets that are not members of themselves can neither contain nor not contain itself. All mathematics asks is this: believe what you have a strong argument for, unless reason shows otherwise, in which case you must not believe.

Mathematics also teaches important skills that are transferable to political and social debates. First, mathematics teaches attention to detail. For example, try solving for x: ax = b. If your answer is x = b/a, then you miss the case in which a = b = 0 (which would mean x can be anything), or the case in which a = 0 but b is not 0 (which would mean there is no value x can be for the equation to hold.) Getting into the habit of considering all the “borderline cases” and scenarios can be useful in discussions about technical issues, such as affirmative action or voter ID law. Second, mathematics teaches basic strategies of argumentation. For example, in calculus, to show the claim that continuous functions must be differentiable somewhere is false, mathematicians created a counterexample called the “Weierstrass function.” In essence, it means that it is conceptually possible for a car to move without speed at any moment. One can also use “proof by counterexample” to show that freedom of speech is not absolute. For example, one cannot falsely shout fire in a movie theater. Third, mathematics teaches how to identify assumptions and gaps in arguments. One famous example in mathematics was the proof of Fermat’s Last Theorem: for any integer n greater than 2, there are no three positive integers x, y, and z such that xn + yn = zn. It took mathematicians 358 years to prove this result, but the first version of the proof contained a fatal error that was discovered during peer review. Eventually, the mathematician Andrew Wiles fixed his proof after a year of hard work, just before he was about to give up. The obsession with the rigor of argumentation in mathematics trains students of mathematics to develop a critical eye for arguments in general, as I have personally experienced. Fourth, mathematics teaches problem solving skills that can be particularly useful in policy-making and nonprofit work. The best proofs in mathematics are those that can be written in relatively few lines but are hard to discover. The best policy ideas are usually the same: easy to state, but hard to come by. Finally, mathematics teaches how to write well. Mathematical arguments are often more technical than arguments in other disciplines. Clarity of writing, therefore, becomes key in the discipline. Évariste Galois, one of the most important mathematicians in the 20th century, developed group theory in a paper that was considered “incomprehensible” by some mathematicians. It was only after he rewrote his mathematical manuscripts and published them the day before he died in a duel that his mathematical contributions became well-known.

Learning mathematical reasoning need not be difficult or stressful. Mathematical proofs should not be included in standardized tests such as SAT. Rather, exposure to mathematical reasoning can take the form of exposition and exploration. Teachers can show the most beautiful and famous proofs in different areas of mathematics, and guide students through each step of the proofs. Teachers can also present students with propositions that they can either prove or disprove. Finally, teachers can include tales from history of mathematics to illustrate how mathematical results can have profound philosophical implications. My favorite example is how Kurt Gödel’s and Alan Turing’s results in mathematical logic have shed light on the nature of human mind. Since this article is partly my opinion on educational reform and partly an advertisement for mathematics, I include here one of my favorite proofs in mathematics. If you like this argument or this way of thinking, you should definitely consider taking one of the mathematics classes offered at Swarthmore, such as Introduction to Mathematical Thinking.

Prove that there are infinitely many prime numbers.

Proof: Assume there are only finitely many prime numbers. Let p1, p2, p3, … , pn be all the prime numbers, arranged from the smallest to the largest. Let p = p1×p2×p3×…×pn + 1 (take the product of all the prime numbers and add 1).  Then p is larger than any of the prime numbers, since it is at least larger than or equal to p1 + 1, p2 + 1, … , pn + 1. Therefore, p is not one of the prime numbers, since our list contains all of them. p is thus a composite number, and it must be divisible by one of the prime numbers. However, the remainder of p divided by any of the prime numbers is 1, meaning that p is not divisible by any of the prime numbers. We have a contradiction. The only possibility, then, is that our assumption at the beginning was wrong. Therefore, there are infinitely many prime numbers. Q.E.D.

 

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