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When mathematical reasoning gets murky

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In a recent piece for the Phoenix, “Why Mathematical Reasoning Should Be a Part of Civic Education,” Zhicheng Fan advocates expanded mathematical education as an antidote to the post-factual political climate into which the U.S. has unfortunately ventured.  The argument, in essence, rests on two assertions: (1) mathematics reinforces the notion of universal truths, leaving “alternative facts” deservedly exposed as the sham that they are, and (2) the precision and logical rigor incumbent on a mathematician trains one to think and act more rationally.  Thus, it appears that by encouraging a broader spectrum of society to study mathematics, we would expect to see a cultural shift toward a greater adherence to logic, truth, and attention to detail.

I absolutely agree that mathematical education should be embraced and enlarged. I particularly support Fan’s suggestions of bringing proofs, both the banal and the beautiful, into the curriculum at an earlier stage and incorporating more math history to contextualize and humanize the material and inspire students. However, I wish to add an additional layer of consideration.  I believe that transferring mathematical reasoning to the “real world” is not so simple as Fan’s quixotic zeal suggests. While the qualities Fan ascribes to mathematical reasoning are indeed staples of the discipline, there is yet another that I have found to be essential though often overlooked: using intuition to guide exploration.  I draw the same conclusion as Fan, that math education should be broadened, and in the same ways he suggests.  However, my premise is a slight variant: I claim that mathematics not only teaches us to write clearly, accurately, and honestly, but also to think creatively, metaphorically, and imprecisely—but in a productive way!

Allow me first to clarify some points for context: (1) Fan invited me to write this response piece, and my goal in doing so is to encourage a continued civil, fruitful discussion where we build on each other’s ideas; (2) Fan is currently my student in Modern Algebra, but I have already learned as much from him, both in that class and in this editorial discussion, as he perhaps has from me as his professor; (3) the other course I am currently teaching is Introduction to Mathematical Thinking, the class Fan recommends toward the end of his article, and I wholeheartedly reaffirm Fan’s suggestion to take it—particularly those of you who might have had a scarring mathematical experience in the past.

Let me begin with the challenges of applying mathematical reasoning to non-mathematical situations.  An example from Fan’s article conveniently illustrates the point.  The mathematical existence of the Weierstrass Function, a graph fluctuating in such an inconceivably dramatic manner that it is impossible to draw, is said to demonstrate that it is “conceptually possible for a car to move without speed at any moment.”  I highly doubt such an occurrence is possible, or that it even makes sense.  Thus, the confidence established by mathematical certainty leads one precipitously off a cliff of plausibility when adapted to a physical setting.  In other words, transferring mathematical fact to the real world leads to what Fan refers to as a “conceptual possibility,” which arguably is none other than our foe, the “alternative fact.”  If such a simple case as interpreting the derivative of a function as the speed of a car is problematic, imagine applying mathematical reasoning and certainty more broadly to the world of beguiling complexity and nuance in which we live. Some believe the 2008 financial crisis was exacerbated by false-confidence resulting from this discrepancy between the precision of mathematical models and the chaotic uncertainty of the real world.  For a fascinating discussion of the over-zealous reliance on mathematical models that our society has in recent years been consumed by, see the recent book “Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy” by Cathy O’Neil.

But hope is not lost. Mathematics is not a useless Platonic realm confined to our imaginations, nor simply a progenitor of pernicious mathematical models.  Mathematics provides the backbone for a great deal of science and technology and has had a tremendous, very real, and often extremely positive impact felt across the planet.  Of course, this includes a positive impact in the classroom as well.

Classroom mathematics can be deceptive. The clean, accurate, polished form with which mathematical facts, or theorems, are presented in textbooks belies their oftentimes controversial histories and messy prior incarnations.  Mathematics continually develops itself by envisioning greater levels of generality and more rigorous foundations and then summoning past knowledge, placing it within modern language and perspective.  In doing so, mathematics acts as a self-cleaning oven, deftly erasing its history of missteps and vagaries while offering a feeling of timelessness and universality.  When the name of a 17th century prodigy comes up in class, for instance, we seldom describe the theorem the way it was stated and viewed at the time of its publication. Instead, with statements that have evolved over centuries, we see only the most recent incarnation.  Thus mathematical ideas percolate through continual modernization, even if the words themselves grow stale.

The mathematical world seamlessly blends irrefutable reality with resplendent fantasy.  When I think about math, I think in terms of metaphor and cartoon simplifications of complex notions.  Details, logic, and precision only enter the thought process at a later stage at which I am ready to probe further the depths of a mathematical thought or to communicate it to others.  Thus, while I agree that math helps train the brain in rigor and truthful properties, what strikes me most about my years of doing mathematics is somewhat the opposite. Rather, math has trained me to embrace the uncertain, to heedlessly leap into vast hinterlands of vague thought while grasping for familiarity, to recognize that creativity arises from uncertainty, and to become cognizant of the difference between an idea and a written or spoken manifestation of it.

Since Fan ended his article with a beautiful theorem demonstrating the enduring elegance of mathematical truths, allow me to end mine with an equally accessible example of a conjecture. That is, a simple statement whose rigorous verification continues to defy the entire mathematical community.

The Collatz Conjecture: Given any positive integer n, divide by two if it is even or multiply by three and add one of it is odd; repeat this procedure and you will eventually end up with the number one.

Try it!  Here is an example for n = 10:

10 —> 5 —> 16 —> 8 —> 4 —> 2 —> 1

Computers have checked unimaginably many values of n yet no general pattern has emerged allowing us to firmly establish this result for the infinitude of possible values.  Mathematics is as much about the unknown as it is about the known, and that’s why I love it.

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