Another Sort of Mathematics:
Selected Proofs Necessary to Finally Acquire an Education in Mathematics
by j. jacob tawney
encounter books, 328 pages, $34.99
The most shining moment of my education as a physics major at UC Santa Barbara came in the final lecture of an upper-division course on electricity and magnetism. We had learned the facts early in the semester and could do the problem sets. Then the instructor, Nobel laureate Bob Schrieffer, seemed to take us backward. He spent a few lectures plumbing the depths of “the wave equation” that can be used to describe all phenomena with periodic motions, from water waves to sound waves. It also describes the pattern that appears when you walk past two fences and their visual elements of light and darkness superimpose—called a “moiré pattern.” (My dad, a physics professor at Berkeley, liked to point these out to me as a kid as we walked past fenced tennis courts toward the swimming pool at the Claremont Hotel, where he would do beautiful dives off the high dive.)
The next several lectures seemed totally unrelated. They were about magnetism, with field patterns and lines of force and whatnot. Professor Schrieffer hinted that some big reveal was coming, but I was losing patience. I couldn’t see the coherence of these disparate things. In the final lecture, without announcing where we were headed or giving anything away, he worked the blackboard furiously. Chalk dust floated in the air and found its way onto his clothes in random smears. About one minute before the end of this last class, he turned and grinned triumphantly. There on the blackboard was something that I “knew” and recognized from early in the course, to be accepted and memorized. But it now appeared as a revelation, something of incomparable beauty and wonder: Maxwell’s Equations. They show the fundamental unity of electrical and magnetic phenomena. We had arrived at them beginning from water waves. I was stunned.
The purely mental activity of doing mathematics can be a mode of access to something that transcends the human mind. That this should be the case would seem to indicate that there is a deep resonance between mind and cosmos—almost as if cosmos participates in mind.
J. Jacob Tawney, in his new book Another Sort of Mathematics, writes that there are “some things from mathematics that you should experience.” What an odd and arresting way to open a book about math. It is not the sort of exhortation we are used to hearing from the STEM-winders-and-grinders who push math for the sake of economic competitiveness, critical thinking skills, or other ends extrinsic to math itself. For Tawney, there is something beautiful and important to be experienced. He comes to us not as an “educator” in the dreary, institutional sense of that word, but as an evangelist.
His book arrives at a time when it has become necessary to rethink education altogether. David McGrogan at News from Uncibal writes about “the coming competence apocalypse.” The likely atrophy of our mental faculties from outsourcing mental tasks to AI will only accelerate the long-running collapse of competence that McGrogan details. Meanwhile, John Carter at Postcards from Barsoom likens the state of the university to that of the monasteries in late-medieval England, where, he writes, “religious vows were more theoretical than daily realities for many monks. Does anyone truly think that Harvard professors take Veritas at all seriously?”
Scholarly careerism, declining curricular standards, the replication crisis, a demented ideological monoculture, administrative bloat—a steady accumulation of chronic cultural entropy has built up inside the organizational tissue of the academy, rendering universities less effective, less trustworthy, less affordable, and less useful than ever before in history.
The function universities have long played is less one of educating than of credentialing. Carter gives us good reason to think the credentialing function of universities is about to collapse, due to AI. But he finds new possibilities, or rather old possibilities, emerging from the wreckage: liberal education in the original sense, as a leisure activity (“scholar” is from schole, leisure) for its own sake; for the love of truth. Unburdened of its current gatekeeping role in the political economy of managerialism and bullshit jobs, and no longer serving as a legitimation operation for unpopular political projects (producing “the Science” that must be “followed”), the successor to the modern university will be something subterranean rather than publicity-seeking, disconnected from power and money, useless, a place where people with the most searching minds gather to pursue truth for the love of it, as literal amateurs.
Which brings me back to Tawney’s book. In Another Sort of Mathematics, mathematical proofs take pride of place. They do so not as a response to that characteristically modern anxiety about grounding truth in a way that will make it a secure possession, a reliable foundation on which to build up an intellectual edifice. Rather, Tawney offers elegant and playful proofs because the activity of doing so answers to a need that is intimate and internal to the nature of mathematics. He speaks of “the effusive nature of truth itself. Upon discovery it demands to be shared, and the proof is the medium through which mathematical truth is communicated.” This seems to suggest that Truth (let’s go ahead and capitalize it) has an overflowing character, a will to commonality, and that human beings somehow participate in this as instruments of its overflow. When you learn something fundamental, it comes with astonishment and elicits a need to share it with others. I recall the grin on the face of Professor Schrieffer.
Truth is a common good, which is to say a good that is enhanced rather than diminished by being shared. As teachers know, the “giving away” of truth tends to inspire an even deeper grasp than simply knowing it on one’s own. This idea about the effusive nature of truth—the intimate connection between finding and communicating it—can be illuminated by considering that our term “mathematics” comes from the Greek expression ta mathemata, which means “what can be learned,” hence also what can be taught. Mathesis means at once “studying and learning” and the content of what is learned. Heidegger begins his essay “Modern Science, Metaphysics, and Mathematics” by considering how “the mathematical” was distinct from other categories for the Greeks. The things of physics are those that “originate and come forth from themselves.” Other things are made by the human hand. Both of these, physica and poioumena, may be “in use and at our disposal,” making them as well chremata, useful things. This is a subset of pragmata, which Heidegger parses as “the things insofar as we have to do with them at all, whether we work on them, use them, transform them, or only look at and examine them.”
What, then, is ta mathemata? And what does it have to do with numbers? Is it that “the learnable” is somehow numerical? Heidegger suggests the relation is better understood the other way around: Mathematics is what is paradigmatically learnable. One recalls the discussion in Plato’s dialogue Meno, in which Socrates is asked to defend his assertion that all learning is remembering. He calls over a slave boy and asks him to attempt a geometric exercise: to double the area of a square. The boy first responds confidently by doubling the length of each side. But this yields four times the area, as Socrates points out. Then he tries increasing the length of each side by half. Wrong again. Socrates leads him through a demonstration that is, in effect, the Pythagorean Theorem. Use the diagonal of the first square as one of the sides of the new square, and you will get double the area. Socrates takes himself to have vindicated his assertion that learning is gaining access to something already inside you.
Yet “math class is tough,” as the 1992 “Teen Talk Barbie” infamously said. Heidegger says Barbie had it right: “The most difficult learning is to come to know actually and to the very foundations what we already know.” A few pages later: “The mathematical is that evident aspect of things within which we are already moving and according to which we experience them as things at all, and as such things.”
Math, in other words, is an adjunct to, or mode of, ontology. Heidegger suggests the aspiration to ontology—to having an adequate grasp of what is—is bound up with something basic to human experience. Tawney says in Another Sort of Mathematics: “There is something unique in the human soul that can only be satisfied by wondering about mathematics.” He says this is because we are oriented to the good, the true, and the beautiful. That human beings should have an “orientation” toward anything at all is the controversial thing here. We call that “teleological thinking,” and it is a bit disreputable. But Tawney insists math is good prior to any application or usefulness or “relevance” it may have, such as the STEMmers emphasize. Might this goodness be due to the resonance that math reveals between mind and cosmos? In Greek, cosmos means “order,” and suggests also “beauty,” as in our word “cosmetic.” Doing math leads one to entertain the idea that the cosmos is itself good and beautiful. Our will to truth seems to derive from a deep attraction to this goodness and beauty. Math is erotic, in other words.
Tawney’s metaphysics are unapologetically Platonic. This is an excellent position from which to begin to recover the possibility of real learning, as the machinery of fake expertise and empty credentials crumbles around us.
