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Tawney, J. Jacob. Another Sort of Mathematics: Selected Proofs Necessary to Acquire a True Education in Mathematics. Encounter Books, 2025.
J. Jacob Tawney is my friend. I talk with him daily about all those things friends discuss, but even perhaps about other things, about the nature of truth, and whether the true and the good are coextensive. Thus I approach this review as one invested, as one with a commitment that the cynic might call bias.
Tawney’s new book, Another Sort of Mathematics: Selected Proofs Necessary to Finally Acquire an Education in Mathematics, proposes to educate the reader in mathematics. I wonder if we understand what that means.
What do you think an education in mathematics is? And why, if we are to take Tawney seriously, have you not acquired it even after twelve or thirteen years of primary schooling? Let us presume the optimal case: you were a straight A student, mastering all that your teachers taught about functions and calculus. Yet, Tawney says, you have not received an education in mathematics. Why is that?
Tawney proposes the following: mathematics concerns itself with some definite, intelligible objects. To be educated in mathematics therefore means to know some basic properties of these objects, to be able to prove these properties, and to know what key questions remain unproven. Tawney’s book, then, is an articulation of these key proofs that reveal core elements of these mathematical objects, as well as key questions that remain unanswered.
To be educated in mathematics, Tawney says, means to know proof. And not just in the abstract, in its logical structure. No, to know specific proofs, proofs about these things.
What do we think the study of mathematics does for the student? A survey of State Standards suggests that government thinks the study of mathematics is necessary for students to become productive members of society. This is math for the sake of utility: with this knowledge we have power over nature and can build amazing things.
But Tawney suggests that this is not the full story of mathematics, and not even the most important story. Tawney quotes me in saying, “There is something unique in the human heart that is only satisfied by the study of mathematics.” What is that? Is that even true? Perhaps your memory of math in middle school suggests this is emphatically not true, at least of you.
The objects of mathematics are things easily understood by the mind, at least in their basic elements. Point, line, surface; number, even, odd. Further, the knowledge of them is certain. The Pythagorean Theorem is always and everywhere true. The study of mathematics, therefore, has as its object that which is always and everywhere true. For the one studying it, for the student of mathematics, this means that they, living here and now, with inadequate air conditioning and indigestible pre-processed food, transcend all time and place and dwell in the absolute.
That thing unique in the human heart that is only satisfied by the study of mathematics is the visceral understanding of truth. Tawney’s book gives, as he says, not a curriculum but a skeleton of a curriculum that would accomplish this in the human heart. To study the proofs in Another Sort of Mathematics is, in a concrete and particular way, to viscerally experience truth.
I think it bears some attention to think about the character of that experience.
The book surveys the major activities of mathematics. Some proofs are intelligible to the amateur and are demonstrated. Some proofs are intelligible only to the expert and are referenced without demonstration. And some proofs have not yet been discovered and are explored as a present mystery.
The book presents proof in natural language, appealing to intuition rather than technical or idiomatic language.
The book presents mathematics as the work of men and women which transcends the power of any algorithm or computer to replicate.
Let us think about that third element some more. The unreflective, common assumption of today is that all human knowing is in principle functionally the same as the work of a computer. That computers have not yet achieved the general intelligence characteristic of humans is simply an accident of time, and will be overcome eventually, perhaps soon.
Against this, Tawney’s work shines like a beacon. In a way, you might say that the final chapter of Part III on Godel’s Incompleteness Theorem is the fulcrum on which this position turns. This proof is the articulation of the power of mind to know truth that transcends the power of algorithm.
But in a more fundamental way, you might say that the whole book is the proof of this thesis. To explore mathematical reality, to make conjectures and devise proofs, is core to being human. Because to be human is to know truth, and that knowing is viscerally experienced in the knowledge of mathematics. There is something unique in the human heart that is only satisfied by the study of mathematics.
Michael Austin graduated from Thomas Aquinas College in 2002 and pursued further studies in Philosophy from the University of Dallas, where he received his master’s degree. He currently serves Great Hearts Academies as Director of Curriculum for the Upper School. Prior to this, he served for many years at Veritas Preparatory Academy as teacher and administrator. He has a wide love for the whole of the Great Hearts Curriculum, from the study of Shakespeare’s Sonnets in Grade 6 to the proof of the Fundamental Theorem of Calculus in Grade 12.