After the trivium (i.e., logic, grammar, and rhetoric), in a classical liberal arts education, comes the quadrivium (i.e., arithmetic, geometry, astronomy, and music). Eventually, as I continue to develop and polish this website, I would like to have a developed quadrivium section.
— George Wick
The term “quadrivium” (four-fold path) comes from the great medieval thinker Boethius (476-525). He linked together the subjects of arithmetic, geometry, astronomy, and music due to their connection to mathematics.
Several philosophers have been great mathematicians. Think of René Descartes with analytical geometry or Gottfried Wilhelm Leibniz with calculus.
As modern mathematics became increasingly formalized, “analytic philosophy” played an important role in setting the foundations of contemporary symbolic logic and set theory. Math has always had a special place for philosophers, going back to Plato.
There’s a deep connection between mathematics and logic!
For centuries, the “gold standard” of axiomatic-deductive reasoning was Euclid’s Elements (around 300 B.C.). Using a handful of definitions, five axioms, and five postulates, he deduced various proofs in what is now called “Euclidean geometry.”
Much later, David Hilbert (1862-1943), author of such books as Principles of Mathematical Logic, cleaned up Euclid’s work with 20 axioms. He saw math as a “formalism” that concerned relations between terms. Euclidean geometry, then, is about the relations between its terms, not about space itself.
When you study higher or “pure” mathematics, it becomes a study of proof making. It becomes quite different than the algorithmic “plug ‘n chug” math that we mostly study in high school, with the possible exception of a geometry course.
In any case, we should approach math conceptually.
To become mathematically literate should be about understanding!
While learning by rote has its place (e.g., how can you get good at long-division without doing it a bunch of times?), math shouldn’t only be taught as algorithmic methods you memorize to solve idealized textbook problems.
Studying the “quadrivium” should get us thinking about math logically. We should understand the rules of algebra, for example, and be able to derive formulas from scratch. And we should develop skills to apply this math to describe quantitative phenomena.
Learn Euclidean Geometry
At the high school level, I highly recommend “Geometry: An Interactive Journey to Mastery” with James Tanton at The Great Courses. You would be hard pressed to find a more enthusiastic and knowledgeable teacher. Professor Tanton makes geometry fun and exciting. He conducts many geometry “experiments” that the student can follow and do at home.
I would call it an “honors” class. It’s a bit more advanced than your average high school geometry course. Note that a few of the reviewers complain that the workbook doesn’t match the lectures. I disagree. They are organically related to the lecture. They force the student to “interact” with the material in such a way to help produce mastery.
Learn Algebra and Pre-Calculus
Discovery, creativity, and “math experiments” are important. Conceptual understanding is important.
For example, rather than only revealing to a student that x^a/x^b =x^(a-b), a teacher or parent can have one explore simple problems to discover that algebraic truth.
When it comes to algebra — which, roughly speaking, is “advanced arithmetic” with “analytical geometry” — I highly recommend the high school books produced by mathematician I. M. Gelfand: The Method of Coordinates, Functions and Graphs, Algebra, and Trigonometry.
These books brilliantly take the student through a step-by-step journey of exploration and discovery. They might not be a complete substitute for an algebra or pre-calculus course, and I wish these books provided solutions to all the problems, yet they will deeply build a student’s mathematical intuition, knowledge, and ability.
We learn math by doing math. I can’t think of better books.
Basic calculus, it’s often (correctly) said, is difficult not so much for the calculus but for its algebra and trigonometry. So, I can basically guarantee this much: anyone who works through these books will have no trouble transitioning to calculus.
Higher Mathematics and Logic
A central focus of this website is on logic. Thus it’s fitting to give a suggestion or two. Be aware, however, that there’s all sorts of exotic logic systems out there, many of which I know too little about (hey, learning is a never ending process!).
How to Prove it: A Structured Approach by Daniel Velleman is commonly assigned to math majors. It’s readable enough for self-study. Propositional and predicate logic, proofs, functions, mathematical induction, and the basics of set theory are covered.
Another book to look at is Book of Proof by Richard Hammack.
Many get their first taste of mathematical logic in “discrete math.” There’s a lot of fair-to-good discrete textbooks out there. Since textbooks are so expensive, I recommend using bookfinder.com to find used copies of older editions.
Physics, Mathematics, and Logic
There’s a deep connection between physics, mathematics, and logic.
In many ways, the language of physics is mathematics. Just as mathematics is logical, so too is the world in which we reside in is logical. There’s a good pedagogical argument to be made that a good way to understand calculus is through physics.
Physics deals with motion and energy.
In part, introductory calculus is a study of rates of change. It’s about changing relations. Differential calculus takes the simple algebraic concept of slope and generalizes it as a member of the broader mathematical concept of derivative.
Physics is often about rates of change. For example, velocity is a rate of change.
Physics also needs philosophy and logic. There’s a lot of “first principles” that make physics intelligible. Anthony Rizzi, in Physics For Realists: Mechanics, explains that physics takes a lot for granted: that things actually exist, the principle of non-contradiction, the principle of causality, the distinction between substance and accident, et cetera.
That’s why physicists need logicians and philosophers, not only mathematicians, for help.
Learn Physics
The Institute for Advanced Physics provides resources to learn physics, on a philosophically rigorous foundation, from an Aristotelian-Thomist perspective. Whereas most physics textbooks flee philosophical considerations, they don’t.
Philosophy (Contemporary Aristotelianism) and Physics
For the relevancy of logic and philosophy, and most especially metaphysics, to physics and quantum mechanics, I suggest this short interview with metaphysician Robert Koons.
See his book Is St. Thomas’s Aristotelian Philosophy of Nature Obsolete?
Nigel Cundy, a physicist who specializes in quantum mechanics, is another scholar to look into, if you want a link between physics and philosophy.
Visit The Quantum Thomist.
Physicist Werner Heisenberg (1901 – 1976) compared quantum mechanics to an Aristotelian metaphysics of act-and-potency. Far from contemporary physics destroying the foundations of ancient and medieval philosophy, or its “liberal arts,” there’s been a recent revival in these schools of thought. They may help explain, and give logical and metaphysical support to, advanced physics.
Nobel laureate Eugene Wigner (1902 – 1995) famously wrote on the “Unreasonable Effectiveness of Mathematics in the Natural Sciences.” How is it possible that mathematics can so very accurately describe things?
This is explored in Matter & Mathematics by Andrew Younan. But also, what does it even mean for something to be a “law of nature”?
This is not a trivial question! “Mathematical ideas apply to the physical world so well because that’s where we learned them in the first place,” argues Fr. Younan.
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A Tangent on Learning Math within the Liberal Arts Tradition
Studying mathematics should be viewed within the larger context.
We can relate the Trivium and the Quadrivium.
It goes without saying that possessing a keen ability to read and write are necessary skills in order to have academic success. It is fundamental to a “liberal education.” Language is an essential property of the human being, since it reflects our rational ability to use signs such to signify or “point to” objects and ideas. It is additionally a communication system, where individuals can exchange information and knowledge between each other. It is obvious that the English language is a language with objects signified via signs, which, qua signs, can be expressed both verbally and non-verbally.
What is not so obvious, perhaps, is that mathematics is also a language. It is, after all, highly symbolic. While mathematics is its own language, we nevertheless learn it within the context of our primary or “native” language such as English. Since language is so tightly linked to our ability to reason and think, intellectual growth in all subjects, including mathematics, depends upon becoming sufficiently literate.
Multiple representation systems reinforce each other. It is presently well-known in the research of pedagogy that, if constructed and presented properly, multiple representations of a complicated idea help us learn better than otherwise. Comprehension increases, ceteris paribus, when a topic is studied both in written and spoken ways. It is also true, mutatis mutandis, that a student learns best when presenting his (developing) understanding using multiple representations, e.g., in speech and in writing.
It is an error to believe that mathematics is properly studied as a hermetically sealed subject from the traditional language arts. For it is obvious that efficaciously studying mathematics presupposes language skills. Math can be viewed as a kind of “language” of its own, with various symbols representing numbers, operators, relationships, grammar, etc.
This specialized language, nonetheless, is learnt with our ordinary (English) language. This additionally makes the mathematical language applicable outside of its seemingly enclosed system. Understanding the technical language of mathematics, despite its abstractive and generalized nature, allows one to apply it to the concrete world.
So-called “word problems” are an important example. Mastering mathematics requires navigating between its specialized language and ordinary language. Without able reading comprehension skills, a student will struggle advancing in mathematics. Perhaps worse, a student with those poor skills may only think of math as a language with arbitrary rules and mechanical algorithms that have no bearing outside of its domain.
Multiple Representations of Math:
Consider learning “slope” for the first time.
Verbal, written, graphical, & numerical representations should be used.
Verbally explain what “slope” is. Write an English paragraph explaining the concept without resorting to use of algebraic equations. (By pointing out, for example, that “slope” is a rate of change.) Symbolically represent slope with an algebraic equation. Visually or graphically show a representation. Numerically show concrete examples with numbers.