“Mathematics,” wrote the agnostic philosopher Bertrand Russell, “is, I believe, the chief source of belief in eternal and exact truth.” Of course, there are lots of other reasons to believe in eternal, exact truth, but Russell’s getting at something really interesting: math has consequences for how we think.

Here’s the story.

## Pythagoras introduces abstract numbers

For the ancient Greeks, math was one with metaphysics. It all started in the sixth century BC, with **Pythagoras**—the first of the Greeks to treat numbers as abstract entities existing in their own right. (Before him, numbering was all about the things being numbered, not the numbers themselves—as David Foster Wallace puts it, “the Babylonians and Egyptians were . . . interested in the five oranges rather than the 5.”) In fact, as Russell explains, Pythagorean numbers and math were more real than sensory reality:

“Geometry [derived from Pythagorean math] deals with exact circles, but no sensible [perceptible] object is

exactlycircular . . . . This suggests the view that all exact reasoning applies to ideal as opposed to sensible objects; it is natural to go further, and to argue that thought is nobler than sense, and the objects of thought more real than those of sense-perception. . . . numbers, if real at all, are eternal and not in time. Such eternal objects can be conceived as God’s thoughts.”

And here’s the important part—for pretty much the first time ever, all this reasoning started spilling over into the observed world. Russell explains:

“Geometry . . . starts with axioms which are (or are deemed to be) self-evident, and proceeds, by deductive reasoning, to arrive at theorems that are very far from self-evident. The axioms and theorems are held to be true of actual space, which is something given in experience.

It thus appeared to be possible to discover things about the actual world by first noticing what is self-evident and then using deduction.” (Emphasis added)

It’s largely thanks to Greek math that we have deductive philosophy, the rigor of logic, and the scientific method. Were it not for Pythagoras, Russell writes, “theologians would not have sought logical *proofs* of God and immortality.” Russell’s conclusion is simple: “I do not know of any other man who has been as influential as he was in the sphere of thought.”

## Plato reimagines abstraction as the theory of forms

The Pythagoreans exerted tremendous influence on **Plato**, whose most important innovation was the theory of forms. Plato held that what’s real in the world is not matter, not individuals, but classes, genres, species. Over two thousand years later, Schopenhauer put it like this: “Whoever hears me assert that the grey cat playing just now in the yard is the same one that did jumps and tricks there five hundred years ago will think what he likes of me, but it is a stranger form of madness to imagine that the present-day cat is fundamentally an entirely different one.”

So here’s the cool part: Plato’s forms are abstract in the same way as Pythagoras’ numbers. As Wallace puts it, “**The conceptual move from ‘five oranges’ and ‘five pennies’ to the quantity five and the integer 5 is precisely Plato’s move from ‘man’ and ‘men’ to Man**.” (Mathematicians who believe that numbers and mathematical relations exist on their own, outside of human conception, are even called Platonists.) Russell made the same connection: “what appears as Platonism is, when analysed, found to be in essence Pythagoreanism. [Plato’s] whole conception of an eternal world, revealed to the intellect but not to the senses, is derived from him.”

And Plato’s forms, of course, influenced pretty much the whole of Western thought. It’s partially thanks to Greek math, then, that we so readily categorize the world.

## Zeno and Aristotle argue about infinity

“There is a concept,” wrote Jorge Luis Borges, “that corrupts and upsets all others. I refer not to Evil, whose limited realm is that of ethics; I refer to the infinite.” The story gets even more interesting with **Zeno**, who, working in Pythagoras’ footsteps, was the first to tease out infinity’s corrupting, upsetting properties. He was the one who argued that fleet Achilles could never catch the tortoise—that, first, Achilles would have to cover half the remaining distance, then three-quarters, then seven-eighths, forever approaching but never passing his competitor. The thrust of the problem: Achilles must occupy every point previously occupied by the tortoise, but as soon as he does, the tortoise has moved on and Achilles has—forever—another vanishingly small point left to occupy.

**Aristotle**, a former star pupil of Plato’s, countered by proposing two senses of the infinite: actual and potential, corresponding to extension and subdivision. No real-life distance, he said, is actually infinite; every distance is potentially so. (An irony: Aristotle also countered Plato’s forms, arguing that if two men are joined by the form Man, the men and Man have something in common—and isn’t there, then, a *third* form comprising men and Man? And a fourth form comprising men, Man, and the third form that joins them? Aristotle rejected Zeno’s infinite regress as merely potential; he rejected Plato’s forms using an infinite regress that is itself potential.)

Satisfied? Me neither. But, though Aristotle’s answer to Zeno isn’t that compelling, it was enormously influential—by relegating infinity’s tricky parts to the merely potential, it basically let math keep functioning in the presence of the infinite.

## Calculus and set theory finish what the Greeks started

Not until Leibniz and Newton invented calculus would Western math develop the tools to start really answering Zeno. And when they did, it was Aristotle’s potential infinities that allowed for infinitesimals—quantities so small they can’t be added, yet somehow big enough to serve as divisors. (**Berkeley**, the famous empiricist and apologist, argued that calculus, no less than religion, comes down to faith—that “he who can digest a second or third [infinitesimal ratio] . . . need not, methinks, be squeamish about [anything] in divinity.”) Calculus’ notion of limits lets us look at a Zenoan infinite sequence—one-half, one-quarter, one-eighth, one-sixteenth—and *prove* that the segments add up not to infinity but to one; this answers the paradox, though not in a way that’s philosophically interesting. After all, by relying on infinitesimals, it relies on Aristotle’s old loophole-esque potential infinities.

More interesting is the work of Georg Cantor, who defined an infinite set as that which can be divided into subsets that are also infinite. (Cantor felt that his insights into the infinite had been directly communicated to him by God.) Because no member of the infinite set {10, 20, 30, 40 . . .} lacks a corresponding number in the infinite set {1, 2, 3, 4 . . .}, there are precisely as many multiples of ten as there are of one. The part, infinitely subdivided, is just as large as the whole; there are as many points on Zeno’s racetrack as there are in the whole universe. So check it out: after Cantor, we can conclude that Achilles, despite the longer distance ahead of him, doesn’t need to cover more points. Since both distances’ points are infinite—*actually* infinite, not just potentially so—the sets are 1:1 matches, and Achilles’ greater speed can win the day. For Russell, this was the first response worthy of being called a true solution.

Thanks to Pythagoras, we can think about numbers as abstract entities existing in their own right. Thanks to Plato, we can apply the same kind of abstraction to forms in general. Thanks to Zeno and Aristotle, we can complete the process of abstraction by thinking about infinity. And thanks to modern calculus (with its Aristotelian infinitesimals) and set theory (with its deeply Zenoan behavior), we can do more than just function in the presence of infinity—we can use it to solve problems.

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## 5 reasons you should study Greek math

If you’ve found this interesting, it’s worth your time to keep learning about Greek math. Here’s why:

**Everyone involved was enormously influential**. Pythagoras was, for Russell, the single most influential person in the sphere of thought. Plato and Aristotle are widely considered the fathers of Western philosophy. Zeno’s infinite regress has become something of a philosophical testing ground—it reappears not only in Aristotle but also in Agrippa, Plotinus, Aquinas, Leibniz, Mill, Bradley, Carroll, James, Cantor, and Russell himself.**Greek math contributed to Platonism**, and Platonism—through Clement, Origen, Augustine, and others—influenced early Christianity.**Greek math is the context for some of modernity’s most interesting thought**. Modern notions of infinity make more sense when you know Zeno’s and Aristotle’s arguments.**These texts represent a remarkable value**. You can get the**Greek Mathematical Works Collection**—which sets you up to study Pythagoras, Zeno, Greek geometry, and more—on Community Pricing for just $14; that’s 58% off. Then add the**Works of Plato**($30 | 83% off), and deepen your study with the**Select Works of Aristotle**($100 | 62% off). For such rich material, that’s a smart investment.**The Logos editions are the most useful—ever**. Math, with its refutations, its shared ideas, and its centuries-long lines of influence, is part of history’s Great Conversation. To study it, you need to be able to make connections. In the past, that would have required flipping through paper books and poring over indexes; not so with**Noet**, Logos’ philosophy and classics division. You’ll study primary texts alongside commentaries, follow lines of thought from author to author, and record your insights with notes and highlights that show up across all your devices.

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