White Rites: Meditations on Mathematics and Materiality

Ὅ τι ἄν σοι συμβαίνῃ, τοῦτό σοι ἐξ αἰῶνος προκατεσκευάζετο.[1] That was how the philosopher-emperor Marcus Aurelius put it nearly two thousand years ago: “Whatever may befall thee, it was ordained for thee from everlasting.” He was elegantly and eloquently expressing a core tenet of Stoicism, the ancient school of philosophy that taught dogged devotion to duty, tireless pursuit of virtue, and unshaken courage in the face of illness, oppression and disaster.

Bright bubbles on black water

But how and why was courage any more admirable than cowardice? Why was virtue worthier than vice? Or devotion to duty better than dereliction? Stoicism is a noble edifice that, in truth, collapses at a pin-drop. Or so some would claim. This is because that core tenet of the philosophy was determinism, the doctrine that the universe is bound by iron and immutable chains of cause and effect, operating from eternity to eternity. If determinism is true, we are bright bubbles on the black river of fate, born willy-nilly, bursting willy-nilly,[2] swirled this way or that between birth and bursting by currents over which we have no control and which hasten us or hamper us at their whim, not ours. Shakespeare said: “All the world’s a stage, and all the men and women merely players.”[3] The Stoics said: “All the world’s a machine, and all the men and women merely cogs therein.” As Aurelius went on: καὶ ἡ ἐπιπλοκὴ τῶν αἰτίων συνέκλωθε τήν τε σὴν ὑπόστασιν ἐξ ἀιδίου καὶ τὴν τούτου σύμβασιν — “and the coherence of causes wove both thy substance from everlasting and all that happens thereto.”[4]

Slime-mold and Stoic: Physarum polycephalum on left Marcus Aurelius on right (images from Wikipedia)

But the elegance and eloquence of Aurelius can’t silence a simple and possibly lethal question. If Stoicism is true, where does that leave the Stoics? Surely they were sawing, not sowing. They thought they were sowing true doctrine into the minds of men; they were in fact sawing off the branch they were sitting on. It was the branch of epistemology, of truth and reason, and determinism is, on some readings, fatal to those weighty things. In a deterministic universe, why should brains and logic have any higher status than stomachs and digestion? Why should the Meditations of Marcus Aurelius have any greater claim to truth and insight than the song of a blackbird in a bush? If everything we humans think, say and do is indeed fixed ἐξ αἰῶνος — “from everlasting” — then we might seem to have the same status as a sunset or a slime-mold. We’re phenomena, never philosophoi.[5] After all, cogs can’t cogitate. And Stoicism tells us that we are cogs in the world-machine. If so, it’s ludicrous to adjure cogs to be calm, courageous and good. Cogs have no control. Cogs do whatever they are compelled to do by external forces.

The whirl of the world

And so crashes into ruin the noble edifice of Stoicism, self-sapped, self-exploded, self-destroyed. Or so some would claim. But does determinism indeed destroy epistemology and the search for truth and insight? That’s too big a question to tackle here and in such a sordid setting. Nevertheless, I want to look at one aspect of it and to argue that, in one way, determinism is vital for epistemology and is, indeed, the only known guarantor of fixed and reliable truth. I also want to emphasize something strange and sublime about human beings. Or about some human beings, at least. I started this essay with a memorable line from the great Marcus Aurelius. I’ll continue it with a memorable line from the great Arthur Conan Doyle (1859–1930): “He shook his two fists in the air — the poor impotent atom with his pin-point of brain caught in the whirl of the infinite.”

Universe — Pin-point — Brain (images of Fireworks Galaxy et al from Wikipedia

That’s from a story called “The Third Generation” (1894), one of Doyle’s “Tales of Medical Life.” It describes the mental agony of a patient diagnosed with hereditary syphilis. The grandfather had sinned; the grandson would now suffer. Doyle himself was steeped in Stoicism and had undoubtedly meditated on The Meditations, thinking deeply about determinism and free will, about the mind and its relation to matter and the body. And he compressed his ideas into a highly memorable metaphor: the human brain is indeed a pin-point by comparison with the Universe. Or far, far less than a pin-point. By comparison with the Earth alone, let alone the Solar System or the Universe, a human brain is considerably smaller than a pin-point is by comparison with the human body.[6] And yet that “pin-point of brain” is, in a sense, far mightier than an entire universe of inanimate, unconscious matter.[7] Our pin-points of brain can contemplate and conquer infinity. Which is a strange and sublime thing. How can mere matter do that?

Primal Potentate

I’m talking about mathematics, a discipline that clearly proves human beings to be philosophoi, not mere phenomena.[8] And it’s not a coincidence that all those abstract polysyllables — mathematics, philosophoi, phenomena — come to us from ancient Greek, the language in which the Roman emperor Marcus Aurelius composed his Meditations. As the oft-remarked dichotomy goes: The Greeks were thinkers; the Romans were doers. The Hispanic Hellenophile Marcus Aurelius was both. And just as Doyle must have read Aurelius, a contemplator of infinity, Aurelius must have read a conqueror of infinity. The Greek mathematician Euclid conquered infinity in his Elements, a textbook of mathematics composed in the third century before Christ and still studied in the twenty-first century after Christ. Here is that conquest of infinity set out in modern English, as Euclid demonstrates[9] the infinitude of prime numbers like 3, 17 and 101, which are evenly divisible only by themselves and 1:

Euclid’s proof that there are an infinite number of primes

(by reductio ad absurdum)

1. Assume there are a finite number n of primes, listed as [p₁, …, p_n].
2. Consider the product of all the primes in the list, plus one: N = (p₁ × … × p_n) + 1.
3. By construction, N is not divisible by any of the p_i.
4. Hence it is either prime itself (but not in the list of all primes), or is divisible by another prime not in the list of all primes, contradicting the assumption.
5. q.e.d.

For example:

1. 2 + 1 = 3, is prime
2. 2 × 3 + 1 = 7, is prime
3. 2 × 3 × 5 + 1 = 31, is prime
4. 2 × 3 × 5 × 7 + 1 = 211, is prime
5. 2 × 3 × 5 × 7 × 11 + 1 = 2311, is prime
6. 2 × 3 × 5 × 7 × 11 × 13 + 1 = 30031 = 59 × 509 (“Euclid’s proof that there are an infinite number of primes,” Susan Stepney, Professor Emerita, Computer Science, University of York, UK)

Euclid conquers infinity in Book IX, Proposition 20 of the Elements (see text at Wikipedia)

That’s simple but sublime. And supremely significant. I think that the proof above was a rite of passage for the human race — an intellectual rite of passage that dwarfs physical achievements like landing on the Moon or splitting the atom. Euclid, with his pin-point of material brain, proved the existence of an infinite number of immaterial entities known as primes. And we, with our pin-points of material brain, can understand and accept his reasoning. Indeed, if we understand his reasoning, we are compelled to accept it. That is the marvel of mathematics. Or one marvel among many. Mathematics is a deterministic system for generating truth. It’s the closest human beings have yet come to infallible knowledge, which is precisely why it doesn’t claim infallibility. That’s the paradox of infallibility: those who overtly claim it thereby prove that they don’t possess it. As Bertrand Russell said:

The most savage controversies are those about matters as to which there is no good evidence either way. Persecution is used in theology, not in arithmetic, because in arithmetic there is knowledge, but in theology there is only opinion. (“On avoiding foolish opinions,” Bertrand Russell)

Yes, there is persecution in theology — and in politics. And there are claims of infallibility in both. The Polish philosopher Leszek Kołakowski wrote in his magisterial Main Currents of Marxism (1978) of how Stalin “laid down the rules of Soviet historiography once and for all: Lenin had always been right, the Bolshevik party was and had always been infallible.” Meanwhile, Stalin’s rival Trotsky “imagined that he was conducting scientific observations with the aid of an infallible dialectical method.” If all art aspires to the condition of music,[10] then all epistemology aspires to the status of mathematics. But never achieves it, because mathematics enjoys the twin advantages of ultimate abstraction and insurmountable incomprehensibility. It’s incomprehensible to non-mathematicians, at least. That’s why mathematicians didn’t suffer under Stalin in the way that many scientists did. As Kołakowski also wrote: “Mathematical studies were scarcely ever ‘supervised’ ideologically in the Soviet Union, as even the omniscient high priests of Marxism did not pretend to understand them; consequently, standards were upheld and Russian mathematical science was saved from temporary destruction.”

Molded by matter

Like Popes and Ayatollahs, Marxists claim infallibility precisely because they don’t have it; mathematicians don’t claim it precisely because they do. Or so I would say. I’m not infallible, of course. Nor am I a mathematician or a philosopher. But I am two things that seem to be of great importance in mathematics and philosophy. That is, I’m White and male. Those are statements about my genetics, that is, statements about my materiality. But mathematics and philosophy are about mind, not matter. How can genetics be important in cognition? It can’t, according to orthodox leftists, who denounce as abhorrently racist and abominably sexist any claim that White men are especially or eminently suited to any field of intellectual endeavor.

Yet it’s obvious in a broader sense that genetics is decisive — indeed, deterministic — in mental matters. Humans can be philosophoi and not mere phenomena because they aren’t sunsets or slime-molds. No, they’re humans, which is a statement about genetics and material bodies. Humans and slime-molds are both products of DNA and the blind forces of evolution, but there has never been a Euclid or an Aurelius among the slime-molds, which are barred for ever from mathematics and philosophy by the mere materiality of their junk-jammed genetics.

Damning Derbyshire

That form of genetic determinism can’t be denied by leftists, who often protest too much in their denial that race and sex have been decisive factors in intellectual fields. This is the Black mathematician Jonathan Farley waxing indignant in the Guardian about the bigotry of a White mathematician:

John Derbyshire, a columnist for the National Review, wrote an essay last week implying that black people were intellectually inferior to white people: “Only one out of six blacks is smarter than the average white.” Derbyshire pulled these figures from a region near his large intestine. One of Derbyshire’s claims, however, is true: that there are no black winners of the Fields medal, the “Nobel prize of mathematics”. According to Derbyshire, this is “civilisationally consequential”. Derbyshire implies that the absence of a black winner means that black people are incapable of genius. In reality, black mathematicians face career-retarding racism that white Fields medallists never encounter. Three stories will suffice to make this point. … The second story involves one of the few black mathematicians whom white mathematicians acknowledge as great — or, I should say, “black American mathematicians”, since obviously Euclid, Eratosthenes and other African mathematicians outshone Europe’s brightest stars for millennia. (“Black mathematicians: the kind of problems they wish didn’t need solving”, The Guardian, Thursday 12nd April, 2012)

Like Euclid, Cleopatra was Greek and White, not a Black “African” (image from Wikipedia)

Guardiancaption: Euclid and other African mathematicians outshone Europe’s brightest stars for millennia.’

Farley was being dishonest in that last line, pretending that geography equates to genetics. Yes, Euclid and Erastothenes were “African mathematicians” in the sense that they lived and worked on one corner of the continent of Africa. But they were not Black Africans. They were White — and worse still, for a leftist like Farley, they were White colonizers, part of the Greek diaspora in the conquered land of Egypt. They cannot accurately or honestly be described as “African mathematicians,” because that suggests that they were something they weren’t, namely, indigenous to Africa and Black.

Euclid’s city of Alexandria, part of a Greek colony on one corner of Africa (image from Wikipedia)

And although Blacks can certainly be good mathematicians, Blacks have never been essential or important in mathematics or any other intellectual field. As I said at the Occidental Observer in 2022:

Here’s an astonishing fact: the White mathematician Claude Shannon (1916—2001) contributed more to STEM (Science, Technology, Engineering and Mathematics) than all Blacks who have ever lived. But then so did the Indian mathematician Srinivasa Ramanujan (1887—1920). And the Jewish mathematician Emmy Noether (1882—1935), which is even more astonishing. Jews have always been a tiny minority of the world’s population and men have always dominated mathematics, yet one Jewish woman in a short lifetime outperformed the teeming masses of Africa and the Black-African Diaspora over millennia. Blacks have never mattered in math or any other cognitively demanding field. But Jews have mattered hugely, in both good and bad ways. (“Rollock’s Bollocks: Interrogating Anti-Racism and Contemplating the Cargo-Cult of Critique,” The Occidental Observer, 13th May 2022)

But it’s in fields invented by goyim that Jews have mattered for good or ill. The words “mathematics” and “philosophy” are ancient Greek, not ancient Hebrew. And although there is some evidence that Black brains were pondering prime numbers 70,000 years ago,[11] it took the White brains of men like Euclid to prove that astonishing and awesome fact about prime numbers — that they never end, that the digits of an infinite number of them could not be written down if all the oceans were ink and all the sky papyrus.[12] I called Euclid’s conquest of infinity a rite of passage for the entire human race. If so, then it was a White rite in some significant way. But I’m not seeking to deify Whites when I say that, only to recognize an important fact that applies to intellectual history just as much as to active history: that Whites have been outliers and achievers there in ways that other races haven’t. Whites are the all-star all-rounders of the human race, capable of great achievements mentally and physically, musically and mathematically, abstractly and athletically.

And so, while mathematics might have been created in Mesopotamia, it burst its chrysalis in ancient Greece, where White men, with their “pin-points of brain,” proved things beyond all bounds of materiality. Men like Euclid weren’t “impotent atoms” “caught in the whirl of the infinite.” No, they were conquerors of the infinite. You’ve seen one marvellous proof by Euclid, one rite of passage for the human race. Now here’s another of his White rites — a stronger and stranger and subtler proof that should captivate and compel everyone capable of understanding it:

An irrational number is a real number that is not rational, that is, cannot be expressed as a fraction (or ratio ) of the form p / q , where p and q are integers.

[Proof] that the square root of 2 is irrational

Pythagorean proof, as given by Euclid in his Elements

proof by contradiction:

1. Assume that √2 is rational, that is, there exists integers p and q such that √2 = p / q ; take the irreducible form of this fraction, so that p and q have no factors in common
2. square both sides, to give 2 = p 2 / q 2
3. rearrange, to give 2 q 2 = p 2
4. hence p 2 is even
5. hence p is even (trivial proof left as an exercise for the reader); write p = 2 m
6. substitute for p in (3), to give 2 q 2 = (2 m ) 2 = 4 m 2
7. divide through by 2, to give q 2 = 2 m 2
8. hence q 2 is even
9. hence q is even

(1) assumes that p and q have no factors in common; (5) and (9) show they they both have 2 as a factor. This is a contradiction. Hence the assumption (1) is false, and √2 is not rational. (“Irrational number,” Susan Stepney, Professor Emerita, Computer Science, University of York, UK)

One consequence of that proof[13] is that the digits of √2 never end and never fall into any repeating or regular pattern. In short, they’re entirely random[14] (while also being entirely deterministic). And one consequence of that randomness is that, represented in suitable format, the digits of √2 somewhere encode the entirety of this essay. And the entirety of the website on which it’s hosted. And the entirety of the internet and of all books in all languages in all libraries that ever existed. But √2 doesn’t just encode all that, it encodes it infinitely often. √2 is Borges’ Biblioteca de Babel, Borges’ infinite “Library of Babel,” with a single, simple, two-symbol label: √2.

If you aren’t awed and astonished by that, I’ve failed in what I’ve written here. With their pin-points of brain, humans haven’t merely contemplated and begun to comprehend the Universe: they’ve transcended the Universe and burst the bonds and the bounds of mere materiality. That’s certainly food for thought and maybe also food for theism. But that’s where, for now, I’ll conclude this White write on White rites, leaving the last word to Edna St. Vincent Millay (1892-1950):

Euclid alone has looked on Beauty bare.

Let all who prate of Beauty hold their peace,

And lay them prone upon the earth and cease

To ponder on themselves, the while they stare

At nothing, intricately drawn nowhere

In shapes of shifting lineage; let geese

Gabble and hiss, but heroes seek release

From dusty bondage into luminous air.

O blinding hour, O holy, terrible day,

When first the shaft into his vision shone

Of light anatomized! Euclid alone

Has looked on Beauty bare. Fortunate they

Who, though once only and then but far away,

Have heard her massive sandal set on stone. — “Euclid alone has looked on Beauty bare” (1923)

[1] The Meditations of Marcus Aurelius, Book X, 5. See translations at Gutenberg and Internet Classics Archive.

[2] “What good is it to the bubble while it holds together, or what harm when it is burst?” Meditations, Book 8, 20.

[3] As You Like It, Act II, scene 7, line 139.

[4] The Meditations of Marcus Aurelius, Book X, 5. See translations at Gutenberg and Internet Classics Archive.

[5]  Philosophoi is the plural of Greek philosophos, “lover of wisdom.”

[6] The Meditations makes a related point: “the whole earth too is a point [by comparison with the Universe].” Book VIII, 21.

[7] But what matters, of course, is not relative size but absolute complexity. The human brain is tiny by comparison with the Universe, but is the most complex object yet known there.

[8] Theories like that of the Jewish physicist Max Tegmark, stating that matter is mathematics, don’t (and aren’t intended to) solve the problem of the relationship between math and matter, or mind and matter, because “mathematics” is used in two different senses: the abstract system used by conscious human minds and the apparently unconscious and extra-rational entities that inspire and underpin that system.

[9]  Or, more precisely, sets out the demonstration of an earlier mathematician. Euclid was a compiler of math, not a creator.

[10] Walter Pater said this in The Renaissance: Studies in Art and Poetry (1877): “All art constantly aspires towards the condition of music. For while in all other works of art it is possible to distinguish the matter from the form, and the understanding can always make this distinction, yet it is the constant effort of art to obliterate it.” See Gutenberg text.

[11] See discussion of the “Ishango Bone,” an ancient African artefact with proto-mathematical markings that may symbolize prime numbers.

[12] “If all the trees on earth were pens and the ocean were ink, refilled by seven other oceans, the Words of Allah would not be exhausted.” — Qur’an, Surah Luqman.

[13] The proof is attributed to Euclid but possibly or even probably not by him. See “Square root of 2” at Infogalactic.

[14] Mathematicians assume that √2 is “normal” in all bases, that is, it contains all possible sequences of digits with the same frequency and probability.