The Two Ways of Knowing the Universe

There is in the universe neither centre nor circumference." For the sixteenth-century Italian philosopher Giordano Bruno, this was as self-evident as the statement that mass and energy are equivalent. He was not alone in thinking it. The same idea had circulated among mystics in a famous maxim: the universe is a sphere whose centre is everywhere and circumference nowhere. Borges would later raise this sphere to an archetype — and reassign its rightful subject from the mystic to the French mathematician Pascal, calling it "Pascal's sphere."

What's worth pausing over is the following. What turned out to be a liberating thought to the mystic and philosopher was ensnaring to the mathematician.

Pascal did indeed speak of the universe as a sphere. And in his manuscripts — as Zacharie Tourneur discovered — he originally used the word effroyable. Frightful. Terrifying. He later crossed it out and replaced it with infinite. Same universe. One man's liberation. Another man's dread.

Two Reactions to Infinity

But is Pascal's effroyable — his apprehension before the infinite — not essentially the sensation of the anxious groom? That strange state somewhere between fear and excitement, which is really neither?

Newton, an analogue of Pascal, knew this sensation. But for Newton, the bride was not so much effroyable as litigious — "such an impertinently litigious lady that a man had as good be engaged in lawsuits as have to do with her." He said that about philosophy. Natural philosophy. What we would call science.

But litigious though she was, Newton never stood her up. He wrote the Principia — chivalrously, and in Latin, the language of the jurisconsultus. The work abounds in Latinisms: laws of motion, objects abiding by or conforming to those laws, phenomena articulated in propositional form, moving case by case. In other words: he litigated, rather than presumed.

The transition from Pascal to Newton holds more significance than it first appears. Pascal's fright is an abject expression of what a more conservative temperament would see as merely problematic. And "litigious," for Newton, is an abject expression of what would normally be called a proof. Two registers. One underlying drive. Spengler would name it.

Spengler Names It

"The Faustian soul, whose prime-symbol is pure and limitless space, and whose 'body' is the Western Culture that blossomed forth with the birth of the Romanesque style… Its direction is not towards a point, but into the infinite."

Oswald Spengler, The Decline of the West

Spengler gives us the clearest vocabulary for the split. The Faustian soul reaches, soars. Its architecture is the Gothic cathedral — pulled upward with primeval force into something it cannot quite grasp. Its direction is never toward a point; it is into the infinite.

Against this, Spengler places the Apollonian soul — classical, Greek — whose prime symbol is the bounded, self-contained, complete form. While the Gothic style soars, the Ionic swings. The interior of the cathedral pulls upward with primeval force; the temple is laid down in majestic rest.

Two architectures. Two cosmologies. And this fissure, Spengler argues, runs all the way to the foundation — to epistemology. We read a passage in which Plato calls the sparks of fire "tetrahedrons" and don't quite know how to frame it. We read Aristotle's definition of motion as "the actuality of a potentiality" and label it "not-physics" — or more specifically "not-kinematics" — and therefore "metaphysics," failing to see that to Aristotle, the qualitative definition is really motion, is really physics. What some cognitive scientists would now describe as "qualitative physics."

The classical world — the Apollonian psyche — remains largely opaque to us. And this is owing to one thing: we have not developed a proper understanding of what the Greeks meant by form.

What the Greek Actually Meant

Plato declares there exists a "world of forms," of which things as they appear on earth are but shadows. That is the Archimedean point. But to understand it, we need to restore to these concepts the materiality we have stripped from them.

Consider the following. You possess five senses, and you transition between them depending on the movement of your attention — you hear a sound here, are touched by a gust of wind there, catch in the wind a scent, think "this smells like such and such," then see the leaves of a nearby tree moving. In the span of about two seconds, you have processed information across all five senses simultaneously.

Substance, to the Greek, is this living receptive faculty. The thing that interacts through the senses, combines them, recombines them, develops ideas about the nature of the smell, senses the direction of the wind as it blows across the skin, and sees evidence of this in the motion of surrounding objects. What the classical thinker does, by default, is transfer this "sensibility" over to the objects of nature themselves.

The substance, as essence, is more physical to them than we have been led to believe. And crucially — the substance is independent of you. It works all by itself. Both the Faustian and the Apollonian share the intuition of the machina mundi — the world-machine — but the Apollonian machine was distinct.

Kinesis, to them, is the motion of this substance — the movement from sight to hearing, to idea, to touch, to basic combinations, to increasingly complex combinations. Matter — hyle — is not the objects one finds in the natural world. It's the sense-perceptions themselves. The same sense-perceptions about which Hume said "they proceed with infinite rapidity." Zeno's arrow never reaches its destination for the same reason Descartes continues to doubt.

Hume vs Aristotle - The Faustian vs Apollonian morphē

Here is what this means more precisely. Our sense-perceptions — hyle as such — are numerous, unceasing, and flow rapidly. Touch, sight, scent, feel, and their countless intersections. A blur, a gush of data. They are initially indistinguishable from one another, and only through the sense-faculty — the sense-organ — do they become differentiated and classifiable as touch, smell, and so on. In other words: the job of hyle is to supply the data, and the job of substance is to classify it.

By hyle, the Greek understood the post-classified object — whereby a sense-datum becomes, in the Aristotelian sense (that is, of moving from potentiality to actuality), a touch or a sight. Now, if substance performs the work of structuring and classifying that makes the world comprehensible to us — according to the morphe that we, or any organism, happen to be — then at some higher level, all those already-classified sense-data must undergo a higher order of classification by a substance of a higher order, whose product is a sense-perception that integrates information from all the sense-organs into one infinitely rich sensory object. This unified sensory object is Aristotle's koinē aisthēsis — the common sense.

Substance, in this scheme, vibrates at multiple levels. Each level is a system unto itself:

When you hear Plato say "idea" — when you hear "platonic" — think of this final product that substance ascends to. That which morphe is approximating. Hume and Plato occupy opposite ends of the same spectrum: Hume claims the pre-classified sense-perceptual world — "That world made of time, of the absolute uniform time of the Principia, an inexhaustible labyrinth, a chaos, a dream"— Borges 'A New Final Refutation Of Time'. Plato claims the perfect unified object, equally unclassified, but which also transcends classification.

Hylomorphism: The Relativity of Form and Matter

It takes some reflection to really see the sense-perceiving faculty as "its own thing." The morphe, or form, in this worldview, is also not the subject — not the ego — but rather the faculty as containing all possible facets, functions, and aptitudes a priori. Aristotle, in De Anima:

"Suppose that the eye were an animal — sight would have been its soul, for sight is the substance or essence of the eye which corresponds to the formula, the eye being merely the matter of seeing; when sight departs, it is no more an eye than the eye of a statue or a painted figure."

Aristotle, De Anima, II.1, 412b18–22

On reading this, one might think that "the eye as the matter of seeing" implies something tangible, physical. But that is not what Aristotle means. The contrast between sight and eye is the contrast between the intangible and its expression. "Matter" is the expression of that which is intangible — that which represents or points to it. Not the fact of its being something we can touch. For remember: touch and feel are themselves hyle, matter, for the Greek. They are expressions of our substance, our sense-perceiving faculty. Hence the second passage:

"We must no more ask whether the soul and body are one than ask whether the wax and the figure impressed upon it are one."

Aristotle, De Anima, II.1, 412b6–8

Wax is matter not insofar as it is tangible, but insofar as it is predicable of the figure. It is only because it points to the figure that it is matter — not because it is concrete. Rather, it is concrete because it is the representation of the figure. As Kant notes: "all we know about matter is simply representation."

Now — and this is where it gets interesting — this relation is relative, not fixed.

The eye is matter relative to the faculty of sight. But suppose the object your retina has picked up begins to move — changes position, changes speed. Your retina picks up those transformations in real time. Those changes, insofar as each is an event the retina registers, are matter — but now with the object as the substance of that matter. The eye, which had been matter before the perception instance, has now become substance relative to the thing perceived.

Matter can become substance, and substance matter, depending on the structure of reference. This is hylomorphism. And to the Greek, it is physics — all the same.

What we call kinematics — the modern, quantitative description of motion — is, to the classical mind, simply the situating of morphe in the external object. Moving the actantial centre of reference from us to the object, so that the object becomes the focus of hylemorphic activity. The classical thinker sees a bee and immediately sees the substantive in it — the dynamical principle — and interprets its transitions, movements, activities as the kinesis of that substance. He sees the subjective in the bee.

They Were Doing It All Along

Semiosis — the speech of morphe — is the selection or syntactic modification of that layer of things we call form. Russell put it this way:

"All sets of terms have, apart from psychological considerations, all orders of which they are capable. Omnipotence itself cannot give terms an order which they do not possess already."

Bertrand Russell, The Principles of Mathematics (1903)

Think of the terms "archetype" or "model" — and the synoptic function they perform, the way they can suddenly swallow an entire field of facts or ideas entirely unknown to us. This semiotic process is that magic: letting the carefully chosen sign do all the work. Leibniz's characteristica universalis spells out this fact — "he sought to versify the universe," as Borges would say — a semiotics that contains and can signify all orders of which things are capable, even if those objects are not yet known.

It should sound familiar. This is what we do in mathematics.

And this is precisely why Planck's "constant" was not a mathematical trick — even if it occurred to him as such. Did he extemporize the solution to the leading problem of modern science, or did he know the solution, just not yet in the mathematical form that came to express it? I am on the latter view. Planck — inasmuch as any physicist or mathematician — was performing a parallel computation: the matrix of standard computational mathematics on one side, and running next to it, in the background, the categories of Aristotle. He did not know he was. But the hylemorphic scheme of semiosis was very much in effect. And insofar as pure mathematics was operative in his calculations, he was also thinking hylemorphically.

The proof of this is found in a work published four decades before Planck's paper — and just four years before Planck's birth. The Laws of Thought (1854), by George Boole. Russell called him "the person who invented pure mathematics":

"In a work which he called the Laws of Thought… His book was in fact concerned with formal logic, and this is the same thing as mathematics."

Bertrand Russell, Mathematics & The Metaphysicians (1901)

Boole's project was to rationalise mathematics — to explain why it is unreasonably effective. He rested his algebra on the Archimedean point of Aristotelian prōtai archai: the three laws of non-contradiction, excluded middle, and identity. In translating those principles into mathematics, he simultaneously translated mathematics into logic. To every mathematical equation, there corresponds a logical or propositional expression — propositional in the classical sense. The classical propositional is an expression of being.

Poincaré described stepping onto a bus, and at that exact moment — "without anything in my former thoughts seeming to have prepared me for it" — the solution to the Fuchsian functions arriving fully formed" .

Heisenberg had his eureka moment at three in the morning on the island of Helgoland, after "an extremely clumsy series of calculations" — climbing out at dawn onto a rock jutting into the sea, breathing slowly, dizzy with the thought that he was looking through the surface of atomic phenomena which he called 'beautiful', a 'sensibility' (beauty) that should be as far away from mathematics as that horizon is from the rock he was standing on, right?, but that horizon in the distance was —at that moment —a rather Husserlian Welt-Horizont (world Horizon). That's the only way I could frankly justify the 'proximity' he felt.

These are not mystical experiences. They are what happens when two parallel computations — the formal mathematical machinery, and the hylemorphic scheme running underneath — suddenly converge.

The "unreasonable effectiveness of mathematics" has a precise reason. We simply haven't been looking in the right place.

Citations

1. Newton — “Philosophy is such an impertinently litigious lady…” Letter to Edmond Halley, 26 June 1686.

2. Spengler — The Decline of the West, Vol. I, Ch. VI §1 (“Apollinian, Faustian and Magian Soul”), trans. Charles Francis Atkinson (Knopf, 1928).

3. Hume — A Treatise of Human Nature (1739–40), Book I, Part IV, §VI: “Of Personal Identity.” [text corrected: “inconceivable rapidity”]

4. Borges — “A New Refutation of Time,” in Other Inquisitions 1937–1952, trans. Ruth L.C. Simms (University of Texas Press, 1964), p. 321.

5. Borges — “The Fearful Sphere of Pascal” 

6. Kant — Critique of Pure Reason, A370–371, trans. Norman Kemp Smith.

7. Aristotle — De Anima, II.1, 412b6–8 and 412b18–22.

8. Boole — George Boole, An Investigation of the Laws of Thought (London: Walton and Maberly, 1854), Ch. I, p. 3; Ch. III, p. 35.

9. Russell — Bertrand Russell, Mathematics & The Metaphysicians (London: George Allen & Unwin, 1919), Ch. I, p. 8.

10. Russell — Bertrand Russell, Principles Of Mathematics (London: George Allen & Unwin, 1919), Ch, 29. §231.

11. Poincaré — Henri Poincaré, Science and Method (1908), trans. Francis Maitland (London: Thomas Nelson, 1914), Pt. I, Ch. III: “Mathematical Discovery,” p. 53.

12. Heisenberg — Werner Heisenberg, Physics and Beyond: Encounters and Conversations, trans. Arnold J. Pomerans (New York: Harper & Row, 1971), Ch. 5, p. 61.

13. Planck — Max Planck, letter to R.W. Wood, 7 October 1931. Reproduced in Armin Hermann, Frühgeschichte der Quantentheorie (Mosbach: Physik Verlag, 1969), p. 31.