Abram Kaplan

Directions for Mathematical Nature


“It’s not like a tree where the roots have to end somewhere / it’s more like a song on a policeman’s radio…” – Richard Siken, “Scheherazade”

What is the mathematization of nature? One thing, many things. Some phenomena seem more amenable than others to such treatment; above all, the stars. The astronomer Ptolemy thought so too. The ancients, our own ancient suspects, “saw that the sun, moon and other stars were carried from east to west along circles which were always parallel to each other,” that “after remaining invisible for some time,” stars “again rose afresh and set; and [that] the periods of these, and also the places of rising and setting, were, on the whole, fixed and the same.” They observed that the stars that never set revolved in circles around a common center, circles of increasing size as the stars’ distance from the center increased, “ever greater in proportion”; that the setting stars closer to the center vanished for less time, and those further from the center vanished for longer, “again in proportion.” They inferred from this, Ptolemy explains, that the heavens are shaped like, and rotate like, a sphere, and “in their subsequent investigation, they found that everything else accorded with it.”

Better not to think of each star on its own, at least, not yet. We suspect that the ancients only suspected that the same stars “again rose afresh and set” because they recognized them in their stable constellations; that the stars they observed revolving about a common center were also fixed in constellations more easily recognizable than any individual star (though some have a recognizable personality). How to recognize that the trajectories of different stars are parallel except by the constancy of their mutual configuration? In other words, would one star alone have done it, tracing its lonely circle, dipping for a season below the horizon to leave the sky empty? But then, if there’s only one star obviously moving in a circle, there’s nothing to explain – the pattern of the explanans would arise directly from the only possible explanandum.1 We use math to explain different things taken together; in nature its patterns describe something else. What is the mathematization of nature? Many things, one thing.

At first methodological, then historical, the center of the first book of Ptolemy’s Almagest (“The greatest”) next establishes the prerequisites of astronomy: a spherically moving spherical heaven, a very much smaller spherical earth fixed at its center, and a second sphere, fixed to the first one at its poles (therefore moved by it), carrying the sun and the wandering stars through the twelve signs of the zodiac – a path Ptolemy calls the ecliptic. The ancients’ early observations discovered the first objects of Ptolemaic astronomy, and the measured tilt of the sun, the length of the year, and the fact of stellar stasis form the backdrop for Ptolemy’s other investigations. The beginning of astronomy was the discovery of its first objects, which – being discovered as circular in path – furnished the explanans that was later applied to the wandering stars.

These postulates thus demarcate astronomy’s methodological starting points; as preliminary formulations of cosmic regularity, they restrict how this regularity can be pictured by constituting it. The heavens as a whole appears in terms of the circles by which the behavior of some of its parts is first noticed as regular. “The earth is as a point to the heavens” – sufficiently far away from even the planets that we cannot ascertain their distance, we are reduced to making measurements of angular displacement and therefore cannot establish any definite orbit.2 The parallel circles of the visible fixed stars, turning tightly around the pole, lend their shape to the orbits of the planets, or rather Ptolemy takes their shape as the pattern for celestial motion. Eccentric circles turning at constant velocity around an off-center point, epicycles carrying planets as they turn their tight circles around a center itself carried on a larger circle: circles built on circles, the back-and-forth motion of the planets is reduced to comprehensibility. But this comprehensibility uses the terms made available by the observations that founded the science: in Ptolemaic astronomy, to comprehend a phenomenon is to understand its motion in terms of compounded circles, in terms of the repetition of the only permissible explanans.

Note that Ptolemy isn’t mathematizing some manifest pre-scientific geometric presence; the fixed path of the sun and the sphere of the fixed stars aren’t there, not exactly. The concentric circles of the fixed stars arise out of memory – memory of the previous night and the previous season and the previous year, but also memory of circles seen in nature or sketched in sand. Seeing a star’s circle is a seeing-as, in which the star’s location today appears as a point on a path apprehended or imagined. So too for geometry: the geometer sees the line he has drawn as perfectly straight, the angle as right, the curve as a semi-circle, and infers accordingly, speaking still about this particular drawing (rather than about eternal – so-to-speak Platonic – figures) seen as a good version of the thing we call it. Likewise, the astronomer sees the orbit as present, in the way geometric figures are present: at any given time, the star is seen to sit on its circular path, a path that is the product of our way of seeing even if it bespeaks something real. He apprehends it as an object of understanding, precisely insofar as understanding it is the business of astronomy. Circularity is a feature of geometrical astronomy, and the act of mind that creates, sees, or gives access to the orbit as an object of understanding is similar to that which sees the diagram as an object of geometry and therefore brings it under the aegis of that science.3

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The astronomical revolution begun by Copernicus and consummated by Newton displaced this general understanding of an orbit no less than it did the specific commitments to circular orbits and a geocentric cosmos (not yet, indeed, a solar system). The roots of Newton’s consummation were present in Copernicus’ De revolutionibus (“On the revolutions”), which first proposed a heliocentric model of the cosmos (a “solar system”) in the early modern period, and probably came no less from Arabic digestions of and advances over Ptolemy than from Christian theological rethinkings of cosmic order. But neither he nor Kepler fully pursued the consequences in the direction common to the three men’s otherwise rather different thought.

When the “movements of the world machine” were set up, Copernicus suggests, the “best and most systematic artisan of all” would hardly have used different mechanisms to move the different planets in such similar paths. This was the great failure of the Ptolemaic model: not its predictive insufficiency but its offensive variety of epicycles and eccentric orbits, which looked “just like some one taking from various places hands, feet, a head, and other pieces, very well depicted, it may be, but not for the representation of a single person; since these fragments would not belong to one another at all, a monster rather than a man would be put together from them.” If his predecessors had only paid more attention to “method” and had “followed sound principles,” Copernicus insists, they would not have failed where he now succeeded.

Where Ptolemy’s astronomy combined astronomical observation with geometry, Copernicus – stressing the “machine” of the world and its artisanal origin – unites these disciplines with mechanics, along the way bypassing physics, the traditional study of natural causes. The mechanical-methodological principles of simplicity and self-sameness, Copernicus says, should govern our geometrical study of the sky. Christian theology had already stripped each wandering planet of its personal motive force; they were not animate. Copernicus, noting the teleological dependency of the cosmos “created for our sake,” subjected its make and its making as well to the deeply human standards of the artisan. Once again the explanation for order – here, the artisan – also provides the criterion for our adequate mapping of it. The Tychonic model that was the solar system’s main competitor in the seventeenth century, although absolutely equivalent in its predictions, lacked this self-sameness: in its compromise model, the earth still stood at the center, and the sun turned around it, and the upper planets turned around the sun.

Order instated by a mechanic – rather than order as self-generated and self-residing principle of natural being – was the order of things in political science no less than in astronomy. Machiavelli’s world, his human world, tended naturally to increased disorder; institutions, never self-sustaining simply through their own principles, needed continual acts of re-founding in order to stave off the inevitable chaos caused by the passage of time. A good prince, Machiavelli argued, using a medical metaphor, had to be a polis-doctor who could recognize the nascent political disease before its symptoms were clear to everyone and who, as consequence, could nip it in the bud. A better prince would build the polis from the ground up, using levees to forestall the flooding of the river that he imagined unpredictable fortune to be. By directing all the motions of its parts – its citizens – along known and controlled channels, by keeping tabs on the new grooves the river cut, by constant attention, the perfect prince could stave off all surprise and thereby ensure control. But such perfection was impossible, and Machiavelli suggested that all human polities would eventually decay even if subjected to frequent acts of renewal.

This is because even the best Machiavellian prince can’t induce flooding; political control is reactive control over what one happens to get rather than active, radically creative, control over what is to come. Control is analytic: through close and continual attention, one assimilates the new unknown to known and controllable schemas, and through quick reaction, one triggers these schemas to behave in predictable ways. At the highest level one maintains one’s system. Any continuing divergence of the new from the known after assimilation is disregarded in the hope that, once the conspirator is shunted onto a well-marked path, he will reproduce that element of his behavior that easily assimilates (because the environment cultivates it) and the remainder will decay. In other words, analytic assimilation is approximate, but in the short run, it’s good enough: its errors don’t add up. The seventeenth century’s great mathematical accomplishment, analytic calculus, was built on the same conviction. Divergence too small to notice and quickly corrected – even with a new, different divergence – was hardly different from no divergence at all, as long as one was continually attentive or had tools whose reiterated application was repeatedly corrective.

Kepler, Newton says, “guessed” that the orbits of the planets were ellipses and that their radii swept out equal areas in equal times. Consider a planet on its path with the sun at its center. A beam reaching from the sun to the planet and perpendicular to the orbit pushes the planet along; the force with which it pushes decreases with distance, so a planet further away is pushed more slowly than one nearby – indeed, so much more slowly that in equal time-intervals a planet always sweeps out same-sized triangles between its path and the sun. Kepler posed both the ellipse and the equal-areas law as hypotheses of his new astronomy based on “physical considerations,” embracing Copernicus’ new criteria while insisting that they bespoke a causal reality. Not merely a mathematical model, Kepler’s cosmos was an answer to the question posed by physics: why, by what cause, were the planets moved? The activity of the sun physically pushed the planet around the ellipse that was its path, as the planet was drawn in and pushed out by a magnetic force.

Despite the dynamic pretensions of the “sun-pushes-the-planet” model, Kepler didn’t derive the elliptical orbits of the planets from more general laws any more than Copernicus derived the circular orbits, and he wouldn’t have been able to had he desired to do so: he lacked the mathematical tools necessary to make sense of instantaneous change. Accordingly, however well the ellipse fit empirically, it (like Copernicus’ circle) was the posit of a mind trying to establish a particular order where order – but not its particular nature – was already evident. Working after Tycho Brahe refuted the nested-sphere, clockwork-mechanism model of the cosmos, Kepler proposed a neo-Pythagorean cosmos in which the increasing distances of the planets from the sun encoded the hidden forms of the Platonic solids. No less than Ptolemy’s, his cosmos was globally ordered with the empty static shapes – solids and orbits – whose presence, deduced painstakingly from phenomena, became the backbone upon which the system was constructed and the track along which phenomena appeared.

Newton’s system of the world was different. In the decades that separated him from Kepler, the mechanist’s world system was displaced by the mechanical-corpuscular world system. Rather than understanding cosmic order in terms of the mechanic who at one point – or, in Machiavelli’s case, continually – set it up and let it run, natural philosophers like Descartes traced the continuity of cosmic and local motion to spatially local interactions, to pushes and pulls, using the simple machines like the wedge and the screw to model the behavior of the tiny corpuscles out of which everything was made. While some attempted to explain planetary orbits in terms of a large vortex or whirlpool generated by the swirling sun, these accounts didn’t try to explain the particularity of individual orbits, preferring instead to depict how something like an orbit might arise from mechanical interactions. Content with plausibility, these philosophers mathematized the motion of bodies on earth, hypothesizing laws – like the pendulum law and the law of free-fall – that governed all trajectories of a certain type rather than the eternal celestial trajectories whose calculation was left to the astronomers.

This project, deeply geometric in its tools and its aesthetics, nevertheless developed the mathematical tools that were geometry’s demise. It did so by using geometry as a tool, by using shapes as patterns for encoding not-necessarily-geometric relations. Geometry was asked to behave non-geometrically, to disregard the shapiness of its shapes in favor of alternative properties. Newton’s treatment of rectilinear descent in the Principia Mathematica (“Mathematical Principles”) epitomizes this: deducing the law of free-fall from his general laws of motion, he takes the limit of an ellipse as its width is “perpetually diminished,” “diminished in infinitum” so that the elliptical orbit, “coinciding” with its own axis, becomes the line along with the orbiting body will “descend in a right line,” perpendicularly.4

Galileo framed his earlier statement of the rate of falling – the distance fallen is as the square of the time of fall – as a proportion between ratios of time and distance for two falls over different lengths of time. Galileo, lacking like Kepler the mathematical tools for addressing instantaneous change and perhaps not even interested in doing so, compared the paths of bodies already fallen; while the proof involved the continual and instantaneous addition of momenta of speed, these moments could only be measured, indeed conceived, in aggregate. The first person to mathematize instantaneous motion with variable acceleration – that is, affected by gravity but moving along a curve rather than in a straight line – was Christiaan Huygens, investigating pendula in the service of making a sea-worthy clock.

Huygens’ discovery depended, like Kepler’s, on a stable path. Knowing that the speed of a pendulum depended on its height of fall, he superimposed a parabola on top of a drawing of a pendulum to represent acquired speeds. But he promptly recognized this pendulum as the cycloid, a curve he had been studying in other contexts and which is “isochronic”: the time it takes a body to fall to its lowest point from anywhere along its length is identical. Huygens turned his attention to other isochronic or harmonic systems, like a massless spring, and drew the inference that instantaneous force of acceleration depended directly on instantaneous displacement. Its velocity at any point, then, was the aggregation of these instantaneous accelerations: “I shall consider…that curve as if made up of an infinite multitude of tangents. Further, for the time of accelerated descent along all these tangents I shall consider the sum of the times in which the individual tangents are traversed by uniform motion at the velocity acquired by fall from the start of descent to the point of tangency.” By representing a curve, a path, as made up of its instantaneous tangents, each a straight line, Huygens reduced motion along it to a series of separable yet combinable displacements.

Huygens’ mathematization of the pendulum, no less than Newton’s of nature, is notable for the mutual assimilation of the continuous and the discrete. As Newton knew, and wrote in his college notebook, Aristotle had denied that this assimilation was possible because no number of points taken together could create a line: either they would all coincide, taking up no space, or they would be separated by a line itself not made of points yet falling at its limits between them. Referring probably to Plato’s Meno, he explained that some people considered a point to be the limit of a line, crediting it with more reality than the line it limited. Thus when Newton, interested in doing precisely this – namely, assimilating the continuous to the instantaneous in order to make sense of a wider range of curves and, eventually, of new phenomena – began to develop the calculus, he did so on new philosophical grounds.

* * *

Newton argued that the geometer, measuring shapes already drawn, would have to become a mechanic, drawing shapes into being and measuring on the go: “Geometry postulates that a beginner has learned to describe straight lines and circles exactly before he approaches the threshold of geometry… To describe straight lines and to describe circles are problems, but not problems in geometry.” They are problems in mechanics, which for Newton becomes foundational for geometry insofar as it – rather than the imagination or the conjunction of the imagination and the mind – is productive of geometry’s objects. These objects are then considered to be real in an immanent, instantiated, even material way. Philosophically speaking, we could say that the ascription of perfection is displaced. No longer an unequivocal description of the noetic object, agreed upon by postulate – that is, the drawn circle seen as a perfect one – it is rather a practical aspiration: “if anyone could work with the greatest exactness, he would be the most perfect mechanic of all.” Elsewhere, Newton identified the key mechanics: “God, nature or any technician.”

The paths drawn by planets through space, no less than those drawn on the page by a skilled artisan, were susceptible to being geometrized and therefore to being measured. Newton said more exactly, distinguishing his conception of geometry from Aristotle’s: “mathematical quantities I here consider not as consisting of least possible parts, but as described by a continuous motion… These geneses take place in the realm of physical nature and are daily witnessed in the motion of bodies.” By identifying the activity of nature with that of the mechanic, Newton justified the application of his new mathematics to the course of nature. But he also undermined the notion, continuous from Ptolemy to Kepler, of stable orbits. Like terrestrial motion, planets’ paths – whatever they were; Newton could geometrize, and measure, many sorts – were drawn instantaneously by planets’ instantaneous motion. “The centripetal force does not always tend to that immovable center,” he explained, “and hence the planets neither move exactly in ellipses nor revolve twice in the same orbit. There are as many orbits of a planet as it has revolutions.”

This philosophical justification for the application of analysis to the study of nature was not entirely reflected in Newton’s own calculations. The proof of the first proposition in the Principia, namely the equal-areas law that Kepler merely postulated, requires for its execution the assumption of some – any – path, upon which the proof can be carried out. Here the analytic position that makes the establishment of mathematical laws possible is the “view from afterward,” the view that sees pending events as already completed and that therefore, like Machiavelli’s prince, tends to assimilate the new to the already known. Just as the order of the solar system is best seen from the position of the sun, entailing a change in perspective through an imaged spatial displacement, so the order of nature no less than of politics is best seen from the position of the effects or consequences, through an imagined temporal displacement. In nature, causes are seen as the effects they produce; in politics, means are seen in terms the ends they effect. Differences of time become intellectually surmountable by obeying the rules of how to envisage. Analysis takes place from an absolute standpoint.

As we’ve seen, in order to make such a standpoint plausible and thereby to justify the mathematical tools he uses in the Principia, Newton turns to the figure of the artisan, whose ongoing practice is carefully controlled locally but also globally. The eyes are fixed on the end, or rather one sees the present through the anticipated end. Meanwhile, the perfection of nature – which Newton says is like God without dominion or providence, a perfect artisan without a plan, local order without global design – promises the synchrony between one’s imagined perspective and the world it images. The eighteenth-century transformation of Newton’s mathematics and physics using Leibniz’s tools reproduces Newton’s physics from the inverse position, deducing future paths from present motions together with the trans-perspectival rules that, like mechanics, act locally. As Leibniz explained, “it is permitted to consider rest as an infinitely small motion (i.e. as equivalent to its contradictory in a sense) and coincidence as an infinitely small distance, and equality as the last of the inequalities, and so on.” Indeed, such “imaginary” entities as infinitely small lines were “useful, and even necessary, for expressing real magnitudes analytically.” The tools of mathematical understanding internalized the absolute assimilation of the future to the present, legitimizing them despite their apparent conceptual problems because only with them were the objects of inquiry available to the understanding. Only with them were the objects of inquiry, as such, available at all.

At its deepest Hobbes’ political philosophy worked this way too. Rejecting the apparent order of the polity as well as its teleological stability, Hobbes’ genetic account of the social contract reduced the activity of the commonwealth to the local motion of its members. Hobbes’ atomic citizens acted continually on the basis of fear – of death, of disorder – and, using the right institutions of governance, this local necessity could be harnessed to produce global stability. In the case of astronomy, the apparent order that lent its shape to the earliest tools for ordering becomes the accidental result of a deeper order that obtains everywhere. In the case of politics, the apparent order of a pre-Hobbesian polity is exposed as real disorder, at least to the degree that it does not derive from agreement contracted on the basis of fear. A firmer and more stable polity would have to be based on this deeper order that, like the new physical order, obtains everywhere. But whereas in astronomy this deeper order is not the correlate of any evident order, that is, any order evident to us, in politics only self-knowledge as a fearful being can provide the reasonable reader with the proof of Hobbes’ account, which (unlike the new astronomy) cannot be predictive until it is productive. Here lies the fundamental divergence between the bases of natural and political science in the early modern period.

In both cases, a different conception of mathematization, one that dovetails with atomism, underlies this critique as it was finally, successfully articulated even in non-mathematical domains. Algebra, the study of the structure of equations with an eye to their more systematic resolution, subjected known order to new orders in order to widen the domain of the orderable. This process was already evident in algebra’s Arabic setting, where, for instance, al-Tusi first identified the general distinction between soluble and insoluble equations as interesting and began to investigate it; it only accelerated in Europe. Things that were originally noticed because of their seemingly unique order (for instance, equations that were soluble or orbits that were regular) became instances of broader orders in which their uniqueness was dissolved, or in which this uniqueness was articulated as a special case of some more broadly applicable order, and expressed in the syntax of that new order. The first instigations to investigation are the epiphenomena of the real order. The stars’ regularity is less striking in a world where regularity – made possible by the discovery and use of the regularity of nature in objects ranging from clocks to cars to computers – is the order of our tools, our schedules, our bodies, our activities. Circular orbits, though at once point useful for the investigation of nature’s regularity, are revealed as a deeply misleading manifestation of that regularity. To paraphrase Wittgenstein, investigation kicks away the ladder after it climbs it, concealing the original motives for investigation by making the first objects invisible, or rather subliming those motives away from the contingent circumstances of their first expression.

On varying grounds, from the valuation of intellectual eros to the conception of reason as a spatiotemporally local manifestation of nous, antique science credited our intuitions as something other than fictions. The seventeenth century rejected this principle, arguing that any order we can see should make us wonder about the deeper patterns schemes that govern it. As Bacon observed, the mind runs too soon to final causes. Our given perspective is thus devalued to the extent that its satisfaction is a cause for suspicion. This is the principle of investigation that underlies critique in all its settings. But it has a strange, unexpected, consequence.

By displacing the site of order from the orbit itself to the local motion of its generating body, Newton raised a possibility that had not previously been considered by his early modern colleagues: perhaps the solar system no more stable than the city. Newton himself, entertaining this possibility, concluded that comets’ exhalations helped restore cosmic order. His eighteenth-century successors had different concerns: how to more correctly assimilate the motion of the moon and that of comets to the Newtonian mathematical model, in order to ensure that it, at least, was absolutely true? No longer able to rely on the seeming regularity of the cosmos as evidence of its underlying regularity, adherents of the new science felt great anxiety in the face of any deviation from the postulated or hypothesized substitute. Critique assumes deeper stability at the expense of affirming the present order or finding security in it.

1 The “explanans” is the explanation or, literally, the “explain thing,” while the “explanandum” is the thing having to be explained. Thus, in English, the same simple object, without any internal differentiation, would have to be used to explain itself.

2 Close one eye and look at the frame of your computer monitor; then open it and close the other. See the monitor move? By observing the monitor from two different points, you can triangulate the distance, comparing the different frames of the same background that your two scopes capture the monitor against. This is called parallax, and it’s essential for measuring the distance of objects one cannot physically approach. No tools available to Ptolemy would have enabled him to measure planetary or stellar parallax.

3 “Creates, sees, or gives access to”: I’m catholic on this one, not because the question’s not important, but because I’m not sure which one is appropriate. A perhaps more Kantian formulation would be: he stipulates it as an object for understanding, precisely because understanding it is the business of astronomy. But whether the orbit is stipulated or projected as a representation or apprehended by the mind observing is itself at issue here. If apprehended, is it apprehended directly? Certainly not correctly; but not entirely incorrectly either. (For instance, the apprehended circular paths of the stars point to the actually circular rotation of the earth.)

4 Some centuries earlier, Nicholas of Cusa had also explained that the circumference of an infinitely large circle was necessarily a straight line, but for a rather different purpose.

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