Friday, July 17, 2009

More Keplerian ruminations - math and nature

Kepler remarks in his Astronomia Nova that he had considered applying the ellipse too simple a solution for earlier astronomers to have overlooked. It is an interesting problem. Did it really never occur to anyone before Kepler to describe the orbit of planets as ellipses? If that is true, then why? None of the major theories of planetary motion devised before 1600 I've reviewed uses ellipses instead of circles to describe orbits; though that doesn't mean none exists. It seems as though the assumption from Aristotle and possibly before is that the celestial bodies must move in circular orbits (whether perfectly or also in epicycles). Why? The math of the conic sections was known since ancient times and developed throughout medieval times in the Arab-speaking world. Why not apply it to the celestial motions?

I don't know the answer to this question, but I think there are three common and wrong answers. The first is that it is the fault of tradition (Aristotle and/or religion). The second is that it contracted immediate experience, to which all ancient and medieval thought was wed. And underlying both these assumptions is a third, namely, that people weren't free to imagine outside the confines of tradition or immediate experience until Copernicus, Galileo, Kepler, et al came along to liberate them, thanks be to their courage to stand up against arbitrary authority.

The mere force of Aristotle, "the tradition", Christianity, Islam, or anything else is too ridiculous to contemplate as an answer to this question. Every single aspect of Aristotlean physics and metaphysics was attacked viciously throughout the middle ages. It's useful to note that the most thoroughgoing criticism took place in the Islamic world, which was in many respects far more tolerant than the Christian world in the era following the collapse of the Roman empire. By the time Galileo and Kepler came on the scene, Avicenna had dismantled Aristotlean physics, metaphysics, and logic, and al-Haytham had already formulated a modern mechanics. In western Europe, Philoponus and Hasdai Crescas had undone Aristotle's concept of "place", and Buridan had already created a forerunner to the concept of inertia. So it's not as though "the tradition" had such a grip on people that alternatives weren't possible.

As for contradicting experience, scientists at Maragheh had already empirically and mathematically proved the motion and axial tilt of the earth by the 13th century, perhaps the most important leap beyond "immediate" experience mankind has ever made. And even the ancient Greek philosophers understood themselves as going beyond experience. What else could be assumed when Thales says "all is water"? It evidently isn't (to the senses), and yet it is. And what else is the trial of Socrates about than the contradiction between who Socrates appears to be and who he is? The idea that the ancients held some kind of naive view of the world is shallow. Things were far more complex than that.

So then why hadn't anyone previously considered that the ellipse might describe the motion of the celestial bodies? It's not because Aristotle said they were circles. It's not because when you look up, things appear to move in circular orbits. What then?

I'm not entirely sure, but I think it has something to do with how people understood the relationship between mathematics and nature generally. Mechanics (a purely mathematical and empirical understanding of motion) only because to become a science in its own right in the 13th century. Before that it was blended imperceptibly into "physics", which was the philosophy of nature. I think in order to apply the conical sections on to observed motion, you first have to be able to "see" nature in the right way. You have to be able to see it as an abstract, geometrical object.

(But, one might object, what else is the application of the epicycles and the equants in the Ptolemaic theory than a free (a too free!) application of geometry to the observed data? I don't know. Is it because Ptolemy and those following him believed the earth to be stationary?)

I think it ultimately comes down to what a person looking out on to the world understands by the word "motion". "Motion" is an elementary concept, a concept by means of which we understand a host of other concepts but which itself is very general and poorly understood from an everyday perspective. In the ancient world, "motion" primarily meant development and growth, the way a thing changes in accordance with its essence. Mere change of place - "locomotion" - was a species of motion-as-growth, and so whatever change of place was, it had to accord with the ancient logic of motion-as-growth. It could not contradict it. This isn't a matter of conception of senses experience. It is a matter of how one lives one life and what the world in general means.

Throughout the Islamic middle ages, largely owing to the attack against Aristotle and the relatively extreme level of intellectual tolerance in the Muslim world, locomotion gets gradually liberated from motion-qua-growth. It doesn't become fully independent from metaphysics. That never happens, not even in modernity. Rather, it acquires a new metaphysical basis. It gets wrapped up in a complex way with mathematics, especially with algebra and trigonometry. (These latter disciplines were not seen as being purely independent in the medieval world. They were seen as separate from logic and language, for example, unlike in modern philosophy.) I think the event that gives the strongest impetus to the transplanting of the study of locomotion—which becomes "mechanics"—from the philosophical account of becoming or growth into an intellectual space which it can share with pure mathematics and geometry, is the discovery that the earth moves. The previous account of becoming—it was called "phusis" or "physics" in the ancient world—was firmly rooted in the experience of an earth that does not move. The discovery in the middle ages that the earth moves created enough of a disturbance that it made sense to study change-of-place separately from becoming. That's what allowed mechanics to start to come into existence.

Obviously I'm not sure of any of the details of this, but in terms of its general features, this appears to make sense. Why it then took an extra 300 years to apply the conic sections to the celestial motions? I don't know. On the historical scale, that's a relatively short amount of time. But there's still a lot to fill in.

(Another consideration: more shapes than just the circle were imputed to the heavens. A crude form of spherical geometrical using triangles was known in ancient times. It was used to predict the positions of the planets and the stars. Two factors led to the invention of a better method. The first is that so many rituals in Islam rely upon the position of the moon and other celestial bodies, it became necessary to come up with a better method. And the second factor is that the method developed in ancient times was so time-consuming and clumsy. So triangles were used in calculating the positions of the celestial bodies. But of course no triangular motions were imputed to them...)

(And another: you can't escape from the fact that the thorough-going application of geometry to nature occurs in the budding mercantilist nations in the 17th century. Classical mechanics doesn't come to fruition in Persia, India, or China. It comes to fruition in the place where people are about to start using machines to make other machines. The practical ability to force nature into geometric configurations does not cause classical mechanics to come into existence. Classical mechanics largely predates that. But the fact that the two things come into existence in such close proximity to one another (in space and time) can't be overlooked. An interesting project would be to trace this development backward. What socio-political transformations were taking place during the golden age of Islam when mechanics first came into existence as a science? In what concrete context did people discover that the earth moves?)


It's the 400th anniversary this year of the publication of Kepler's Astronomia Nova. While thinking about his contributions, I realized Kepler probably made a larger contribution to the scientific revolution in astronomy than Copernicus did. While Copernicus made use of the epicycle, Kepler made it obsolete. More important, though, before Kepler, the systems of Ptolemy, Copernicus, and Brahe had roughly the same power to predict celestial motion. It's easy to forget, but the mathematics of Copernicus' heliocentric system was still firmly rooted in Ptolemy and methods employed by Arab astronomers in the middle ages, especially al-Tusi. Kepler's decision to employ the geometry of conic sections is really what made the qualitative leap from a geocentric to a heliocentric system possible. Or I should say it was a necessary but not a sufficient condition. Galileo's observations of the phases of Venus and the moons of Jupiter provided the other pillar necessary to make the transition. But while Galileo's observations made a transition to a heliocentric model necessary, it was Kepler's mathematics of planetary motion that gave such a model the predictive edge over competing theories.

Friday, February 20, 2009

Kicilof and Starosta on Rubin

I’m reading through Kicilof’s and Starosta’s essay on I. I. Rubin called “On Materiality and Social Form” and came across the following rather dense passage:
In our view, Rubin’s confusion (or rather, inversion) stems from the fact that he reads Marx’s passages where he states that exchange (as a necessary mediating form of the essentially private character of the direct process of production in capitalism) manifests outwardly the inner determinations borne by the direct process of production, as implying that it brings those determinations into existence. In other words, he confuses the qualitative determination of those more abstract forms (hence, of the social objectivity of value) with its concrete mode of realisation. For Rubin, then, abstract labour has no existence prior to the exchange process but comes into being through it, by subjecting concrete labour to a ‘social transformation’.
it’s worth it to unpack this passage a little.

Rubin's view is that systematic exchange of the product of labor transforms labor from concrete labor into abstract labor. This means that abstract labor only exists in those societies where there is systematic exchange of the product of labor. But systematic exchange of the product of labor only occurs in a commodity or a capitalist-commodity economy. Of course exchange of the product of labor occurs in non-commodity or non-capitalist societies, but it is usually only the surplus product that is exchanged, not the entire product of labor. The laborers get to keep some of the product—usually the means of subsistence. But obviously in a capitalist economy, we don't keep any of the product of labor. We're given a wage, instead. In this consists the difference between mere incidental exchange of the product of labor and “systematic” exchange. In distinction from incidental exchange, this process of systematic exchange reacts back on labor, transforming it from concrete labor into labor with a dual character: labor that is both concrete and abstract. This is because in order to systematically exchange the product of labor, we need to treat the products of labor as generally exchangeable (one quantity of one product for x quantity of any other product). But this requires us to treat the qualitatively different labors as also equal, since only equal labors could produce equal values.

Kicilof and Starosta think this is an error. They believe abstract labor exists in all societies (but it is not for this reason the substance of value, since value does not yet exist as a real social phenomenon in societies where there is no systematic exchange). They think Rubin makes this error because he confuses one actual function of exchange with another imagined function. Exchange is "a necessary mediating form of the essentially private character of the direct process of production in capitalism". Since all labor in capitalist society is carried on privately and independently—since production isn't planned—the only way different labors are coordinated with one another is through the exchange of the product of labor. One doesn't know if he made too many or too few beds until he brings them to market and tries to sell them. The only way we find out how much society needs of this or that is by bringing it to market with a price on it and seeing if we can sell it at that price. There exists no other force in capitalist society capable of directing the production process. Producers are "free" of any other kind of coercion.

This is the real function of exchange in a capitalist society. It "manifests outwardly the inner determinations borne by the direct process of production". Exchange plays the role it does in capitalist society because of the specific, historically conditioned characteristics of the production process. Because production is carried on by formally free, independent producers, uncoerced by any tradition, custom, or plan for production, the process of exchange must coordinate and transform their labors. Indeed, in the absence of any custom or plan, exchange is the only thing that can fulfill that role. This is what Kicilof and Starosta mean when they say that exchange outwardly manifests production. It outwardly manifests labor the same way, since a society in which there is systematic exchange is also one where there is labor with a dual character.

Rubin’s error consists in his belief that this actual function of exchange implies “that it brings those determinations [the characteristics of production] into existence.” Exchange can realize or embody characteristics of the production process without bringing those characteristics into existence. On Kicilof’s and Starosta’s reading, Rubin confuses the thing itself, which can take many outward forms, with its “concrete mode of realization”, and for this reason, according to Rubin, “abstract labour has no existence prior to the exchange process but comes into being through it”.

And yet while this critique is elegant, I’m not sure it hits the mark precisely. Compare Kicilof’s and Starosta’s account with the following passage from Rubin:
In a strictly enforced caste system, the physiological homogeneity of human labor cannot be expressed to a significant extent. In a small community based on a division of labor, the physiological homogeneity of labor is manifested in a small circle of people, and the human character of labor cannot be expressed. Only on the basis of commodity production, characterized by a wide development of exchange, a mass transfer of individuals from one activity to another, and indifference of individuals towards the concrete form of labor, is it possible to develop the homogeneous character of all working operations as forms of human labor in general. The physiological homogeneity of human labor was a necessary presupposition of the social division of labor, but only at a determined level of social development and in a determined social form of economy does the labor of the individual have the character of a form of manifestation of human labor in general. We would not be exaggerating if we said that perhaps the concept of man in general and of human labor in general emerged on the basis of the commodity economy. This is precisely what Marx wanted to point out when he indicated that the general human character of labor is expressed in abstract labor.
Rubin appears to say that there is an abstract equality in human labor in all times and in all epochs of production; however, it can only manifest itself as equal labor in a capitalist economy. Before that, it must be treated as unequal labor. So the work of a slave is not the same as the work of a free man. It is only with the advent of a specific mode of production—capitalist production—that this real abstract equality of labor can manifest itself as an actual social phenomenon. Only at that point can it become the substance of value.

I haven’t finished reading the essay yet, but I’m curious to see how Kicilof and Starosta will deal with this. Likely they will point out that Rubin makes a distinction between physiologically equal labors and abstract labor. Rubin believes physiologically equal labor precedes capitalism as its material presupposition, but physiologically equal labor only manifests itself as a real social fact under the capitalist mode of production. The social manifestation of this material presupposition is abstract labor, so while physiologically equal labor precedes capitalism, abstract labor does not. I think Kicilof and Starosta want to argue that abstract labor precedes capitalism, and that Rubin has no good reason to make the distinction (which they see as being merely conceptual) between physiologically equal labor and abstract labor.