Learning to Think New Thoughts

Most of what passes for education, remarked Bergson, is merely putting things in drawers. Or, to use a different metaphor, much of what we think of as learning is the ingesting of a sequence of conclusions — the ones accepted as correct today, with brief nods to conclusions that have been jettisoned. Textbooks are designed for smooth, painless ingestion. There is nothing in itself wrong about conclusions, but if our educational process convinces us that the matters we study have somehow been “concluded” then our learning becomes something like a polite lie, quietly bolstering the orthodoxies of our day or of our department. In contrast, there is a different kind of education, one based on questions and not conclusions. All important intellectual work —  mathematical, scientific, philosophical — is generated by questioning. An original thinker or discoverer reaches the limits of what has so far been understood, and then finds a way to think beyond these limits — often by coming up with a powerful new question that opens a hole in the established order.

Work like this occurs within a larger conversation of thought, sometimes spanning many centuries and even languages — for instance, the two-thousand year discussion of the nature of curves from the Greek geometers via Arabic mathematicians to Descartes and beyond; or the conversation that we call “the atomic theory” that began with Democritus, achieved a kind of climax with Mendeleev’s Periodic Table, and goes on vigorously to this day. These are unconcluded conversations, and the richest way to study them is to read the works of major thinkers in the order of questioning that arose in the course of the investigation. At each stage of the work we watch thinkers going as far as they can with what they have, and then being interpreted, qualified, superseded, re-visioned by subsequent thinkers; there are dead ends, red herrings, wild goose chases, wrong trees — and part of the delight of such study is to learn how to recognize these things and move through them. Even when a great thinker is wrong, his thinking is fruitful and alive; and error is the best testimony to the marvel of the imagination. A textbook approach moves in a straight line from conclusion to conclusion, and is much swifter — but it shelters the student from the actual process of thought and discovery, as if we have to be protected from the agonizing struggles of doubt and error.  And yet it is only in the course of struggling that we grow. Perhaps the most valuable benefit of studying a great book is the opportunity to participate in the generating of a new thought, with all its obstetric pain and hardship; and doing this every day is surely the best possible way to hone and invigorate the mind.

When Johannes Kepler (1571-1630) began the investigation of planetary movements that would culminate in his New Astronomy (1609), he found himself confronted with a legion of questions generated by Copernicus’ organization of the cosmos into a single system of planets orbiting around the sun, with distances and periods that could now be determined. The cosmos was no longer divided between sublunary bodies, with earth as the center, and heavenly bodies beyond the moon. Now Earth — one planet among several –was set in motion, and the angles at which we viewed the other planets had to take into account our motion as well as theirs. The switch in perspective is vertiginous. Prior to this, astronomical theorizing was grounded in the evidence of our eyes: we are stationary, and sun and planets revolve around us. This view of things is not only comforting, but also empirically sound: it is what we, in our bodies, experience. But when we become aware that our empirical experience might in fact be the product of a very different arrangement that we cannot directly experience, we find ourselves in the odd position of trusting our reason and imagination more than our senses in comprehending old quotidian faniliarities, such as the sun in the sky and the earth beneath our feet. For example, I recently witnessed a lunar eclipse: as the sun was going down, the moon was coming up in opposition to the sun, and since the earth was between both of them its shadow covered the moon. Even this is an interpretation, for all I saw was the moon magically darkening; it is an interpretation that assumes a stationary earth, and ourselves unshakeable upon its solid surface. But what if we are moving? — flying around the sun at 67,000 miles an hour? At the same time, the moon is orbiting us at 2,288 miles an hour, and in addition, the appearance of the moon’s rising is created by a combination of its own movement and the earth’s rotation at 1000 miles an hour. Our senses have never experienced such speeds, but each moment — if the Copernicans are right —  we are living them. How could all this be happening,so grand and sweeping and yet imperceptible to us? The ancient geocentrists had the advantage with his question: the earth is simply stationary , as it seems to be; and the celestial bodies, because they are eternal and beautiful, trace the path that is most appropriate to the eternal and the beautiful, namely circles.

But if the earth is a body, and if it is kindred to the other planets in orbiting the sun together, then the other planets are bodies too — and perhaps even the sacred sun is a body. If they are bodies, what keeps them at a predictable distance from the sun, especially in all that infinity of empty space — and why do they go around in circles? The investigation becomes more complicated if the sun isn’t at the exact center of the orbits and if those centers are only mathematical points: in all those millions of miles of empty space, how can the planets know where those mathematical points are and judge their exact distances from those points? Are the orbits governed by precisely mathematical intelligences, or are the planets obeying physical or mechanical principles?  If we are going to understand what is happening — where we are, what kind of world we live in — we have to tackle the question of what bodies are, such that the motions of planetary bodies make sense to us in the same way that the bodies we experience every day seem to make sense to us. This of course is a less interesting question if we refuse to accept the claim that all planets are bodies.

In the exciting, fertile introduction to his New Astronomy, Kepler daringly asserts, The true theory of gravity rests upon the following axioms. We, children of Newton, take gravity for granted as a “fact,” and rarely question what justifies it. Here Kepler gives birth to it, decades before Newton, as a theory resting upon axioms, beginning points that cannot be proven — yet it is the true theory, because all the others fail to explain the phenomena. The axioms that follow are startling:

 Every corporeal substance, to the extent that it is corporeal, has been so made as to be suited to rest in every place in which it is put by itself, outside the orb of power of a kindred body. 

(New Astronomy, tr. Donahue, pp.24-25)

In other words, a body — in so far as it is body (and not mixed with something non-bodily) will stay where it is and not move if it is “outside the orb” of influence of a body that is of the same type as it. Kepler’s phrasing is based on phenomena like magnets and magnetic bodies, which do not move until they are brought close enough to other magnetic bodies. This we can see — but Kepler’s boldness is in extending this to every corporeal substance. He says in the next axiom that magnetism is one example of the tendency of bodies to clump:

 Gravity is a mutual corporeal disposition among kindred bodies to unite or join together; thus, the earth attracts a stone much more than the stone seeks the earth. (The magnetic faculty belongs to this order of things.)

Thus objects thrown into the air do not seek the ground; objects attract one another. We are so used to this way of talking about physics that we have to pause to reflect that hardly anyone before Kepler would have considered falling objects as being  attracted, and that really the language of attraction is not less mysterious than the language of seeking; we are just more used to one of them. The attraction, he goes on to say, is not to centers, but it is part of the way bodies relate to one another as bodies, and in the case of large spheroid bodies like planets, attracted bodies will tend to go straight down, that is, towards the center, because they are spheroid bodies and not because there is something wonderfully interesting about centers.

 Heavy bodies (most of all if we establish the earth in the center of the world) are not drawn towards the center of the world because it is the center of the world, but because it is the center of a kindred spherical body, namely the earth. Consequently, wherever the earth be established, or whithersoever it be carried by its animate faculty, heavy bodies are drawn towards it.

 If the earth were not round, heavy bodies would not everywhere be drawn in straight lines towards the middle point of the earth, but would be drawn towards different points from different sides.

With the next axiom, Kepler’s audacity reaches a new height, as he gives birth to a shocking new thought:

 If two stones were set near one another in some place of the world outside the sphere of influence of a third kindred body, these stones, like two magnetic bodies, would come together in an intermediate place, each approaching the other by an interval proportional to the bulk [moles] of the other. 

The protasis of this conditional sentence describes something that we could never experience with our senses; we will never witness two stones placed somewhere far, far away from another, kindred body. This is an imagined event — perhaps suggested by magnetism, but even so we will never see two isolated magnetic bodies without a third within range of influence. More brazenly, Kepler asserts that they will move towards each other in proportion to something like their respective masses; that is, the attraction can be mahematically expressed. Now we have a conception that might help us to understand how bodies somehow hang together in space by a mutual harmony of attraction — although we have to wonder why all the planets haven’t already lumped into one with the sun instead of continuing to revolve around it. Kepler will tackle this later. Here he gves us a dazzling demonstration of the power of mathematical fantasy by extrapolating the previous axiom to the relation of earth and moon:

 If the moon and the earth were not each held back in its own circuit by an animate force or something else equally potent, the earth would ascend towards the moon by one fifty-fourth part of the interval, and the moon would descend towards the earth about fifty-three parts of the interval, and there they would be joined together; provided, that is, that the substance of each is of one and the same density.

This would not really be an axiom by itself but an example of the preceding one, if Kepler did not suggest that there is some counter-force to gravity, equally potent, that maintains distance between bodies. Again, none of this can be experienced with our senses, so why does it seem reasonable? — if indeed it does. I have known intelligent people to whom these axioms of gravity seem bizarre and extravagant, and no axiom more so than the following:

 If the earth should cease to attract its waters to itself, all the sea water would be lifted up, and would flow onto the body of the moon.

This wild vision epitomizes the imaginative power of Kepler’s book: the conceptions can seem intensely mad,while at the same time also grounded in strict, abstract mathematical principles. If the previous axioms are sound, then this should follow — and Kepler goes on with an extended imaginative meditation on the effect of the moon on the seas in various parts of the world. I will quote this passage at length, because of its sustained phantasmagorical power as it follows the moon and tides over specific portions of the globe.

 The orb of the attractive power in the moon is extended all the way to the earth, and calls the waters forth beneath the torrid zone, in that it calls them forth into its path wherever the path is directly above a place. This is imperceptible in enclosed seas, but noticeable where the beds of the ocean are widest and there is much free space for the waters’ reciprocation. It thus happens that the shores of the temperate latitudes are laid bare, and to some extent even in the torrid regions the neighboring oceans diminish the size of the bays. And this when the waters rise in the wider ocean beds, it can happen that in its narrower bays, if they are not too closely surrounded, the moon being present, the water might even seem to be fleeing the moon, though in fact they are subsiding because a quantity of water is being carried off elsewhere.

   But the moon passes the zenith swiftly, and waters are unable to follow so swiftly. Therefore, a westward current of the ocean arises beneath the torrid zone, which, when it strikes upon the far shores, is thereby deflected. But when the moon departs, this congress of the waters, or army on the march towards the torrid zone, because it is abandoned by the traction that had called it forth, is dissolved. But since it has acquired impetus, it flows back (as in a water vessel) and assaults its own shore, inundating it. In the moon’s absence, this impetus gives rise to another impetus until the moon returns and submits to the reins of this impetus, moderates it, and carries it around along with its own motion. So all shores that are equally exposed are flooded at the same time, while those more remote are flooded later, some in different ways because of the different approaches of the ocean.

  I will point out in passing that the sand dunes of he Syrtes [in north Africa] are heaped up in this way; that thus are created or destroyed countless islands in bays full of eddies (such as the Gulf of Mexico); that it seems that the soft, fertile, and friable earth of the [East] Indies was thus at length broached and penetrated by this current, this perpetual inundation, with help from a certain all-pervading motion of the earth. For it is said that India was once continuous from the Golden Chersonese [Malay Peninsula] towards the east and south, but now the ocean, which was once further back between China and America, has flowed in, and the shores of the Moluccas and of other neighboring islands, which are now raised on high because of the subsidence of the surface of the sea, suppress the credibility of this matter. (Pp.25-26)

From what point of view can the mind of a mortal being say such things? These paragraphs are astounding imaginings of a world: Kepler’s naked, meager fundamental axioms have expanded to swallow the entire known world with its gulfs and continents, even though those gulfs and continents are drawn from incomplete explorations and distorted ancient accounts. The big thing here is how a deep insight into principles can move in one step to the formation of the world. Although wrong in specifics, Kepler was right about the action of the moon — and it is because he was right about the nature of bodies. He had never visited the places he mentions in these paragraphs; he hadn’t even seen the sea — yet through his imagination he was able to create it and watch it in its workings.

When in studying a book like this we give ourselves over to slow contemplation of its thinking and its vision, for a few moments at least we find ourselves expanded and able to see not only farther but also deeper. Behind the world of our immediate senses is an actual order of things that makes sense not only to reason but to vision. The greatest wonder is that we can experience the sea with our ordinary eyes and also see with our mind’s eye what would happen if we switched gravity off. In ways like this studying is the opposite of folding old thoughts into cedared drawers; it is a wild, intoxicating journey in which each day we visit our own world and our own lives as things new to us.

[All references to Kepler are from William H. Donahue’s translation of the New Astronomy (Green Lion Press, 2015). It is a difficult, often frustrating, always brilliant and thrilling book of genius, one of the most honest and also subtle books I have had the honor to struggle through. It is beautifully translated and beautifully produced.]