BOILING GEOMETRY

If I boil water in a kettle on a stove, the operation and the objects that support it are, in reality, bound up with a multitude of other objects and a multitude of other operations; in the end, I should find that our entire solar system is concerned in what is being done at this particular point of space. But, in a certain measure, and for the special end I am pursuing, I may admit that things happen as if the group water-kettle-stove were an independent microcosm. That is my first affirmation. Now, when I say that this microcosm will always behave in the same way, that the heat will necessarily, at the end of a certain time, cause the boiling of the water, I admit that it is sufficient that a certain number of elements of the system be given in order that the system should be complete; it completes itself automatically, I am not free to complete it in thought as I please. The stove, the kettle and the water being given, with a certain interval of duration, it seems to me that the boiling, which experience showed me yesterday to be the only thing wanting to complete the system, will complete it tomorrow, no matter when tomorrow may be. What is there at the base of this belief? Notice that the belief is more or less assured, according as the case may be, but that it is forced upon the mind as an absolute necessity when the microcosm considered contains only magnitudes. If two sides of a triangle and the contained angle are given, the third side arises of itself and the triangle completes itself automatically. I can, it matters not where and it matters not when, trace the same two sides containing the same angle: it is evident that the new triangles so formed can be superposed on the first, and that consequently the same third side will come to complete the system. Now, if my certitude is perfect in the case in which I reason on pure space determinations, must I not suppose that, in the other cases, the certitude is greater the nearer it approaches this extreme case? Indeed, may it not be the limiting case which is seen through all the others and which colors them, accordingly as they are more or less transparent, with a more or less pronounced tinge of geometrical necessity? In fact, when I say that the water on the fire will boil today as it did yesterday, and that this is an absolute necessity, I feel vaguely that my imagination is placing the stove of yesterday on that of today, kettle on kettle, water on water, duration on duration, and it seems then that the rest must coincide also, for the same reason that, when two triangles are superposed and two of their sides coincide, their third sides coincide also. But my imagination acts thus only because it shuts its eyes to two essential points. For the system of today actually to be superimposed on that of yesterday, the latter must have waited for the former, time must have halted, and everything become simultaneous: that happens in geometry, but in geometry alone. Induction therefore implies first that, in the world of the physicist as in that of the geometrician, time does not count. But it implies also that qualities can be superposed on each other like magnitudes. If, in imagination, I place the stove and fire of today on that of yesterday, I find indeed that the form has remained the same; it suffices, for that, that the surfaces and edges coincide; but what is the coincidence of two qualities, and how can they be superposed one on another in order to ensure that they are identical? Yet I extend to the second order of reality all that applies to the first. The physicist legitimates this operation later on by reducing, as far as possible, differences of quality to differences of magnitude; but, prior to all science, I incline to liken qualities to quantities, as if I perceived behind the qualities, as through a transparency, a geometrical mechanism. The more complete this transparency, the more it seems to me that in the same conditions there must be a repetition of the same fact. Our inductions are certain, to our eyes, in the exact degree in which we make the qualitative differences melt into the homogeneity of the space which subtends them, so that geometry is the ideal limit of our inductions as well as of our deductions. The movement at the end of which is spatiality lays down along its course the faculty of induction as well as that of deduction, in fact, intellectuality entire.

It creates them in the mind. But it creates also, in things, the “order” which our induction, aided by deduction, finds there. This order, on which our action leans and in which our intellect recognizes itself, seems to us marvelous. Not only do the same general causes always produce the same general effects, but beneath the visible causes and effects our science discovers an infinity of infinitesimal changes which work more and more exactly into one another, the further we push the analysis: so much so that, at the end of this analysis, matter becomes, it seems to us, geometry itself. Certainly, the intellect is right in admiring here the growing order in the growing complexity; both the one and the other must have a positive reality for it, since it looks upon itself as positive. But things change their aspect when we consider the whole of reality as an undivided advance forward to successive creations. It seems to us, then, that the complexity of the material elements and the mathematical order that binds them together must arise automatically when within the whole a partial interruption or inversion is produced. Moreover, as the intellect itself is cut out of mind by a process of the same kind, it is attuned to this order and complexity, and admires them because it recognizes itself in them. But what is admirable in itself, what really deserves to provoke wonder, is the ever-renewed creation which reality, whole and undivided, accomplishes in advancing; for no complication of the mathematical order with itself, however elaborate we may suppose it, can introduce an atom of novelty into the world, whereas this power of creation once given (and it exists, for we are conscious of it in ourselves, at least when we act freely) has only to be diverted from itself to relax its tension, only to relax its tension to extend, only to extend for the mathematical order of the elements so distinguished and the inflexible determinism connecting them to manifest the interruption of the creative act: in fact, inflexible determinism and mathematical order are one with this very interruption.

It is this merely negative tendency that the particular laws of the physical world express. None of them, taken separately, has objective reality; each is the work of an investigator who has regarded things from a certain bias, isolated certain variables, applied certain conventional units of measurement. And yet there is an order approximately mathematical immanent in matter, an objective order, which our science approaches in proportion to its progress. For if matter is a relaxation of the inextensive into the extensive and, thereby, of liberty into necessity, it does not indeed wholly coincide with pure homogeneous space, yet is constituted by the movement which leads to space, and is therefore on the way to geometry. It is true that laws of mathematical form will never apply to it completely. For that, it would have to be pure space and step out of duration.

We cannot insist too strongly that there is something artificial in the mathematical form of a physical law, and consequently in our scientific knowledge of things. Our standards of measurement are conventional, and, so to say, foreign to the intentions of nature. But we may go further. In a general way, measuring is a wholly human operation, which implies that we really or ideally superpose two objects one on another a certain number of times. Nature did not dream of this superposition. It does not measure, nor does it count. Yet physics counts, measures, relates “quantitative” variations to one another to obtain laws, and it succeeds. Its success would be inexplicable, if the movement which constitutes materiality were not the same movement which, prolonged by us to its end, that is to say, to homogeneous space, results in making us count, measure, follow in their respective variations terms that are functions one of another. To effect this prolongation of the movement, our intellect has only to let itself go, for it runs naturally to space and mathematics, intellectuality and materiality being of the same nature and having been produced in the same way.

If the mathematical order were a positive thing, if there were, immanent in matter, laws comparable to those of our codes, the success of our science would have in it something of the miraculous. What chances should we have indeed of finding the standard of nature and of isolating exactly, in order to determine their reciprocal relations, the very variables which nature has chosen? But the success of a science of mathematical form would be no less incomprehensible, if matter did not already possess everything necessary to adapt itself to our formulae. One hypothesis only, therefore, remains plausible, namely, that the mathematical order is nothing positive, that it is the form toward which a certain interruption tends of itself, and that materiality consists precisely in an interruption of this kind. We shall understand then why our science is contingent, relative to the variables it has chosen, relative to the order in which it has successively put the problems, and why nevertheless it succeeds. It might have been, as a whole, altogether different, and yet have succeeded. This is so, just because there is no definite system of mathematical laws, at the base of nature, and because mathematics in general represents simply the side to which matter inclines.

Put one of those little cork dolls with leaden feet in any posture, lay it on its back, turn it up on its head, throw it into the air: it will always stand itself up again, automatically. So likewise with matter: we can take it by any end and handle it in any way, it will always fall back into some one of our mathematical formulae, because it is weighted with geometry.

Henri Bergson, Creative Evolution
Read the whole book here.

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s