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From the Closed World to the Infinite Universe, by Alexander Koyré, [1957], at

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III. The New Astronomy Against the New Metaphysics

Johannes Kepler's Rejection of Infinity

The conception of the infinity of the universe is, of course, a purely metaphysical doctrine that may well—as it did—form the basis of empirical science; it can never be based on empiricism. This was very well understood by Kepler who rejects it therefore—and this is very interesting and instructive—not only for metaphysical, but also for purely scientific reasons; who even, in anticipation of some present-day epistemologies, declares it scientifically meaningless.1

As for the metaphysical reasons for which Kepler denies the infinity of the universe, they are derived chiefly from his religious beliefs. Indeed, Kepler, a devout though somewhat heretical Christian, sees in the world an expression of God, symbolizing the Trinity2 and embodying in its structure a mathematical order and harmony. Order and harmony that cannot be found in the infinite and therefore perfectly formless—or uniform—universe of Bruno.

Yet it is not this conception of God's creative action,

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but a conception of astronomical science, as based upon, and limited by, the phenomena that Kepler opposes to Bruno and to those who share his views. Thus, discussing the interpretation to be given to the appearance of a new star in the foot of the Serpentarius, Kepler raises the question whether this amazing and striking phenomenon does not imply the infinity of the universe. He does not think so, yet he knows, and tells us that,3

. . . there is a sect of philosophers, who (to quote the judgment of Aristotle, unmerited however, about the doctrine of the Pythagoreans lately revived by Copernicus) do not start their ratiocinations with sense-perception or accommodate the causes of the things to experience: but who immediately and as if inspired (by some kind of enthusiasm) conceive and develop in their heads a certain opinion about the constitution of the world; once they have embraced it, they stick to it; and they drag in by the hair [things] which occur and are experienced every day in order to accommodate them to their axioms. These people want this new star and all others of its kind to descend little by little from the depths of nature, which, they assert, extend to an infinite altitude, until according to the laws of optics it becomes very large and attracts the eyes of men; then it goes back to an infinite altitude and every day [becomes] so much smaller as it moves higher.

Those who hold this opinion consider that the nature of the skies conforms to the law of the circle; therefore the descent is bound to engender the opposite ascent, as is the case with wheels.

But they can easily be refuted; they indulge indeed in their vision, born within them, with eyes closed, and their ideas and opinions are not received by them [from valid experience] but produced by themselves.

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This general criticism may be sufficient. Yet Kepler does not content himself with it and continues:4

We shall show them that by admitting the infinity of the fixed stars they become involved in inextricable labyrinths.

Furthermore we shall, if possible, take this immensity away from them: then, indeed, the assertion will fall of itself.

Kepler knows quite well that this particular opinion concerning the infinity of the world goes back to the ancient heathen philosophers, criticized—rightly, according to him—by Aristotle.5

This particular school of the ancient heathen philosophers is chiefly refuted by the argument by which Aristotle demonstrated the finitude of the world from motion.

As for the moderns, he tells us that the infinity of the world6

. . . was defended by the unfortunate Jord. Bruno. It was also asserted in a by no means obscure way, though he expressed himself as if he doubted it, by William Gilbert in the otherwise most admirable book De magnete. Gilbert's religious feeling was so strong that, according to him, the infinite power of God could be understood in no other way than by attributing to Him the creation of an infinite world. But Bruno made the world so infinite that [he posits] as many worlds as there are fixed stars. And he made this our region of the movable [planets] one of the innumerable worlds scarcely distinct from the others which surround it; so that to somebody on the Dog Star (as, for instance, one of the Cynocephals of Lucian) the world would appear from there just as the fixed stars appear to us from our world. Thus according to them, the new star was a new world.

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Neither Bruno's enthusiasm for the infinity of the universe, nor even Gilbert's desire to enhance God's infinite power, is shared by Kepler. Quite the contrary, he feels that7

This very cogitation carries with it I don't know what secret, hidden horror; indeed one finds oneself wandering in this immensity, to which are denied limits and center and therefore also all determinate places.

[paragraph continues] From the purely religious point of view, it would be sufficient, perhaps, to make an appeal to the authority of Moses. Yet the question we are discussing is not a dogmatic one; it has to be dealt with not by recourse to revelation, but by scientific reasoning,5

But because this sect misuses the authority of Copernicus as well as that of astronomy in general, which prove—particularly the Copernican one—that the fixed stars are at an incredible altitude: well then we will seek the remedy in astronomy itself.

[paragraph continues] Thus by the same means which seem to those philosophers to enable them to break out of the limits of the world into the immensity of infinite space, we will bring them back. "It is not good for the wanderer to stray in that infinity."

Kepler's refutation of the infinitist conception of the universe may appear to the modern reader unconvincing and even illogical. Yet, as a matter of fact, it is a perfectly consistent and very well-reasoned argument. It is based on two premises, which, by the way, Kepler shared with his opponents. The first one is a direct consequence of the principle of sufficient reason and consists in admitting

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that, if the world has no limits and no particular, determined, structure, that is, if the world-space is infinite and uniform, then the distribution of the fixed stars in this universe must be uniform, too.9 The second premise concerns the science of astronomy as such. It postulates its empirical character; it tells us that astronomy, as such, has to deal with observable data, that is, with the appearances (φαινόμενα); that it has to adapt its hypotheses—for instance, the hypotheses concerning the celestial motions—to these appearances, and that it has no right to transcend them by positing the existence of things that are either incompatible with them, or, even worse, of things that do not and cannot "appear." Now these "appearances"—we must not forget that Kepler is writing in 1606, that is, before the enlargement of the observable data by the discovery and the use of the telescope—are the aspects of the world that we see. Astronomy therefore is closely related to sight, that is, to optics. It cannot admit things that contradict optical laws.

Let us now turn back to Kepler:10

To begin with, it can most certainly be learnt from astronomy that the region of the fixed stars is limited downwards; . . . moreover it is not true . . . that this inferior world with its sun differs in no way in its aspect from any one of the fixed stars; that is, [that there is no difference] of one region or place from another.

For, be it admitted as a principle that the fixed stars extend themselves in infinitum. Nevertheless it is a fact that in their innermost bosom there will be an immense cavity, distinct and different in its proportions from the spaces that are between the fixed stars. So that if it occurred

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to somebody to examine only this cavity, even [if he were] ignorant of the eight small bodies which fly around the centrum of this space at a very small distance from it, and did not know what they are, or how many; nevertheless from the sole comparison of this void with the surrounding spherical region, fitted with stars, he certainly would be obliged to conclude that this is a certain particular place and the main cavity of the world. Indeed, let us take, for instance, three stars of the second magnitude in the belt of Orion, distant from each other by 81´, being, each one, of at least 2 minutes in diameter. Thus, if they were placed on the same spherical surface of which we are the center, the eye located on one of them would see the other as having the angular magnitude of about 2¾°; [a magnitude] that for us on the earth would not be occupied by five suns placed in line and touching each other. And yet these fixed stars are by no means those that are the nearest to each other; for there are innumerable smaller ones that are interspersed [between them]. Thus if somebody were placed in this belt of Orion, having our sun and the center of the world above him, he would see, first, on the horizon, a kind of unbroken sea of immense stars quasi-touching each other, at least to the sight; and from there, the more he raised his eyes, the fewer stars would he see; moreover, the stars will no longer be in contact, but will gradually [appear to be] more rare and more dispersed; and looking straight upward he will see the same [stars] as we see, but twice as small and twice as near to each other.

Kepler's reasoning is, of course, erroneous. But only because the data available to him are faulty. In itself it is quite correct. Indeed, if we assume that the fixed stars, or at least the equally bright ones, are at an approximately equal distance from us, if we assume, moreover,

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that their visible diameter corresponds to their real one, we are bound to admit that the two big stars in the belt of Orion, separated by the angular distance of 81´, will be seen from each other as covering more surface of the sky than five suns put together; the same will be the case for a great number of the other fixed stars, and therefore the visible aspect of the sky will be, for the observer placed on the fixed stars, quite different from its aspect for us. This implies, of course, a variation in the pattern of the real distribution of the fixed stars in space, that is, the negation of the homogeneity and the uniformity of the universe. Once more, let us not forget that Kepler wrote before the invention of the telescope and did not—and even could not—know that the visible diameter of the fixed stars is a pure optical illusion that gives us no information about their size and distance. Not knowing it, he was entitled to conclude:11

For us the fact of the sky is quite different. Indeed we see everywhere stars of different magnitude, and [we see them] also equally distributed everywhere. Thus around Orion and the Twins we see many of them big and closely packed together: the eye of the Bull, the Capella, the heads of the Twins, the Dog, the shoulders, the belt and the foot of Orion. And in the opposite part of the sky there are equally large ones: the Lyre, the Eagle, the heart and the brow of the Scorpion, the Serpentarius, the arms of the Balance; and before them Arcturus; the head of the Virgin; also after them the last star of the Water Bearer and so on.

I have just pointed out that Kepler's discussion of the astronomical data that enabled him to assert the particular, unique structure of our site in the world-space was based on the assumption of the equidistance—from us

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—of the fixed stars. Couldn't this conclusion be avoided if we admitted that the stars are so far away from us—and therefore from each other—that, seen from each other, they will not appear as big as we have calculated? Or couldn't we go even farther and admit that our fundamental assumption could, possibly, be incorrect and that stars which appear to be near each other could, in point of fact, be separated by an enormous distance, the one being near us and the other exceedingly far away? As we shall see, even if it were so, it would not change the fundamental fact of the singularity of our world-space. But the objection has to be dealt with. Kepler, therefore, proceeds:12

When, some time ago, I advanced these views [just developed] some people, to try me, vigorously defended the cause of infinity, which they had taken from the above-mentioned philosophers. They asserted that, granting infinity, it was easy for them to separate the pairs of fixed stars (which we on the earth perceive as being very near each other) by as great a distance as that which separates us from them. Yet this is impossible. Even admitting that you can arbitrarily elevate13 the double fixed stars [that are] equally distant from the center of the world, it must be remembered that, if we elevate the fixed stars, the void which is in the middle, and also the circular envelope of the fixed stars, increase at the same time. Indeed, [these people] assume thoughtlessly that, the fixed stars being elevated, the void will remain the same.

[paragraph continues] As it will not, the singular character of our site will be maintained.14

But what, they say, if, of the two stars of the belt of Orion, we assume one to remain in its sphere, because the

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theory of parallaxes does not admit an inferior position,15 and the other to be higher by an infinite distance? Shall we not, in this way, obtain that, seen from each other, they appear as small as they appear to us? and that there will be a distance between them, void of stars, equal to the distances between us and them?

I answer that, perhaps, one could use this method if there were only two stars, or only a few of them, and if they were not dispersed and disseminated in a circle. Indeed, you either alternately remove the stars to a greater distance and let them stay where they are or [you remove them] all together. If alternately, you do not solve the problem, though you decrease somewhat the difficulty. For, concerning those that will remain near, the affirmation [made by us] will still be just as valid. The pairs of stars will be nearer to each other than to the sun, and their diameters, as seen from each other, larger [than they are as seen by us]. But those that are removed higher will, of course, be more distant [from each other], yet nevertheless they will be comparatively large [as seen from each other]. And I would even easily concede, without endangering my cause, that all the fixed stars are of the same magnitude; and that those which to us appear large are near to us, and those [which appear small] are so much farther. As sings Manilius:15a 'Not because less bright, but because they are removed to a greater altitude.'

I say: I will concede not: I will assert. For it is just as easy to believe that [the stars] differ really in brightness, in color and also in magnitude. And it is possible that both [opinions] are true, as is the case with the planets, of . which some are really larger than others, whereas some others only appear to be larger though in themselves they are smaller, namely because they are nearer to us.

[paragraph continues] The consequences of these hypotheses will be seen later.

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[paragraph continues] For the moment we have to discuss the implications for the φαινόμενα of a really uniform distribution of the fixed stars in the world-space, that is, of a distribution according to which they would be separated from each other by equal distances, namely by the same distance that separates them from us.16

But let us pass to the other member [of the argument], and say what would result if all the stars were separated from each other by the same distance, in such a way that the nearest ones would retain the propinquity which astronomy imposes as a limit to all [stars], not allowing any one to be nearer, and all the others would be elevated in respect to it, and removed to an altitude equal to the distance of the nearest one to us.

As a matter of fact nothing will result from all this. It will never be the case that the [starry heavens] would appear to those whom we may imagine observing them from these stars as they appear to us. From which it follows that this place, in which we are, will always have a certain peculiarity that cannot be attributed to any other place in all this infinity.

Once more, in order to understand Kepler's reasoning, we have to remember that we are not discussing the abstract possibility of a certain distribution of stars in the world-space, but the concrete distribution of stars corresponding to the appearance of the sky; that is, we are dealing with the distribution of visible stars, of those that we actually see. It is their distance from us that is in question, and it is for them that the possibility of a uniform distribution, which would place most of them at very great, and regularly increasing, distances from us, is denied.17

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For, if the state of affairs were such as has been said, it is certain that those stars that are two, three, a hundred times higher will also be two, three, a hundred times larger. Indeed, let a star be as elevated as you wish, you will never obtain that it would be seen by us as having a diameter of two minutes.18 Thus the diameter will always be two thousandth, one thousandth, or so of the distance from us; but this diameter will be a much larger part of the mutual distances between two fixed stars (since these distances are much smaller than their distance from us). And though from a star near us the face of the sky will appear nearly the same as it does to us; yet from the other stars the aspect of the world will be different, and all the more different in that they are farther. Indeed, if the intervals of the pairs of stars (which to us appear as nearest each other) remain constant, their aspects [dimensions], as seen from each other, will increase [with their distance from us]. For the more you remove the stars to an infinite altitude, the more monstrous you imagine their dimensions, such as are not seen from this place of our world.

An observer starting from the earth and moving upwards to the outer spaces would, therefore, find the "appearance" of the world constantly changing, and the fixed stars always increasing in their real as well as visible dimensions. Besides,19

The same must be said concerning the space that for such a traveler increases continuously, every time he transfers the stars from one order to the next and moves them higher. You may say that he is building the shell of a snail, which becomes ever wider towards the exterior.

You cannot, indeed, separate the stars [by moving them] downward; the theory of the parallaxes does not allow it

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because it puts a certain limit to the approximation; you cannot separate them sideways, as they possess already their places determined by sight; it remains thus to separate the stars by moving them upward, but in this case the space that surrounds us and in which are found no stars whatever except the eight small globes in the very centre of this void, grows at the same time.

Thus it is obvious that we may assume the world to be as large as we like; still the disposition of the fixed stars as seen by us will be such that this our place will appear as possessing a certain particularity and as having a certain manifest property (the absence of fixed stars in the vast void) by which it is distinct from all other places.

Kepler is perfectly right. We can make the world as big as we wish, and yet, if we have to restrict its contents to the visible stars, which moreover appear to us as finite, measurable bodies—not points of light—we will never be able to assign to them a uniform distribution that would "save" the phenomena. Our world will always be distinguished by a particular structure.20

It is certain that, on the inside, toward the sun and the planets, the world is finite and, so to say, excavated. What remains belongs to metaphysics. For, if there is such a place [as our world] in this infinite body, then this place will be in the center of the whole body. But the fixed stars which surround it will not, in respect to it, be in a position similar [to that of our sun] as they should be if there were everywhere worlds similar to ours. But they will form a closed sphere around this [void]. This is most obvious in the case of the Milky Way which passes through [the heavenly sphere] in an uninterrupted circle, holding us in

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the middle. Thus both the Milky Way and the fixed stars play the role of extremities. They limit this our space, and in turn are limited on the exterior. Is it, indeed, credible that, having a limit on this side, they extend on the other side to infinity? How can we find in infinity a centrum which, in infinity, is everywhere? For every point taken in the infinity is equally, that is, infinitely, separated from the extremities which are infinitely distant. From which it would result that the same [place] would be the center and would not be [the center], and many other contradictory things, which most correctly will be avoided by the one who, as he found the sky of the fixed stars limited from inside, also limits it on the outside.

Yet, can we not assume that the region of the fixed stars is boundless and that stars follow upon stars, though some, or even most of them, are so far away that we do not see them? Assuredly we can. But it will be a purely gratuitous assumption, not based on experience, that is, on sight. These invisible stars are not an object of astronomy and their existence cannot in any way be demonstrated.

In any case there cannot be stars—especially visible ones—at an actually infinite distance from us. Indeed, they should necessarily be infinitely large. And an infinitely large body is utterly impossible because it is contradictory.

Once more Kepler is right. A visible star cannot be at an infinite distance; nor, by the way, can an invisible one:21

If there were an infinite altitude of the sphere of the fixed stars, that is, if some fixed stars were infinitely high, they would also be in themselves of an infinite corporeal bulk.

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[paragraph continues] Imagine, indeed, a star, seen under a certain angle, for instance, 4´; the amplitude of such a body is always a thousandth part of its distance, as we know from geometry. Consequently if the distance is infinite, the diameter of the star will be the thousandth part of the infinite. But all the aliquot parts of the infinite are infinite. Yet at the same time it will be finite, because it has a form: all form is circumscribed by certain bounds, that is, [all form] is finite or limited. But we have given it a form when we have posited it as visible under a certain angle.

The impossibility of a visible star's being at an infinite distance thus demonstrated, there remains the case of an invisible one.22

But what, you will ask, if it were so small as not to be seen? I answer that the result is the same. It is necessary, indeed, that it occupy an aliquot part of the circumference that passes through it. But a circumference of which the diameter is infinite is itself infinite. Thus it follows that no star, either visible, or having vanished because of its smallness, is separated from us by an infinite distance.

It remains only to ask ourselves whether an infinite space without stars can be posited. Kepler replies that such an assertion is utterly meaningless, since wherever you put a star you will be at a finite distance (from the earth) and if you go beyond, you cannot speak of a distance.23

Finally, even if you extend the place without stars to infinity, it is certain that wherever you put a star into it, you will have a finite interval and a finite circumference determined by the star; thus, those who say that the sphere of the fixed stars is infinite commit a contradiction in adjecto.

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[paragraph continues] In truth, an infinite body cannot be comprehended by thought. For the concepts of the mind concerning the infinite are either about the meaning of the term " infinite," or about something which exceeds all conceived numerical, visual, tactual measure: that is, something which is not infinite in actu, as an infinite measure can never be thought of.

Kepler, once more, is perfectly, or at least partially, right. It is quite certain that wherever you put a star you will find yourself at a finite distance from your starting point, as well as from all other stars in the universe. A really infinite distance between two bodies is unthinkable, just as an infinite integer is unthinkable: all integers that we can reach by counting (or any other arithmetical operation) are necessarily finite. Yet it is perhaps too rash to conclude therefore that we have no concept of the infinite: does it not mean precisely—as Kepler tells us himself—that it is what is "beyond" all number and all measure?

Furthermore, just as in spite of—or because of—the finiteness of all numbers we can go on counting without end, can we not also go on putting stars in space, all, of course, at finite distances, without ever coming to an end? Certainly we can, provided we abandon Kepler's empirical, that is, Aristotelian or semi-Aristotelian, epistemology which precludes this operation, and replace it by another: an a priori Platonic or at least semi-Platonic one.


In my analysis of Kepler's objections to the infinity of the world I have pointed out that they were formulated several years before the great astronomical (telescopical) discoveries of Galileo. These discoveries, which so tremendously

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enlarged the field of observable stars and so deeply modified the aspect of the celestial vault, discoveries which Kepler accepted and defended with joy, and which he supported not only with the weight of his undisputed authority but also by establishing the theory of the instrument—the telescope—used by Galileo, obliged him, of course, to modify some of the views he had expressed in his treatise on the new star. However, and this seems to me extremely interesting and significant, they did not lead him to the acceptance of the infinitist cosmology. On the contrary, they seemed to him to confirm his own finitistic world-view and to bring new data in favor of the unicity of the solar system and of the essential distinction of our moving world and the motionless congeries of the fixed stars.

Thus in his famous Dissertatio cum nuntio sidereo he tells us that at first, before having in hand the publication of Galileo, he was somewhat disturbed by the conflicting reports about the latter's discoveries, namely, whether the new stars were new planets moving around the sun, new "moons" accompanying the solar planets, or, as his friend Mattheus Wackher believed, planets revolving around some fixed stars: a strong argument in favor of Bruno's conception of the uniformity of the world. In this case, indeed,24

. . . nothing could prevent us from believing that numberless others would be discovered later on, and that either this our world were infinite as Melissos and the author of magnetic philosophy, William Gilbert, held, or that there was an infinity of worlds and earths (besides this one) as was believed by Democritus and Leucippus and, among the moderns, by Bruno, Brutus, Wacherus and, possibly also, by Galileo.

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The perusal of the Nuntius tranquillized Kepler. The new stars were not planets: they were moons, Jupiter's moons. Now, if the discovery of planets—whether revolving around fixed stars or around the sun—would have been extremely disagreeable for Kepler, the discovery of new moons did not affect him at all. Why, indeed, should the earth be the only planet to possess a moon? Why should the other ones not be similarly endowed with satellites? There is no reason why the earth should have this privilege. Nay, Kepler thinks that there are good reasons why all the planets—with the exception perhaps of Mercury, too near the sun to need one—should be surrounded with moons.

It could be said, of course, that the earth has a moon because it is inhabited. Thus, if the planets had moons, they should be inhabited too. And why shouldn't they be? There is, according to Kepler—who, for our world, accepts the teachings of Nicholas of Cusa and Bruno—no reason to deny this possibility.

As for the other discoveries of Galileo, namely, those concerning the fixed stars, Kepler points out that they enhance the difference between the stars and the planets. Whereas the latter are strongly magnified by the telescope and appear as well-defined discs, the former hardly increase their dimensions for, seen through the telescope, they are deprived of the luminous haze that surrounds them,25 a fact of tremendous importance because it shows that this haze belongs not to the seen stars but to the seeing eye, in other words, that it is not an objective but a subjective phenomenon and that, whereas the visible dimensions of the planets have a determinate relation to their real ones, this is not the case for the fixed stars.

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[paragraph continues] Thus we can calculate the dimensions of the planets, but we cannot do it, at least not as easily, for the fixed stars.

The explanation of this fact is easy: whereas the planets shine by the reflected light of the sun, the fixed stars shine by their own, like the sun. But if so, are they not really suns as Bruno has asserted? By no means. The very number of the new stars discovered by Galileo proves that the fixed stars, generally speaking, are much smaller than the sun, and that there is in the whole world not a single one which in dimensions, as well as in luminosity, can be equal to our sun. Indeed, if our sun were not incommensurably brighter than the fixed stars, or these so much less bright than it, the celestial vault would be as luminous as the sun.

The very existence of a tremendous number of fixed stars which we do not see, but which observers placed upon one of them would, is a proof, according to Kepler, that his fundamental objection to the infinitist cosmology, namely, that for no observer in the world would the aspect of the sky be the same as it is for us, is even better grounded in the facts than he had imagined. Thus the conclusion formerly drawn from the analysis of the phenomena accessible to the unassisted eye finds itself confirmed by the adjunction to them of the phenomena revealed by the telescope: our moving world, with its sun and planets, is not one of many, but a unique world, placed in a unique void, surrounded by a unique conglomeration of innumerable fixed—in the full sense of the term—stars.

Kepler thus maintains his position. Of the two possible interpretations of the telescopic discoveries of Galileo, that the new (fixed) stars are not seen by the unassisted eye

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because they are too far, and that they are not seen because they are too small, he resolutely adopts the second.

He is wrong, of course; and yet, from the point of view of pure empiricism, he is blameless because there are, for him, on the one hand, no means of determining the intervals that separate us from the stars and no reason therefore to assume that they are not very different in size; all the more so as there are, on the other hand, some examples—the "Medicean" planets, in fact—of celestial objects imperceptible because they are too small to be seen.

Let us turn now to the Epitome astronomiae Copernicanae, the last, and the most mature, great work of Kepler. We shall find the rejection of the infinity of the world presented just as vigorously, or perhaps even more vigorously, than ever before. To the question26

What is to be held concerning the shape of the sky?

the reply is given:

Though we cannot perceive with our eyes the matter of the etheric aura, there is nothing, however, to prevent us from believing that it is spread through the whole amplitude of the world on all sides surrounding the elementary sphere. That the army of the stars completely encircles the earth and thus forms a certain quasi-circular vault is clear from the fact that, while the earth is round, men, wherever they go, see the stars above their heads, as we do.

Thus if we turned around the earth, or if the earth turned around with us, we would see the whole troop of the stars

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arranged in a closed circuit. But that is not an answer to the question asked, as nobody doubts that the earth is surrounded by stars. What we have to find out is something quite different, namely, whether this quasi vault is more than a simple appearance, that is whether21

the centres of the stars are placed on the same spherical surface.

[paragraph continues] At this stage of the discussion Kepler does not want to commit himself. Thus he gives a rather cautious answer:

This is rather uncertain. As some of them are small, and others big, it is not impossible that the small ones appear such because they are far away in the high ether, and the large [do so] because they are nearer to us. Nor is it absurd that two fixed [stars] of different apparent magnitude be distant from us by the same interval.

As for the planets, it is certain that they are not in the same spherical surface as the fixed stars; indeed they eclipse the fixed stars but are not eclipsed by these.

[paragraph continues] But in this case, that is, if we can neither determine the intervals that separate us from the fixed stars nor decide whether their apparent magnitude is a function of their real size or only of their distance, why should we not admit that their "region" is unlimited or infinite? Indeed,28

If there is no more certain knowledge concerning the fixed stars, it would seem that their region is infinite; nor will this our sun be anything other than one of the fixed stars, larger and better seen by us, because [it is] nearer to us than the fixed stars; and in this case around any one of the fixed stars there may be such a world as there is around us; or, which is exactly the same, among the innumerable places in that infinite assembly of the fixed stars

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our world with its sun will be one [place] in no way different from other places around other fixed stars, as [represented] by the adjoined figure M.

The supposition seems reasonable or, at least, admissible. Yet Kepler rejects it, and does so for the same reasons he had twelve years before: from the hypothesis of infinity, that is, of a uniform distribution of the fixed stars in space, would follow an aspect of the sky that is not in accordance with the phenomena. For Kepler, indeed, the infinity of the world necessarily implies a perfect uniformity of its structure and contents. An irregular, irrational scattering of fixed stars in space is unthinkable; finite or infinite, the world must embody a geometrical pattern. But whereas for a finite world it is reasonable to choose a particular pattern, the principle of sufficient reason prevents the geometrically minded God of Kepler from doing it in an infinite one. As already explained by Bruno, there is no reason (or even possibility) for God to make a distinction between the "places" of a perfectly homogeneous space, and to treat them in a different way. Kepler thus states:29

This [the infinity of the world] indeed [was asserted] by Bruno and some others. But [even] if the centers of the fixed stars are not on the same spherical surface, it does not follow that the region in which they are dispersed is everywhere similar to itself.

As a matter of fact, in the midst of it [the region of the fixed stars] there is assuredly a certain immense void, a hollow cavity, surrounded in close order by the fixed stars, enclosed and circumscribed as by a wall or vault; it is in the bosom of this immense cavity that our earth with the sun and the moving stars [planets] is situated.

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FIGURE 3<br> The figure M of Kepler<br> (from the Epitome astronomiae Copernicanae, 1618)
Click to enlarge

The figure M of Kepler
(from the Epitome astronomiae Copernicanae, 1618)

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In order to demonstrate this assertion, Kepler gives us a detailed description of the aspect that the sky would have in the case of a uniform distribution of the fixed stars (which, moreover, in this case would have to be assumed as being, all of them, of the same size), and opposes this hypothetical picture to the actual one.30

If the region of the fixed stars were everywhere similarly set with stars, even in the vicinity of our movable world, so that the region of our world and of our sun had no peculiar outline compared to the other regions, then only a few enormous fixed stars would be seen by us, and not more than twelve (the number of the angles of the icosahedron) could be at the same distance from us and of the same [visible] magnitude; the following ones would be scarcely more numerous, yet they would be twice as distant as the nearest ones; the next higher would be three times as far, and so on, always increasing their distance [in the same manner].

But as the biggest of all appear so small that they can hardly be noted or measured by instruments, those that would be two or three times farther off, if we assume them to be of the same true magnitude, would appear two or three times smaller. Accordingly we should quickly arrive at those which would be completely imperceptible. Thus very few stars would be seen, and they would be very different from each other.

But what is seen by us in fact is quite different. We see, indeed, fixed stars of the same apparent magnitude packed together in a very great number. The Greek astronomers counted a thousand of the biggest, and the Hebrews eleven thousand; nor is the difference of their apparent magnitudes very great. All these stars being equal to the sight, it is not reasonable that they should be at very unequal distances from us.

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Thus, as the general appearance of the fixed stars is everywhere nearly the same in respect to their number and magnitude, the visible sky is also everywhere raised above us by nearly the same distance. There is therefore an immense cavity in the midst of the region of the fixed stars, a visible conglomeration of fixed stars around it, in which enclosure we are.

In the belt of Orion there are three big stars which are distant from each other by an interval of 83´; let us suppose the visible semidiameter of each to be only of one minute; accordingly it will appear to the sight as being of 83´, that is, nearly three times the breadth of the sun, and as for the surface, it would be eight times larger than the sun itself. Consequently the appearance of the fixed stars as seen from each other is not the same as it is from our world, and accordingly we are farther away from the fixed stars than the neighbouring fixed stars are from each other.

As we see, the telescope did not change the pattern of Keplerian reasoning: it only made him diminish somewhat the visible dimensions of the fixed stars. And, of course, as long as this visible dimension is not completely removed from the objective sphere to the subjective one, Kepler's deduction can be upheld.

Yet, it may be objected, its second premise, that of the uniform size of the fixed stars, is gratuitous. It seems that,31

The strength of this argument can be weakened by assuming that the stars are so much larger as they are higher [farther] from the earth. For, if among the so numerous stars that are seen under nearly the same angle, some were assumed to have small bodies, and others enormous ones, it would follow that the former are near us and the latter exceedingly far; and thus, in this case, stars which

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are seen by us as very near [to each other] could in point of fact be very distant.

This is a possible assumption, but, as we know, a rather improbable one, since it would imply an extremely unlikely star distribution, a distribution, moreover, completely incompatible with our fundamental assumption of a homogeneous, uniform universe:32

In this case, this region would be conspicuous if not by its vacuity then by the smallness of the stars in the neighbourhood of our moving world, and thus the very minuteness of the stars would present a kind of void, whereas the increasing magnitude of the stars on the exterior would play the role of the vault. In the universe there would be less stellar matter in this cavity in which our moving world is located, and more matter in the circumference which contains and limits it. Thus it would still remain true that this place is singular and notable compared to all the remaining parts of the region of the fixed stars.

Moreover, it is more probable that those [stars] that are nearly of the same sensible magnitude are separated from us by nearly the same distance, and that a kind of hollow sphere is formed by the packing closely together of so many stars.

The arguments already developed are more than sufficient to enable us to maintain the unicity of this our moving and sun-centered world, and to oppose it to the realm of the fixed stars. We can, however, supplement them by more direct ones, and show that the phenomena clearly point out our (the solar system's) central position in the midst of the peripheral accumulation of stars. The appearance of the Milky Way—in spite of its resolution

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by Galileo into an innumerable multitude of stars—still seems to Kepler to preclude any other conclusion. Thus, elaborating the demonstration outlined in the De stella nova, Kepler continues:33

Do you have any other argument demonstrating that this place in the midst of which are the earth and the planets is particularly distinguished in respect to all other places in the region of the fixed stars?

The way called by the Greeks the Milky Way and by us the Road of St. Jacob is spread around in the middle of the orb of the fixed stars (as the orb appears to us), dividing it into two apparent hemispheres; and though this circle is of unequal breadth, still it is, all around, not very dissimilar to itself. Thus the Milky Way conspicuously determines the place of the earth and of the moving world in relation to all other places in the region of the fixed stars.

For if we assume that the earth is on one side of the semidiameter of the Milky Way, then this Milky Way would appear to it [the earth] as a small circle or small ellipse . . . it would be visible at one glance, whereas now not more than half of it can be seen at any moment. On the other hand, if we assumed that the earth were indeed in the plane of the Milky Way, but in the vicinity of its very circumference: then this part of the Milky Way would appear enormous, and the opposite part, narrow.

Thus the sphere of the fixed stars is limited downwards, towards us, not only by the stellar orb but also by the circle of the Milky Way.

Still, in spite of being thus limited "downwards," the sphere of the fixed stars could nevertheless extend indefinitely "upwards"; the walls of the world-bubble could be indefinitely, or infinitely, thick. Once more we see

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[paragraph continues] Kepler reject this supposition as groundless and perfectly unscientific. Astronomy, indeed, is an empirical science. Its field is coextensive with that of observable data. Astronomy has nothing to say about things that are not, and cannot, be seen.34

But then is not the region of the fixed stars infinite upwards? Here astronomy makes no judgment, because in such an altitude it is deprived of the sense of seeing. Astronomy teaches only this: as far as the stars, even the least ones, are seen, space is finite.

Kepler does not mention Galileo in this discussion, and we can understand why: the telescope does not change the situation. It allows us to see more stars than we did before its invention; it enables us to transcend the factual limitation of our sense of seeing; but it does not remove its essential structure. With as without the telescope, things at an infinite distance cannot be seen. The optical world is finite.

Thus to the question:35

But is it not possible for some of the visible stars to be separated from us by an infinite distance?

[paragraph continues] Kepler replies:

[paragraph continues] No; because everything that is seen, is seen by its extremities. Consequently a visible star has limits all around. But if the star receded to a really infinite distance, these limits too would be distant from one another by an infinite space. For everything at once, that is, the whole body of the star, would participate in the infinity of this altitude. Therefore, if the angle of vision remained the same, the diameter of the star, which is the line between its limits, would be increased proportionally to the distance; thus the

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diameter of a [star] twice as distant will be twice as large as the diameter of the nearer one, the diameter of a [star] distant by a finite space will be finite, but when a body is assumed to acquire an infinitely increased distance [its diameter] also becomes infinitely great.

Indeed, to be infinite and to be limited is incompatible, just as it is incompatible to be infinite and to have a certain, that is, determinate, proportion to something finite. Consequently, nothing that is visible is separated from us by an infinite distance.

So much for the visible world. But can we not assume that outside and beyond the world, or the part of the world that is seen by us, space, and stars in space, continue to exist without end? It may be meaningless from the point of view of astronomy, it may be metaphysics. . . . But is it a good one? Not according to Kepler, who held that this concept—that of modern science—is bad, as a really infinite number of finite bodies is something unthinkable, even contradictory:36

But what if there were in reality stars, of finite body, scattered upwards in the infinite spaces, [stars] which, because of so great a distance, were not seen by us?

First, if they are not seen, they in no way concern astronomy. Then, if the region of the fixed stars is at all limited, namely downwards, towards our mobile world, why should it lack limits upwards? Third, though it cannot be denied that there can be many stars which, either because of their minuteness or because of their very great distance, are not seen, nevertheless you cannot because of them assert an infinite space. For if they are, individually, of a finite size, they must, all of them, be of a finite number. Otherwise, if they were of an infinite number, then, be they as small as you like, provided they are not infinitely so, they

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would be able to constitute one infinite [star] and thus there would be a body, of three dimensions, and nevertheless infinite, which implies a contradiction. For we call infinite what lacks limit and end, and therefore also dimension. Thus all number of things is actually finite for the very reason that it is a number; consequently a finite number of finite bodies does not imply an infinite space, as if engendered by the multiplication of a multitude of finite spaces.

Kepler's objection against infinity is, of course, not new: it is essentially that of Aristotle. Yet it is by no means negligible, and modern science seems rather to have discarded than to have solved the problem.37 Now, even if we deny that there is an infinite number of stars in space, there still remains, for the infinitist, a last possibility: that of asserting a finite world immersed in an infinite space.38 Kepler does not accept this, either, and his reasons for rejecting it reveal the ultimate metaphysical background of his thinking:39

If you are speaking of void space, that is, of what is nothing, what neither is, nor is created, and cannot oppose a resistance to anything being there, you are dealing with quite another question. It is clear that [this void space], which is obviously nothing, cannot have an actual existence. If, however, space exists because of the bodies located in it [it will not be infinite as] it is already demonstrated that no body that can be located is actually infinite, and that bodies of finite magnitude cannot be infinite in number. It is therefore by no means necessary that space be infinite on account of the bodies located in it. And it is also impossible that between two bodies there be an actually infinite line. For it is incompatible to be infinite and to have limits in the two individual bodies or points that constitute the ends of the line.

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Space, void space, is just "nothing," a non-ens. Space, as such, neither is—how, indeed, could it be if it is nothing?—nor has it been created by God, who assuredly has created the world out of nothing, but did not start by creating "nothing."40 Space exists on account of the bodies; if there were no bodies, there would not be space. And if God should destroy the world, there would be no void space left behind. There would be simply nothing, just as there was nothing at all before God created the world.

All that is not new, nor specific to Kepler: it is the traditional teaching of Aristotelian scholasticism. Thus we have to admit that Johannes Kepler, the great and truly revolutionary thinker, was, nevertheless, bound by tradition. In his conception of being, of motion, though not of science, Kepler, in the last analysis, remains an Aristotelian.

Next: IV. Things Never Seen Before and Thoughts Never Thought