From the Closed World to the Infinite Universe, by Alexander Koyré, , at sacred-texts.com
Johannes Kepler's Rejection of Infinity
The conception of the infinity of the universe is, of course, a purely metaphysical doctrine that may wellas it didform the basis of empirical science; it can never be based on empiricism. This was very well understood by Kepler who rejects it thereforeand this is very interesting and instructivenot 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 formlessor uniformuniverse of Bruno.
Yet it is not this conception of God's creative action,
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
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.
This general criticism may be sufficient. Yet Kepler does not content himself with it and continues:4
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, criticizedrightly, according to himby Aristotle.5
As for the moderns, he tells us that the infinity of the world6
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
[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
[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
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 hypothesesfor instance, the hypotheses concerning the celestial motionsto 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 telescopeare 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
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
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,
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 notand even could notknow 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
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 equidistancefrom us
of the fixed stars. Couldn't this conclusion be avoided if we admitted that the stars are so far away from usand therefore from each otherthat, 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
[paragraph continues] As it will not, the singular character of our site will be maintained.14
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.
[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
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
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
You cannot, indeed, separate the stars [by moving them] downward; the theory of the parallaxes does not allow it
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 bodiesnot points of lightwe 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
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 starsespecially visible onesat 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
The impossibility of a visible star's being at an infinite distance thus demonstrated, there remains the case of an invisible one.22
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
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 preciselyas Kepler tells us himselfthat it is what is "beyond" all number and all measure?
Furthermore, just as in spite ofor because ofthe 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
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 instrumentthe telescopeused 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
The perusal of the Nuntius tranquillized Kepler. The new stars were not planets: they were moons, Jupiter's moons. Now, if the discovery of planetswhether revolving around fixed stars or around the sunwould 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 planetswith the exception perhaps of Mercury, too near the sun to need oneshould 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 Keplerwho, for our world, accepts the teachings of Nicholas of Cusa and Brunono 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.
[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 fixedin the full sense of the termstars.
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
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 examplesthe "Medicean" planets, in factof 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
the reply is given:
Thus if we turned around the earth, or if the earth turned around with us, we would see the whole troop of the stars
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
[paragraph continues] At this stage of the discussion Kepler does not want to commit himself. Thus he gives a rather cautious answer:
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
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
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.
Click to enlarge
The figure M of Kepler
(from the Epitome astronomiae Copernicanae, 1618)
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
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.
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
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
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 Wayin spite of its resolution
by Galileo into an innumerable multitude of starsstill seems to Kepler to preclude any other conclusion. Thus, elaborating the demonstration outlined in the De stella nova, Kepler continues:33
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
[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
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
[paragraph continues] Kepler replies:
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 conceptthat of modern scienceis bad, as a really infinite number of finite bodies is something unthinkable, even contradictory:36
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
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
Space, void space, is just "nothing," a non-ens. Space, as such, neither ishow, 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.