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

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IV. Things Never Seen Before and Thoughts Never Thought:


Galileo & Descartes

I have already mentioned the Sidereus Nuncius1 of Galileo Galilei, a work of which the influence—and the importance—cannot be overestimated, a work which announced a series of discoveries more strange and more significant than any that had ever been made before. Reading it today we can no longer, of course, experience the impact of the unheard-of message; yet we can still feel the excitement and pride glowing beneath the cool and sober wording of Galileo's report:2

In this little treatise I am presenting to all students of nature great things to observe and to consider. Great as much because of their intrinsic excellence as of their absolute novelty, and also on account of the instrument by the

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aid of which they have made themselves accessible to our senses.

It is assuredly important to add to the great number of fixed stars that up to now men have been able to see by their natural sight, and to set before the eyes innumerable others which have never been seen before and which surpass the old and previously known [stars] in number more than ten times.

It is most beautiful and most pleasant to the sight to see the body of the moon, distant from us by nearly sixty semidiameters of the earth, as near as if it were at a distance of only two and a half of these measures.

[paragraph continues] So that

Any one can know with the certainty of sense-perception that the moon is by no means endowed with a smooth and polished surface, but with a rough and uneven one, and, just like the face of the earth itself, is everywhere full of enormous swellings, deep chasms and sinuosities.

Then to have settled disputes about the Galaxy or Milky Way and to have made its essence manifest to the senses, and even more to the intellect, seems by no means a matter to be considered of small importance; in addition to this, to demonstrate directly the substance of those stars which all astronomers up to this time have called nebulous, and to demonstrate that it is very different from what has hitherto been believed, will be very pleasant and very beautiful.

But what by far surpasses all admiration, and what in the first place moved me to present it to the attention of astronomers and philosophers, is this: namely, that we have discovered four planets, neither known nor observed by any one before us, which have their periods around a certain big star of the number of the previously known ones, like Venus and Mercury around the sun, which sometimes precede

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it and sometimes follow it, but never depart from it beyond certain limits. All this was discovered and observed a few days ago by means of the perspicilli invented by me through God's grace previously illuminating my mind.

To sum up: mountains on the moon, new "planets" in the sky, new fixed stars in tremendous numbers, things that no human eye had ever seen, and no human mind conceived before. And not only this: besides these new, amazing and wholly unexpected and unforeseen facts, there was also the description of an astonishing invention, that of an instrument—the first scientific instrument—the perspicillum, which made all these discoveries possible and enabled Galileo to transcend the limitation imposed by nature—or by God—on human senses and human knowledge.3

No wonder that the Message of the Stars was, at first, received with misgivings and incredulity, and that it played a decisive part in the whole subsequent development of astronomical science, which from now on became so closely linked together with that of its instruments that every progress of the one implied and involved a progress of the other. One could even say that not only astronomy, but science as such, began, with Galileo's invention, a new phase of its development, the phase that we might call the instrumental one.

The perspicilli not only increased the number of the fixed, and errant, stars: they changed their aspect. I have already dealt with this effect of the use of the telescope. Yet it is worth while quoting Galileo himself on this subject:

First of all, this is worthy of consideration, namely that

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stars, as well fixed as errant, when they are seen through the perspicillum, are never seen to increase their dimensions in the same proportions in which other objects, and the moon itself, increase in size. Indeed in [the case of] the stars this increase appears much smaller, so that a perspicillum which, for instance, is powerful enough to magnify all other objects a hundred times will scarcely render the stars four or five times larger. But the reason for it is this: namely the stars, when seen by our free and natural eyesight, do not present themselves to us with their real and, so to say, naked size, but are surrounded by a certain halo and fringed with sparkling rays, particularly so when the night is already advanced; therefore they appear much larger than [they would] if they were stripped of these adventitious fringes; for the angle of vision is determined not by the primary body of the star; but by the brightness that surrounds it.

According to Galileo, this "adventitious" and "accidental" character of the halo surrounding the stars is clearly demonstrated by the fact that, when they are seen at dawn, stars, even of the first magnitude, appear quite small; and even Venus, if seen by daylight, is hardly larger than a star of the last magnitude. Daylight, so to say, cuts off their luminous fringes; and not only light, but diaphanous clouds or black veils and colored glass have the same effect.5

The perspicillum acts in the same way. First it removes from the stars the accidental and adventitious splendours, and [only] then enlarges their true globes (if indeed they are of a round shape), and therefore they appear to be magnified in a smaller proportion [than other objects]. Thus a starlet of the fifth or the sixth magnitude seen through a perspicillum is shown only as of the first magnitude.

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This, indeed, is extremely important as it destroys the basis of Tycho Brahe's most impressive—for his contemporaries—objection to heliocentric astronomy, according to which the fixed stars—if the Copernican world-system were true—should be as big, nay much bigger, than the whole orbis magnus of the annual circuit of the earth. The perspicillum reduces their visible diameter from 2 minutes to 5 seconds and thus disposes of the necessity to increase the size of the fixed stars beyond that of the sun. Yet the decrease in size is more than compensated by an increase in number:6

The difference between the appearance of the planets and of the fixed stars seems equally worthy of notice. Planets indeed present their discs perfectly round and exactly delimited, and appear as small moons completely illuminated and globular; but the fixed stars are not seen as bounded by a circular periphery, but like blazes of light, sending out rays on all sides and very sparkling; and with the perspicillum they appear to be of the same shape as when viewed by the natural sight, and so much bigger that a starlet of the fifth or sixth magnitude seems to equal the Dog, the largest of all the fixed stars. But below the stars of the sixth magnitude, you will see through the perspicillum so numerous a herd of other stars that escape the natural sight as to be almost beyond belief; for you may see more than six other differences of magnitude; of which the largest, those that we may call stars of the seventh magnitude or of the first of the invisible ones, appear with the aid of the perspicillum larger and brighter than stars of the second magnitude seen by natural sight. But in order that you may see one or two examples of their nearly inconceivable number, we decided to make out two star-pictures, so that

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FIGURE 4<br> <i>Galileo's star-picture of the shield and sword of Orion</i><br> (from the <i>Sidereus Nuncius</i>, 1610)
Click to enlarge

Galileo's star-picture of the shield and sword of Orion
(from the Sidereus Nuncius, 1610)

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from these examples you may judge about the rest. At first we determined to depict the entire constellation of Orion, but we were overwhelmed by the enormous multitude of stars and by lack of time, and have deferred this attempt to another occasion; for there are adjacent to, or scattered around, the old ones more than five hundred [new ones] within the limits of one or two degrees.

As a second example we have depicted the six stars of Taurus, called the Pleiades (we say six, because the seventh is scarcely ever visible), which are enclosed in the sky within very narrow boundaries, and near which are adjacent more than forty other visible ones, none of which is more than half a degree distant from the aforesaid six.

We have already seen that the invisibility for the human eye of the fixed stars discovered by Galileo, and, accordingly, the role of his perspicillum in revealing them, could be interpreted in two different ways: it could be explained by their being (a) too small to be seen, (b) too far away. The perspicillum would act in the first case as a kind of celestial microscope, in enlarging, so to say, the stars to perceivable dimensions; in the second it would be a " telescope " and, so to say, bring the stars nearer to us, to a distance at which they become visible. The second interpretation, that which makes visibility a function of the distance, appears to us now to be the only one possible. Yet this was not the case in the seventeenth century. As a matter of fact both interpretations fit the optical data equally well and a man of that period had no scientific, but only philosophical, reasons for choosing between them. And it was for philosophical reasons that the prevailing trend of seventeenth century thinking rejected the first interpretation and adopted the second.

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There is no doubt whatever that Galileo adopted it too, though he very seldom asserts it. As a matter of fact he does it only once, in a curious passage of his Letter to Ingoli where he tells the latter that:7

If it is true, as is commonly held,8 that the highest parts of the universe are reserved for the habitation of substances more pure and perfect [than ourselves] they [the fixed stars] will be no less lucid and resplendent than the sun; and yet their light, and I mean the light of all of them taken together, does not come up to the tenth part of the visible magnitude and of the light that is communicated by the sun; and of the one as well as of the other of these effects the sole reason is their great distance: how great therefore must we not believe it to be?

[paragraph continues] Indeed, in the debate about the finiteness or the infinity of the universe, the great Florentine, to whom modern science owes perhaps more than to any other man, takes no part. He never tells us whether he believes the one or the other. He seems not to have made up his mind, or even, though inclining towards infinity, to consider the question as being insoluble. He does not hide, of course, that in contradistinction to Ptolemy, Copernicus and Kepler, he does not admit the limitation of the world or its enclosure by a real sphere of fixed stars. Thus in the letter to Ingoli already quoted he tells him:9

You suppose that the stars of the firmament are, all of them, placed in the same orb: that is something the knowledge of which is so doubtful that it will never be proved either by you or by anybody else; but if we restrict ourselves to conjectures and probabilities I shall say that not even four of the fixed stars . . . are at the same distance

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from whichever point of the universe you may want to choose.

[paragraph continues] And, what is more, not only is it not proved that they are arranged in a sphere but neither Ingoli himself,10

. . . nor any one in the world, knows, nor can possibly know, not only what is the shape [of the firmament] but even whether it has any figure at all.

Consequently, once more in opposition to Ptolemy, Copernicus and Kepler, and in accordance with Nicholas of Cusa and Giordano Bruno, Galileo rejects the conception of a center of the universe where the earth, or the sun, should be placed, "the center of the universe which we do not know where to find or whether it exists at all." He even tells us that "the fixed stars are so many suns." Yet, in the selfsame Dialogue on the Two Greatest World-Systems from which the last two quotations are taken, discussing ex professo the distribution of the fixed stars in the universe, he does not assert that the stars are scattered in space without end:11

Salv.—Now, Simplicius, what shall we do with the fixed stars? Shall we suppose them scattered through the immense abysses of the universe, at different distances from one determinate point; or else placed in a surface spherically distended about a center of its own, so that each of them may be equidistant from the said center?

Simp.—I would rather take a middle way and would assign them a circle described about a determinate center and comprised within two spherical surfaces, to wit, one very high and concave, the other lower and convex betwixt which I would constitute the innumerable multitude of stars, but yet at diverse altitudes, and this might be called

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the sphere of the universe, containing within it the circles of the planets already by us described.

Salv.—But now we have all this while, Simplicius, disposed the mundane bodies exactly according to the order of Copernicus. . . .

We can assuredly explain the moderation of Salviati, who does not criticize the conception presented by Simplicio—though he does not share it—and who accepts it, for the purpose of the discussion, as agreeing perfectly with Copernican astronomy, by the very nature of the Dialogue: a book intended for the "general reader," a book which aims at the destruction of the Aristotelian world-view in favor of that of Copernicus, a book which pretends, moreover, not to do it, and where, therefore, subjects both difficult and dangerous are obviously to be avoided.

We could even go as far as to discard the outright negation of the infinity of space in the Dialogue—which had to pass the censorship of the Church—and to oppose to it the passage of the letter to Ingoli where its possibility is just as strongly asserted. In the Dialogue, indeed, Galileo tells us, just as Kepler does, that it is:12

. . . absolutely impossible that there should be an infinite space superior to the fixed stars, for there is no such place in the world; and if there were, the star there situated would be imperceptible to us.

Whereas in the Letter to Ingoli he writes:13

Don't you know that it is as yet undecided (and I believe that it will ever be so for human knowledge) whether the universe is finite or, on the contrary, infinite. And, given that it be truly infinite, how would you be able to say that

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the magnitude of the stellar sphere would be proportionate to that of the orbis magnum, if this one, in respect to the universe, were rather smaller than a grain of millet in respect to it?

[paragraph continues] We must not forget, however, that in the selfsame Dialogue where he so energetically denied the infinity of space, he makes Salviati tell Simplicio—just as he himself had told Ingoli—that:14

Neither you nor any one else has ever proved that the world is finite and figurate or else infinite and interminate.

[paragraph continues] Moreover, we cannot reject the testimony of Galileo's Letter to Liceti, where, coming back to the problem of the finiteness and the infinity of the world, he writes:15

Many and subtle reasons are given for each of these views but none of them, to my mind,' leads to a necessary conclusion, so that I remain in doubt about which of the two answers is the true one. There is only one particular argument of mine that inclines me more to the infinite and interminate than to the terminate (note that my imagination is of no help here since I cannot imagine it either finite or infinite): I feel that my incapacity to comprehend might more properly be referred to incomprehensible infinity, rather than to finiteness, in which no principle of incomprehensibility is required. But this is one of those questions happily inexplicable to human reason, and similar perchance to predestination, free-will and such others in which only Holy Writ and divine revelation can give an answer to our reverent remarks.

It is possible, of course, that all the pronouncements of Galileo have to be taken cum grano salis, and that the fate of Bruno, the condemnation of Copernicus in 1616,

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his own condemnation in 1633 incited him to practise the virtue of prudence: he never mentions Bruno, either in his writings or in his letters; yet it is also possible—it is even quite probable—that this problem, like, generally speaking, the problems of cosmology or even of celestial mechanics, did not interest him very much. Indeed he concentrates on the question: a quo moventur projecta? but never asks: a quo moventur planetae? It may be, therefore, that, like Copernicus himself, he never took up the question, and thus never made the decision—though it is implied in the geometrization of space of which he was one of the foremost promoters—to make his world infinite. Some features of his dynamics, the fact that he never could completely free himself from the obsession of circularity—his planets move circularly around the sun without developing any centrifugal force in their motion—seem to suggest that his world was not infinite. If it was not finite it was probably, like the world of Nicholas of Cusa, indeterminate; and it is, perhaps, more than a pure contingent coincidence that in his letter to Liceti he uses the expression also employed by Cusa: interminate.


Be this as it may, it is not Galileo, in any case, nor Bruno, but Descartes who clearly and distinctly formulated principles of the new science, its dream de reductione scientiae ad mathematicam, and of the new, mathematical, cosmology. Though, as we shall see, he overshot the mark and by his premature identification of matter and space deprived himself of the means of giving a correct solution to the problems that seventeenth century science had placed before him.

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The God of a philosopher and his world are correlated. Now Descartes’ God, in contradistinction to most previous Gods, is not symbolized by the things He created; He does not express Himself in them. There is no analogy between God and the world; no imagines and vestigia Dei in mundo; the only exception is our soul, that is, a pure mind, a being, a substance of which all essence consists in thought, a mind endowed with an intelligence able to grasp the idea of God, that is, of the infinite (which is even innate to it), and with will, that is, with infinite freedom. The Cartesian God gives us some clear and distinct ideas that enable us to find out the truth, provided we stick to them and take care not to fall into error. The Cartesian God is a truthful God; thus the knowledge about the world created by Him that our clear and distinct ideas enable us to reach is a true and authentic knowledge. As for this world, He created it by pure will, and even if He had some reasons for doing it, these reasons are only known to Himself; we have not, and cannot have, the slightest idea of them. It is therefore not only hopeless, but even preposterous to try to find out His aims. Teleological conceptions and explanations have no place and no value in physical science, just as they have no place and no meaning in mathematics, all the more so as the world created by the Cartesian God, that is, the world of Descartes, is by no means the colorful, multiform and qualitatively determined world of the Aristotelian, the world of our daily life and experience—that world is only a subjective world of unstable and inconsistent opinion based upon the untruthful testimony of confused and erroneous sense-perception—but a strictly uniform mathematical world, a world of geometry made

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real about which our clear and distinct ideas give us a certain and evident knowledge. There is nothing else in this world but matter and motion; or, matter being identical with space or extension, there is nothing else but extension and motion.

The famous Cartesian identification of extension and matter (that is, the assertion that "it is not heaviness, or hardness, or color which constitutes the nature of body but only extension,"16 in other words, that "nature of body, taken generally, does not consist in the fact that it is a hard, or a heavy, or a colored thing, or a thing that touches our senses in any other manner, but only in that it is a substance extended in length, breadth and depth," and that conversely, extension in length, breadth and depth can only be conceived—and therefore can only exist—as belonging to a material substance) implies very far-reaching consequences, the first being the negation of the void, which is rejected by Descartes in a manner even more radical than by Aristotle himself.

Indeed, the void, according to Descartes, is not only physically impossible, it is essentially impossible. Void space—if there were anything of that kind—would be a contradictio in adjecto, an existing nothing. Those who assert its existence, Democritus, Lucretius and their followers, are victims of false imagination and confused thinking. They do not realize that nothing can have no properties and therefore no dimensions. To speak of ten feet of void space separating two bodies is meaningless; if there were a void, there would be no separation, and bodies separated by nothing would be in contact. And if there is separation and distance, this distance is not a length, breadth or depth of nothing but of something, that is, of

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substance or matter, a "subtle" matter, a matter that we do not sense—that is precisely why people who are accustomed to imagining instead of thinking speak of void space—but nevertheless a matter just as real and as "material" (there are no degrees in materiality) as the "gross" matter of which trees and stones are made.

Thus Descartes does not content himself with stating, as did Giordano Bruno and Kepler, that there is no really void space in the world and that the world-space is everywhere filled with "ether." He goes much farther and denies that there is such a thing at all as "space," an entity distinct from "matter" that "fills" it. Matter and space are identical and can be distinguished only by abstraction. Bodies are not in space, but only among other bodies; the space that they "occupy" is not anything different from themselves:17

The space or the interior locus, and the body which is comprised in this space are not distinct except in our thought. For, as a matter of fact, the same extension in length, breadth and depth that constitutes space, constitutes also body; and the difference between them consists only in this, that we attribute to body a particular extension, which we conceive to change place with it every time that it is transported, and that we attribute to space an [extension] so general and so vague, that after having removed from a certain space the body which occupied it, we do not think that we have also transported the extension of that space, because it seems to us that the same extension remains there all the time, as long as it is of the same magnitude, of the same figure and has not changed its situation in respect to the external bodies by means of which we determine it.

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[paragraph continues] But that, of course, is an error. And,18

. . . it will be easy to recognize that the same extension that constitutes the nature of body constitutes also the nature of space so that they do not differ in any other way than the nature of the gender or of the species differs from the nature of the individual.

[paragraph continues] We can, indeed, divest and deprive any given body of all its sensible qualities and19

. . . we shall find that the true idea we have of it consists in this alone, that we perceive distinctly that it is a substance extended in length, breadth and depth. But just that is comprised in the idea we have of space, not only of that which is full of bodies, but also that one which is called void.

[paragraph continues] Thus,20

. . . the words "place" and "space" do not signify anything which differs really from the body that we say to be in some place, and denote only its magnitude, its figure and the manner in which it is situated among other bodies.


. . . there cannot be any void in the sense in which philosophers take this word, namely as denoting a space where there is no substance, and it is evident that there is no space in the universe that would be such, because the extension of space or of the interior locus is not different from the extension of the body. And as from this alone, that a body is extended in length, breadth and depth, we have reason to conclude that it is a substance, because we conceive that it is not possible that that which is nothing should have an extension, we must conclude the same about the space

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supposed to be void: namely that, as there is in it some extension, there is necessarily also some substance.

The second important consequence of the identification of extension and matter consists in the rejection not only of the finiteness and limitation of space, but also that of the real material world. To assign boundaries to it becomes not only false, or even absurd, but contradictory. We cannot posit a limit without transcending it in this very act. We have to acknowledge therefore that the real world is infinite, or rather—Descartes, indeed, refuses to use this term in connection with the world—indefinite.

It is clear, of course, that we cannot limit Euclidean space. Thus Descartes is perfectly right in pursuing:22

We recognize moreover that this world, or the entirety of the corporeal substance, has no limits in its extension. Indeed, wherever we imagine such limits, we always not only imagine beyond them some indefinitely extended spaces, but we even perceive them to be truly imaginable, that is, real; and therefore to contain in them also the indefinitely extended corporeal substance. This because, as we have already sufficiently shown, the idea of this extension which we conceive in such a space is obviously identical with that of the corporeal substance itself.

There is no longer any need to discuss the question whether fixed stars are big or small, far or near; more exactly this problem becomes a factual one, a problem of astronomy and observational technics and calculation. The question no longer has metaphysical meaning since it is perfectly certain that, be the stars far or near, they are, like ourselves and our sun, in the midst of other stars without end.

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It is exactly the same concerning the problem of the constitution of the stars. This, too, becomes a purely scientific, factual question. The old opposition of the earthly world of change and decay to the changeless world of the skies which, as we have seen, was not abolished by the Copernican revolution, but persisted as the opposition of the moving world of the sun and the planets to the motionless, fixed stars, disappears without trace. The unification and the uniformization of the universe in its contents and laws becomes a self-evident fact23—"The matter of the sky and of the earth is one and the same; and there cannot be a plurality of worlds"—at least if we take the term "world" in its full sense, in which it was used by Greek and mediaeval tradition, as meaning a complete and self-centered whole. The world is not an unconnected multiplicity of such wholes utterly separated from each other: it is a unity in which—just as in the universe of Giordano Bruno (it is a pity that Descartes does not use Bruno's terminology)—there are an infinite number of subordinate and interconnected systems, such as our system with its sun and planets, immense vortices of matter everywhere identical joining and limiting each other in boundless space.24

It is easy to deduce that the matter of the sky is not different from that of the earth; and generally, even if the worlds were infinite, it is impossible that they should not be constituted from one and the same matter; and therefore, they cannot be many, but only one: because we understand clearly that this matter of which the whole of nature consists, being an extended substance, must already occupy completely all the imaginary spaces in which these other

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worlds should be; and we do not find in ourselves the idea of any other matter.

The infinity of the world seems thus to be established beyond doubt and beyond dispute. Yet, as a matter of fact, Descartes never asserts it. Like Nicholas of Cusa two centuries before him, he applies the term "infinite" to God alone. God is infinite. The world is only indefinite.


The idea of the infinite plays an important part in the philosophy of Descartes, so important that Cartesian-ism may be considered as being wholly based upon that idea. Indeed, it is only as an absolutely infinite being that God can be conceived; it is only as such that He can be proved to exist; it is only by the possession of this idea that man's very nature—that of a finite being endowed with the idea of God—can be defined.

Moreover, it is a very peculiar, and even unique, idea: it is certainly a clear and positive one—we do not reach infinity by negating finitude; on the contrary, it is by negating the infinite that we conceive finiteness, and yet it is not distinct. It so far surpasses the level of our finite understanding that we can neither comprehend nor even analyse it completely. Descartes thus rejects as perfectly worthless all the discussions about the infinite, especially those de compositione continui, so popular in the late Middle Ages, and also in the xviith century. He tells us that:25

We must never dispute about the infinite, but only hold those things to which we do not find any limit, such as the extension of the world, the divisibility of the parts of matter, the number of stars, etc., to be indefinite.

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Thus we shall never burden ourselves with disputes about the infinite. Indeed, as we are finite, it would be absurd for us to want to determine anything about it, to comprehend- it, and thus to attempt to make it quasi-finite. Therefore we shall not bother to answer those who would inquire whether, if there were an infinite line, its half would also be infinite; or whether an infinite number would be even or odd; and such like; because about them nobody seems to be able to think except those who believe that their mind is infinite. As for us, in regard to those [things] to which in some respects we are not able to assign any limit, we shall not assert that they are infinite, but we shall consider them as indefinite. Thus, because we cannot imagine an extension so great that a still greater one could not be conceived, we shall say that the magnitude of possible things is indefinite. And because a body cannot be divided into so many parts that further division would not be conceivable, we shall admit that quantity is indefinitely divisible. And because it is impossible to imagine such a number of stars that we should believe God could not create still more, we shall assume that their number is indefinite.

In this way we shall avoid the Keplerian objections based upon the absurdity of an actually infinite distance between ourselves and a given star, and also the theological objections against the possibility of an actually infinite creature. We shall restrict ourselves to the assertion that, just as in the series of numbers, so in world-extension we can always go on without ever coming to an end:28

All these [things] we shall call indefinite rather than infinite: on the one hand that we may reserve the concept of infinity for God alone, because in Him alone we not only do not recognize any limits whatsoever, but also understand

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positively that there are none; and on the other hand because, concerning these things, we do not understand in the same positive way that, in certain respects, they have no limits, but only in a negative way that their limits, if they had any, cannot be found by us.

The Cartesian distinction between the infinite and the indefinite thus seems to correspond to the traditional one between actual and potential infinity, and Descartes’ world, therefore, seems to be only potentially infinite. And yet . . . what is the exact meaning of the assertion that the limits of the world cannot be found by us? Why can they not? Is it not, in spite of the fact that we do not understand it in a positive way, simply because there are none? Descartes, it is true, tells us that God alone is clearly understood by us to be infinite and infinitely, that is absolutely, perfect. As for other things:27

We do not recognize them to be so absolutely perfect, because, though we sometimes observe in them properties that seem to us to have no limits, we do not fail to recognize that this proceeds from the defect of our understanding and not from their nature.

But it is hard to admit that the impossibility of conceiving a limit to space must be explained as a result of a defect of our understanding, and not as that of an insight into the nature of the extended substance itself. It is even harder to believe that Descartes himself could seriously espouse this opinion, that is, that he could really think that his inability to conceive, or even imagine, a finite world could be explained in this way. This is all the more so as somewhat farther on, in the beginning of the third part of the Principia Philosophiae, from which

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the passages we have quoted are taken, we find Descartes telling us that in order to avoid error,28

We have to observe two things carefully: the first being that we always keep before our eyes that God's power and goodness are infinite, in order that this should make us understand that we must not fear to fail in imagining His works too great, too beautiful or too perfect; but that, on the contrary, we can fail if we suppose in them any boundaries or limits of which we have certain knowledge.

The second of these necessary precautions is that,29

We must always keep before our eyes that the capacity of our mind is very mediocre, and that we must not be so presumptuous as it seems we should be if we supposed that the universe had any limits, without being assured of it by divine revelation or, at least, by very evident natural reasons; because it would [mean] that we want our thoughts to be able to imagine something beyond that to which God's power has extended itself in creating the world. . . .

which seems to teach us that the limitations of our reason manifest themselves in assigning limits to the world, and not in denying outright their existence. Thus, in spite of the fact that Descartes, as we shall see in a moment, had really very good reasons for opposing the "infinity" of God to the "indefiniteness" of the world, the common opinion of his time held that it was a pseudo-distinction, made for the purpose of placating the theologians.

That is, more or less, what Henry More, the famous Cambridge Platonist and friend of Newton, was to tell him.

Next: V. Indefinite Extension or Infinite Space