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The Arguments of Philoponus

Ancient Beliefs About Creation

The early Medieval cosmology referred to in the last chapter shows how easily the human mind was able to slip back from the sophisticated ideas necessary to penetrate the appearances of nature. But there is never any escape from the alternatives, either that reality is only intelligible if we see through its appearances, or that reality is not intelligible if it is taken to be simply what it appears to be. The latter option is easy, but always a dead-end. The changes which took place in the sixth century were occasioned by the fact that the infinite and the indefinite were not very well understood at that time, which would excuse the reaction against them.

However, the after-effects of this situation have continued to shape some of the cosmological ideas which orthodoxy uses to this day, where the finitude or otherwise of creation is concerned. But in the light of what is known about infinity today, this subject should no longer be a stumbling-block in doctrinal matters. Much of the original problem was owing to the prestige of Aristotle, whose idea of the infinite was so defective that it could scarcely be distinguished from a very large finite quantity. His bias in favor of the finite is the negative side of an appreciation of the finite which is peculiarly Greek. For Greek thought, God was not equated with the infinite, and justifiably, because the Divine nature cannot be comprehended by any single concept, even that of infinity. Given that God transcends conceptual polarities like that of finite and infinite, and of universal and particular, the infinite and the finite can be conceived as a duality on a lower level, where the finite balances the infinite as quality balances magnitude. Similarly, the microcosm is a counterpoise to the macrocosm.

The Christian doctrine of the Incarnation depends on this insight, so that it could hardly have taken root in cultures where these ideas had not been developed. However, in its original context, this understanding of the finite went so far as to exclude the irrational and the unknown, an imbalance which Christianity was to correct. Nevertheless, finitist views still prevailed for a long time in areas where they were not relevant.

The Infinity Arguments

Because the thought of pagan antiquity was not at ease with the idea of infinity, it often failed to distinguish between it and the indefinite. Clear and distinct knowledge was sought, which requires finite objects and not the kind at which the mind staggers. One indication of this attitude can be seen in the Pythagorean table of opposites, where the infinite is placed on the same side as Evil, and the finite on the same side as Good. Philoponus saw how this negative view of the infinite could be exploited in relation to theories about the age of the world. If the world was without any temporal beginning, it could only mean that an infinite length of time must already have passed, and that this must be an actual and no merely potential infinity.

The importance of the latter point derives from Aristotle’s treatment of this subject, which was accepted without question by both sides of the issue. For Aristotle, the infinite was to be conceived on a finitist basis, as an ever-extendable finitude, built up by a continual addition of finite increments. If a truly actual infinity were to exist, this process of addition would have to have been completed, but in practice this could never be, if endless addition was all that was involved. Where it is a question of dividing up a line, there is no limit to the number of points on it which can be constructed, so that one can say that the line amounts to a potential infinity. Because the infinitude of points in it is only a potential one, the traversal of the line presents no problem. But where there is an actual infinity the situation is quite otherwise, because an actual infinity could only be traversed through all its successive elements, which would amount to making a full count of them. Such a count was held to be impossible, and therefore it was also held to be impossible to traverse or go through an actual infinity, which was supposed to be unrealizable in itself.

According to Aristotle, in the de Caelo, the infinite was conceived as a maximum quantity to which nothing could be added; any increase in it would be contradictory. But elsewhere, he treats it as a very large finite quantity which could always be increased. Neither of these conceptions is adequate to separate this idea of the infinite from that of the finite, and for this reason Aristotle took issue against Anaxagoras who held that the infinite could be made up of lower-order infinities. Such a possibility would mean that the infinite could in some real sense be multiplied, since the absolute infinite would thus be the product of infinity by the number of its infinite component groups. This would be even more opposed to its unique nature than being subject to addition. The Aristotelian idea of infinite quantity made different orders of infinity contradictory, as also additions to it.

At this time, no one was believed to know more about this than Aristotle, and the idea that infinite quantities could be capable of operations analogous to those of finite quantities was still unknown. But the Aristotelian idea of infinity existed alongside the conception of creation discussed above, according to which the world had existed over beginningless time. Now it was Philoponus who first realized that there was a contradiction between these received ideas concerning the infinite and the supposedly infinite age of the world, and his new insight was to enable Christian thought to take the initiative against the domination of Aristotelian and Platonic ideas over the intellectual life of the time.

If there was no time at which the world began, must it not mean that an infinite length of time has passed up to the present time? This, moreover, must be a fully actual infinite, realized by the accumulation of an unlimited number of years. The present era has been reached by the passage of time over an infinite multitude of years, therefore, in which case an actual infinity must have been traversed in finite increments. Yet other apparent impossibilities were involved: although an infinite length of time may have passed up to the present, time is still passing, and so the infinite time-span is continually having new days and years added to it. But according to Aristotle, no such additions to the infinite were possible.

Moreover, an infinite time must mean various multiplications of the infinite as well. In an infinite time there must have been an infinite number of human beings, but then there must have been an infinite number of horses as well, and the total number of humans and horses must be two times infinity. If there had also been an infinite number of dogs, the result would make three times infinity, and so on. Similarly, in an infinite length of time, Saturn must have completed an infinite number of revolutions, but in the time for each one of these, the sun will have completed about thirty revolutions, so that in total the sun will have made thirty times infinity revolutions. At the same time, the moon revolves about twelve times for every one of the sun, so that it must have revolved twelve times infinity in relation to the sun, to say nothing of three hundred and sixty times infinity in relation to Saturn.

Since an actual infinity, its traversal, and its being added to or multiplied were all held to be impossible by general agreement, the only alternative seemed to be the conclusion that the world could only have existed for a finite time. Clearly the Proclan arguments did not meet this kind of attack, and the pagan side of the issue had no satisfactory reply. Simplicius, one of the last philosophers of the Academy, argued that the humans and other creatures which had existed over infinite time had all ceased to exist with the passage of time, so that they could not be said to make up infinite quantities. However, as Sorabji points out, while they as individuals do not at any given time make up an infinite quantity, their time-sum does, and in any case the non-existence of past things is only true in the weak sense of no longer being present to us.

Because no new developments arose in connection with the idea of the infinite, therefore, Philoponus’s infinity arguments prevailed, and left the Christian side free to date the beginning of creation by a literal reading of biblical chronology, to a date around 4000 bc, which remained without any serious challenge until well into the nineteenth century, when geologists began to work out the age of the earth. During the intervening period, the arguments of Philoponus continued to be repeated in the context of a geocentric idea of creation by Christian, Muslim, and Jewish theologians right up to the thirteenth century, when they were still being used by Saint Bonaventure, as this account of his work shows:

If the world had existed from eternity, it would follow that it is possible to add to the infinite. . . but it is impossible to add to the infinite. . . if one considers simply the past, then one would have to admit an infinite number of lunar revolutions. But there are twelve lunar revolutions to one solar revolution. Therefore we are faced with two infinite numbers, of which one is twelve times the other, and this is an impossibility.1

As this quotation indicates, even the same examples as those of Philoponus were still being used. The idea of the infinite had not developed, and would not begin to develop until the fourteenth century, when new advances were made in logic beyond Aristotle. But even when the Aristotelian conception was known to be wrong it did not bring about a reassessment of the belief that the world had a beginning in time, because the idea of creation required an expression in temporal form.

Corrections to the Idea of Infinity

The belief that there cannot be an actual infinite quantity was owing to a failure to see that infinity is a reality in its own right, independent of the finite. There is in any case something contradictory in setting up the idea of the infinite and making it a result of adding up finite increments. One is not far from saying that infinity is self-contradictory. However, the infinite is now known to be a quantitative reality in an order of its own, where it is distinguished from finite quantity in essential ways, one of which is that the relation of whole and part has a quite different meaning. In finite quantity, the part is in every way less than the whole, while the size of the whole is determined by the number and sizes of the parts. In infinite quantity, these relations do not hold, because it can gain or lose finite parts without alteration, and just as importantly it can have parts which are themselves infinite, since the infinite natural number series is made up of others, such as the odd and even number series, which are both infinite as well. As finite quantities are made up of finite parts, infinite quantities are made up of infinite parts.

A clear example of an infinite quantity which is made up of infinite parts is the series of natural numbers. It can be divided first into two parts comprising the odd and even numbers, both of which are infinite, despite the fact that they each contain only half the numbers which make up the whole. Similarly, an unlimited number of infinite series can be set up on the basis that each pair of numbers in them differ by the same ratio. There is no limit to the number of infinite series which can thus be drawn from the natural number series, since any series containing any systematic selection from the natural numbers will also be infinite. Innumerable further possibilities result when these common differences and common ratios themselves change according to numerical laws. To these may be added infinite series of fractions with the same possibilities of development. The ancients were familiar with the series of triangular, square and pentagonal numbers, which were clearly infinite while containing fewer numbers than does the natural number series, but nevertheless the conclusion that an infinity can be made up of infinite parts was never generally realized.

The natural number series is thus an infinite collection which is not merely made up of an infinite number of finite members, but also of an infinite number of infinite members, in the form of all the different infinite series. On this basis, the existence of actual infinities and the multiplication of infinities are established beyond question. The objection that it is not possible to add to or subtract from an infinite quantity is similarly groundless as soon as the question is considered outside the bounds of Aristotelian definitions. The length of an infinite series cannot be affected by the addition of finite quantities to it. If it could be so affected, there would have to be a common measure between the finite increment and the infinite series itself, which could only mean that the latter was in fact finite and not infinite at all. Thus the fact that new time units are being added to a supposedly infinite time span does not mean that it was in any way ‘less infinite’ before the addition, or ‘more infinite’ after it.

As for the argument that the past cannot be of infinite length because it ends at a determined point, the present moment, this needs no refutation beyond the fact that the natural number series begins from unity (or from zero, if we make allowance for the fact that this beginning is not specific to number as such). Here, an infinity obviously proceeds from a fixed point.

There is even a spurious argument against the immortality of the soul which results from this belief that an infinite series cannot have a terminus. Could the soul be immortal if it begins from a fixed point like that of birth or conception? In fact it can be so, similarly to the way in which the natural number series is infinite while starting from unity or zero.

Such is the mathematical context of Philoponus’s argument that an actual infinite quantity cannot be traversed. Despite appearances, this is based intentionally or not on an argument used by Proclus in Elements of Theology, Prop. 206. (This no doubt made it so much the more effectual when used against the Neoplatonic position.) Here, Proclus argues that a series with no beginning can have no end, and vice-versa; that there must always be a symmetry in this respect. For this reason he states in the same place that the soul can only reincarnate without end, since there would have to be an end to this beginningless process if the soul were finally to escape it. While this property of having neither beginning nor end is true for some kinds of infinite quantity, Proclus’s mistake was to conclude that it must be true for all such quantities. The existence of infinite series ending or beginning at one fixed point refutes both this argument of his, and Philoponus’s argument that an actual infinite cannot be traversed.

As the present time must needs be an ‘end’ of an infinite length of past time, it could of course never have been reached if no beginning had always to mean no end. But in view of the number series, starting from or ending with 1, there is no reason to believe this. Thus it was ironic that one of Proclus’s very few erroneous arguments should have been considered effective against his philosophy. The idea of infinity involved in this argument is all too dubious for more reasons than those just given. If the word ‘traversed’ was just a synonym for ‘counted’, this statement would be nothing more than a repetition of the original assertion that there cannot be an actual infinite. If, however, it should refer to the action of a real agent, one need only ask ‘By whom?’ to reveal the confusion involved. If the answer was ‘By God,’ it would be obviously untrue, since omnipotence cannot be so incapable. But if the answer was ‘By any finite being,’ it would be just a statement of the obvious, like declaring that one cannot measure the depth of the ocean with a six-inch ruler.

The idea of traversal springs from a half-instinctive belief that the temporal continuation of the universe somehow depends on whether an imaginary agent can live through it all. To think in this way is to presuppose that there must have been a first day of time, an infinite number of days ago, and that from this Day Zero an eternal wanderer begins his career, having to live through an infinite number of days, that is, an actual infinity, in order to reach the present time. The argument then goes that no such result can be reached by this method, so that the present could only have been reached over a finite length of time. Such a finite past time means that there must have been a first day of time or Day Zero, of course, but this was precisely what had to be assumed in order to rule out an infinite traversal. If an infinite length of time were to follow a first day of time, the time series could never conclude now or at any other time without ceasing to be infinite (two fixed points have to mean finitude). Thus the traversal argument is really one of question-begging; to argue from a first day of time is the same as to say that all past time exists between two fixed points (the first day and today), i.e., that it is finite. It should hardly now be necessary to add that the existence of an endless time series cannot really depend on the activities of imaginary time travelers, and that to think otherwise would be a subjectivist delusion like thinking that Mount Everest only began to exist after it was first climbed.

The fact that this argument continued to be quoted like an oracle for centuries is a sign that it pointed to a result which was too desirable to allow room for criticism. The resulting finite view of the universe fulfilled a spiritual need which could not otherwise be met except by means of metaphysical ideas which were not popularly explicable. Unless man is a significant actor on the stage of the universe, spiritual values will seem to have no more than a purely subjective meaning; they can only have a cosmic role if man has one. Accordingly the world must be seen to be finite, if man is not to appear to be nothing in relation to it. A valid alternative to this is that the relation of man to the world should be understood more profoundly in terms of microcosm and macrocosm, an idea not suitable for popularization in the present era.

The Mature Creation Paradox

If the arguments of Philoponus are compared with those of Proclus, it is easy to see why they are only of historical interest. Their essential content consists of controversial devices relevant only in late antiquity. The Proclan arguments, on the other hand, have most of them a timeless quality. They express a set of universal realities which stand by themselves above the flux of cultural conditions, despite Proclus’s uniformitarian bias; and however they may be criticized, most of them are not liable to be shown not to be arguments at all. Nevertheless, a lack of substance in Philoponus’s counter-arguments does not mean that one can ignore the position which they were used to support. It could still be true that time as such is of finite length, for reasons which so far have never been determined. If this were the case, a number of important results would follow, not least where our ability to relate to a real past is concerned.

This is because, contrary to common sense, an absolute beginning of time and movement brings with it the possibility that it could have begun at any moment, even up till a moment ago. In the face of this paradox, it is not relevant to argue that the world contains written records and artifacts produced in the past, or persons with memories. If time and the world were of recent origin, it would contain all such things from the start, and human faculties would be deceived. Even if the time of the beginning was as remote as is usually believed, so that it contained only Adam and Eve, they, being complete human beings in early maturity, would be in that state without having lived through the previous twenty-odd years of development that their mature state would require in the course of nature. The equivalent of this would apply to the earth, sun, moon, and stars which made up the world in which they were created. Insofar as creation has to mean mature creation, therefore, it must begin from a state which manifests the outcome of a history which has not happened. If we allow that time and the world began at some moment in time, we cannot assume that we have the right to relegate that moment to some period in remote antiquity which is too far off to concern us.

In practice, no one is willing to sacrifice common sense to this paradox and believe that the world began a moment ago, or even a few centuries ago, even though there is no logical basis for dismissing such an idea. The reasons for this are of more than psychological interest, although on a basic level the rejection of the paradox is no doubt owing only to the shock it gives to customary ideas of reality. But more significantly, the general disbelief results from something most minds feel instinctively about their knowledge of time. This is an intuition that the temporal process is per se an endless continuum, in which continuity is an intrinsic property. The intuition is that if we do not know that much about time, we do not know time at all. There is a connection here with the Principle of Plenitude, which concerns not only the self-extension of the totality of being, but of each modality of being, such as time. This does not exclude any number of relative limits which arise from external causes, such as those of world-ages, but it does exclude the idea that time may be limited of itself. If this idea of the temporal continuum embodies a true insight, there need be no doubt that historical events have actually happened, even if not always in the ways supposed.

On the other hand, the idea that time began at a moment in time would mean that we could have no such assurance. There would in fact not be a single historical event which need actually have happened, no matter how much evidence there was for it. Besides the shock to common sense, this is an impossible conclusion from the point of view of the monotheistic religions, which are so largely based on historical revelation, and consequently it is paradoxical in another way that the idea of a temporal beginning of time should be so much associated with orthodox beliefs. The alternatives thus seem to be either that time is, if not infinite, an indefinitely-extensible continuum, or that our knowledge of the past should be theoretically unfounded.

The Question of Coherence

I have added the phrase ‘at a moment in time’ in connection with the beginning to emphasize the fact that the very idea of an absolute beginning of time in this world would make time itself begin (and end) just like any series of events in time. There is something deeply incoherent in this, as can be seen when it is understood that time is definable as the condition subject to which all things capable of beginning and ending do in fact begin and end. On this basis, to speak of a beginning or an ending of time in an absolute sense is as much a category mistake as to speak of a ‘history of time’. The mistake involved can only appear subtle because of the intangible nature of time, but its crassness is clear enough when one sees how it invites comparison with an idea that a cinema might be wrecked by an explosion which takes place in one of the films shown in it. Such a view of time is of a piece with Cosmas’s idea of cosmic space, which was said to be contained within four walls, as if they could be anything but boundaries in an even greater spatial extension.

In case this form of the above argument should seem to concede too much to the conventional distinction between time and its contents, it may also be expressed in a way which shows it does not rely on this distinction. If the world of time were of finite extent, it would have to start at one moment and end at another. But neither starting nor stopping are ever known except as contained in a time-process. Thus ‘starting’ presupposes a continuous passage of time before, during, and after the event, and so likewise does ‘ending’. Therefore if the world-time process itself started and stopped, this could only take place in a higher-order time or ‘meta-time’. But the introduction of the latter would clearly defeat the object of reducing time as we know it to a finite quantity. If we were to argue that this higher-order time was also finite, the question of its starting and stopping would similarly require an even higher order of time, and so on in an infinite regress. In this way, a supposedly finite time gives rise to its opposite, so that there is no possibility that time could be overcome on its own level.

If then there were no intermediate reality between eternity and time, the idea of time as such ‘beginning’ could only be a confusion, for the reasons given. However, the continuity of being requires that the interval between eternity and time should be filled by at least one kind of ‘meta-time’, one which possesses attributes both of finite time and of eternity. Such a higher kind of time was known to the Scholastics as aevum, and it is conceived as having a duration greater than historical time by whole orders of magnitude, so that cosmic cycles or world-ages would only be successive moments relative to it while, possibly, the whole extent of time on our level of being would amount to just one cycle of aevum. If time and the world as we know it were to ‘begin’ and ‘end’ in aevum, therefore, this would imply finitude for one special kind of time, not for time as such; endless duration would still be there in a different form.2

Time and the world have been treated as being inseparable in the above discussion, not only from Platonic premises, but because the idea that there was an ‘empty time’ before and after the duration of the cosmos would extend the idea of time into a realm in which it could have no meaning. The fact that such time before or after this world was made up of successive moments would be completely unverifiable. There would be nothing at any one of its moments to distinguish that moment from any other before or after it, and by the Identity of Indiscernibles, such moments, being indistinguishable, would thus be all one and the same moment, and therefore there would be no temporal extension. Some such confused idea of a ‘time before time’ enters into the supposition that somehow time was already passing when God created the world at a certain moment, and it shows how little substance there is in the idea of an absolute beginning in this world.

Curiously enough, the doctrine that the world was created ex nihilo provides yet another reason for excluding the idea that time and the world are of limited extent, in a way which is hardly ever noticed. If the material world were created out of nothing, not only would there be nothing before and after it, there would be no other cosmic reality related to it in any way. In this case, the world is bounded by nothing, which is to say unbounded. This argument requires only that the word ‘nothing’ should mean what it says, and not signify some formless quasi-something which may fill up all the unwanted space and time beyond a supposedly limited world. However, Aquinas claims that this way of thinking serves to make the world eternal in the same way as God:

The assertion of the world’s eternity; the assertion of the eternity of the world’s matter, out of which at a certain time the world began to be formed. . . . For in all these cases, something beside God is claimed to be eternal; and this is incompatible with the Catholic faith.3

The assumptions made here are that there is only one way in which a being can be eternal, and that the only alternative to eternity is time of finite length, whereas it has already been shown that we are not confined to these alternatives. Eternity comprises the whole of reality in a totum simulwhich has no need of change for the realization of its possibilities, since they are all realized together once for all. Between this and finite time, the natural intermediary is that of endlessly extended duration, which we have just encountered as aevum.

From an orthodox point of view, Aquinas’s argument here cannot be acceptable without considerable qualification because the Christian faith also teaches that the saints in heaven participate in the Divine eternity, so that they too must effectively be eternal, even if in a dependent relation; they cannot be supposed to exist only a finite time in the hereafter just because only God is eternal per se, an elaboration of the idea of the eternal which Aquinas seems to ignore. Elsewhere, however, he argues implicitly in the opposite sense, where he argues that creation must have the maximum possible diversity if it is to correspond to the power of the Creator:

Now by the fact that the active power is actualized the effect receives the likeness of the agent. Hence there would not be a perfect likeness of God in the universe if all things were of one grade of being. For this reason, then, there is distinction among created things: that, being many, they may receive God’s likeness more perfectly than by being one.4

Here it is frankly allowed that there should be a correspondence between the power of the Creator and the range of content in creation, and it is therefore all the more strange that no account is taken of the way in which this also applies to the quantitative extent of the creation in time. This must be requisite if ‘the passive potentiality of matter’ is to be ‘completely actualized.’5

Aquinas’s argument in the above, and the quantitative extension of it I am suggesting, are logically connected by the idea that the realization of all the possible species and grades of beings should naturally require an unlimited expanse of time. Such a conception need involve no more than the idea of indefinite extension, without applying terms like ‘eternity’ and ‘infinity’ which cause confusions because they apply primarily to Divine attributes. In conclusion, therefore, the theological reasoning against this idea is indecisive, in view of the different possible meanings for ‘the world’ which I referred to at the beginning of Chapter XII; they cannot all have the same consequences. While the time in which human experience takes place may be finite in relation to higher orders of being, it can only be perceived as finite on its own level by a confusion of thought.

1. Frederick Copleston, A History of Western Philosophy, vol. 2, pt. v, chap. 27 (3).

2. The idea of time as self-extensive by its very nature expresses a property it has in common with all modalities of being. All of time’s limits, like those of cycles or world-ages, are only imposed on it by other realities, as spatial extension is accidentally limited by some kinds of objects in it.

3. SCG, vol. ii, chap. 38, [I6].

4. Ibid., chap. 45, [3].

5. Ibid.

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