Eternity and Infinity in the Scholarship of Al-Ghazālī and Aquinas

The definitive starting point for deciding if there are infinites, given a medieval context, is to understand that in asking the question, Aristotelians are asking if anything without bounds exists (Moore 2001, 35). Since this seems vague, various allocutions should be given to track different senses of things that are boundless. These senses of boundless are those things which are invisible, processes not readily having termination, and endless supplies (Aristotle III, 4). Whatever technical addendums which are going to be made by Aristotle will take these into account. This is done especially with care for describing infinity in sensible physical terms, per Aristotle’s preference for more of an empirical argumentative style.

Aristotle’s infinite may be split into two sorts of taxonomical distinctions. The first is between divisible and extended infinities (Aristotle III, 4). A way to draw this distinction is to see the difference between taking a finite object and extending it onward, via addition, or taking a finite object and breaking it down into infinitesimal parts, via division. This may not alert one to the existence of such fables but instead gives a good account of what kind of physical things are being discussed, really small and really large. The second distinction which is made is between potential infinity and actual infinity (Aristotle Phys. III, 6). The more general distinction of something existing potentially versus existing in actuality is made elsewhere in Book III of Aristotle’s Physics. “The same thing, if it is of a certain kind, can be both potential and fully real, not indeed at the same time or not in the same respect,” (Aristotle III, 1). For example, a deflated basketball could exist as simply a piece of rubber which also has the potentiality of being a ball. The ball thing and the piece of rubber thing are there but only at different times. The same distinction is said to plague the debate of infinities, except for the fact that there are no actual infinities.

Eternity, being a temporal-like[1] infinity, is one of the infinities which concern this discussion. Specifically if one is going far back in time, one is said to be considering infinities of addition i.e. big infinities. One claim which was understood is that, “Time indeed and movement are infinite, and also thinking, in the sense that each part that is taken passes in succession out of existence,” (Aristotle iii 8). Thus from this we may understand that Aristotle took time to have no single beginning and no single end (Duhem 1985, 5). The way to explain this as consistent with the denial of actual infinities is to explain how a potential temporal infinity of addition exists. The solution is interpreted by Moore in his reading of how Aristotle relates time with motion (there being no time without motion). Moore’s diagnosis of Aristotle is that a potential temporal infinity is one that does not occur all at once, as Moore details (Moore 2001, 39).[2] Leaving this up to Moore’s reading, we obtain time going back and forward for a potential infinity which is to say that time occurs infinitely but not all at once.

The other sort of infinity which concerned Aristotle was infinities of magnitude, whether it was the magnitude of stuff or space.  He claims that, “Magnitude is not infinite either in the way of reduction or of magnification in thought,” (Aristotle iii 8). This makes sense when we consider that infinity in magnitude is an actual infinity which cannot exist by Aristotle’s reasoning.

Given these Aristotelian details, a medieval problem of eternity may be acquired. “There is some antagonism between these two propositions of Peripatetic philosophy:

1.) The universe has existed or could have existed from all eternity.

2.) Something infinitely large in actuality is impossible,” (Duhem, 80).

This issue arises from the works of Richard of Middleton, as cited by Duhem. Eternity is taken to be potentially infinite of something time-like going backward. The argument which Duhem cites from Richard sums up as: assuming that there is an eternity, then god can realize and then create an actual infinite magnitude of any given object, souls, or space itself specifically such that there is actually an infinite amount of that thing created by god. However it has been shown that there can be no actual infinite magnitude of anything so there is no eternity. Furthermore, this makes acts of creation by a god limited to infinite in actuality mixed with potentiality (the actuality being god’s realization of the magnitude but only creating it so at least corporeally it is only a potential infinite).[3]

Al-Ghazālī’s Account of Infinite

The response of some medieval thinkers of the Arabic speaking tradition was to keep eternity and defend that actual infinities do exist, but only certain sorts. The most notable of these authors was al-Ghazālī, who followed in the intellectual footsteps of Avicenna. In the incoherence of the philosophers, al-Ghazālī criticizes those philosophers which propose that time had no beginning and will have no end, (Al-Ghazālī 2000, 47). This critique in no way means that he didn’t accept eternity. In fact, al-Ghazālī frames the issue such that his opponents who believe in the past being infinitely back accept the world being “pre-eternal,” (Al-Ghazālī 2000, 12). This situates time as an entity which relates to corporeal things, (Al-Ghazālī 2000, 83). In this way there is a beginning of time which originated from a god.  Eternity is, by al-Ghazālī’s lights, a sort of non-corporeal part of god’s existence wherein this god existed before it created the world and after it annihilates the world. In this sense, eternity is treated like god’s time that overlaps with actual time.

However the problem states that other infinities arise out of the eternal. In discussion four, al-Ghazālī harkens to Avicenna when he claims that there is no problem in certain infinities, such as those of souls (Al-Ghazālī 2000, 83). This connection comes as a result of al-Ghazālī stating, “Souls have no connection to one another and have no order either by nature or position,” (Al-Ghazālī 2000, 80). For Avicenna, and assumably al-Ghazālī, actual infinities have two conditions. The first of these is recapitulated by al-Ghazālī above. Jon McGinnis, a contemporary scholar, refers to this as an ‘ordering condition’ and attributes it to Avicenna (McGinnis 2010, 20). Just like any set of numbers[4] there is a logical necessity to the arrangement of where each value goes, ordered from certain numbers being between others as in the case of natural numbers. The other condition for an actual infinity, reported by Jon McGinnis as the ‘wholeness condition’, is that infinities are, ”predicated of sets all of whose members exist together,” (Jon McGinnis 2010, 20). This however is consistent with Aristotle’s notion that potential temporal infinities, as cited of Moore above, do not occur all at once.[5] When reading these two conditions for actual infinity, one reads them conjunctively as necessary conditions for non-existing actual infinites. But when read with an exclusive disjunctive, these are sufficient conditions for existing actual infinities. One then gets an argument from Avicenna, what is called the mapping argument, that the conjunctions, as stated before, of these conditions constitute actual infinities which do not in fact exist (McGinnis 2010, 20). Whereas what al-Ghazālī aforementioned harkened from Avicenna, was that infinity of souls is an existing actual infinity which meets the wholeness condition but does not meet the ordering condition.

Going back to Duhem’s recitation of how certain historical figures handled the tension between eternity and there being no actual infinities he implicates that, “Avicenna and al-Gazali, if not denied, at least restricted the second [proposition],” (Duhem 1985, 81). The second proposition as stated before is that there are no existing actual infinities. What one takes to be the restriction made is that in order to qualify as a non-existing actual infinity are wholeness and ordering. If one looks at the three other infinities actualized by eternity; infinite in magnitude of any given object, souls, or space itself one can formulate utilizing the sufficient conditions in such a way that each are existing actual infinites. Al-Ghazālī already recites above which condition does not hold for creating infinity of souls, the ordering condition. The same is true for any other object, infinity of widgets which god makes for eternity has no order that is not also arbitrary such as time. Space fits the wholeness condition in that any given part of space exists at the same time as any other part of space. The ordering however is not made all that intuitive as one may choose any starting point of space and be able to order outward, ergo infinite space does not meet the ordering condition. Consistent with this one may read al-Ghazālī defending god’s power to make infinite space (Al-Ghazālī 2010, 37-39).

Aquinas’ Account of Infinity

Later on, Thomas Aquinas expounded upon non-existent actual infinity and eternity, found primarily in Summa Theologica, in a manner which bears on the tension between these two propositions. Aquinas first and foremost agrees with Aristotle on the impossibility of the infinite magnitude of objects, souls, and limits of space. In response to the question of whether an actual infinity can actually exist, he states, “the accidents follow upon the substantial form, it is necessary that determinate accidents should follow upon a determinate form; and among these quantity,” (Aquinas 1 VII 3). Here one may take his allocution to be speaking in terms of various Aristotelian metaphysical classifications. The basic point is that it is the case that for something to exist in nature it needs a specific accident, like quantity, such that the accident is determined. Since infinities are indeterminate things that exist in nature, they cannot actually be constituted with infinities.  He states that the same is true for the motion of bodies, defending on the finitude of space. In defending this physical version of infinity elsewhere he defends an account of god as an infinite being (Aquinas 1 vii 1). It is not pertinent to the current argument but important to keep in mind that the notion of actual metaphysical infinities, which Aristotle never discusses, is a position which Aquinas holds.

The way in which Aquinas deals with the tension between the propositions of eternity and there being no actual infinity is by denying that eternity exists in a manner which would cause an issue. From the start, it was taken as an article of faith, “that the world did not always exist,” (Aquinas 1 xlvi 2). So in his view time had a beginning. However he does claim a difference between time and eternity. This point is most obscurely hidden in Summa Theologica when he puts forward, “we may say that [prior] signifies the eternity of imaginary time, and not of time really existing; thus, when we say that above heaven there is nothing, the word “above” signifies only an imaginary place,” (Aquinas 1 xlvi 1). In describing how god is ‘prior’ Aquinas gives the contemporary reader what may seem like a spooky idea of imaginary time and place. In fact this is consistent with, and evidence of, Aquinas believing in a B-theory of time (Brenner 2010, 18). That is to say, Aquinas probably held that god is outside a finite space-time manifold. Although it is unlikely Aquinas held to this exact position as there was not enough mathematics to come up with something as sophisticated as a manifold. Instead, picture Aquinas’ view more rudimentarily, like a marionette stage, sans control of the subjects, where god is simply outside the space-time of the show. Notice however that given there being no argument for a beginning of time and the fact that Aquinas calls upon the reader to think of god’s eternity like imaginary time, one may obtain that there is not an argument for eternity. One is meant to take on faith that eternity is real in this way. This leaves Aquinas with all of his infinities as either metaphysical or not existing, which is contra Aristotle’s intuition of infinity as purely a physical notion.

The major break in the tension between eternity and there being no existing actual infinities for Aquinas, is harmonizing a sort of eternity which does not meddle in existent actual infinities. When god exists for eternity, his creation is meant to begin and end with time and not with him. Stated by Aquinas, “Therefore from the eternal action of God an eternal effect did not follow,” (Aquinas 1 xlvi 1). This essentially collapses the magnitude of objects, souls, and space into a determinate effect that is not infinite. As Duhem rightly points out, Aquinas “attempted to establish an agreement, or to mitigate the disagreement between the two propositions,” (Duhem 1985, 81). Aquinas can at least be described as trying to save both propositions.

A Hypothetical Dialogue between Philosophers

Although no such dialogue took place it would be of interest to the issue at hand as to how the medieval philosophers discussed above would argue against with the others. For instance, since Aquinas dates his text after al-Ghazālī, he is able to respond directly to some of his arguments. In the reply to the question of whether an infinite multitude can exist, Aquinas says, “[accidental infinity] is impossible; because it would entail something dependent on infinity for its existence; and hence its generation could never come to be,” (Aquinas 1 vii 4). Aquinas throws down the gauntlet about what he finds wrongheaded about al-Ghazālī’s, and furthermore Avicenna’s, affirmation of there being actual infinite magnitudes. The trouble for these two Arabic scholars is to show the traversal of infinity. Were they to do so, they would be ignoring the Aristotelian forbearing to not demonstrate that an infinite magnitude of objects, souls, and space can be traversed.

In al-Ghazālī’s defense, one may go back to Avicenna through the scholarly work of McGinnis. “Avicenna believes that there are actual infinities… Avicenna does not believe this based on unanalyzed intuitions about the objects of God’s knowledge,” (McGinnis 2013, 20). The reason for citing this is to stake out the claim that the acceptance of actual infinities existing may go against one’s Aristotelian intuitions. By defining an actual existing infinity as either exclusively whole or exclusively ordered, one obtains infinity of magnitude where it does not need to be possible for them to be traversed. In the case of an infinite magnitude of objects, and souls for that matter, since there is no ordering the traversal of them, is an arbitrary state of affairs to constrain them. One should say this constraint is not only not met, but does not need to be met. The same goes for space, the infinite magnitude is out there and not here on earth so why should it be conceivable to humans that space is to be traversed?

The second point which Aquinas would, and did in fact raise, is that the metaphysics offered of souls is wrong. Since this disagreement is less on the point of infinity, a distinction should be made between these two views for clarity’s sake. Aquinas’ view of souls is taken from Augustine i.e. “that souls separated from their bodies return again thither after a course of time,” (Aquinas 1 xlvi 2). The general thesis expounded elsewhere is that the soul is immaterial and subsists so as not to be a part of corporeal life, after death (which has no actual infinities) (McInerny et al. 2014, 12-13). Al-Ghazālī instead defends that, “[souls] have singular entities and a possibility preceding [their] origination,” (Al-Ghazālī 2000, 45). Within the discourse al-Ghazālī would say that given the eternity of god, infinity of souls were made before they were placed, the key term being origination, their corporeal bodies. Therefore souls are, to al-Ghazālī, also immaterial but do more than subsist, they additionally predate their being in bodies. This marks the distinction for the two authors between souls being formed around the time a corporeal thing was fashioned, per Aquinas, and the souls being formed at eternity, per al-Ghazālī.

To shift the discussion, al-Ghazālī may wish to also question Aquinas’ view of eternity. The difference is that for al-Ghazālī, eternity is on a similar line of time, or time as a line segment at least overlaps on top of the line that is eternity, pointing forward and backward. In contrast, eternity for Aquinas is apart from time such that god looks on from eternity to time, which at least has a beginning focus. Since Aquinas’ eternity relies perfectly on faith the best al-Ghazālī, can rebut with is pinning him as a heretic. “They have entirely cast off the reins of religion through multifarious beliefs,” (Al-Ghazālī 2000, 2). Thereafter al-Ghazālī cites the Qur’ān to deem those pronounced as disbeliever’s in his god. This response is not very substantive and only is meant to contradict Aquinas’ claim to a revealed truth. The problem here is that each of the two parties claims that the truth has been revealed, through religious texts in this way, makes the other man’s claim moot. This is a non-starter to analysis but opens up questions concerning methodology.

Methodological Issues and Conclusion

Al-Ghazālī went through the historically contingent process of restating Avicenna’s clarification of what it ought to take to be an actual infinity. In this clarification, Avicenna makes clear two conditions of actual infinities; wholeness and order. With this clear, one can come to two distinct sorts of actual infinities; existent actual infinities, wherein each condition is sufficient such that the other does not arise and non-existent actual infinities where these conditions are necessary and sufficient. With this distinction doing all of the explanatory work, one can generate the actual infinities which are at odds with eternity, potential time-like extension, such that those actual infinities exist. This distinction leaves no room for an excess of existing actual infinities but instead just exist to make sense of the actual infinities which arise from eternity.

Aquinas had only required the philosophical distinctions and baggage of Aristotle to solve this tension between the non-existence of actual infinity and eternity. Since Aquinas sealed his fate by accepting wholly all of Aristotle’s arguments against actual infinities Aquinas would have to explain how eternity can exist but time not be infinitely going backwards. The first step was to make a distinction between the eternal and time; the former being corporeal and the latter being incorporeal. Claiming then that god exists in an eternal realm so as to be outside time and space. This explanation introduces a starting limit to god’s creation, beginning with Genesis. This of course departs from Aristotle’s potential temporal infinity.

Methodologically, al-Ghazālī and Aquinas begin on equal footing, each with their respective theological commitments. However one of the resources which Aquinas lacks is the ability to consider Avicenna as an authority close to being on par with Aristotle. While this was a mere historical contingency Aquinas was on record as reading the arguments, of what he calls accidental actual infinities, of Avicenna on the topic of infinity (Aquinas 1 vii 4). Instead of arguments of the sort where necessary and sufficient conditions come into play, Aquinas makes proclamations on faith, like time having a beginning, then sets forth to show that eternity is different from time in a metaphysical versus physical distinction. al-Ghazālī’s solution contrasts by obtaining a mathematical distinction from Avicenna’s arguments and uses it to get out all of the goodies expected from the tension of eternity and non-existent actual infinities.

The more preferable methodology is that of al-Ghazālī, obviously this arises from the help of Avicenna. It is more advantageous in an argument to make a clarification to strengthen the adoption of a prior author’s position, in this case Aristotle’s denial of some infinities. However more in terms of Aristotle’s original point is lost when al-Ghazālī and Avicenna’s argument is put forward. Aristotle’s position of infinity not belonging to magnitude contains more content than the position of past eternity, such that al-Ghazālī and Avicenna lose a lot of the original theory. However what is lost by Aquinas is the credibility of reasoning. Instead of changing his position to that which fits the argument at hand, Aquinas tries to harmonize some already clear concepts, in an ad-hoc fashion to boot. Aquinas agrees with Aristotle’s reasoning for the position that a potentiality of past time exists but this does not bring him to assent to the argument. Instead, Aquinas takes it on faith that time had a beginning and that time is separate from eternity. The question of method, which is reflective of this article’s contemporary prejudice, is less to do with how well one may salvage the content of a theory but instead, how one develops the contents of new theory. That is in this case, one had to develop a new theory of infinity on pain of the problem raised by Richard of Middleton. Aquinas develops a new theory where eternity just has an altered meaning, but without good reason. In contrast, al-Ghazālī, from Avicenna, has a new theory of infinity which clarifies that certain mathematical concepts are used, wholeness and ordering, which gives him the upper hand.

Aristotelian infinity in conjunction with the idea of a creative god may be solved in at least two ways. The first was al-Ghazālī and Avicenna’s use of there being a distinction between existing and non-existing actual infinities which permitted the existence of infinite magnitude from meeting one condition, of wholeness or order, and not both. The second solution was Aquinas’ defense that eternity was god’s realm outside of space-time which Aquinas saw, on pain of doctrine, as being created by god. I find it apparent that al-Ghazālī and Avicenna could competently defend their view from Aquinas. Furthermore al-Ghazālī and Avicenna’s method for arriving toward a solution may not have saved an Aristotelian theory of infinite nor assumed to get the concept right but was of higher quality reasoning and innovation than Aquinas’. This result however is only from my inspection and I may not have as keen of an eye on medieval scholarship as most.

 

Endnotes:

[1] The reason for calling this time-like is that the philosophers, Al-Ghazālī and Aquinas, have different definitions for what they deem eternity to be (especially Aquinas).

[2] The phrase all at once is gleaned from Aristotle’s description of motion “Similarly, too, in the case of any alteration whatever if that which suffers alteration is infinitely divisible it does not follow from this that the same is true of the alteration itself, which often occurs all at once, as in freezing,” (Aristotle viii 3).

[3] To find something like this from an original source (Richardus 1963) is recommended. However realize that there is no readily available print copy of this work translated into English, so read up on Latin.

[4] Avicenna is said to treat infinity as numbers when he refers to them as sets (McGinnis 2010, 16). This mathematical reasoning is innovative to the Aristotelian framework which only admitted physical proof.

[5] That is considering actual and potential to be the dichotomy of Aristotle’s making.

Bibliography

Al-Ghazālī , and Michael E. Marmura trans.. The Incoherence of the Philosophers = Tahāfut Al-falāsifah: A Parallel English-Arabic Text. Provo, Utah: Brigham Young University Press, 2000.

Aristotle, R. P. Hardie trans., and R.K. Gaye trans.. Physics. Raleigh, North Carolina: Alex Catalogue, 199 BCE.

Aquinas, Thomas. Summa Theologica. New York, New York: Benziger Bros., 1947.

Brenner, Andrew. “Aquinas on Eternity Tense, and Temporal Becoming.” Florida Philosophic Review X, no. 1 (2010): 16-24.

Duhem, Pierre Maurice Marie, and Roger Ariew. Medieval Cosmology Theories of Infinity, Place, Time, Void, and the Plurality of Worlds. Chicago, Illinois: University of Chicago Press, 1985.

McGinnis, Jon. “Avicennan Infinity: A Select History of the Infinite through Avicenna.” Documenti E Studi, no. 21 (2010): 199-222. http://www.umsl.edu/~philo/People/Faculty/McGinnis Works/Avicennan infinity.pdf.

McInerny, Ralph, and John O’Callaghan. “Saint Thomas Aquinas.” Stanford University. July 12, 1999. Accessed May 8, 2015.

Moore, A. W. The Infinite. 2nd ed. London: Routledge, 2001.

Richardus, De Mediavilla. Super Quatuor Libros Sententiarum Petri Lombardi Quaestiones Subtilissimae. Frankfurt: Minerva, 1963.