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Medieval Thought and Infinity

Medieval thought begins in late antiquity growing out of the fall of the Roman empire. The main influence on medieval ideas about the infinite is the synthesis between ancient ideas and frameworks and the growing power of Christianity and Islam over the societies of what used to be the Greek and Roman worlds. In general, medieval thinkers agree that an actual infinity must be real. However, like most of the Greeks before them they refuse to associate the infinite with the world of nature. For the medieval mind God was the one infinite spiritual being and out of his goodness is created the finite natural world. Some attempts to secure this single infinite power include the early neoPlatonic theology of Christianity (exemplified by Saint Augustine) and the later Aristotelian transformation of the monotheistic worldview beginning with the 're-discovery' of Aristotle - first in the Islamic world and then in the Christian west. Al-Kindi and Thomas Aquinas will serve as examples of medieval Aristotelians.

Augustine Bishop of Hippo (c. 354-430 C.E.)

augustine in his study

Augustine of Hippo, also known as Saint Augustine, was an early Christian thinker (the first medieval thinker really) who is both greatly influenced by Plotinus and also contributes significantly to the Neoplatonic theory and understanding of infinity. Like Plotinus, Augustine was born in the North African provinces of the Roman Empire; unlike Plotinus he spoke and wrote in Latin. Again, unlike Plotinus, instead of remaining a pagan, Augustine converts to Christianity at the age of 35. He ends his life as a Bishop of the Catholic Church. We will be looking at selections from two writings of Augustine’s: the first is The Confessions, and the second is his most important philosophical work The City of God. Where necessary, and in order to better understand the concepts he makes use of, we will also turn to other writings for clarifying and establishing the Augustinian philosophical framework.

In his Confessions (probably one of the first autobiographies), Augustine tells us how it was through the writings of the Platonists (Plotinus, Porphyry, Proclus, Simplicius, Iamblycus, etc.) that he was able to take Christianity seriously as an intellectual framework (Conf. Book VII, chp. 15). The Confessions describes a man measuring the accomplishments of his life and examining the meaning of his existence. Augustine takes a highly introspective approach to present this analysis of the Soul. Like Plato and Plotinus before him (who take a turn inwards to understand reality –an intellectual turn in the case of Plato and a full blown mysticism in Plotinus), Augustine examines the spiritual side of being and undertakes a philosophical exploration of reality within a basically Christian Neoplatonist framework. In the Confessions Augustine also writes of how by turning away from the senses, by introspecting, he eventually “awoke in [God] and saw [God] to be infinite, but in a different sense; and that vision in no way derived from the flesh” (Con. VII, 16). In Augustine's Confessions Book XI, Chapter 7, we find the correlation of God with what is unending clearly expressed. “…of Your Word nothing passes or comes into being, for it is truly immortal and eternal” . More importantly, against the very Greek and earlier Neoplatonic understanding of the One as infinite spiritual presence, Augustine understands God as the ultimate power in the Universe and also as a divine creator.

“Thus it is by a Word co-eternal with Yourself that in one eternal act You say all that You say, and all things are made that You say are to be made. You create solely by thus saying. Yet all things You create by saying are not brought into being in one act from eternity”.

The important innovation for the notion of infinity here is that God is eternal, according to Augustine, in the sense that he is present all at once. God’s being therefore transcends existence in any normal sense. He is like the Neoplatonic One of Plotinus and the NeoPlatonists, but superior even to the One since he is a Creator. Augustine also had problems in accepting the One of the Neoplatonists due to the ethical focus of Christianity which, as a religion, demands obedience to God as supremely good. While, it is true, Plotinus had made the One a principle and the origin of Goodness, the Neoplatonists tended to be intellectuals and had a strange theory about the problem of evil.

The Problem of Matter and Evil

Since Augustine believes in an absolutely good God who also creates the world, he had a problem with the Neoplatonic theory of evil. For the Neoplatonists, God is not a creator but they instead believed that the One emanated and with its eternal emanation we get, first, the Forms and then – following from the Forms-Nature. Augustine, after his conversion to Christianity, believes that (1) God creates nature directly out of His divine essence. (2) The Bible tells us that God saw that what He had created "was good". This conflicts with Neoplatonic doctrine, since (in Neoplatonism) Goodness is associated only with the One and evil with the privation of Form and farthest kind of being from the One- namely: matter. Therefore, following Plato (and making Forms the principles of being and goodness), Plotinus equated matter with both non-being and evil. Taking this further, Plotinus equated evil with the body and goodness with the soul, but, he still thought that everything (including nature) ultimately emanated from the One. This leads Plotinus into a real problem in explaining where matter (as a principle of evil), and evil itself for that matter, comes from. If the One is all good, then the One cannot bring forth or be responsible for evil. Plotinus only has one option: evil comes from the One. But this makes evil and matter both an emanation of the One. The One, however, was supposed to me a principle and source of all Goodness. The problem is quite pronounced because the Neoplatonists cannot resolve the mytery of how the principle and source of all Goodness becomes the origin of evil

Augustine therefore, seeing the above inconsistency, rejects the attribution of limits on God and any sense of evil attributed to Him. God will be a Divine Creator and any evil in the universe is simply a privation of the Good that cannot be associated directly either with nature or matter (– since a perfectly Good God created them both). Since God is also omniscient and all knowing, he knows that the fall from grace (in the garden of Eden) will happen, but he still endows us with free will and gives us dominion over the Earth because he is a personal creator who loves us. In the Confessions, moreover, Augustine explicitly takes issue with the limitations set on God within the Neoplatonic framework. In Book VII, Chp. XII, [section 18], Augustine begins by expressing the Platonic insight that being is good, he then writes of how corruptible/material things are also good, but if they were supremely good they could not be corrupted. And if they were not good at all then there would be nothing in them to corrupt.

Augustine's informal argument can be formalized as follows:

The upshot is the following: Whatever exists is good, and evil is not a substance (in either a Neoplatonic or Aristotelian sense). Evil is a privation of the good of being. Since God is perfectly good, he will save us from ultimate corruption. Richard Niebuhr explains Augustine’s succinctly explains position in his book Christ and Culture, when he writes, “Man is by his created nature made to obey, to worship, to glorify, and depend on the Goodness which made him and made him good; on God, who is his chief good. As his primary goodness consists in adhering to God, so his primal sin lies in turning away from God to himself or to some inferior value” (cf. Niebuhr 1975, 211). For Augustine this means that God is a highest, infinite being, in a new sense. God is simultaneously omnipotent and eternal. To say God is eternal is to say that he is outside of time, like Plato’s Forms, but more precisely Augustine thinks that this renders God actually and fully infinite in a transcendent sense, God is the creator of time itself. Since God also creates us willingly out of his divine essence (though the divine will), God becomes a personal God. This gives Augustine’s ethics a moral dimension derived from Judaism and Christianity that is missing in Plato and Aristotle and it connects the infinite to us in very direct and personal way. We do not become infinite, but we can participate in God’s infinity. Mary Garvey describes the Christian God as Augustine understands him by stating, “The God to Whom the Christian adheres is a personal God, a Creator, a Being supreme in order of existence… A Divine decision stands between Him and it (the creation of the universe)” (cf. Garvey 1939, 52-3). She goes on to contrast this with the ideas of Plotinus (a Neoplatonic thinker) and his notion of the One, which is his interpretation of the Form of the Good. Garvey explains his idea of the One when she says, “All things proceed from Him (the One) but not as a result of free choice… He is the source of the essence of things but not the creator of their being.” (Garvey 1939, 53). Although Plotinus’s idea of the One differs from Plato’s original Form of the Good, the basic idea that this highest goodness is not the creator of the universe so much as the “source of the essence of things” is the same. Therefore, this results in an irreconcilable difference arising between Augustine’s personal divine creator of the universe and Platonic and Neoplatonic notions of a perfect existence that is Goodness (the One) from which our reality emanates.

It is nonetheless to Plato and Neoplatonism that Augustine will turn to make sense of the infinite and of Christian cosmology. Like the Neoplatonists, he accepts an actual infinity. Unlike the Neoplatonists, Augustine’s infinity is an ultimate power transcendent of nature, who both creates us and gives our lives meaning. In merging these two traditions, Augustine focuses on Plato and Neoplatonic ideas to the extent that he can be described as a Christian Neoplatonist. Augustine’s admiration of Plato, for example, is present in the City of God, where he states: “among the disciples of Socrates, Plato was the one who shone with a glory which far excelled that of the others, and who not unjustly eclipsed them all” (City of God, 310). Augustine’s appropriation of Plato’s doctrine of the Forms can be seen in the Book of Eighty-Three Various Questions. There we see how Augustine takes Plato’s transcendent Forms and adapts them to fit into Christian philosophy. Of the Forms, which he calls the “Ideas”, he writes, in questio 46, that they are changeless and eternal entities which provide the model or basis through which everything is fashioned. Augustine also states that because Platonic Forms are everlasting and not subject to any change, these ‘Ideas’ are “contained in the Divine Mind”. By adapting the Forms from entities that exist in a transcendental realm beyond reality (as Plato described them) to mental configurations of God’s Divine Mind, Augustine adapts Platonic philosophy to Christianity. His strategy strongly parallels the earlier approach of Plotinus who made the Forms united in the absolute infinite One.

In Christianity, God is a divine, transcendental being who is perfect in every regard, and thus it can be argued that he exemplifies the supreme perfection of every concept, like Justice, Beauty, Health, etc. So it is not a hard stretch for a Christian to define the Forms, which are described by Plato as also being the epitome of these concepts, as entities which exist through God. In The Confessions, Augustine shows his belief in God as the highest good when he states, “My God, who is not only good, but goodness itself” (Confessions 147). This is very similar to the Platonic notion of the Form of the Good as the perfect manifestation of Goodness in itself which is the source of all other inferior goodness as it is manifested in reality. In Christianity, there is also a very similar belief that the highest form of goodness, God, is the source of all worldly goodness. This is shown in the first passage in the Bible which states that after God created the universe, he claimed that everything he created was good (in the Book of Genesis). Augustine states a similar sentiment in The Confessions when he states, “Thou (God) madest all things good” (Confessions 160). As the source of all goodness, Augustine interprets this to mean that not only was every object in nature that God created good, but also all human beings have this good in their nature because they too are a creation of God. This position gives Augustine a framework to understand reality, one that makes sense of the cosmos as a whole and informs the philosophical understanding of humanity and nature. Humans are finite, nature is finite, God is infinite. Like the ancient authors, Augustine’s worldview is hierarchical. Unlike these earlier authors, Augustine thinks that nature as a whole serves a purpose given to it from the outside by God. God’s will enforces what comes to be known as “the Great Chain of Being”.

God’s Infinity

In the Confessions, Chapter 9, we find Augustine stressing the omnipotence and high divinity of God. God is a creator, the creator of the world that he brings into being out of nothing. “This then, O God, was the Beginning in which You created heaven and earth, marvelously creating in Your Word, who is Your Son and Your Strength and Your Wisdom”. Also, “You made heaven and earth”. This notion of creation from nothing: ex nihilo, is the one idea that the Greek tradition never had. Not even Plotinus could attribute true creation to the One, nor did the Vedantic tradition give this power to Brahman (there is no ex nihilo creation in Hinduism). In attributing this quality and power of creation to God however, Augustine’s entire conception of the universe and time and infinity is transformed in comparision to earlier frameworks. According to the pagans the view that time had a creation and nature came from into being out of a divine will was so unusual and irrational that they tended to mock Christians for holding it. Augustine writes, in Chapter 10 of the Confessions, about the ‘ancient error’ that leads many to say: “What was God doing before He made heaven and earth?”, if God was doing nothing except waiting, then why did he ever decide to create the world after an eternity of non-creation? If he just felt bored and decided to create (after an eternity) how could it have been a true eternity that he existed, and why would a perfect, all knowing, God change his mind or become bored in the first place? Augustine answers:

The will of God is not a creature: it is prior to every creature, since nothing would be created unless the will of the Creator first so willed. The will of God belongs to the very substance of God which was not there before, that substance could not rightly be called eternal; but if God’s will that creatures should be is from eternity, why are creatures not from eternity?.

This is a good question. But the mistake the critics make is twofold: (1st) to think that God exists in time, the way that we do, and (2nd) to believe that God’s will or mind is anything like our finite wills and minds. The past and future moments of things is not true Being. Here Augustine takes an Aristotelian view of time and nature, reducing them to a special kind of potentiality that is never fully as real as substance. But like Plotinus, Augustine thinks that true Being, or Platonic perfection, exists only in God and God exists beyond time and nature. God is eternal, actually infinite and in Chapter 11 of the Confessions, Augustine writes of how “in eternity nothing passes but all is present, whereas time cannot be present all at once”. However the question of the eternity of the world and the cyclical pagan theory of time is only addressed by Augustine in the City of God. Augustine’s point about God not needing the world and being eternal is that the true infinity of God cannot be captured by time or in time. If God is actually infinite, then He is superior to time and space- he is transcendent of nature. The Neoplatonic theory of the One and its Tri-fold emanation into reality [that we saw in Plotinus] is also seen by Augustine to both parallel and anticipate the revealed nature of the one true faith, i.e. Christianity. Thus, the Christian conception of the one God as three persons, claims Augustine, was foreshadowed and anticipated by the Neoplatonist tradition. “There is then one sole Good, which is simple, and therefore unchangeable; and it that is God. By this Good all good things were created; but they are not simple, and for that reason they are changeable. They are, I say, created, that is to say, they are made, not begotten”. Augustine distinguishes what is made from what is begotten (emanated?) by saying that what is begotten by a simple good is also equally simple as its source, but what is made can be changeable and a lower kind of being. What is begotten in God, we are told, is the Son and the Holy Spirit. These are not made they are equally eternal. This notion of a Triune but simple God is philosophically developed in Augustine’s essay De Trinitate (“On the Trinity”) and is taken up again in the City of God, Book XI, Chapter 10. There Augustine tries to reach some kind of intellectual understanding of what it means to have One ultimate transcendent, personal and infinite substance, that also is (or has) three essential aspects – Father, Son and Holy Spirit. On the Trinity begins (Books 1-4) with some Biblical exegesis, ends with an articulation of some original and influential thoughts on human cognition and the nature of mind (Books 8-15) but includes, at its heart, some stimulating discussions in epistemology and metaphysics (Books 5-7). The metaphysical books raise the main problem. In the classical tradition, things are said by reference to substance if they are real beings. Things are also said by way of relation –however, as Aristotle makes clear, relations are dependent upon substances. Taking the infinite nature of God, the triad is first God and then metaphorically the Father but also the Holy Spirit and the Son. But the Father, Son and Holy Spirit are not in the triad as parts with the triad being one substance. There are three substances - three modes of being - that are also one. How can the substance contain these three natures? The answer is that the first substance as the primary being of God is infinite. The Holy Spirit is thus an inexpressible communion or fellowship of the Father and Son (De Trin. Bk V, chp. 3), and the Father is an inexpressible communion of the Holy Spirit and the Son is an inexpressible communion of the Father and Holy Spirit. Three substances in One. This is a problem because it goes against reason. But, Augustine tells us, the kind of being that God has is radically different from other kinds of beings/substances. For one thing, God is his own essence, divinely simple, non-composite, omnipotent, purely Good and purely intelligent (De Trin. Bk VII, 3, 10). This doctrine of divine simplicity seems difficult to reconcile with the Triad that is God. For a clue to the resolution of the problem we can turn to the De civitate Dei (City of God). Keeping in mind a key point in all of Augustine’s thought, namely that faith should seek understanding, i.e. whatever is not immediately clear to the intellect may be maintained by the firmness of faith. As Augustine famously writes: “unless you believe, you will not understand” (On the Trinity, Book 7, chp. 4). In City of God, Augustine writes about what he calls the Imago Dei: “...it is from the one true and supremely good God that we have that nature in which we are made in the image of God, and that doctrine by which we know Him and ourselves, and that grace through which by cleaving to Him, we are blessed” (City of God, Bk VIII, chp. 10). Granting that the above holds Augustine is now free to reason from analogy. Because we are made in God’s image we can therefore know something (albeit in a vague, dim and confused manner) about our Creator. In this way, Augustine says, our mind has a plurality of acts within it but is yet still one unified thing. The way we perceive, apprehend, and judge all in one act of mind is viewed as a suitable analogy for how the three persons of the Trinity are really one infinite substance, i.e. God. This “psychological analogy” of the Trinity presents Augustine’s answer to the problem of the Three divine persons of the Triad constituting the One infinite Christian God.

The Medieval Islamic world

Like Christianity, Islam is a monotheistic faith and based on holy-scriptures. The founding of Islam goes back to the Seventh Century C.E. Muhammad (circa 570-632) is the central figure and is considered its main prophet. By 900 C.E. the Islamic world had extended its influence throughout most of the modern Near East and even into the Mediterranean. Falsafa, is the Arabic term derived from the Greek philosophia, and was the standard term used by intellectuals in the medieval Near East to describe what we call philosophy. The Arabic translation of philosophia, h.ikma or “wisdom,” is sometimes also used to designate this activity as well (cf. McGinnis & Reisman, 2007, p. xvii). As can be inferred from the vocabulary used above, these are all loan words. That’s because the Islamic society arose independently of the Greco-Roman society that sustained the Greek and Latin world. The Romans never colonized Saudi Arabia, but they colonized many other parts of the Near East (such as Syria). In the Middle-Ages the Islamic faith expanded and its traditions came into contact with the remains of the Roman Empire eventually absorbing large parts of what were earlier Roman and Greek speaking regions. In doing so, the Moslem philosophers are able to obtain a virtual treasure-trove of Greek works that were simultaneously lost in the West. In this way the Islamic thinkers continue the Greek tradition of philosophy and science and, in many cases, add to it. By contrast, a sharp intellectual decline affects the Western world after the death of St. Augustine. The Islamic tradition therefore establishes an interpretation of the Greek works –both Platonic and Aristotelian— and then establishes a tradition of commentary that later is re-introduced into Europe and developed by European scholars starting in the 12th century.

Al-Kindi (800-801 C.E.)


Born in Basra around 801 C.E. (current day Iraq), died c. 870. Al Kindi was educated in Baghdad and is widely considered to be the first important philosopher to write in Arabic. He wrote nearly two hundred and fifty treatises on philosophy and science, of which less than forty survive. Known as “the philosopher of the Arabs,” he was associated with the Abbasid courts of the caliphs al-Ma’ mūn (reigned from 813–833), al-Mu tasim (reigned from 833-842), whose son-Ahmad, Al-Kindi would tutor. Al-Kindi flourished during the period of the Arabic translation movement of Greek philosophical and scientific texts, in which he played a limited translating role—most likely simply advising on philosophical and scientific content rather than directly undertaking any translations himself. Al-Kindi is therefore very important as the historical individual who helps to first establish a vocabulary for philosophy in the Islamic world. Philosophically he is a blend of Neoplatonist and Aristotelian. And it should go without saying that al-Kindi was a Moslem, but he was a Moslem whose life’s work was devoted to translating Aristotle and Neoplatonic texts.

The Aristotelian scientific outlook on the cosmos informs al-Kindi’s writings. Aristotle’s concepts of act/potency, form/matter, substance/accident and the four causes are all central for understanding al-Kindi’s texts. Neoplatonism is another influence. In On First Philosophy, his most important philosophical work, he discusses the “One” and the “many” and supports the notion of the “One True Being.” But, al-Kindi does not blindly follow the Greeks. He notably rejects (as Augustine did) the eternity of the world, a doctrine held by most Greek philosophers and even other Islamic philosophers (e.g. Al-Farabi, Avicenna and Averroes). Among al-Kindi’s scientific achievements are works on mathematics, optics, medicine, and music. Again, although Greek scientists such as Hippocrates, Euclid, and Ptolemy influence his thought, his work also showed some originality. This was especially true in the realm of optics and medicine. Unlike later Islamic thinkers in the Neoplatonic and Aristotelian tradition, like Al-Gazzari or Avicenna, al-Kindi is notable for the limited direct engagement with theology. This didn’t mean he avoided trouble, at one point his library was taken away. The same problem would affect Christian thinkers who studied Aristotle. It took a long time for Aristotle to be integrated into the Judeo-Christian framework- in Europe that would happen with Thomas Aquinas. But Aquinas will be influenced by the Arabic tradition of natural philosophy and it is this tradition that al-Kindi both belongs to and, in an important way, actually begins. To understand al-Kindi therefore, we really only need to understand that he’s reading Plotinus into an Aristotelian framework and his reading is tempered with a theory of a finite cosmos created by an infinite transcendent God- just like we saw in Augustine. His great contribution to understanding the concept of the infinite therefore, is that- like Augustine- he will reinterpret earlier Greek and pagan traditions against the context of a monotheistic worldview. Unlike Augustine, al-Kindi’s inspiration (mostly present in the background) is not the Bible but the Koran. According to the Koran, God exists and has certain attributes. He is perfect, unchanging, omnipotent, and omniscient. The God described in the Koran, just like that of the Bible, is a creator and there is a strong suggestion, in al-Kindi’s understanding of creation, that God created the world out of nonbeing at some first moment in time. As in Christianity, therefore, a Creator God is taken as a grantor of salvation in the form of eternal life to the human being. The human soul, also created by God, is taken to be immortal and like Christianity many Moslems accept the doctrine of bodily resurrection.

Therefore, from the Koran, al-Kindi absorbs similar ideas as those that Augustine takes from the Old and New Testaments of the Bible. Like Augustine, al-Kindi views God as supremely infinite. But al-Kindi is writing from a more Aristotelian than Neoplatonist context, and because of this he places a sharper focus on the natural world. Al-Kindi is at the forefront of this endeavor. He starts a movement focusing on the translation of ancient Greek and Neoplatonic writings. His esteem for Aristotle can be judged by the central focus he placed on trying to understand him. His work On the Quantity of Aristotle’s Books, is an overview of all of Aristotle’s works and meant to guide philosophers and help them understand Aristotle.

Al-Kindi's On First Philosophy

Al-Kindi’s most important philosophical work is called: Fī al-Falsahfah al-Ūlā, or in English, “On First Philosophy”. Al-Kindi’s audience for this work would have been scholars and aristocrats, members of the family of the caliph of Baghdad, students and theologians. All of them would have been familiar with the Koran but not with Aristotle. The text On First Philosophy is short, consisting of Four Chapters. The main topic and central focus of the work is to describe not truth or being primarily but Oneness. It is representative of al-Kindi’s philosophical program in general since it combines Aristotle and Neoplatonism. Also, since First Philosophy, and only First Philosophy, can study the divine, al-Kindi thinks that philosophy is perfectly compatible with faith and the belief in God. Al-Kindi begins the book by expressing his thanks to the caliph al-Mu tasim. Then, in Chapter One of his book, he introduces philosophy which he calls the “greatest and most noble human art”. He writes that philosophy seeks to “know the true nature of things” including the nature of the true One (Divine Being)”. (Al-Kindi 1974, 8). This is all very Aristotelian. Aristotle called the study of the highest substances (divine being) metaphysics or ‘First Philosophy,’ and the study of sensible substances was said to depend on his physics, which was also known as ‘Second Philosophy’. Al-Kindi is concerned with metaphysics which he talks about as equivalent to ‘first philosophy’ because it takes knowledge of God as its fundamental principle. However, he also accepts the Neoplatonic stress of the One. Unlike the Neoplatonists, al-Kindi will read the One as capable of effecting what Aristotle called ‘change’ and therefore of being a first efficient cause of the world. In Chapter Two, al-Kindi compares sensory perception with thought. Here we learn that, like Aristotle, he thinks the physical universe and nature is finite. Al-Kindi uses spatial and logical examples to illustrate this point. We will examine these examples below. In Chapter Three he examines the concept of auto-causation (that things cause themselves to come into being) and he rejects it. He also discusses the concept of the essence of a thing and how this is different from its existence. Chapter Three closes with a study of the Platonic notion of the One and the Many. In Chapter Four, al-Kindi argues that absolute unity (simply being) cannot be present in anything that has magnitude (size) or quantity since these are relative notions. He argues that numbers do not have actual reality- so that the number One is not a thing but depends on a relation to other numbers (which again are abstract entities). Finally, we are told that the True One is not a genus and has no genus- it is eternal and absolutely One- which means that it can only be described negatively – and we can’t know it even through analogy. Unity exists in things we can experience accidently because of the emanative power of the One (Al-Kindi 1974, 11). But emanation is understood differently by al-Kindi than how Plotinus understood it.

Analysis of Al-Kindi's treatment of the Infinite

In "On First Philosophy" Part Two, al-Kindi begins by talking about a body, or what Aristotle would have called a “sensible substance” (Al-Kindi 1974, 266) “Now inasmuch as a body has a genus and species, while the eternal has no genus, a body is not eternal; and let us now say that it is not possible, either for an eternal body or for other objects which have quantity or quality, to be infinite in actuality, infinity being only in potentiality” Here al-Kindi sounds exactly like Aristotle, except that above he is talking only about the natural world. There is no actual infinity in nature. Certain logical concepts can also help us establish the above claim. Al-Kindi writes: “…among the true first premises which are thought with no mediation are: all bodies of which one is not greater than the other are equal; equal bodies are those where the dimensions between their limits are equal in actuality and potentiality; that which is finite is not infinite…” Here al-Kindi seems to be saying that we have (A) True First Premises These can be (A’) Immediate and (A’’) mediated. The immediate truths of principles, in the Aristotelian tradition, are often said to be grasped immediately with the mind (nous). Later philosophers call this “intuition” Let’s list under A’ the concepts of: Equality, Unity, Part-Whole dependence (this is known as mereology in the context of logic, meros in Greek means part, share, portion, etc.) Under A’’ we can list: non-intuitive knowledge, knowledge that needs the senses and is dependent. Sight, hearing, colors, tastes, etc. Thinking about the concept of body, a surface with a boundary, something containing mass, qualities, powers, etc. We can affirm that equal bodies are equal in a finite way. This seems to be something we have to understand a priori. We can’t see equality itself, we see two things and just know equality. Thinking about the concept of body, a surface with a boundary, something containing mass, qualities, powers, etc. We can affirm that equal bodies are equal in a finite way. To qualify this claim, al-Kindi uses the example of two finite bodies (bodies of finite magnitudes). They can be joined together. When two bodies are joined, the resulting amalgamation of bodies is also necessarily of finite magnitude (On First Philosophy, Chp. 4). If there were an infinite body, then:

“…whenever a body of finite magnitude is separated from it, that which remains of it will either be a finite magnitude or an infinite magnitude. If that which remains of it is a finite magnitude, then whenever that finite magnitude which is separated from it is added to it, the body which comes to be from them both together is a finite magnitude; though that which comes to be from them both is that which was infinite before something was separated from it. It is thus finite and infinite, and this is an impossible contradiction”


In other words, suppose, we are told, there is an infinite body: Then we could separate a finite part of it, and we will be left with either a still infinite body or a finite one. But- If that which remains is finite, then when we add it back to the original body, how can we still say that the original is infinite? A= we can’t. We end up in “an impossible contradiction” To strengthen this claim, al-Kindi says, imagine that we have an infinite body, take away a part and still end up with an infinite magnitude remaining. Then “whenever that which was taken from it is added to it, it will either be greater than or equal to what it was before addition” If it is greater, then the new infinite body will be greater than the old one. Defending the notion of the equal, al-Kindi argues that when two bodies are joined, the resulting amalgamation of bodies is also necessarily of finite magnitude. By the axioms of mereology, if A is a part of B, then A must be smaller than B, or A=B. This mereological analysis studying the parts and the wholes of things, leads al-Kindi to the position that “it is impossible for a body to have infinity, and in this manner it has been explained that any qualitative thing cannot have infinity in actuality.” This orthodox Aristotelian claim is quickly followed by a very unorthodox one: Now time is quantitative, and it is impossible that time have infinity in actuality, time having a finite beginning.

Infinity and Time

Although Aristotle would agree that time is not actually infinite in nature, he would find the claim that time has anything like a beginning impossible to accept. How does al-Kindi argue against the Aristotelian and Greek theory of an eternal, cyclical, time? Take a temporal series, T1-----------------------------------T2 We know that at any instant before T2 happens, we can divide the now into an infinite series. We learned this from Zeno. Al-Kindi follows Aristotle and views time as a continuum. He agrees with the Aristotelians that “If time is limited, then the being of the body [of the universe] is limited, since time is not an existent, and there is no body without time, since time is the number of motion.” Since al-Kindi has already provided a proof that no body (not even the body of the universe) can be infinitely large and he also thinks that any feature of a finite body must itself be finite. He is now saying that, if time measures motion; and motion applies to bodies – neither motion or time can ever be infinite if there is no infinite body. When he writes: “if a definite time cannot be reached until a time before it is reached, nor that before it until a time before it is reached, and so on to infinity;” What he means is that the infinite can never be traversed so as to reach a definite time (i.e. the present). But time is finite necessarily, and so the present divides the past, T1----T2 from the future, T3----T4 (let’s say). If time is viewed as a continuum and therefore less real than motion and bodies, then since bodies are finite, time is also finite. But al-Kindi wants to argue that, although by its nature, time is capable of both infinite addition and infinite division, taken as a process time has one duration which we can conceptualize into units. He writes: “Body is not prior to time, so it is not possible that the body of the universe have no limit, because of its being. So the being of the body of the universe is necessarily limited”. Al-Kindi is arguing that these units point to a finite beginning, a moment when time starts and begins. Aristotle would disagree. Aristotle explicitly distinguished between the infinity of a body in nature, what he called an actual infinity that would be present all at once, and the kind of infinity present in time. The kind of infinity that time has is not actual but potential – like the infinity of numbers and mathematical series. By “a definite time” al-Kindi means the present, but it is not clear when this dividing present would be. It is also not clear why the infinity of time should be on a par with the infinity of the physical universe. Aristotle would say that the two kinds of infinity are nothing alike and in treating of time as a magnitude in the way he does, al-Kindi is dealing with an illusion (the present moment ‘now’) and taking it to be as real as a body that is present. Al-Kindi writes: “Every definite time has two limits: a first limit and a last limit” Here it seems that the validity of his argument depends on how we interpret ‘definite time’. If this is different from continuous time or time as dependent on the motion of bodies, then al-Kindi has proven the finiteness of time, but other Aristotelians argued that his terms are equivocal, there is no definite time that can actually divide the past from the future….