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Medieval Thought and Infinity (Part 2)

St. Thomas Aquinas (1225-1274 C.E.)

St. Thomas Aquinas: Scholastic Aristotelianism and infinity

Later medieval thought is called "Scholastic" and is typified by an abstruse argumentative style known as the questio. Amongst the greatest of the high medievals is Saint Thomas Aquinas. Thomas was born in Southern Italy in 1225 C.E. Just before his birth, the study of Aristotle had been banned by decree at the University of Paris (the center of Christian theology in Western Europe). Soon after his death in 1274, Thomas’s writings were already being put to use by the Catholic Church and Aristotelianism was viewed as the foundation for western science and theology. In large part, this was due to the accomplishment of Thomas and his 'baptism' of Aristotle. Today Thomas is considered one of the most important philosophers in the Catholic tradition. Thomas’s thought eventually came to be viewed as so important that fifty years after he died, in 1323, he was canonized and declared a Saint by Pope John XXII.

Thomas Aquinas is a systematic thinker (as systematic as Plato or Aristotle), 'Thomism' is the adjective often applied to his system and position. Metaphysically, Thomism can be said to consist of a synthesis of different aspects of the thought of Plato,Neoplatonism, Islamic philosophy, St. Augustine, St Bonaventure and other Church Fathers. However it is Aristotle, who is cited regularly by Thomas and often simply called “The Philosopher”, who most decisively influences his system. Other important names that Thomas frequently mentions are the Islamic thinkers Averroes (called “The Commentator” by Thomas) and Avicenna, and the Greek mystical thinker Dionysius the Areopagite. In general, philosophy for Thomas is a term that covers a whole wide set of sciences (theoretical and practical) but at the heart of Thomism are the notions of existence and the hierarchy of being used together to establish an investigation into human nature and the world. Like Plato, Thomas believes that humans are spiritual creatures who exist between heaven and earth. But like Aristotle he accepts that the soul needs an intimate connection to an individual and unique body in order for personhood to emerge. Regarding the physical universe, Thomas starts from Aristotle. However, like earlier medieval thinkers, he embraces a transcendent infinity, namely God, and in this way he is motivated to link Aristotelian natural philosophy to Neoplatonic metaphysics when talking about the nature of the infinite.

The Summa Theologica: Thomas’s ideas are developed in the Summa Theologica, a treatise on theology and philosophy that he published over the course of many years. Principally written between: 1265-1274 the Summa Theologica [ST] was Thomas’s second Summa- (‘Summa’ in medieval Latin meant ‘summary’ or ‘complete overview’). The ST was written after, and should not to be confused with, the Summa Contra Gentiles. The ST is primarily a theological work but there is a lot of metaphysics and philosophy present within its pages as communicated through the high medieval disputatio or questio structure. In the ST Question 7 Articles 1-4 we see how Thomas addresses the problems related to infinity that go back to the ancient Greek tradition. Ancient Greek thought ends with Aristotle, who thought an actual infinity in nature was impossible. Early Christianity, which had taken a decidedly Neoplatonic turn to try to establish Greek philosophy as compatible with Biblical revelation (cf. Augustine's framework), was under attack from the potential heresies stemming from Aristotelianism. Among these heresies would be the failure to treat God as actually infinite in a full, transcendental, sense. Aristotle would say God is infinite, but mainly in negative terms- his infinity cannot be made fully actual without upsetting the autonomy of the natural world (since there can be no full or actually infinite bodies in nature).

Question 7, Article 1. Whether God is infinite?

Thomas begins by exploring the Greek/Aristotelian position (as mentioned, Aristotle is referred to throughout the reading as simply: “The Philosopher”) After asking the above question, we are given an initial objection:

It seems that God is not infinite. For everything infinite is imperfect, as [Aristotle] says; because it has parts and matter, as is said in Phys. iii. But God is most perfect; therefore He is not infinite.

Aristotle had argued that logical contradictions emerge if we accept the actual infinite. Thomas rightly concludes that therefore God is not actually infinite for Aristotle, but instead pure form which is most perfect (i.e. most fully intelligible). Then we get a second objection: Aristotle tells us that finite and infinite as categories belong to the class of ‘quantity’. Thomas writes:

But there is no quantity in God, for He is not a body, as was shown above (Q3, a1). Therefore it does not belong to Him to be infinite”


Finally, a further [third] objection is the following:

…what is here in such a way as not to be elsewhere is finite according to place. Therefore that which is a thing in such a way as not to be another thing, is finite according to substance. But God is this, and not another; for He is not a stone or wood. Therefore God is not infinite in substance.


This last objection touches on an important theological point. If we say that God is an infinite substance then what is to stop us from finding God in a tree or a stone (i.e. an idol). The latter was a form of heresy in Christianity –originally adopted from Judaism. There were actually Christians who fell into this heresy, for example some early Christians argued that Jesus was wholly man and that therefore God had become nature for a time. Others argued that God exists as part of the physical universe. Aquinas now quotes an Church authority Damascene (De Fide Orth. i, 4) God is infinite and eternal, and boundless. This is the position of St. John Damascene who was chief councilor of Damascus and a Church Father.

Thomas’s position:

Thomas begins by explaining how all the ancient philosophers attribute infinitude to the first principle This is taken verbatim from Aristotle’s Physics, where he examines the thoughts of the Pre-Socratics and Eleatics (Phys. iii). Thomas adds: the early Greeks thought that things flowed forth infinitely from this first principle. They were wrong, especially the Ionians, because they thought the first principle was matter. Aristotle, recall, argues that matter is finite and, as Thomas adds: they [the ancient Greeks] attributed to the first principle a material infinity to the effect that some infinite body was the first principle of things. Whatever is infinite, according to Thomas, is so-called because it is not finite. Matter is, in a way, however made finite by form, as is form by matter (this is Aristotle’s theory of hylomorphism, i.e. that whatever exists is a matter-form hybrid unless it is a special celestial substance - such as God). Matter is then said by Thomas to be a principle of potentiality: Matter is indeed made finite by form, inasmuch as matter, before it receives its form, is in potentiality to many forms; but on receiving a form, it is terminated by that one [form]. But form is equally made finite by matter. All alone, taken in and by itself, a form is a universal common to many. When a form is received in matter it is thereby determined to be one particular thing. This is an anti-Platonic or anti-realist position regarding universals. Plato thought the forms existed as real and separate transcendent of matter. Thomas, following Aristotle, is a moderate realist. He thinks forms are real (so they exist) but not that they transcend the natural world- they are instead real as immanent principles in nature. Since for most of the ancient Greeks, and especially for Aristotle, the most basic sense of infinite meant unbounded and undetermined, Thomas writes that matter without any form has the nature of something imperfect since it is mere potency and doesn’t even have a determined existence. Thomas’ reasoning can be understood as follows:

Thomas's position amounts to a Neoplatonic twist on Aristotle. The Prime or Unmoved ‘First’ Mover was said to be pure form and highest divine being in Aristotle’s metaphysics, but he equivocated as regarded the First mover’s relationship to matter. While true that the Prime Mover had no matter as part of its nature, and was not a material or efficient cause (for which reason Aristotle’s God does not create), nonetheless The Prime Mover influences the motion of the cosmos (it is the reason why motion exists) and so is viewed by Aristotle as the transcendent formal and final cause of reality causing motion by being an object of love. Aristotle, however, never explains how God exists in any greater detail or what the precise relationship is between God and matter. Thomas, a medieval Christian, wants to say the following:

God is pure and transcendent form (as was suggested by Aristotle and described by Plotinus) but he can create the world as is written in the Bible and was argued to be an essential part of God’s nature by Augustine.

The replies to the Objections further clarify Thomas’s position: Reply to Objection 2- Thomas stresses that: Quantity is terminated by its form, which can be seen in the fact that a figure which consists in quantity terminated, is a kind of quantitative form. Hence the infinite of quantity is the infinite of matter; such a kind of infinite cannot be attributed to God Thomas retains Aristotelian physics as a defense for why matter and nature cannot be infinite, but does not extend this model to God.

And, Reply to Objection 3- the being of God is self-subsisting, according to Thomas this is not true of any other being. Thomas takes this theory from the Islamic philosopher Avicenna. All particular existing beings are contingent or accidental (they exist, but they might not have existed and need not exist necessarily), but since contingent beings need a reason why they exist, and no contingent being alone is necessary, there must be a necessary being to condition what can be called ‘the set of contingent beings’ or the universe. This is God a self-subsisting (read: necessary) being. Thomas adds, God is also: distinguished from all other beings, and all others to be apart from Him. If anything could exist like God, it would be a perfect Platonic form, and be distinguished from everything else. Since Thomas is an Aristotelian however, he rejects Platonic forms as separate from things (whiteness can ONLY exist in particular white things for an Aristotelian) but since he is a Christian, he makes an exception for God. God is a perfect and infinite necessary being.

Question 7, Article 2. Whether anything but God can be essentially infinite?

The objections to disallowing another infinite being are the following:

  1. God himself is an infinite power and can create. Therefore God can create another infinite thing. As Thomas writes: if the essence of God is infinite, His power must also be infinite. Therefore He can produce an infinite effect, since the extent of a power is known by its effect.
  2. Our souls and minds are created by God. Our mind’s can grasp infinite: every created intellectual substance is infinite
  3. Primary Matter –pure potentiality, is not God [He is pure Form]. Since it can be anything, it is infinite and the natural world can therefore be infinite.

On the contrary: The authority this time is Aristotle: The infinite cannot have a beginning, as said in Phys. iii. But everything outside God is from God as from its first principle. Therefore besides God nothing can be infinite.

Thomas's answer is as follows:

Things other than God can be relatively infinite, but not absolutely infinite. Why does Thomas say this? Matter seems to be potentially infinite, but ultimately (metaphysically) it is determined by being limited to a specific existence. Matter must take a form to be an actual thing. By contrast: …if we speak of the infinite in reference to form, it is manifest that those things, the forms of which are in matter, are absolutely finite, and in no way infinite. Forms in matter are definite and actualize a primary substance. The form, says Thomas, is the substantial form making a thing essentially what it is. The angels are cited as an exception. Thomas was known as "the Angelic Doctor" for a reason. No other scholastic philosopher discussed the angels in more detail than Thomas. According to Thomas the angels were divinely created beings with no bodies. That meant that each angel was a self-subsistent form (which also means – in Thomistic Angelology -the systematic study of angels- that each individual angel is its own species!) Thomas writes: …if we speak of the infinite in reference to form, it is manifest that those things, the forms of which are in matter, are absolutely finite, and in no way infinite. If, however, any created forms are not received into matter, but are self-subsisting, as some think is the case with angels, these will be relatively infinite, inasmuch as such kinds of forms are not terminated, nor contracted by any matter

As pure forms, the angels don’t seem to be limited [‘contracted’] by any matter. Nonetheless, the angels are created (by God); they are not their own beings (as God is), and so angels have a received being [Elsewhere Thomas writes that angels = divine form whose being is primarily intellectual and can therefore be seen as combinations of intellect and will. Angels are given existence by God and created to serve him]. As created and subsisting (as forms) the angels have a nature (as does the human soul) and so cannot be absolutely infinite. The replies are important for the insight they shed into the Thomistic system and its classification of infinity:

  1. Existence and essence must be distinguished. Again, this is a position Thomas adopts from Avicenna. It is latent in Aristotle, but not developed explicitly in detail by the latter in any of his surviving writings. Thomas’s point is that no created thing can exist as an essential being (whatever has come into being or been caused to be, can go out of being and is therefore contingent or ‘accidental’ in its essential mode of being). And: God, although He has infinite power, cannot make a thing to be not made (for this would imply that two contradictories are true at the same time), so likewise He cannot make anything to be absolutely infinite.
  2. Our minds and angelic beings are a non-material power and seemingly infinite because of this. But even our minds have a relative infinity since if they were infinite we would be gods. And we are not: there are things we do not and may never know and our intellect’s also need a finite body to act- this limits them.
  3. Primary matter is co-created with the world and so is not absolutely infinite. Just as God creates the world, so he also creates possibilities along with the created world. Amongst these ‘concreated’ possibilities Thomas includes the potential of pure matter to be whatever it can and will be since it depends on nature which God (the absolutely infinite) creates.

Question 7, Article 3. Whether an actually infinite magnitude can exist?

Here is a familiar question from ancient Greek science that is still important today. Can something actually infinite exist in nature as a material body? Here Thomas’ Aristotelian bias is strongly present. His objections are actually arguing in favor of such an infinity- but he quickly takes a more orthodox Aristotelian approach after the four objections are given:

Objection 1: mathematics seems to posit infinite series. Mathematics never lies: it is reliable knowledge and a strict science according to Aristotle. If we can mathematically posit infinite lines extending throughout space, then the physical universe might be infinite. This is actually exactly what is said in Euclid’s parallel postulate (one of the axioms of Euclidean geometry).

Objection 2: “to be infinite is not against the nature of magnitude; but rather both the finite and the infinite seem to be properties of quantity”. There is no contradiction in saying that matter or space are infinite.

Objection 3: Zeno showed us that magnitude is infinitely divisible and Aristotle agreed (this is the basis of his view of mathematics as infinite in a potential and qualified sense). Why not say that an infinite series can exist in adding existing quantities, as well as in dividing things?

Objection 4: Movement and time, according to Aristotle have no beginning and no end. Every instant of time is both and every instant of movement is both. Therefore they are actually infinite.

On the contrary, Thomas sates, every body has a surface. But every body which has a surface is finite; because surface is the term of a finite body. Therefore all bodies are finite. The same applies both to surface and to a line. Therefore nothing is infinite in magnitude.

This is another Aristotelian proposition.

Thomas answers as follows: infinity can relate to essential truth or being and actual existing things. In the latter case, nothing that exists can be essentially infinite. All existence is determined in some way by material and formal causes. Only God the creator is infinite in essence. Thomas’s actual position is that God IS His essence and the essence of God is pure Being. “We must therefore observe that a body, which is a complete magnitude, can be considered in two ways; mathematically, in respect to its quantity only; and naturally, as regards its matter and form”.

Regarding material or natural existence, everything is a hylomorphic compound of matter and form. What Thomas calls the ‘substantial’ form of a primary substance limits it to a finite existence. Everything exists as something, and “every natural body has a greater or smaller determinate quantity”. We can criticize Thomas’s Aristotelian arguments against movement here, since they rely on the concept of natural place (which is invalid according to modern physics), but Thomas thinks that mathematical infinites are not actual but only potential: “For if we imagine a mathematical body actually existing, we must imagine it under some form, because nothing is actual except by its form; hence, since the form of quantity as such is figure, such a body must have some figure, and so would be finite; for figure is confined by a term or boundary”.

His replies are as follows,

  1. geometry is not about real existing lines or points. It abstracts from finite bodies and the mind creates idealized shapes that can be infinite but not actually so.
  2. Actual infinity is against the nature of any species. All species are definite in nature and that’s why they are called species (species means to be a specific thing different from others).
  3. the infinite can never be of an actual nature, it belongs to division or mathematical and potential series alone in nature or the world of magnitude.
  4. Movement and time are also at best potentially infinite. Since they are wholes and continuums and this also means Al-Kindi was wrong in his analysis of time as a collection of units of different now points. This can only be true for matter and Thomas thinks matter is created and finite.

Question 7, Article 4. Whether an infinite multitude can exist?

The last article is similar to the previous but instead of an infinite magnitude Thomas is asking about an infinite number of individual finite things. Objection 1- “it is not impossible for a potentiality to be made actual”, potentially the number of existing things can be multiplied, why can we not have an infinite number of things? Objection 2- the individuals of any species exist as a plurality. The species of some things, like figures, are infinite, therefore there might come to be an infinite number of things Objection 3- things that are “not opposed to each other”, Thomas says, “do not obstruct each other”. The reasoning here seems to be along the lines of: If a block of ice exists and there is no fire nearby, we can produce more and more ice. The same with plants, humans, minerals, etc... Next the Sed Contra:

On the contrary, It is written, “Thou hast ordered all things in measure, and number, and weight” (Wisdom 11:21). Thomas’ Reply: Thomas begins with outlining two different opinions on this matter as held by previous important natural philosophers. He writes: “Some, as Avicenna and Algazel, said that it was impossible for an actually infinite multitude to exist absolutely; but that an accidentally infinite multitude was not impossible”

Accordingly: “A multitude is said to be infinite absolutely, when an infinite multitude is necessary that something may exist. Now this is impossible; because it would entail something dependent on an infinity for its existence; and hence its generation could never come to be, because it is impossible to pass through an infinite medium”. This is basically an Aristotelian argument. If infinity were actual, nothing could pass through it- so it couldn’t be a cause or a part of the delimited natural world.

Thomas continues: “A multitude is said to be accidentally infinite when its existence as such is not necessary, but accidental. This can be shown, for example, in the work of a carpenter requiring a certain absolute multitude; namely, art in the soul, the movement of the hand, and a hammer; and supposing that such things were infinitely multiplied, the carpentering work would never be finished, forasmuch as it would depend on an infinite number of causes.” An accidently infinite multitude in the sense of a carpenter deciding how to build something and in what way to actualize their skill, cannot be made actually infinite or the carpenter would never actually finish work- or possibly even start. This is an example very similar to Zeno’s paradox trying to eliminate motion. Then Thomas writes: “But the multitude of hammers, inasmuch as one may be broken and another used, is an accidental multitude; for it happens by accident that many hammers are used, and it matters little whether one or two, or many are used, or an infinite number, if the work is carried on for an infinite time. In this way they said that there can be an accidentally infinite multitude”. Even if the decision to use a hammer cannot be infinitely extended without ever actually getting to work, there might be an actually infinite number of hammers or nails, and then the carpenter would only have to pick one. This is a valid interpretation of “an accidentally infinite multitude”. Thomas, however, rejects the above as impossible for the following reason: “…every kind of multitude must belong to a species of multitude. Now the species of multitude are to be reckoned by the species of numbers. But no species of number is infinite; for every number is multitude measured by one.”

Although the natural numbers, for example, might be infinite- there can be only one type of “two’s” or “three’s” just infinitely tokened. Two apples, two birds, two hammers, but one species or form of each kind of set; therefore:

“…it is impossible for there to be an actually infinite multitude, either absolute or accidental. Likewise multitude in nature is created; and everything created is comprehended under some clear intention of the Creator; for no agent acts aimlessly. Hence everything created must be comprehended in a certain number. Therefore it is impossible for an actually infinite multitude to exist, even accidentally.”

Thomas’s Aristotelianism is then stressed very forcefully and his answer about an infinite multitude is consistent with his answer above rejecting an infinite magnitude:

“But a potentially infinite multitude is possible; because the increase of multitude follows upon the division of magnitude; since the more a thing is divided, the greater number of things result. Hence, as the infinite is to be found potentially in the division of the continuous, because we thus approach matter, as was shown in the preceding article, by the same rule, the infinite can be also found potentially in the addition of multitude”.

The reading ends by replying to the original three objections that claimed an infinite multitude was possible. Thomas states: (1) Whenever a potency is made actual the deciding factor is the ‘mode of being’ of the potency. Somethings, like days or journeys, are processes and cannot be given all at once. Others, like God, are purely actual. Multitudes are things produced in succession like a process and the kind of infinity they can have is merely potential. (2) Figures and species of figures are mathematical kinds of entities, like a triangle. Thomas thinks that “an infinitely numerable multitude is not all at once reduced to act, so neither is the multitude of figures”. (3) “Although the supposition of some things does not preclude the supposition of others, still the supposition of an infinite number is opposed to any single species of multitude. Hence it is not possible for an actually infinite multitude to exist.”.