Mathematics and Religion:
An Augustinian Synthesis 

                                                                                                                                                                                                         Keith Luoma
Augusta State University

Saint Augustine, Bishop of Hippo, was a prolific writer who produced numerous books and maintained a voluminous correspondence.  He wrote on a wide variety of subjects, including but not limited to philosophy, theology, education, music, history, and Biblical commentary.  Two of his works, the Confessions and The City of God, are considered classics in world literature, and a number of others are still read for their theological insights.

Not as well known is the extent of Augustine’s appreciation for mathematics, or his frequent use of mathematical arguments and metaphors.  The purpose of this article is to examine several passages of a mathematical nature from Augustine’s works, and to consider their significance to religion, philosophy, and mathematics.

Historical background^[i]

Augustine was born in 354 A.D. at Thagaste, a city in what is now eastern Algeria.  His parents, although of modest means, saw to in that he received an education.  After gaining a mastery of Latin literature and rhetoric, he held teaching posts at Carthage, Rome, and Milan.

A passion for knowledge led him to embrace a number of different philosophical and religious systems, including Manicheanism, Academic Skepticism, and the mystical Neoplatonism of Plotinus and Porphyry.  He eventually found peace in the Catholic Church, and was baptized by St. Ambrose in 387.  Within five years, he was ordained Bishop of Hippo, a position which he held until the end of his life (430).

Educational background

Education in Augustine’s day consisted of the seven liberal arts: grammar, rhetoric, logic, arithmetic, geometry, music, and astronomy.  For our purposes, we may restrict our attention to Augustine’s mathematical education.  As he never quotes from any surviving mathematical texts, a precise reconstruction of his studies is impossible.  Nevertheless, a probable one is as follows: As a young boy, he learned the rudiments of calculation.  In his autobiography he writes, “The initial elements, where one learns the three Rs of reading, writing, and arithmetic, I felt to be no less a burden and an infliction than the entire series of Greek classics.”^[ii]  (Augustine always preferred Latin to Greek.)  And again, “to me it was a hateful chant to recite ‘one and one is two’, and ‘two and two are four’.”^[iii]

As he progressed, he probably found his studies more to his liking.  “I learnt about the art of speaking and disputing, and about the dimensions of figures and music and numbers, with no great difficulty...”^[iv]  It is especially interesting to note his inclusion of music with figures and numbers.  In liberal learning, music was a theoretical subject in the Pythagorean tradition, stressing ratios and other numerical concepts.

Augustine’s school texts may have included the Greek text Introductio Arithmetica of Nicomachus of Gerasa (ca. 100 A.D.), which contains a study of the properties of numbers and their supposed mystical significance; as will be seen, Augustine’s writings share these interests.  But regardless of the actual textbook used in his instruction, Augustine never developed a deep knowledge of mathematics; his liberal arts education stressed breadth rather than depth.

Philosophical background

Although Augustine mentions a number of philosophers in his writings, it was the philosophy of Plato that would affect him the most.  Plato taught that ultimate reality exists outside of space and time.  The “Forms” of this perfect realm are reflected--albeit imperfectly--in the material world.  The goal of the philosopher is to contemplate these unblemished, divine objects.  Augustine was exposed to Platonic doctrine through the writings of the Neoplatonists, a group of philosophers who preserved and expounded on Plato’s teachings.  Chief among these was the philosopher Plotinus.

Another thread running through Platonism and Neoplatonism was the number mysticism of the Pythagoreans.  Pythagoras taught that mathematics was the key to understanding the universe--as a result his cult of followers virtually worshiped numbers.  Plato apparently concurred with this teaching, and the gate to his Academy reputedly warned that no one “ignorant of geometry” should enter within.

As will be seen, the influence of these philosophers on Augustine was enormous.   

Mathematics and scripture

Augustine searched for parallels in scripture to support his philosophical beliefs.  As a biblical counterpart to the Pythagorean maxim “all is number,” he frequently quoted Wisdom 11:21, “...you have ordered all things in measure and number and weight.”^[v]  While other Christian theologians  discounted the value of secular knowledge, including mathematics, Augustine defended its use, claiming “...the good...Christian should understand that wherever he may find truth, it is his Lord’s.”^[vi]

Although Augustine at times made use of literal interpretations of scripture, he was also quick to search for symbolic and allegorical meanings.  Since certain numbers, such as seven, ten, and twelve appear repeatedly throughout scripture, Jews and Christians had long sought to decipher their hidden meanings.  Augustine delighted in this tradition of number mysticism and his writings display a number of his own interpretations.  For example, ten was viewed as symbolic of a relationship between creature and Creator: three represents the triune God, while seven represents the life (heart, soul, mind) and body (made up of the four elements.)  The scriptural number forty is equal to ten times four, where four represents time (four periods to the day, four seasons to the year.)  Thus forty represents a period of communion between God and man, as in the forty day fasts of Moses, Elijah, and Jesus.^[vii]

In the City of God we find this oft quoted numerical exposition:

These works are recorded to have been completed in six days (the same day being six times repeated), because six is a perfect number... [T]he perfection of the works was signified by the number six. For the number six is the first which is made up of its own parts... And, therefore, we must not despise the science of numbers, which, in many passages of holy Scripture, is found to be of eminent service to the careful interpreter.”^[viii]  [A perfect number is a number which is equal to the sum of its proper factors.  Thus, 6=1+2+3.]

Admittedly, numerical interpretations such as these are no longer popular--not even in religious circles.  But it should be remembered that number mysticism is at least as old as Pythagoras, and can count among its practitioners a number of scientists of the first rank.

Philosophy of Mathematics

Augustine’s own brief flirtation with skepticism probably provided the impetus to examine his own philosophical attitude toward mathematics.  The skeptics taught that it was impossible for humans to attain certain knowledge about anything.  Augustine liberated himself from this philosophy with three arguments.  He first established--as Descartes would do over 1000 years later--his own existence.

I was brought up into your light by the fact that I knew myself both to have a will and to be alive.  Therefore when I willed or did not will something, I was utterly certain that none other than myself was willing or not willing.^[ix]

He then confirmed, by an appeal to arithmetic, the existence of truth outside of himself.

I wanted to be as certain about things I could not see as I am certain that seven and three are ten.  I was not so mad as to think that I could consider even that to be something unknowable.^[x]

Finally, he convinces himself of the existence of God in a manner similar to St. Anselm’s Ontological Argument.

And since all those who think of God think of something living, only they can think of Him without absurdity who think of Him as life itself.^[xi]

Satisfied that he had refuted skepticism, Augustine gave greater attention to mathematics.

I particularly noted the rational, mathematical order of things, the order of seasons, the visible evidence of the stars.^[xii]

He noted that although mathematical objects were represented crudely in the physical world, their natural realm was the mental.

 But numerical principles are neither Greek nor Latin nor any other kind of language.  I have seen the lines drawn by architects.  They are extremely thin, like a spider’s web.  But in pure mathematics lines are quite different.  They are not images of the lines about which my bodily eye informs me.  A person knows them without any thought of a physical line of some kind; he knows them within himself.  I am also made aware of numbers which we use for counting on the basis of all the senses of the body.  But they are different from the numbers by which we are able to think mathematically.  Nor are they the images of numbers as mental concepts, which truly belong to the realm of being.  A person who does not see that mental numbers exist may laugh at me for saying this, but I am sorry for the person who mocks me.^[xiii]

Was the origin of mathematics to be found, then, in the human mind?

It is perfectly clear to the most stupid person that the science of numbers was not instituted by men, but rather investigated and discovered.  Virgil did not wish to have the first syllable of Italia short, as the ancients pronounced it, and it was made long.  But no one in this fashion because of his personal desire arranges matters so that three threes are not nine, or do not geometrically produce a square figure, or are not the triple of the ternary, or are not one and a half times six, or are evenly divisible by two when odd numbers cannot be so divided.  Whether they are considered in themselves or applied to the laws of figures, or of sound, or of some other motion, numbers have immutable rules not instituted by men but discovered through the sagacity of the more ingenious.^[xiv]

Here Augustine ably defends the Platonist philosophy of mathematics.  But although this aspect of his philosophy is hardly original, he was willing to apply it to problems where few thinkers would dare venture.  One of these problems concerns the nature of infinity.

On Infinity

The infinite had long been a source of controversy among the world’s great thinkers, and its paradoxical nature led many to avoid or even reject its existence.  The Universe of Plato and Aristotle was a finite one, and the paradoxes of Zeno had shown the danger of allowing infinite processes into mathematics. As a result, mathematicians generally accepted, as Aristotle had, that mathematical quantities could be potentially infinite, but not actually infinite.  For example, the number sequence {1,2,3,...} was “potentially infinite” in that it could always be further extended.  However, since the number sequence can never be finished, it never becomes “actually” infinite.  Thus, the infinite has only an imaginary, metaphorical existence.

In the 3^rd century A.D., Plotinus would insist that God was beyond the bounds of number, and therefore infinite.^[xv]  Augustine, who was considerably influenced by Plotinus, also allowed for aspects of God’s creation to be infinite.  Writing in a now famous passage from The City of God, Augustine provides a justification for the existence of actually infinite quantities:
 

As for their other assertion, that God’s knowledge cannot comprehend things infinite, it only remains for them to affirm ...that God does not know all numbers.  For it is very certain that they are infinite; since, no matter what number you suppose an end to be made, this number can be...increased by the addition of one more...and while they are simply finite, collectively they are infinite.  Does God therefore, not know numbers on account of this infinity; and does His knowledge extend only to a certain height in numbers, while of the rest He is ignorant?  Who is so left to himself as to say so?  Yet they can hardly pretend to put numbers out of the question...for Plato, their authority, represents God as framing the world on numerical principles.^[xvi]

Thus, Augustine argues that God’s powers take Him beyond the potentially infinite to the actually infinite, since He can surely grasp the entire number sequence all at once, as a finished, completed collection.  Furthermore, since the created world is “framed” on “numerical principles,” infinity is an aspect of the real world.  Thus we have that God can think infinite thoughts, and infinity is an aspect of His creation.

It is interesting to note that Augustine’s defense of the infinite was largely ignored by the philosophers, theologians, and mathematicians who came after him.  The difficulty of dealing with the infinite in a consistent manner overwhelmed most thinkers, and was typified by St. Thomas Aquinas: “although God’s power is unlimited, he still cannot make an absolutely unlimited thing, no more than he can make an unmade thing.”^[xvii]  It was not until the nineteenth century that the German mathematician Georg Cantor developed a theory which would allow actually infinite collections to be used in mathematics.  In light of this, the following passage, also from The City of God, is especially illuminating:
 

The infinity of number, though there be no numbering of infinite numbers, is not yet incomprehensible by Him whose understanding is infinite.  And thus, if everything which is comprehended is defined or made finite by the comprehension of him who knows it, then all infinity is in some ineffable way made finite to God, for it is comprehensible by His knowledge.  Wherefore, if the infinity of numbers cannot be infinite to the knowledge of God, by which it is comprehended, what are we poor creatures that we should presume to fix limits to His knowledge?”^[xviii]

We note in particular Augustine’s claim that the infinite collection of all numbers must somehow be “made finite to God,” since “everything...comprehended is..made finite by the comprehension of him who knows it.”  Fifteen hundred years later, Cantor would reach a similar conclusion about his transfinite numbers, claiming that they “are clearly limited, subject to further increase, and thus related to the finite.”^[xix]  Cantor was well aware of Augustine’s work, a fact that he acknowledged on several occasions.^[xx] 

Towards a philosophy of mathematics

If mathematical truth concerns itself with abstract objects that are independent of the physical world--although imperfectly reflected in it--how do humans come by this knowledge?  Plato had taught that all knowledge is an act of remembering--that the human soul is exposed to truth before its embodiment, and that this truth must be drawn back out.  The dialogue in Meno between Socrates and the slave boy illustrates this viewpoint.^[xxi]  At first Augustine seemed to have adopted this perspective.

Such are those who are well educated in the liberal arts.  Doubtless in learning them they draw them out from the oblivion that has overwhelmed them, or dig them out as it were.^[xxii]

But later Augustine was to change his view.  Referring to the above statement, he later wrote in the Retractions:

I do not approve this...a more credible reason is that they have according to their capacity the presence of the light of Eternal Reason.  Hence they catch a glimpse of immutable truth.^[xxiii]

And elsewhere:

No bodily sense makes contact with all numbers, for they are innumerable...How can any phantasy or phantasm yield such certain truth about numbers which are innumerable?  We must know this by the inner light, of which the bodily sense knows nothing.^[xxiv]

Augustine’s belief, therefore, is that mathematical knowledge is a gift of God, known by an act of divine “illumination”, and it offers a glimpse at the eternal truth in the mind of God.  Mathematical knowledge, according to Augustine, is a form of revelation.

This theological solution to the epistemology of mathematics may leave the modern reader feeling a bit uncomfortable.  But for Augustine, philosophy is always subordinate to theology.  What we may perceive as a weakness in his philosophy is to  Augustine a strength: as God is the ground of all being, He is also the ground of all mathematical truth.  In this Augustinian scheme of things, mathematics without God is impossible.

Conclusion

How are we to assess the significance of Augustine’s mathematical thought? 

On the one hand, the number mysticism that flavored many of his writings is fairly typical for the time period, and can be found in the writings of numerous authors of Christian and non-Christian persuasion.  The best that can be said is that theological and philosophical fashions change, and that number mysticism is now out of vogue.

It is more difficult to assess Augustine’s views on the nature of mathematics because even today there seems to be no consensus on these matters.  There is a version of Platonism, still very popular among mathematicians, which finds expression in the belief that mathematics is an objective study of preexisting entities, that mathematics is discovered rather than created.  Others prefer to avoid the difficulties of Platonism by retreating to a humanistic philosophy of mathematics; one such account refers to mathematics as a “social construction.”^[xxv]  Jettisoned are the absolutes of mathematics, or as Augustine would say, the certainty that “seven and three are ten.”

One mathematician who proposes a humanistic account of mathematics is Reuben Hersh.  Commenting on Platonism, he makes these interesting and revealing remarks: “What a beautiful solution to the ontology of mathematics!  We needn’t ‘break our heads’ figuring out how numbers have their existence independent of human thought.  The thought is God’s!”  But, Hersh continues, “The present trouble with the ontology of mathematics is an after-effect of the spread of atheism.”^[xxv]

But developing a philosophy that would be satisfactory to atheists was the last thing on Augustine’s mind.  As he had already abandoned skepticism, Augustine’s main concern was that his philosophy be consistent with his Christian belief.  Although Plato’s spiritual realm of abstract Forms appealed to Augustine’s sensibilities, it was only logical for him to abandon Plato’s “knowledge as remembrance” in favor of his own “knowledge as revelation.”  After all, if God is the God of all truth, then why not assume that He would reveal this truth to His creation?