The reception and modification of ancient education by our Christian ancestors in early Christendom show a commitment to the seven liberal arts: the trivium of grammar, logic, and rhetoric, and the quadrivium of arithmetic, geometry, music, and astronomy. In addition to (and higher than) the arts was the study of Scripture. Is there any relation between the two disciplines of the arts and the Scripture? Do we find themes or echoes of the seven arts in Scripture? Or better, would the original hearers of Scripture have studied these arts, and would that have any bearing on how they heard or read the divinely-inspired text? In antiquity, the Jews were notable for their literate culture and no doubt would have spent much time on grammar, logic (as evidenced by the disputation traditions), and rhetoric. While much could be said for the trivium, in this article I want to focus on the mathematical arts of the quadrivium, whose presence may be less obvious to readers of Scripture.
Histories surveying the liberal arts begin with the Greco-Roman tradition, and not without reason. The father of the quadrivium is Plato. He proposed a mathematical education for philosopher-kings in the Republic (arithmetic, [planar] geometry, stereometry, astronomy, and music; the four of the quadrivium joins planar geometry with stereometry ). In the text, Socrates also emphasized a particular mode of mathematical study: geometry should not merely be the calculation of shapes and areas in physical objects but rather a theoretical pursuit (as later seen in Euclid). Likewise, arithmetic proper is not counting or the handling of quantity but a study of numbers, their properties, and relations. The Platonic tradition produced canonical texts for each of these disciplines: Nicomachus for arithmetic, Euclid for geometry, various writings for music (Nicomachus, Ptolemy, and Aristides Quintillianus), and Ptolemy for astronomy. Later authors would also provide commentaries and handbooks that pointed out the connections between mathematics and Plato’s philosophy and even provided the motivations of studying the former to understand the latter.
Yet these authors were also quick to note peoples of greater antiquity in the study of mathematics: Egypt for geometry, Babylon for astronomy, and perhaps Assyria for arithmetic. The feature that makes Greek mathematics unique is the generalization of theorems and the formulation of their proofs. Further, the unique feature of the mode of study in the quadrivium was the organization of these ideas into one grand educational vision: to see the order of the cosmos, to order the soul, and to contemplate the divine. The connection of mathematics to these topics was typically made in the form of image, analogy, or symbol.
Given the concession by the Greeks that yet-more-ancient cultures had begun mathematical explorations, we may wonder if these other cultures continued to develop mathematical ideas and connections along lines similar to the Greeks. In this article, I will particularly focus on whether ancient Israel participated in such an endeavor. I will draw on Scripture itself but will also explore Christian interpretations that have perhaps been influenced by this “quadrivial” thinking. I will treat each of the four disciplines separately and then address more comprehensive questions about using mathematics symbolically.
Arithmetic
Numbers are an integral part of creation. Evening and morning make “one day,” forming the unit by which days are counted. There are six of these units in God’s active work of creation, and God rested on the seventh day. The creation narrative sets the tone for the rest of Israel’s ordering of time with seven-day weeks and seven-year festival cycles. Just as Genesis provides an account for night and day on Day 1 prior to the creation of celestial bodies on Day 4, numbers and counting likewise precede the existence of physical objects. This is notable in that the ancients, ranging from the Babylonians to Plato, understood counting as originating from astronomical observation. The act of counting is ubiquitous in the ancient world, whether its object is days, animals, monies, or peoples. Of the last category, so much counting was included in the fourth book of the Pentateuch that the translators of the Septuagint called it Arithmoi, from which we get the English title, “Numbers.” Where there is counting to be done, there must be people to do the counting. It may seem obvious, but the recording of so many numbers showcases a people who has studied at least elementary arithmetic.
Perhaps even more notable is the set of particular numbers which I will refer to as ‘symbolic,’ in the literary sense.1 Readers of Theopolis need not be reminded of all the uses of 3, 7, 12, 40, as well as others throughout Scripture. The presence of these numbers has meaning behind them when they appear. Clearly, the ancient Israelites were quite familiar with numbers. Does familiarity imply education on the topic? One might speculate it to be the case.
Geometry
Next, we turn to geometry. Practical use of geometry was critical for the partitioning of land and for architecture. The former was of great importance to Israel. If the Jubilee was kept, it would require the return of land with the former ancestral boundaries (Lev 27:24). If the borders changed through acquisitions during the fifty-year period (note the numerical symbolism), it is speculative but reasonable to suppose that some form of practical geometry would have been required to mark the original boundaries. Would the Israelites have utilized some technique comparable to that of the Egyptians who had to repeatedly redraw land boundaries due to the flooding of the Nile, techniques which were further developed in the abstract theorems of Euclid’s application of areas (Elements 1.44)? It is again speculative but not unreasonable to think that they could have learned such techniques during their captivity.
As for architecture, the building of city walls and domestic residences surely required skilled technicians. But as the builders of Gothic cathedrals later made evident, one need not be versed in geometric theorems to make proportional measurements even though applied architectural techniques are geometrical in nature. So the prophets’ references to the plumbline (e.g., Is 28:17; Am 7:7–8; Zech 4:10) have geometric connotations. It communicates a people’s familiarity with applied geometry while also symbolizing moral uprightness.
Yet the richest theme of geometry in Scripture is found in the various passages related to sacred buildings: the tabernacle, the temple of Solomon, the temple-vision of Ezekiel, and the New Jerusalem. Here we find shapes such as the square altar surface and the circle of the bronze laver that have elemental connotations: the land and the sea, respectively. We also find proportions that may have symbolic meaning (the cubic dimensions of the new Jerusalem is perhaps a reference to equal relations of a triple dimension). We also find specific whole-number magnitudes, which have been a source of exploration for allegorical interpretation in Christian tradition (see below). In sum, we find the use of practical geometry as well as themes of shape, dimension, and proportion, which a people cognizant of basic geometric principles would have noticed.
Music
The topic of music as it was studied in the quadrivium needs a brief preface. “Music” today usually refers to instrumental and vocal music. Its study takes the form either of training as a practitioner or of music theory. The ancient use of the term was far more encompassing, and its study within the liberal arts entailed a study of ratio, proportion, and generally how things fit together.2 This was applied in vocal and instrumental music by setting concords and intervals to specific mathematical ratios. Music performance was a reflection of the higher “musical” order. In Greece, the Pythagoreans initiated this mathematical pursuit with the discovery of whole-number ratios matching the perfect concords (i.e., 2:1, 3:2, and 4:3 for the octave, perfect fifth, and perfect fourth). This discovery is one of the few generally accepted to be genuinely Pythagorean, even if the circumstances (the pinging of blacksmith hammers) are legendary.
Scripture ascribes the origin of music to Cain’s descendent, the metal-worker Tubal-cain. However, the greatest attention in the Hebrew Scriptures regarding music is focused on the house of David. When Samuel anointed Saul, he told him that he will meet prophets coming down from a high place with an array of musical instruments (1 Sam 10:5). Israel’s king was supposed to organize the musical worship of YHWH. Saul failed to do this, and his anointment was given to another more worthy: David. He established orders of singers from the tribe of Levi who were devoted to the playing of music, being “free from other obligations” (1 Chr 9:33–34). The head musician was chosen because “he understood music” (15:22), perhaps indicating the study of music theory.
The music of Israel’s worship is meaningful, not merely enjoyable. When Jehoiada overthrows the worship of Baal, he reestablishes the Davidic order of music (2 Chr 23:18). What makes David’s order of music proper to the worship of the true God? Does the distinction extend beyond lyrics into harmonics and rhythm? Perhaps. But the role of music and the affections it aroused was long-known, as when Moses recognized the revelry in the camp by the music he heard (Ex 32:18).
While the importance of music in worship, political order, and the influence on the affections is prominent, there are also some technical curiosities we can deduce from Scripture. While the Greeks had standard stringed instruments of four, seven, and eight strings, David calls for the playing of the ten-stringed harp (Ps 33:2). How long were the strings, and what were their intervals? We can only speculate. (Augustine treats the number allegorically: the ten strings represent the ten commandments.)
In any case, the novel Pythagorean application of mathematics to music post-dates the ancient Israelite kingdom by several centuries; we would not expect to find it in the Hebrew Scriptures. What of a higher music which encompasses the harmony of the cosmos or the well-balanced soul? In the New Testament, the epistle to the Romans calls for the community to live in harmony (Rom 12:16; 15:5), an idiom that is now in common parlance but which was originally an application of music to ethics.
Astronomy
Finally, we consider astronomy. The celestial lights are part of the creation account and were made for “signs and seasons.” God’s promise to Abraham that his descendants would be as many as the stars has led some to believe that, because of Abraham’s Mesopotamian origins and their traditions of stargazing, he was versed in astronomy. The festivals outlined in the Mosaic Law followed a calendar based on the sun and moon. Presumably, the priesthood, which organized the ceremonies, would have had astronomers to track the sun and moon. Astronomy was a priestly activity in other cultures, and we have no reason to think the case was any different here. Of course, the priests of the nations worshipped such objects in the sky, a practice that the Law of Moses expressly forbade (Deut 4:19). The ancient Babylonians had quite accurate tables for the stars and those wandering objects we now call planets (planetai means “to wander”). The Greeks are credited with generating mathematical models of the planets, so the epicycles and eccentrics would not have been not studied in ancient Israel. However, there was a general awareness of the circularity of the paths of the stars and their constancy. Scripture refers to specific constellations (Job 9:9) and also speaks of the circuit of the Sun (Ps 19). The order and constancy of the cosmos represented God’s enduring covenant (Jer 31:35–36). So, we can conclude that the human authors of Scripture had some understanding of order in astronomy, although the exact mathematical nature of it may have been unknown.
The Primacy of Numerology
From the testimony of Scripture, we have indications of both astronomical study and musical training in Israel’s priesthood. Neither was likely to be of the mathematical form that was developed centuries later by the Greeks in their formulation of the quadrivium. However, we do see that the heavens are a sign of cosmic order; their regularity is a token of God’s faithfulness. We also see that there is not only an order to music which would have required musical study for Levites leading the worship of Israel but that the music of worship ordered the soul of the nation. Geometry has practical uses as well as its higher modes of symbolic expression.
While each of the arts are represented in biblical symbolism, numbers play the largest symbolic role in Scripture. Scripture records many aspects and instances of counting, and numbers frequently reveal the deeper meaning and significance of events and ideas.
In order to better draw out the numerological significance in Scripture, I’d like to pivot back to the Greeks and the quadrivium for a moment to see if we can gain additional insight into the Hebrew use of mathematical symbolism by considering it in parallel with the Greek treatment of the topic. We’ll look at distinctions between the modes of numerology and finally look at how the Greek understanding of numbers has been used in the Christian tradition.
Note, Plato’s philosophy of the forms was mathematical in nature, and his promotion of mathematical education was likely for the purpose of studying the highest things (metaphysics and theology).3 As a result, those who wrote what are regarded as the “canonical texts” of the quadrivium were deeply influenced by Plato and interwove themes of metaphysics, psychology, and ethics into their mathematical work.4 These connections between philosophy and mathematics in the tradition of the quadrivium are most prominent in the applied topic of the ordered cosmos and the higher topic of numerological symbolism.
An example may be helpful here. The section on arithmetic in Martianus Capella’s Marriage of Philology and Mercury recounts various decad-deity relationships drawing on traditions in the Greek quadrivium. Early in the text, it is “demonstrated” that Mercury and Philology were a good match based on an arithmetic procedure that took their respective names, computed their name-numbers (a type of gematria), and divided them out by a common measure to find the remainders of three and four, respectively. As these are the numbers of male and female,5 it makes for the archetypal match.
This may seem obscure, nonsensical, or thoroughly pagan. But I use this example to illustrate ancient thought about the higher meanings of numbers. These two individuals are known truly by their names, and their true names are mathematical (or at least mathematically symbolized). Treating mathematics both symbolically as well as quantitatively reinforced a larger intellectual framework of seeing a world as not only quantitatively ordered but one wherein there are layers of order. This layered symbolism, if properly understood, calls us to look beyond the tangible appearances (transcending rather than abandoning the tangible) to “other-worlds” that make ours the way it is.
What does all this have to do with ancient Israel and Scripture? Let’s return to Scripture and look at several numerological passages regarding temple geometry as well as gematria. We may be tempted to think of these themes as something that mattered to ancient peoples but are quaint or trite to our modern and enlightened minds. Yet I would say that such an approach to the world, names, and the construction of sacred spaces is not only a curiosity but something we should recover. We should agree with ancient peoples, especially our ancestors in the faith, that the world around us is filled with signs and that these signs have meaning. The Platonic excursion therefore is only a secondary corroborative account of mathematical symbolism, which we could have inferred solely from embracing a world-hermeneutic that the human writers of Scripture under divine inspiration also held.
A specific and more direct way to make the point is to pose a question that I often ask to my students in our arithmetic class: Do the tabernacle numbers matter? The text doesn’t simply say that the people of Israel gathered to make an aesthetically proportional place of worship under engineering and architectural constraints. Rather than mere general guidelines, God gave mathematical specification. For example, God told Moses to make the tabernacle with curtains measuring 28 cubits by 4 cubits, 5 together and with 50 loops in them (Ex 26:1–6); the courtyard was to be 50 cubits by 100 cubits with curtains 5 cubits high (27:9–19). Every portion of the complex was numerically specified, and God seemed to care about those specific numbers. Such care only makes sense if the numbers carry symbolic meaning beyond the mere quantitative.
Interpretive modes like gematria require a bit more support since we do not have direct command for its utilization. Yet once the symbolic nature of numbers in Scripture is granted, it is far easier to embrace the idea of this feature within the text. While its use can be in excess if interpreted without care, being aware of it allows for fascinating insights as Jim Jordan has shown (see “Abram’s 318 Men” and “153 Large Fish”).
Quadrivium in Christian Tradition
If this mode of thinking is valid, then the study of the quadrivium—even through Hellenic texts—offers significant promise for the Christian. In God’s providence, the study of the Greco-Roman texts of the liberal arts nurtured the minds of many in the Christian intellectual tradition. Augustine explains the role of signs, especially in the figurative passages of scripture in On Christian Teaching. Throughout Book 2, he affirms the studies of the arts and sciences to aid in our study and interpretation of Scripture. With respect to mathematics, he goes so far as to say, “Ignorance of numbers, too, prevents us from understanding things that are set down in Scripture in a figurative and mystical way” (2.16.25). He gives further praise of mathematics that the science of number is discovered and not created, its teaching unchangeable and true because of its source in God (2.38.56).
I propose, then, that study of the quadrivium particularly facilitates insights into Scripture. I will mention a few such insights here as representative of the tradition. In City of God, Augustine asks which is the more perfect number: six (on the grounds of it being the sum of its proper divisors) or seven (on account of completeness and rest) (De civitate Dei 11.30–31). He later reflects on the dimensions of the ark, citing Origen and alluding to Acts 7:22 to claim that Moses knew Egyptian geometry, and that therefore the cubits listed are “geometrical cubits which are six to one in our reckoning of cubits”; he affirms at the end of the passage that events such as the flood are historical and that they also have symbolic meaning.
A few centuries after Augustine, Isidore’s section on arithmetic in his Institutiones covers the standard topics of Nicomachus’ arithmetic (translated by Boethius) before giving a short list of the symbolic references of the first seven numbers as they pertain to Scripture. Not content with this work, he later wrote an entire treatise on numbers in scripture.
Still later, Bede took an allegorical approach to the numbers used in the tabernacle and temple; he linked the four feet of the table in the tabernacle to the four gospels and the four ways of reading Scripture.
Finally, Hugh of St. Victor, in his work on Noah’s ark, repeats Augustine’s argument regarding Moses’ learnedness in Egyptian geometry before adding some numerological interpretations of his own: fifty cubits stands for the church as seven times seven is the completeness of all believers plus one who is Christ, the Head of the Church.
This mode of symbolism abounds in the tradition. Some of these allegories are stretched. Some have little or no reference to the mathematical properties of the numbers being utilized. Still, all of these authors bear witness to a mode of thinking that treats number not merely as quantity but as quality or symbol. It is a way of thinking that should be recovered: to see a world of signs and symbols which are showcased in the mathematical arts, and to put its insights to use in the course of the life we walk.
Phillip R. Johnson is a Teaching Fellow at New College Franklin. He lives in Franklin, TN with his wife and four children.
NOTES
- To distinguish from symbolic number in the mystical sense or esoteric use. ↩︎
- Thus, for Boethius, the well-ordered man is properly the true musician even if he is untrained in vocal or instrumental music. ↩︎
- The analogy of the Divided Line in Republic VII advocated studying mathematics as preparatory for the study of dialectic, or metaphysics. Aristotle asserts in Metaphysics A.6 that Plato held that “besides sensible things and Forms he says there are the objects of mathematics, which occupy an intermediate position, differing from sensible things in being eternal and unchangeable, from Forms in that there are many alike, while the Form itself is in each case unique.” See Lloyd P. Gerson, From Plato and Platonism (Cornell University Press, 2017), for more on how Aristotle and the successors of the Academy sought to develop or critique these ideas. ↩︎
- Nicomachus’ Introduction to Arithmetic stands out. Similar themes can be found in Nicomachus’ Handbook of Music, Ptolemy’s Harmonics, and Aristides Quintillianus’ On Music. Promoters of the quadrivium in the Greek tradition are almost always Platonists, with Iamblichus and Proclus being the two exemplars of the later movement. ↩︎
- Three is the first odd and four the first even, so odd is associated with sameness and maleness, and even is associated with otherness and femaleness. The text of Theologoumena Arithemeticae is perhaps the earliest formulation of the idea which is then repeated by mathematically-inclined Platonists. ↩︎