The Theology of Arithmetic/Introduction

The Theology of Arithmetic is a curious, but valuable work. It is preserved for us in the corpus of works of the eminent Neoplatonist Iamblichus, but it is almost certainly not the arithmological treatise he promised to write (On Nicomachus’ Introduction to Arithmetic, 125.15 ff.): the author of our treatise is unknown.

This is, as far as I know, the first translation of the text out of its original Greek, let alone into a modern language. There are two main reasons for the neglect of the work. In the first place, Neoplatonism in general, and Neopythagoreanism in particular, are not popular studies in today's universities. Secondly, the text itself is undeniably scrappy.

The treatise is, in fact, a compilation and reads like a student's written-up notes. Whole sections are taken from the Theology of Arithmetic of the famous and influential mathematician and philosopher Nicomachus of Gerasa, and from the On the Decad of Iamblichus' teacher, Anatolius, Bishop of Laodicea. These two sources, which occupy the majority of our treatise, are linked by text whose origin is at best conjectural, but some of which could very well be lecture notes—perhaps even from lectures delivered by Iamblichus. At any rate, the treatise may tentatively be dated to the middle of the fourth century A.D.

The value of the treatise is threefold. First, from an academic point of view, given the author's eclecticism, he has preserved, or he reflects, work which would otherwise be lost to us: that of Nicomachus is especially important (Anatolius' book is independently preserved), but there is also, for instance, a long fragment of Speusippus, Plato's successor as head of the Academy in Athens.

The second, and related merit of the book is simply that we have little extended evidence of the vast tradition of arithmology (as distinct from the hard science of mathematics) among the Greeks; The Theology of Arithmetic is a welcome, if sometimes obscure, addition to the slender corpus of such texts.

The third chief merit of the treatise is harder to explain. Pythagoras of Samos lived towards the end of the sixth century B.C. Our knowledge of his work and that of his successors is usually tantalizingly scanty, but enough is preserved to be sure that, while in exoteric areas such as mathematics there was considerable development, the esoteric and religious side of their work remained firmly based on the same principles and fully deserves to be called the Pythagorean tradition, which lasted for at least a thousand years. At any rate, to anyone familiar with the fragments of Pythagoreanism and Neopythagoreanism, it is immediately clear that a great deal of what The Theology of Arithmetic says could equally well have been said, and in many cases was, by a Pythagorean of the fifth or fourth centuries B.C.

I shall say a little more about this Pythagorean arithmological tradition shortly; here I need only say the following about it. Like Kabbalists, they recognized ten principles of all things. These ten principles are equated with, or described as, the first cycle of ten numbers. Thus our treatise simply takes each of the numbers from one to ten, and spends a few pages describing its qualities and giving reasons for those qualities. These qualities are primarily mathematical, and then by extension allegorical. So, for instance, the number one is present in all numbers and makes them what they are. This is a straightforward mathematical proposition; an extension is to call the number one Intellect, Artificer, Prometheus.

This approach has often been called superstitious mumbo-jumbo, but it is really only a different theological approach to ones with which we are more familiar. Mathematics is the most abstract language known to mankind; to base a theology upon it is to attempt to describe the abstract laws, as distinct from their applications, which govern God's universe. The outstanding nature of The Theology of Arithmetic, and its third chief merit, is that at one level it is unexceptionable, and fairly simple, Greek mathematics (since the complexity of mathematics lies in proving theorems, not merely in using the terminology, axioms or conclusions)—but the simplest mathematical statement is simultaneously imbued with religious import. The text makes it clear that for the Pythagoreans mathematics was more than a science: God manifests in the mathematical laws which govern everything, and the understanding of those laws, and even simply doing mathematics, could bring one closer to God.

In the Western world, arithmology is inevitably associated with the name of Pythagoras and his followers. It should not, however, be forgotten that long after the demise of the Pythagorean schools number mysticism was still flourishing among Jewish and Christian mystics. The study and use of gematria by Jewish Kabbalists is relatively well known; more often forgotten is the fact that even such seminal Christian writers as John of Ruysbroeck (1293-1381) were perfectly prepared to see numbers as authoritative symbols and paradigms of divine principles.

Among the Greeks, arithmology enjoyed a long history, as has already been remarked. It was practiced and studied concurrently with the science of mathematics, and by the same practitioners. As far as we can tell, no sooner had the Greeks been prompted (probably by contact with the Middle East) to elevate mathematics out of the market-place and into a pure science, than they also began to perceive philosophical and religious connotations to what they were doing. It is not too misleading a generalization to say that the Greek philosophers were impressed by the apparent orderliness of the universe and that this was for them the chief argument for some Intelligence being behind the universe. The Pythagoreans simply attributed this orderliness to the presence of number: numbers contain and manifest the laws of God's universe. Since all numbers after ten are repetitions of the first cycle, the first decad of numbers alone manifest these principles.

In passing, it should be noted that the idea that the laws which govern the universe are numerical is not in itself at all silly. It is not just that we are happy with formulae such as E = mc²: these are perhaps different in that they are supported by scientific proof. More significant is the fact that we are still happy to accept more mysterious mathematical phenomena, like the Fibonacci series of natural growth, or Bode's law of the distances between planets, and a host of other series and constants. Do we, I wonder, have the right to praise these as scientific, but condemn the Pythagoreans as irrational?

It should also be noted that while the Pythagorean attempt to give meaning to peculiar properties of number is unfashionably mystical, such peculiar properties have not been explained or explained away by modem mathematicians. They exist, and one either ignores them and gets on with doing mathematics, or gives them significance, which is what arithmologists do.

The beginnings of arithmology among Pythagoras and his immediate successors are unknown. Both mathematics and its mystical counterpart belonged at that time to the oral tradition. By the end of the fifth century B.C., however, it is certain that Philolaus of Croton was writing arithmology down; perhaps his great fame depends upon his having been the first to do so systematically. A number of fragments of his work are extant, but the genuineness of many of them is disputed. Enough remains, however, to afford glimpses into an arithmology which is not vastly different, in content or in quality, to what we find in our treatise.

The two great names from the beginning of the fourth century B.C. are Archytas of Tarentum and, of course, Plato. It is certain that Archytas made advances in the science of mathematics; but my opinion is that he left arithmology pretty much as he found it, though he certainly wrote on the subject. As for Plato, whose enthusiasm for mathematics (and arithmology, especially in Timaeus, certain parts of Republic, Epinomis and in his unwritten doctrines as reported by Aristotle and others) cannot be overestimated, it is an insoluble argument whether he borrowed from a tradition or was innovative in the field. Those who do not wish his greatness to be tarnished by the suggestion of plagiarism should remember that in the ancient world the borrowing of material was not theft but a sign of respect. It is likely that he both drew on and developed the tradition. Plato's Academy certainly nurtured both mathematics and arithmology, as we can tell from the titles of lost books written by Speusippus and Xenocrates, his successors as heads of the Academy, and by the presence in the Academy of the eminent mathematician Eudoxus of Cnidos and others.

In the centuries following Plato, the science of mathematics was developed in technical works such as those of Euclid, Archimedes, Heron and Pappus, but the history of arithmology becomes obscure. It is not until the first century A.D. that we again start to have extant texts. The fragmentary evidence from the intervening centuries allows us to be sure that arithmological tenets were used in various commentaries on Plato's dialogues (especially Timaeus, of course); but the mainstream of progress can only be conjectured. That there was progress is clear: in the first place, arithmologists would have borrowed useful developments of mathematics, as corroborative evidence for some property of some number within the decad, or more generally as confirmation of the importance of number in the universe; in the second place, the extant texts of later centuries assign more properties to numbers within the decad than we can safely say were assigned in earlier centuries, which suggests that the intervening period was when these properties began to be assigned; and thirdly, as philosophical fashions came and went, arithmology absorbed the language and some of the doctrines of Platonism, Aristotelianism and Stoicism. More importantly, the later extant texts often show a high degree of unanimity of thought and even of language. This suggests that there was some common seminal writer, whose name is unknown to us (it used to be thought to be the polymath Posidonius (c. 135-c. 50 B.C.), but who can be dated to the second century B.C.

This unknown teacher seems to have given a new impetus to Pythagoreanism. The name of Pythagoras was no doubt never far from the lips of all the arithmologists of these centuries; but in the first century B.C., in Rome and Alexandria, thinkers again began to claim direct descent from the master, and we nowadays call them the Neopythagoreans. Many of these philosophers, such as Publius Nigidius Figulus (first century B.C.), Apollonius of Tyana (first century A.D.) and Numenius of Apamea (second century A.D.) are little more than names to us now, though the work of others like Theon of Smyrna and Nicomachus of Gerasa, who were contemporaries at the beginning of the second century A.D., is better preserved. It is unlikely that we are dealing with a unified school of thought, but nevertheless the influence of this Pythagorean revival was great. It was especially strong over the Neoplatonists, but the early Neopythagoreans also paved the way for Greek arithmology to enter the Jewish tradition via the works of Philo of Alexandria, and the Christian tradition via the works of Clement of Alexandria.

Now, this potted history of Greek arithmology is meant only to illuminate our treatise by putting it in context. What I have called the 'lecture-note' style of the book already warns us, and the context of the vast arithmological tradition confirms, that the best approach would be to regard nothing in the book as original to the anonymous compiler, and to remember that far more ancient arithmology is lost than is extant.

It is not just that our author liberally quotes from Nicomachus and Anatolius, nor that much of what our treatise says can be found here and there in other books. It is also that, apart from one or two more extended and discursive passages (such as the discussion of the relation between the pentad and justice), one constantly gets the impression that we are being shown the tip of an iceberg—that far more arithmological speculation was available than we know from treatises such as The Theology of Arithmetic or occasional remarks in other philosophers.

As Keith Critchlow's foreword suggests, the most convincing way to demonstrate this is to practice a little arithmology our selves. In the last pages of the book, we find Anatolius associating the numbers 36 and 55 in various way. However, he omits one context in which these numbers may be related, and this context is so obvious that it must have been known to the Greeks. It would have been obvious because it stems from Plato, whose authority was revered.

In Timaeus, Plato declared that the primary sequence of numbers by which the universe gains life is 1,2,3,4,9,8,27; these numbers are displayed on a lambda diagram as follows:

A natural addition to this diagram, which completes it in the sense that it becomes a tetraktys (see Keith Critchlow's foreword), is to insert the geometrical means:

Three things immediately become clear: that the three means we have inserted add up to 36; that the three apexes of the triangle add up to 36; and that the diagram now contains all the factors of 36, which, as our author says, add up to 55. (A fourth point, that the separate double and triple series down the arms of the lambda add up to 55, is mentioned by Anatolius.)

This slight example of such speculation is deliberately intended to encourage others to do the same, as well as to support my contention that we are faced in The Theology of Arithmetic with the tip of an iceberg. An academic cannot acknowledge such speculation as evidence: his or her discipline is necessarily and rightly rooted in the available textual evidence—otherwise it fails to be the valuable mental discipline that it is. But I think that such speculation, provided it is not too extravagant, can afford personal insights, at any rate, into what the Pythagorean tradition was concerned with.

The Greek text I have used in preparing this translation is that of V. de Falco (Teubner series, Leipzig, 1922; additional notes by U. Klein, 1975.) De Falco's text is the most recent edition, and this was only the second time that the Greek had been critically edited: the first such edition was by F. Ast (Leipzig, 1817). Nevertheless, because of the lecture-note style of the text, and because of its inherent difficulties, over the centuries the Greek has become corrupted in many places: much more work even than that of Ast, de Falco and others remains to be done. Possibly some passages will never be recoverable: at any rate, I have often felt constrained to differ from de Falco's extremely conservative text (those who are interested are referred to my article in Classical Quarterly 1988). For those who want to compare the translation with the Greek text, I have inserted de Falco's page numbers in square brackets throughout the translation for ease of cross-reference.

A convention I have used in the translation needs mentioning. Our author often indicates by various Greek grammatical devices when he is changing from one source (say, Nicomachus) to another (say, Anatolius, or even a different bit of Nicomachus). I have indicated such changes by leaving an extra line space in the text (a single centered asterisk is used when such a break coincides with the end of a page). Sometimes he jumps alarmingly from one source to another: on p. 39, for instance, he attributes the content of one sentence to Anatolius, then skips to an unattributed doctrine (which is probably from Nicomachus), only to revert, without attribution, to Anatolius a couple of lines later. For those who read German, there is an excellent account of this aspect of our treatise by H. Oppermann, in the course of his review of de Falco's edition in Gnomon 1929, pp. 545-58.

It was tempting to clutter up the pages with many footnotes, but I have tried to keep these to a minimum. I have footnoted only those passages where I felt that the meaning might not be clear even on a second or third reading. There is a high degree of subjectivity about this, of course: what I think readers will find difficult to understand is not necessarily what any reader will find difficult to understand. It may be assumed, anyway, that those passages without footnotes are ones where I felt that a little thought could reveal the meaning. Flowever, there are one or two sentences which lack footnotes not for this reason, but because I completely fail to understand them!

I owe a particular debt of gratitude to Keith Critchlow and Kairos for being easily persuaded that it would be worthwhile to translate and publish The Theology of Arithmetic. Several others have helped me to understand one or more passages of the treatise, and I would like to acknowledge such assistance from the following people: Professor John Dillon, Dr. Vivian Nutton, Dr. David Sedley, Rod Thom, Annie McCombe, Peter Thomas and Keith Munnings. It goes without saying that none of them are responsible for any remaining defects in quality and tone of my translation or commentary.

— Robin Waterfield