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.