What do you think?
Rate this book
The Theology of Arithmetic
Attributed to Iamblichus (4th cent. AD), The Theology of Arithmetic is about the mystical, mathmatical and cosmological symbolism of the first ten numbers. Its is the longest work on number symbolism to survive from the ancient world, and Robin Waterfield's careful translation contains helpful footnotes, an extensive glossary, bibliography, and foreword by Keith Critchlow. Never before translated from ancient Greek, this important sourcework is indispensable for anyone intereted in Pythagorean though, Neoplatonism, or the symbolism of Numbers.
136 pages, Paperback
First published November 1, 1988
16 people are currently reading
505 people want to read
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 - 12 of 12 reviews
September 6, 2017
This was definitely an interesting book. It's one of the few that we still have from that era that breaks down Pythagorean numerology in detail.
While this book has historically been attributed to Iamblichus, it isn't certain who the author, or rather, compiler, was. I say "compiler" because like a lot of works from this period, it seems to be more of a doxographical compendium. There are excerpts from Nicomachus, Anatolius, Speusippus, etc. It is very possible that some portions of this work are notes from Iamblichus' lectures, but it's difficult to say for certain. One can say though that much of the material is repeated in a number of analogous sources, i.e. in Nicomachus, Theon, Plato, Euclid, etc.
The overall structure of the book is based on the Pythagorean Tetraktys. Chapter one is on the Monad, chapter 2 is on the Dyad… all the way to chapter 10 on the Decad. Some of the chapters were a bit more interesting than others; some of the natural philosophy would be considered archaic and obsolete by today's scientific standards. That being said, there's some really great discussions regarding the properties of the tetraktean numbers that goes beyond just mundane mathematics, e.g. even numbers are considered female and odd numbers male. I also thought that the chapter on the triad was quite intriguing; while one is the monad, and two the dyad, three is considered to be the synthesis of the thesis of the monad and antithesis of the dyad. Those not interested in the mystical interpretation of numbers in ancient Greek philosophy, probably won't be interested in this work and should just stick to Euclid. It's surprising that this work, or works like it, were not included in Guthrie's Pythagorean library. It really is a great addition to the works found in that collection.
Taking into account the importance of numbers in Pythagorean philosophy, and the importance of the Tetraktys specifically, this is definitely an essential read for those interested in that tradition; but it also serves as a great example of Neo-Platonism as well.
While this book has historically been attributed to Iamblichus, it isn't certain who the author, or rather, compiler, was. I say "compiler" because like a lot of works from this period, it seems to be more of a doxographical compendium. There are excerpts from Nicomachus, Anatolius, Speusippus, etc. It is very possible that some portions of this work are notes from Iamblichus' lectures, but it's difficult to say for certain. One can say though that much of the material is repeated in a number of analogous sources, i.e. in Nicomachus, Theon, Plato, Euclid, etc.
The overall structure of the book is based on the Pythagorean Tetraktys. Chapter one is on the Monad, chapter 2 is on the Dyad… all the way to chapter 10 on the Decad. Some of the chapters were a bit more interesting than others; some of the natural philosophy would be considered archaic and obsolete by today's scientific standards. That being said, there's some really great discussions regarding the properties of the tetraktean numbers that goes beyond just mundane mathematics, e.g. even numbers are considered female and odd numbers male. I also thought that the chapter on the triad was quite intriguing; while one is the monad, and two the dyad, three is considered to be the synthesis of the thesis of the monad and antithesis of the dyad. Those not interested in the mystical interpretation of numbers in ancient Greek philosophy, probably won't be interested in this work and should just stick to Euclid. It's surprising that this work, or works like it, were not included in Guthrie's Pythagorean library. It really is a great addition to the works found in that collection.
Taking into account the importance of numbers in Pythagorean philosophy, and the importance of the Tetraktys specifically, this is definitely an essential read for those interested in that tradition; but it also serves as a great example of Neo-Platonism as well.
December 16, 2025
A great explication of the theological elements of arithmetic in the Neoplatonist and Pythagorean traditions.
Read
May 16, 2019Somewhat ridiculous to give something like this a star rating. It's an ancient textbook on Neopythagorean Theology. Often attributed to the Syrian mathematician Iamblichus, but not many scholars think it's a work by him. At most it might be partially based on notes of a lecture by him (or another guess I had was, it could be notes of Iamblichus' in preparation for a book he was writing on the subject). So what we have is a collection of notes from various sources, many now lost, by an anonymous 4th century author on Neopythagorean Theology. That's Neo-pythagorean because this is written about 900 years after the life of Pythagoras. Yes, that's the Pythagoras you learned about in geometry class who (allegedly) invented the Pythagorean Theorem. It seems that Pythagoras wasn't just a mathematician but a mystic and philosopher. His ideas influenced Plato and Aristotle and subsequently all of western philosophy. Essentially, he and his followers believed that numbers were the fundamental element of which the entire universe is made. All the properties of numbers such as even, odd, prime, etc. (and much more you probably never thought about), they ascribed theological significance to. They were particularly interested in music and are attributed with first identifying the harmonic ratios. They were the ones who were first obsessed with the "golden ratio", which influenced art and architecture for centuries (or millennia really, considering that the renaissance was 2000 years after the life of Pythagoras). Their cosmology may have been completely wrong but they are one of the earliest if not first to start thinking about the universe as being describable with math and geometry, in short the line of thinking that has lead to all the advances in understanding we have today. Kepler and Copernicus were greatly influenced by Pythagorean ideas.
Anyway, I could go with that line of thought for quite a while but back to the point. There are many ancient biographies of Pythagoras by Porphyry, Diogenes Laertes, Iamblichus, and an epitome by Photius of a lost biography, and they all make some reference to the Pythagorean's beliefs but this book is the only source we really have that describes them in detail. Here is a really brief introduction to their ideas:
1.) The Monad is the fundamental building block of Number. All other numbers are made of monads. “Everything has been organized by the monad, because it contains everything potentially: for even if they are not yet actual, nevertheless the monad holds seminally the principles which are within all Numbers.” Also, “the monad is the non-spatial source of number” because it is a single point, so it doesn’t have any dimension. The Pythagoreans placed the monad in the "fire" or "hearth" in the center of the universe.
2.) The Dyad is the first extension of the Monad. With two points you now have a line which makes a single dimension. Pythagoreans felt that “whereas the monad seems to give rise to generation, the dyad seems to admit destruction.” And they weren’t able to fully commit to the idea that it was an even number because, “every even number is divisible into both equal and unequal parts, but the dyad alone cannot be divided into unequal parts, and also, when it is divided into equal parts, it is completely unclear to which class its parts belong, as it is like a source.”
3.) “The triad has a special beauty and fairness beyond all numbers, primarily because it is the very first to make actual the potentialities of the monad — oddness, perfection, proportionality, unification, limit.” The triad makes 3 points, which defines a 2 dimensional plane. It is the first number to be odd. It is "perfect" because it is the first, after the monad, in a series of numbers that are equal to the sum of the previous numbers. 3, 6, and 10.
4.) They introduce the tetrad with the following statement, “Everything in the universe turns out to be completed in the natural progression up to the tetrad, in general and in particular, as does everything numerical — in short, everything whatever its nature.” They are referring to the fact that 1+2+3+4 = 10 aka the decad which represents the universe. With the tetrad you have 4 points and are able to create a 3 dimensional solid.
5.) The Pentad is made of 5 points so it also can make a 3 dimensional shape but with a point left over. Conceptually this point can be thought of as the something extra that gives the spark of life, or the heart. So we have moved from an inert solid into the lowest stage of life. “In the realm of embodiment there are, according to natural scientists, three life-engendering things--vegetative, animal and rational. The [...] pentad is the minimal extreme of life: the vegetative, hexad is the next: the animal, and the hebdomad is the next: the rational.”
6.) “The hexad is the first perfect number; for it is counted by its own parts [1+2+3=6], as containing a sixth, a third and a half. When squared, it includes itself, for 6x6=36; when cubed, it no longer maintains itself as a square, for 6x36=216, which includes 6, but not 36.” (attrib. Anatolius) If you count three numbers at a time starting with 1, 2, 3, and add them together, the individual integers of the sum always add together to equal six (until you get up to 60 it seems).
1, 2, 3, = 6
4, 5, 6 = 15
7, 8, 9 = 24
10, 11, 12 = 33
13, 14, 15 = 42
16, 17, 18 = 51
19, 20, 21 = 60
7.) The Heptad is associated with all the stages of life, the development of the embryo, and the viability of birth. After seven years the child loses their baby teeth, at 14 they undergo puberty, at 21 they are done growing in height, at 28 done growing in breadth, at 35 you can no longer gain anymore strength, athletes usually retire at this age and the best states end conscription at this age, saying that they "after this point allow people to be officers, but not to serve in the ranks any more." Then at 70 you should be "released from all tasks" and enjoy yourself. A lunar quarter is 7 days. 1+2+3+4+5+6+7 = 28, the length of a lunar month. Incidentally, women menstruate monthly! It is associated with Chance in myth.
8.) “We describe the octad [or ogdoad] as the first actual cube, and as the only number within the decad to be even-times-even, since 4 appears to combine the characteristics of being odd-even and even-times-even in admitting only two divisions up to the monad, one of itself, the other of its parts.”
“Philolaus says that after mathematical magnitude has become three-dimensional thanks to the tetrad, there is the quality and 'color' of visible Nature in the pentad, and ensoulment in the hexad, and intelligence and health and what he calls 'light' in the hebdomad, and then next, with the ogdoad, things come by love and friendship and wisdom and creative thought.”
9.) “The Ennead is the greatest of the numbers within the decad and is an unsurpassable limit.”
10.) The Theology of Arithmetic explains the importance of the decad thus: “the creative mind wrought the construction and composition of the universe and everything in the universe by reference to the likeness and similarity of number, as if to a perfect paradigm. [...] However, a natural equilibration and commensurability and wholeness existed above all in the decad. [...] And this is why, both in general and in particular, things from heaven to Earth are found to have been organized by it.”
“The spheres of the universe are ten and fall under the decad.” Speusippus says, the decad has equal numbers of prime and composite numbers. “the decad generates the number 55, which encompasses wonderful beauties. For in the first place, this is formed by doubling and trebling the systematic sequence of numbers — the doubles are 1, 2, 4, 8 (i.e. 15), the triples are 1, 3, 9, 27 (i.e. 40), and the addition of these makes 55.” (attrib. Anatolius)
“Moreover, the sequence of the first five triangular numbers generates 55 (3, 6, 10, 15 and 21 make 55) and again, the sequence of the first five squares generates 55 (1, 4, 9, 16 and 25 make 55); and according to Plato the universe is generated out of triangle and square. For he constructs three figures out of equilateral triangles — pyramid, octahedron and icosahedron, which are the figures respectively of fire, air and water — and the cube, the figure of earth, out of squares.”
Anyway, I could go with that line of thought for quite a while but back to the point. There are many ancient biographies of Pythagoras by Porphyry, Diogenes Laertes, Iamblichus, and an epitome by Photius of a lost biography, and they all make some reference to the Pythagorean's beliefs but this book is the only source we really have that describes them in detail. Here is a really brief introduction to their ideas:
1.) The Monad is the fundamental building block of Number. All other numbers are made of monads. “Everything has been organized by the monad, because it contains everything potentially: for even if they are not yet actual, nevertheless the monad holds seminally the principles which are within all Numbers.” Also, “the monad is the non-spatial source of number” because it is a single point, so it doesn’t have any dimension. The Pythagoreans placed the monad in the "fire" or "hearth" in the center of the universe.
2.) The Dyad is the first extension of the Monad. With two points you now have a line which makes a single dimension. Pythagoreans felt that “whereas the monad seems to give rise to generation, the dyad seems to admit destruction.” And they weren’t able to fully commit to the idea that it was an even number because, “every even number is divisible into both equal and unequal parts, but the dyad alone cannot be divided into unequal parts, and also, when it is divided into equal parts, it is completely unclear to which class its parts belong, as it is like a source.”
3.) “The triad has a special beauty and fairness beyond all numbers, primarily because it is the very first to make actual the potentialities of the monad — oddness, perfection, proportionality, unification, limit.” The triad makes 3 points, which defines a 2 dimensional plane. It is the first number to be odd. It is "perfect" because it is the first, after the monad, in a series of numbers that are equal to the sum of the previous numbers. 3, 6, and 10.
4.) They introduce the tetrad with the following statement, “Everything in the universe turns out to be completed in the natural progression up to the tetrad, in general and in particular, as does everything numerical — in short, everything whatever its nature.” They are referring to the fact that 1+2+3+4 = 10 aka the decad which represents the universe. With the tetrad you have 4 points and are able to create a 3 dimensional solid.
5.) The Pentad is made of 5 points so it also can make a 3 dimensional shape but with a point left over. Conceptually this point can be thought of as the something extra that gives the spark of life, or the heart. So we have moved from an inert solid into the lowest stage of life. “In the realm of embodiment there are, according to natural scientists, three life-engendering things--vegetative, animal and rational. The [...] pentad is the minimal extreme of life: the vegetative, hexad is the next: the animal, and the hebdomad is the next: the rational.”
6.) “The hexad is the first perfect number; for it is counted by its own parts [1+2+3=6], as containing a sixth, a third and a half. When squared, it includes itself, for 6x6=36; when cubed, it no longer maintains itself as a square, for 6x36=216, which includes 6, but not 36.” (attrib. Anatolius) If you count three numbers at a time starting with 1, 2, 3, and add them together, the individual integers of the sum always add together to equal six (until you get up to 60 it seems).
1, 2, 3, = 6
4, 5, 6 = 15
7, 8, 9 = 24
10, 11, 12 = 33
13, 14, 15 = 42
16, 17, 18 = 51
19, 20, 21 = 60
7.) The Heptad is associated with all the stages of life, the development of the embryo, and the viability of birth. After seven years the child loses their baby teeth, at 14 they undergo puberty, at 21 they are done growing in height, at 28 done growing in breadth, at 35 you can no longer gain anymore strength, athletes usually retire at this age and the best states end conscription at this age, saying that they "after this point allow people to be officers, but not to serve in the ranks any more." Then at 70 you should be "released from all tasks" and enjoy yourself. A lunar quarter is 7 days. 1+2+3+4+5+6+7 = 28, the length of a lunar month. Incidentally, women menstruate monthly! It is associated with Chance in myth.
8.) “We describe the octad [or ogdoad] as the first actual cube, and as the only number within the decad to be even-times-even, since 4 appears to combine the characteristics of being odd-even and even-times-even in admitting only two divisions up to the monad, one of itself, the other of its parts.”
“Philolaus says that after mathematical magnitude has become three-dimensional thanks to the tetrad, there is the quality and 'color' of visible Nature in the pentad, and ensoulment in the hexad, and intelligence and health and what he calls 'light' in the hebdomad, and then next, with the ogdoad, things come by love and friendship and wisdom and creative thought.”
9.) “The Ennead is the greatest of the numbers within the decad and is an unsurpassable limit.”
10.) The Theology of Arithmetic explains the importance of the decad thus: “the creative mind wrought the construction and composition of the universe and everything in the universe by reference to the likeness and similarity of number, as if to a perfect paradigm. [...] However, a natural equilibration and commensurability and wholeness existed above all in the decad. [...] And this is why, both in general and in particular, things from heaven to Earth are found to have been organized by it.”
“The spheres of the universe are ten and fall under the decad.” Speusippus says, the decad has equal numbers of prime and composite numbers. “the decad generates the number 55, which encompasses wonderful beauties. For in the first place, this is formed by doubling and trebling the systematic sequence of numbers — the doubles are 1, 2, 4, 8 (i.e. 15), the triples are 1, 3, 9, 27 (i.e. 40), and the addition of these makes 55.” (attrib. Anatolius)
“Moreover, the sequence of the first five triangular numbers generates 55 (3, 6, 10, 15 and 21 make 55) and again, the sequence of the first five squares generates 55 (1, 4, 9, 16 and 25 make 55); and according to Plato the universe is generated out of triangle and square. For he constructs three figures out of equilateral triangles — pyramid, octahedron and icosahedron, which are the figures respectively of fire, air and water — and the cube, the figure of earth, out of squares.”
February 8, 2015
For certain synesthetes, all numbers have allusions beyond simple quantification. Thus, "936" might be a restaurant in the French countryside that specializes in dishes seasoned with tarragon, while "1474" is an aging trailer park with metalized domiciles bleaching under a fierce Arizona sun. Indeed, these images are attempts at describing something more elusive: at bottom "936" is that which feels like 936.
For the Pythagoreans, numbers evoke an entire metaphysics that moves well beyond the charms of synesthesia. They believe that numbers embody reality and their allusions represent universal truths. They give special emphasis to the first ten numbers. This book, attributed to Iamblichus but reading more like a student's lecture notes, outlines Pythagorean numerology. As an example, the pentad embodies justice since it is the center of the run of numbers between one and nine as well as the sum of the first odd and even numbers (one, the monad, is not considered either even or odd).
The text is terse throughout and often obscure, but it is a fascinating glimpse of minds attempting to make sense of a chaotic world. The simple mathematics of Greek arithmetic can inspire a complex and sophisticated mysticism.
For the Pythagoreans, numbers evoke an entire metaphysics that moves well beyond the charms of synesthesia. They believe that numbers embody reality and their allusions represent universal truths. They give special emphasis to the first ten numbers. This book, attributed to Iamblichus but reading more like a student's lecture notes, outlines Pythagorean numerology. As an example, the pentad embodies justice since it is the center of the run of numbers between one and nine as well as the sum of the first odd and even numbers (one, the monad, is not considered either even or odd).
The text is terse throughout and often obscure, but it is a fascinating glimpse of minds attempting to make sense of a chaotic world. The simple mathematics of Greek arithmetic can inspire a complex and sophisticated mysticism.
May 29, 2025
great. I will come back to it again in the future when I go further into platonism.
April 26, 2012
An excellent translation (indeed, the only English translation) of an exceedingly interesting set of notes on Neopythagorean arithmology. The explication of the Neopythagorean idea of the relationship between the tetrad and the decad has much to tell students of the Qabalah about the true origins of the Tree of Life.
January 11, 2020
A profound elucidation of Tetractys components, from the Monad to the Decad, and weaving them into a whole of the Monad again. If we take a transcendent view in which mathematics has its own Pythagorean-Platonic realities that co-arise with the visible world (phenomenal, or Demiurgic world), and combine it with the Chaldean views of thrice-transcendent worlds divided by the veils of Hekate, it gives a moderate view of how the ancients viewed proportions, or ratio and logos between things and how they related their existence to the penetration in the generative world. Most likely, I will return to this book for more, as I progress with my studies, to round and 'wholize' certain finer aspects that I haven't caught earlier. It must be noted, that scientific knowledge of the modern day goes together with this scientia, it is a question of extending the base foundations and extending them towards the treasuries of modern astronomy. For example - although we discovered Uranus, Neptune, Pluto, Eris etc. The system presented is wholly consistent within the seven spheres, and thus is finite and perfected according to means and tools. It takes merely a step of imaginative theology to link these mysteries into one splendid whole once again.
December 16, 2025
Saw someone say that this is a must read for anyone slightly schizophrenic and Id have to agree. Loved this translation and found this ancient text on Neopythagorean Theology and Arithmetic to be extremely profound and eye-opening. Recommend for anyone interested in approaching arithmetical concepts from a more metaphysical, symbolic, and cosmological point of view, or for anyone wanting to delve deeper into Platonic/Pythagorean number mysticism, Qabalah, and the qualitative nature of number, arithmetic, harmony, and ratio. Truly a blessing to be able to read something like this.
October 26, 2021
Such a brilliant read, though not an easy one at all!
March 12, 2022
Very interesting book!
September 25, 2024
A must-read for anyone vaguely schizophrenic
March 22, 2021
It came as no surprise to me that a religious identity was established with numbers.
"The Theology of Arithmetic" follows the thought-pattern of the Pythagoreans, a numerically obsessive cult of individuals whom were instructed to: take no notes, commit everything to memory, and that "numbers are life" and "numbers provide the basis of reasoning for everything".
The group was so devoted to their instructor (Pythagoras of Samos) that they formed a bridge of bodies that he safely could walk across, to escape a fire that was intentionally set within a study of his, with the intention to assassinate him.
"The Theology of Arithmetic" follows the thought-pattern of the Pythagoreans, a numerically obsessive cult of individuals whom were instructed to: take no notes, commit everything to memory, and that "numbers are life" and "numbers provide the basis of reasoning for everything".
The group was so devoted to their instructor (Pythagoras of Samos) that they formed a bridge of bodies that he safely could walk across, to escape a fire that was intentionally set within a study of his, with the intention to assassinate him.
Displaying 1 - 12 of 12 reviews