Pythagorean Triples and the Generation of Space

The traditional formulation of the theorem of Pythagoras is that the square on the hypotenuse of a right angled triangle is equal to the sum of the squares on the other two sides. This is the Greek formulation, but we know that some of the properties of Pythagorean triangles were known in earlier cultures, such as in Babylonia.

The triangles of most interest were those constructed using integral values, such as the 3,4,5 triangle. These values were commensurable, and not irrational.

If we consider this in terms of the construction of the version of the triangle constructed using the square values, we can see that it is not a geometrical figure in two dimensions at all, but a line, built out of the three squared values (9, 16, and 25), exactly 25 integral units in length, with the three sides subsumed into one dimension.

This understanding of the geometrical arrangement perhaps tells us something about the ancient perception of space, and why it exists.

There are several possible Pythagorean triangles which exist, all of which exhibit the same properties, but with different combinations of integral values.

There are 16 primitive Pythagorean triples with c ≤ 100:

(3, 4, 5)  (5, 12, 13)  (8, 15, 17)  (7, 24, 25)

(20, 21, 29)  (12, 35, 37)  (9, 40, 41)  (28, 45, 53)

(11, 60, 61)  (16, 63, 65)  (33, 56, 65)  (48, 55, 73)

(13, 84, 85)  (36, 77, 85) (39, 80, 89) (65, 72, 97)

Additionally these are the 29 primitive Pythagorean triples with 100 < c ≤ 300:

(20, 99, 101) (60, 91, 109) (15, 112, 113) (44, 117, 125)

(88, 105, 137) (17, 144, 145) (24, 143, 145) (51, 140, 149)

(85, 132, 157) (119, 120, 169) (52, 165, 173) (19, 180, 181)

(57, 176, 185) (104, 153, 185) (95, 168, 193) (28, 195, 197)

(84, 187, 205) (133, 156, 205) (21, 220, 221) (140, 171, 221)

(60, 221, 229) (105, 208, 233) (120, 209, 241) (32, 255, 257)

(23, 264, 265) (96, 247, 265) (69, 260, 269) (115, 252, 277)

(160, 231, 281)

All of these Pythagorean triples when squared, represent mathematical and geometrical constructions which do not exist in Euclidean space as geometries: they are lines existing in one dimension.

We tend to look at the phenomenon of Pythagorean triangles as just that: something which has practical usage, but no further implications for the nature of the world in which we live, or our understanding of it. This is not likely to have been the case in the ancient world, in which number was venerated as something belonging to the divine and knowledge of the divine. The triples meant something profound.

What might be the meaning of the existence of these triangles and their properties?

In antiquity, it was obvious to anyone interested in number, mathematics and geometry, that there were several aspects of the physical world that involved irrationality, long before it was possible to provide logical proof of such irrationality. One of these irrationalities was the relationship between the diameter and the circumference of the circle. We know that irrationality (understood as an absence of commensuration) was a major concern in antiquity, since the existence of it seemed to undermine the idea that the world was rational, and constructed by the divine on rational principles. In other words, the existence of irrational things served to undermine the idea that the world made sense, and that it was good.

What we understand as Pythagoreanism is actually a way of approaching the world and reality on the basis of number, mathematics and geometry. We have lost a grasp of this, particularly since the close of the ancient world. Pythagorean ideas are not the creation of Pythagoras in the sixth century B.C.E., but a range of ideas about the world, focussing particularly on numbers and geometry, and the puzzles which the study of these throws up (the Greek name for these puzzles is ‘aporia’). As such, these ideas and puzzles belong to any culture which chooses to address the divine in terms of how the universe is constructed. As already suggested, the Babylonians had a sense of this, though they were also interested in the practical applications. It is also the case that the inhabitants of Britain in the late Neolithic and the early Bronze Age had such a sense.

The engineer Alexander Thom surveyed many of the megalithic circles across Britain from the 1930s into the 1970s, and established that the circles were constructed on the basis of a number of different Pythagorean triangles, and that these circles were not in fact circular.  The circumferences of these circles were modified in order to make their lengths commensurate with the length of the sides of the underlying triangles.  These modifications testify to the contemporary idea in ancient times that the incommensurate nature of diameter and circumference shouldn’t be the case.

I’ve written elsewhere that Pythagoreanism, whether in the sixth century or long before, was a transcendentalist view of the world. Meaning that the world of physics and appearance in which we live, is not reality itself, but simply a presentation of it. And the presentation of it is, in a number of ways, crooked. So some aspects of physical reality are not rational. 

This does not mean that the ancient Pythagoreans were pitching themselves against the workings of the divine, but rather that they were trying to understand why what they saw, experienced and understood, was not rational. The answer was that their place of refuge was not reality itself, but a false representation of it.

In the physical world, they could therefore not expect rationality to be woven all through it. Thom identified the obsessive concern of the ancient Britons with whole numbers, and as a consequence (though this was not understood at the time he was studying the megaliths), we know that they were looking to a world beyond the puzzles and paradoxes, in which the relationships of one thing to another were rational in nature.

The theorem of Pythagoras, however it was articulated in the late Neolithic and the early Bronze Age, provided the answer to this. The relationship between the sides of a 3, 4, 5 triangle is irrational in nature, but by squaring the sides, the result is rational and commensurate. This would have been understood to point to a world which transcended space, in that it indicated a one-dimensional reality.

 In that world, some things which are incommensurate here,  were commensurate. Which they might have taken to indicate that, beyond that limited form of reality, there was another reality with no dimensions at all, in which all irrational values existed as commensurate with one another.

Plato echoed a range of Pythagorean ideas in his work, including that reality itself exists in no particular place, has no form or shape or colour. He also suggested that forms existed beyond geometrical figures existing in space, and that these were to be accessed in the mind alone.

The Pythagoreans may have understood physical reality to have been generated as the square root of mathematical values in a higher reality. The resulting incommensuration would necessarily generate space. We could not possibly live in a reality which embraced only one dimension, or even none at all. In which case physical reality might have been understood by the ancient Pythagoreans as a compromise of sorts, which made it possible for mankind to live.

Our Lady Underground: Aporia and Religious Thought

This is a short notice of a forthcoming article.

The article argues that the outcome of philosophical debate in classical Athens was often thought to have implications for our knowledge of the nature of the reality which we inhabit, and consequently, for the meaning of religion. Apparent failure to reach a satisfactory conclusion to an argument, was sometimes a conclusion which provided insight into the complex nature of reality.

Looked at in this way, philosophical argument about difficult and sometimes intractable problems is about developing an understanding of the nature of the reality in which the argument is intelligible, and which has implications for the meaning of religious thought and practice.

Not all philosophical questions have this kind of importance of course, but the practice of precise and forensic discussion of philosophical questions of any sort prepares the student for those questions which do. This was the main business of the Academy.

'Our Lady Underground: Aporia and Religious Thought' is a companion paper to 'Patterns of thought in Late Neolithic and Early Bronze Age Britain', which is expected to be published in the Spring. The paper is discussed here. https://t.co/Nh0xxtvAeT

TY January 12, 2018.

Being and Eternity in the Neolithic

For the past three weeks or so I've been writing a paper on aspects of mind during the late Neolithic and the early Bronze Age in Britain, during the period when most of the megalithic circles were built. I wrote a paper in 2016 (Stone Circles, Phenomenology, and the Neolithic Mind) which explored the possibility that we might have enough information from the structures to infer thought and motive of their builders. 

In that article I wrote: 

The main objection of the archaeological community to the acceptance of this view of the circles as sophisticated tools for the observation of the heavens and its movements is two-fold. The first is that there is nothing (beyond what these objects seem to show) which suggests to us that the megalith builders and their contemporary culture was concerned at all with precision. The second is that, even if the circles can be shown to mark certain astronomically precise points, we have no clue at all about why the megalith builders would want to do this. We have no understanding of the function and purpose of such observations, and so we can say virtually nothing about either the purpose of the monuments, or about the culture which gave rise to them. In other words we are no further forward in our understanding.

This two-fold objection is sound, at least from the archaeologists point of view. We have no contemporary description of anything at all from the time and culture of the megalith builders. We have no understanding of their intentions, and no understanding of the function of the circles, and it is unlikely that archaeological interpretation by itself will reveal these things to us. It seems to be gone altogether, and all we are left with is insupportable speculation. Perhaps it would be best to draw a veil over this apparently anomalous aspect of the past, since we can have no understanding of it.

The upshot of the article was not as negative. I argued that:

[in] both Greece and Assyria, the heavens, and what the heavens represented, were of particular importance for cult practice. Plato is clear that the heavens represent an image of divine Being, also spoken of as the ‘Living Animal’, created out of the materials of other gods, by the demiourgos (the ‘Living Animal’ is created by the Demiourgos rather than by God directly, so that there should be not too much of the divine present in the world). He is also quite explicit that the body of the ‘Living Animal’ was created with precision.

In which case astronomy is important for the understanding of divine Being. An infamously unintelligible passage in Plato’s Timaeus talks about the perception of images being possible only because the soul already contains exemplars of these. If the heavens represent an image of divine Being, then all other images are poor imitations, and we should direct our attention principally to the heavens. The soul represents the heavens most closely, and in antiquity was notionally the part of us which is most connected to the divine. It is the soul which recognizes divine knowledge when it is presented to it.

We already know the importance of the soul in Britain in the late 1st millennium BCE, from several sources, including Julius Caesar.  It may be that these ideas were similarly tied together in Britain, and perhaps at a much earlier period. I explored a Mesopotamian parallel:

In ancient Assyria, part of the ritual for creating an image of a god (thereby conferring divinity on the object) involves the image being exposed to the sight of the heavens. As a god, it needs to have perception of Being, of which it is an aspect. We have this ritual from the reign of Esarhaddon (7th century BCE), who was the predecessor of Ashurbanipal, one of the last Assyrian kings, and the owner of the famous palace library, much of which is now located in the British Museum.

So the point is the establishment of contact - even an identity with - what is divine. It is possible to see this concern with connecting Heaven and Earth as a parallel to what Plato describes more abstractly in his Republic. I wrote:

...Plato argues that the philosopher should ascend to the idea of the Good, which is another way of referencing the idea of transcendent Being, by a series of connected images, without specifying what those images might be. These images, as imperfect representations of Being, are subject to change. Matters are complicated by the fact that the things which are imaged are also subject to change. Thus both subjective impression and objective reality are subject to change and uncertainty.

So the process is problematic, and complex. To some extent it is a matter of conjecture, as well as knowledge.

A key characteristic of ancient religion was that it entertained conjecture regarding knowledge of the gods. Plato refers to this in his discussion of the divided line in The Republic. We cannot know Being itself directly. We cannot know the lesser gods directly either, but we can understand some of the characteristics of these gods, though full knowledge is necessarily beyond our understanding. We can approach some limited understanding of Being via the images and descriptions of the lesser gods however, which is one reason why they were deemed to have some kind of reality.

In which case knowledge of divine things is a journey through the problematic and sometimes paradoxical aspects of reality as it presents itself to us. You need a priesthood for dealing with that. They were dealing with ontological questions, and did the best they could within the limitations of human knowledge.

This limited set of characteristics and properties emerges from thorough and precise discussion of the nature of ontology (Being), as the focus of human conjecture. This is quite clear in both Plato and also in what we know of the teachings of Pythagoras. The paradox of knowledge is the consequence of the idea that all knowledge is present in the divine, and that we only have knowledge because we have a soul. In other words, we can have knowledge as the result of the divine having a presence in us already.

I suggested that Plato, rather than exploring a pattern of ideas created in the Academy in Athens, was in fact transmitting and refurbishing some

...key aspects of a Bronze Age philosophy, which has its roots in the Neolithic in both Europe and the near East. The transcendental doctrine which is enshrined in his work is actually in plain sight to anyone who can follow his reasoning that all the merely likely stories are just that: no more than likely. The single argument which is not a mere likelihood, is counterintuitive, paradoxical, and beyond common sense. And it remains a matter of conjecture. This argument about the nature of the world used to be the apex of learning and knowledge. Reality is transcendental, and neither subjective or objective. Nothing lay beyond this fundamental understanding of the world.

If this argument is correct, then we can know what Neolithic man was up to, and in some detail. In short, for the intellectually sophisticated in the Neolithic period, the heavens represented, as for Plato much later, a moving image of eternity. To measure the parameters of this moving image of the divine was to know God, and to have knowledge of divine things.

This represents a radical departure in our understanding of the development and the history of human thought. We know that the heavens were important in the neolithic and the early Bronze Age in Britain, and if that importance stemmed from an understanding that the heavens represented a moving image of eternity, then the people who built the megalithic circles had a concept of Being, and of a reality which transcended most human experience. Such an idea is the most abstract of all. It transcends all images of what is real. It implies a culture built on responses to questions about the nature of reality, and many conjectures about what can be known.

The Pythagoreanism which the surveyor of many of the megalithic sites in Britain, Alexander Thom, detected in his findings, was very basic in nature, and based on statistical analysis of his data, and the geometrical construction which could be found in the monuments. Which means the observance of the importance of the Pythagorean right angled triangle, and the concern with whole numbers.  Both of these were of importance to the later Pythagoreans, as we know. A further clue was the statements by a number of ancient authors to the effect that the religious belief of the Britons in the late 1st millennium BCE was akin to Pythagoreanism.

But there is very little detail about what Pythagoreanism in Britain might have entailed, if the identification is correct. This is rather surprising, since we have a great deal of information about Pythagoras and Pythagorean doctrine available to us from sources other than Julius Caesar, and the authors whose opinions have survived as a result of being quoted by Eusebius (though often inconsistent, and sometimes quite obscure in meaning). We have a biography of Pythagoras from both the neoplatonist Iamblichus, and also from Porphyry. We also have an extensive and rather good account from the author of the Lives of the Philosophers, Diogenes Laertius. Plus we have the works of Plato, who is constantly referencing the doctrines of the Pythagoreans.

So this new paper looks at the material we have concerning Pythagoras, in order to know a fuller range of what Pythagoreanism implied in the late sixth century BCE, both in Italy and in Greece. As it turns out, certain key ideas in Pythagoreanism arise as the logical consequence of the kind of discussions which they entertained, mainly concerning number, mathematics and geometry. And these key ideas shape their ideas about religion, divinity, the nature of reality, of Being, and of Eternity.

The paper concludes with the suggestion that, if  key religious ideas emerge in the Pythagoreanism of the 1st millennium BCE, in the course of logical argument concerning number, mathematics and geometry, then we should entertain the notion that parallel discussions may have taken place in the late Neolithic and the early British Bronze Age, on account of their own evident concern with number, mathematics and geometry.

***

The title of the paper is: 

Patterns of thought in Late Neolithic and Early Bronze Age Britain

Abstract: Pythagorean elements detected in megalith circles in ancient Britain have no easy explanation, and precede 1^st millennium Pythagoreanism by an extraordinary period of time. This paper explores the idea that there is a connection between some core Pythagorean mathematical and geometrical concerns, and ideas of divinity and eternity.  On the basis of a close examination of Pythagorean ideas in the 1^st millennium, for which we have extensive documentation, it is suggested that this connection is based on a series of logical inferences. It is therefore possible that similar conclusions were arrived at in the Late Neolithic.

Key words: Megalith, Pythagoras, Philosophy, Religion, Mathematics

Paper sections are:

1 The Longevity of Ideas

2 Pythagoreanism in 1^st Millennium Britain

3.The Principal Sources for Pythagoreanism

4 The Core of Pythagorean Doctrine 

5 Diogenes Laertius on Pythagoreanism

6 Pythagorean Thought in Italy

7 The Existence of Irrational Numbers

8 Religious Aspects of Pythagoreanism

9 The Pattern of Eternity

10  Pythagorean Syncretism

11. Transcendentalism in Late Neolithic and early Bronze Age Britain

12 Walking back the Insight into Ancient Mind

13 Pythagoreanism and the Deep

TY, December 10, 2017.

The paper was reviewed a short time ago, and rejected for publication in Time and Mind (for reasons I understand, but disagree with). Thanks to Paul Devereux, who thought it was worth the shot. Rather than waste a lot of time looking at the very few options available for publishing this kind of article in a scholarly journal, I've opted to make it available here, at: Patterns of Thought in Late Neolithic and Early Bronze Age Britain.

TY, March 8, 2018.

I changed the name of this post to 'Thought in Late Neolithic and Early Bronze Age Britain' on January 26 2018, in order to avoid confusion with the actual paper intended for journal publication. The title was changed again, to 'Being and Eternity in the Neolithic', on May 6 2019, while the foregoing text was being edited for inclusion in Mirrors of the Divine, along with 'Patterns of Thought in Late Neolithic and Early Bronze Age Britain',

TY May 6, 2019.