Exploring Connections: Music, Cosmology and Mathematics

In the Tao Te Ching we are told, ‘True words seem paradoxical’.  This is a prediction for science and philosophy. As such it is a fabulously generous hostage to fortune.  It can be explained in philosophy by reference to Buddhism’s doctrine of ‘Two Truths’ or ‘Two Worlds’.  Its explanation could never be called easy, even though it can be made quite brief, but it may be no more difficult to roughly comprehend than quantum theory, and it would almost certainly be no more difficult to roughly comprehend than number theory. It is interesting to explore the connections between these areas of knowledge, and it may shed some light on Lao Tsu’s uncompromising statement.    

Number theory is the study of the whole  numbers.. It may also be called ‘higher arithmetic’. Mostly it is about the prime numbers.  It is usually considered the most difficult area of mathematics. A million dollar prize awaits whoever solves one of its most famous problems, and so difficult is it that the prize has been on offer for over a century.  So difficult is number theory that I am not embarrassed to admit that many years after first setting out to win the prize I still cannot understand the problem. All the same, from my bumbling investigations some interesting connections came to light. 

The Tao Te Ching goes on to tell us, 

Tao begot one,
One begot two,
Two begot three,
And three begot the ten thousand things.*

What Lao Tsu  (or whoever it was) does here is explain the reason why prime numbers greater than three can always be found adjacent to a number divisible by six, or, in a more mathematical language, why primes >3 occur at 6n+/-1.  That they do behave in this way becomes obvious once we know what to look for.  

5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, …

The reason why there is this immediate connection between the behaviour of the primes and the description of cosmogony given in the Tao Te Ching is that number and form would be artefacts of isomorphic processes, perhaps even the same process. That it to say, the description of the emergence of form given by Lao Tsu may also be read as the process whereby the numbers come into being. In mathematical terminology his description would be an ‘axiomatisation’ of the number line.  First there is Tao, which implies one, which implies two, and so forth.  Then it becomes a genesis, a self-sustaining creative process of multiplication by which the ten thousand things emerge naturally, inevitably. The number line would populate itself spontaneously. Forms would multiply ineluctably.  

In order to make sense of this we can attempt to axiomatise the number line for ourselves. As a necessary first step we would have to imagine a state in which there are no numbers. Either there are no numbers or we are not aware of them. Not even one phenomenon could exist in such a state. The idea of ‘one’.would not be present. This would be a state of mind or the world prior to all division. This is where we must begin, in the same place that Lao Tsu begins. In trying to define or imagine this state we are drawn immediately into philosophy and mysticism, for this would be Hegel’s Absolute Idea, the phenomenon defined by Kant as ‘not an instance of a category’ and which he calls, with profound implications for consciousness studies, the ‘proper subject for rational psychology’. We cannot imagine such a state. This is easy to verify. But we can imagine that such a state is possible without actually trying to imagine it, and it is not difficult to define. We can even talk about it, albeit that we we cannot quite know what we are talking about. The Tao that is eternal cannot be talked, says Lao Tsu, in so doing proving himself wrong and thus obeying his rule for true words.    

Out of this numberless axiomatic or fundamental state we must create the number line. As there are no numbers this is an undifferentiated void. It would not be ‘a’ void, as if it were one thing and there could be another, nor could it be extended in time or space, for extension requires number, more than one location for instance. We must define this void as being unmanifest, unimaginable, an empty concept, something that cannot be conceived, an absence of concepts, a phenomenon that does not seem to be there. When we try to imagine it, as we naturally would, we are likely to imagine it as Zero or Nothing. But ‘zero’ is one number, and by creating ‘zero’ we cannot help but also create the number ‘one’ as its implied counterpart, and this is two numbers, and ‘two’ begets ‘three’ and so on.  To proceed in this way would be cheating. In creating the number line from scratch we cannot begin with a number even if it is zero or one. If we imagine Tao, or whatever preceeds the numbers for our axiomatisation, as a quantifiable phenomenon, unambiguously ‘one’ or ‘something’, then we will conceive of it as being the opposite of ‘zero’ or ‘nothing’, and that is two numbers, and so on.  For a fundamental  theory we must begin with a true void and not assign a numerical value to what is supposed to be prior to numerical values. We must begin at the beginning. This numberless void, which is a true void, or what the mathematician Hermann Weyl calls the ‘intuitive continuum’, must play the role of Tao for the number line we are creating in our heads, a phenomenon that we can define but not imagine.

Kant explains that this undifferentiated state would be more profound than the famous axiom from which Descartes attempted to derive the world.  Körner summarises Kant’s view.

In the Analytic of Concepts Kant has drawn a sharp distinction between the ‘I think which must be capable of all my presentations,’ thereby giving them synthetic unity, and the empirical, introspective, self which is itself a presentation. To be truly a priori rational psychology must have for its subject the former, i.e. the self of pure self-consciousness. This however is not, according to Kant, an object of experience and so of the applicability of the Categories. It is not an instance of any Category. (Korner.S, Pelican (1955)).

How can the numbers arise from such a state? To create the numbers we would have to imagine a different kind of void, one we can actually imagine. This will give us a space within which to create the numbers. This is the void that we would naturally imagine when we try to imagine a void. This would be the void of Democritus and probably of most physicists, an infinitely extended empty space. Not a true void, for we cannot think what is prior to thought, but the best our imagination can do. This phenomenon  can and must be imagined as ‘one’, since there is just one of it, and also as ‘zero’, since it is empty of phenomena. Now we have the space in our heads necessary to draw the number line, the time to do it, and two numbers to work with.

How did they arise? How can we justify our creation of this pseudo-void and the atoms it contains? We began with Weyl’s intuitive continuum. There are no numbers here. There is no thought. By what means can a universe of number and form emerge from this seemingly arid state? This is not a question that can be answered in metaphysics or mathematics. We cannot calculate how the imaginable comes from the unimaginable. Either this can be known empirically, by walking in the footsteps of Lao Tsu. or it cannot be known at all.  When constructing theories there is only one thing to do in this situation, which is to make the extended pseudo-void of our own creation an axiom, a given, a permanent question mark, something that our theory does not explain but on which it depends. Just as long as we do not assume that this created void is the final origin of the numbers, as opposed to being just the beginning point for our explanation of them, then we need we need not be troubled by  the problems of self-reference that this axiom causes for fundamental theories, most notably the problem that defeated Russell’s attempt to axiomatise mathematics.

Our axiomatisation will start where it must, with the first number, but will not reify the numbers by assuming that there is no prior state. In physics this axiomatic pseudo-void is usually seen as fundamental as opposed to emergent, a view that drives theoretical physicists into the arms of intractable metaphysical problems equivalent to those met by Russell in the foundation of mathematics, At a foundational level the distinction between mathematics, physics, psychology, metaphysics and mysticism becomes arbitrary, for all of them face the same problems of self-reference in the end. Mathematics cannot be axiomatised by Russell’s method, for his method included the rejection of Lao-Tsu’s mysticism and thus the whole notion of a state prior to number. As a neutral monist Russell was openly committed to the idea that number and form are fundamental, and could not understand why this idea is paradoxical.in logic, mathematics, physics, psychology, metaphysics and even theology.         

It seems to me that we cannot justify the creation of the numbers from the original void. We can concede their emergence, but we can say nothing about them until they have emerged. We have no choice but to begin our theory with a nonreductive axiom, a void with the properties of time, space, form and number. A rabbit from a hat. The only choice we have would be whether to assume that this void is the fundamental state or whether the phenomenon spoken of by Kant and Lao Tsu is prior to this.  

Let us now draw a line across this infinitely extended pseudo-void of our imagination, as a musician might draw a stave on a blank piece of paper, then mark a point on the line and call it ‘zero’.  We cannot in fact pick a point on a continuous line, but we can conceive of a finite and distinct sub-region of the line that we can mark as ‘zero’. Where did ‘zero’ come from again? We invented it when we imagined one infinite space with zero phenomenon in it. We cannot help but conceive of such an extended void as one space containing zero phenomena, because this is what it is.  The very existence of this concept of a void requires the presence of the numbers ‘one’ and ‘zero’.  It cannot be considered a true void or continuum precisely because its existence as a concept is dependent on the concept of number. It is not a conceptual void, it is our concept of a void.

By adding ‘zero’ to the number line, which is one number, we create the number ‘one’.  We can now add ‘one’ to the line. As we add the number ‘one’ to the line little happens. One and zero multiplied together create no new numbers. However, since there are now two numbers we have created the number ‘two’ as a by-product. When we add the number ‘two’ to the line there is some spontaneous activity, for we immediately create the series;  4, 8, 16, 32, 64, 128…. , which is an infinity of numbers. The powers of two are immediately implied by its existence as a number. If two exists then so does two times two and so forth. This is very few numbers though, as infinities go, and in the grand scheme of things it might as well be none at all.

It is only when we add the number ‘three’ that the process properly explode into life.  This is because the multiples of two and three account for four numbers in every six all the way to infinity. Two-thirds of all the integers are created all at once when we add the number ‘three’ to the number line. The ten thousand things are begotten. The only numbers that are not created in this initial evolutionary explosion are those that occur on either side of 6n. These locations remain as gaps in the line, null points in the combination wave of multiples produced by the interaction of the numbers two and three. Into these gaps go the prime numbers >3 and their products.  

In something like this way the universe of Lao Tsu would emerge by a process of symmetry-breaking, division and multiplication. The process can be interpreted as psychological, mathematical  or metaphysical. If we say that Tao is ‘zero’ or ‘one’, ‘Something’ or ‘Nothing’, then not only will we have created the ‘ten thousand things’ but we will have reified them. Tao would then be a number and it would be irreducible. We will have substituted our own idea of Tao for the real phenomenon, in so doing rendering Taoist doctrine incomprehensible. If, however,  we say that Tao is prior to number, then number, the concept of number, becomes an artefact of a process that is essentially a misunderstanding of Tao, a process that does not involve Tao. Tao begets one but would not create anything. Lao Tsu does not endorse the reification of form and number. We found no causal link between the true void and the pseudo-void we required for the existence of number. We just made it up. The begetting process was dependent on a dualistic conception of the void, the idea of a void that is ‘one’ or ‘zero’ and extended in time and space, in order that we could begin the process of  inferring the rest of the numbers. Hegel’s Absolute Idea, Kant’s ‘proper subject for rational psychology’ and Lao Tsu’s ‘Tao’ would be prior to this. For Kabbalism it would be prior to God. For Nagarjuna and Rumi it would be prior to the ‘two worlds’, or prior to the division of the world into two. It would be prior to intentional consciousness. This would be is why it does not seem to be there when we look. It would be prior to our intellect. It cannot appear on the number line or anywhere else in our extended void. Our extended void was emergent, contingent, and thus also all it contains. We created the number line in our heads. We created a void with a numerical value and went on to derive the rest of the numbers.  Zero and one emerged in mutual dependence from an empty concept, a phenomenon that we are unable to imagine but which, when we tried to imagine it, became zero and one. At the start of our attempted axiomatisation, rather than our being in a state of having no awareness of the concept of number we tried to imagine a void in which there were no numbers, and to us, observing it from the outside, it looked like zero or one depending on which way we looked at it. But we cannot look at it.  Tao stands aloof from number and form, which would be dependently-arisen and have no intrinsic existence.

In this way Taoism can acknowledge the conventional reality of the numbers and the multiplicity of phenomena without having to reify anything except Tao. The numbers, all mental and corporeal phenomena, would be reducible to a phenomenon prior to number that is unmanifest and inconceivable. If we take a different view and say that number must be there right from the start, then we can never axiomatise mathematics or have a fundamental cosmological view.  Rather, we will be forever plagued by paradoxes of self-reference. We will have assumed that number and form, time and space are non-contingent, and on analysis this assumption does not make sense.    

How would any of this shed light on our opening quotation? Why would it follow from all this that true words must seem paradoxical?  Tao would be prior to the number line. It would lie beyond all distinctions.  It would not be this or that in any case.  It would not be an instance of any category. It would be the ultimate, first or original phenomena and all else would be emergent and exist only in dependence on this original phenomenon. In this case there would always be two ways of describing the world. Firstly,  a conventional description which respects the world as it appears, by which the numbers are real things, existent right from the start, and thus also the extended space-time void in which they exist. Secondly, an ultimate description, for which all that seems to exist would reduce to the unmanifest, from which  ‘zero’ and ‘one’ and all instances of Yin and Yang would emerge in mutual dependence as complementary opposites and in this sense would be unreal, epiphenomenal or dependently arisen. The world of change would reduce to a conceptual imputation. Thus Heraclitus can state ‘We exist and do not exist’, since there are these two ways of  looking at our situation. True words, which to be rigorously true must take account of both ways of looking, will seem paradoxical. It is remarkable that Lao Tsu adds no proviso to his statement and that none is needed.  

These two ways of creating the number line, one that begins with a true void and carefully distinguishes between this and the pseudo-void of our imagination, the other that begins with the conception of a pseudo-void as fundamental and is thus nonreductive, where the former cannot be represented by a number and the latter may be, represent two very different ways of looking at the world, two very different metaphysical schemes or philosophies, and two very different cosmological models.  Roughly speaking, we could say that the former represents the nondualist view of Lao Tsu and ‘eastern’ philosophy in general, mysticism and the ‘perennial’ philosophy, and the latter the dualistic and nonreductive traditional western view. 

These issues are explored in a book by George Spencer Brown called Laws of Form  (1969).  This was highly praised by Bertrand Russell, partly because in it Brown solves the famous paradox that had defeated Russell’s ten year attempt to axiomatise mathematics.  Russell found that Lao Tsu’s axiomatisation as sketched out in the verse quoted earlier was a solution for his paradox, once Brown had translated it into a simple calculus.

Rather implausibly, it was while designing electrical switching systems for the railway network that Brown found himself having to invent and use a form of logic that turned out to be the same as that used by Lao Tsu and his like. Later Brown became a friend of the advaita master Wei Wu Wei and has even claimed to be a buddha, but at the time he was just trying to solve an engineering problem. His book explains the emergence of form from formlessness, the arising of the ‘ten thousand things’ from the unmanifest, a phenomenon that Brown likens to a blank piece of paper. This would not be one or zero. One and zero would emerge when we draw a circle on the blank piece of paper and thus create the first set-theoretic distinction. From there on the process is unstoppable. The rest of the numbers would follow ineluctably according to the laws of form by a simple process of division and multiplication. 

Here is Robin Robertson, President of the American Jungian Society, who has written at length about Brown’s ideas and has a useful website. 

Anyone who thinks deeply about anything eventually comes to wonder about nothingness, and how something (literally some-thing) ever emerges from nothing (no-thing). A mathematician, G. Spencer-Brown (the G is for George) made a remarkable attempt to deal with this question with the publication of Laws of Form  in 1969. He showed how the mere act of making a distinction creates space, then developed two “laws” that emerge ineluctably from the creation of space. Further, by following the implications of his system to their logical conclusion Spencer-Brown demonstrated how not only space, but time also emerges out of the undifferentiated world that preceeds distinctions. I propose that Spencer-Brown’s distinctions create the most elementary forms from which anything arises out of the void, most specifically how consciousness emerges.

It is not surprising, given his position, that Robertson endorses Spencer Brown’s view. Here is Carl Jung from VII Sermones ad Moruos (Seven Sermons to the Dead).  He restates Lao Tsu’s view, that we cannot call Tao no-thing even though it cannot be conceived as some-thing. It would be prior to zero and one. It would be non-numerical, undifferentiated, unlimited.      

Nothing is the same as fullness. In the endless state fullness is the same as emptiness. The Nothing is both empty and full. One may just as well state some other thing about the Nothing, namely that it is white or that it is black or that it exists or that it exists not. That which is endless and eternal has no qualities, because it has all qualities.

Here is Brown introducing Laws of Form and discussing the importance of the connection between mathematics, metaphysics and mysticism. The ‘act of severance’ he speaks of here may be the act of conceiving of ourselves apart from the whole, the act by which ‘Tao begot one’.   

The theme of this book is that a universe comes into being when a space is severed or taken apart. The skin of a living organism cuts off an outside from an inside. So does the circumference of a circle in a plane. By tracing the way we represent such a severance, we can begin to reconstruct, with an accuracy and coverage that appear almost uncanny, the basic forms underlying language, mathematical, physical, and biological science, and can begin to see how the familiar laws of our own experience follow inexorably from the original act of severance. The act is itself already remembered, even if unconsciously, as our first attempt to distinguish different things in the world where, in the first place, the boundaries can be drawn anywhere we please. At this stage the universe cannot be distinguished from how we act upon it, and the world may seem like shifting sand beneath our feet.
    Although all forms, and thus all universes, are possible, and any particular form is mutable, it becomes evident that the laws relating to such forms are the same in any universe. It is this sameness, the idea that we can find a reality which is independent of how the universe actually appears, that lends such fascination to the study of mathematics. That mathematics, in common with other art forms, can lead us beyond ordinary existence, and can show us something of the structure in which all creation hangs together, is no new idea. But mathematical texts generally begin the story somewhere in the middle, leaving the reader to pick up the thread as best he can. Here the story is traced from the beginning.
    Unlike more superficial forms of expertise, mathematics is a way of saying less and less about more and more. A mathematical text is thus not an end in itself, but a key to a world beyond the compass of ordinary description.

Brown’s  friendship with Wei Wu Wei is partly explained by the correspondence between the view endorsed by Lao Tsu  and the advaita doctrine of the Hindu Upanishads.  Advaita translates as ‘not-two’. It is the claim that there are not two real things. Tao would be real and all else would be not so much created as begotten .  Created things would be unreal.  The unconditioned Brahman would not be a numerical quantity.  It would not be zero, for if it is zero it is one, and if the advaita doctrine stated that the ultimate is one then it would not be called ‘not-two’, it would be called ‘one’. Where Monism posits a ‘One’ as opposed to a ‘Two’ or a ‘Many’ then it is dualism. Advaita might as well be called ‘not-dualism’.    

What has music got to do with this? The prime numbers follow rules that would be recognisable to most musicians. One of the skills a pianist or percussionist must learn is to play different rhythms with each hand at the same time. For beginners there is a useful trick for getting the feel of this. If we keep repeating the phrase ‘Nice cup of tea’ in a natural rhythm and tap out the words with our hands as follows: Together, Right, Left, Right; TRLR,  TRLR, and so on, then we will be playing threes with our right hand against twos with our left.  Tapping different sounding surfaces with each hand will make this more obvious. This is a bar of 6/8 where ‘Together’ is the first beat of the bar. There are two rests in each bar, one on either side of the first beat. These are the only possible locations on the number line for primes greater than three. If we had enough hands we could add in the rest of the ten thousand things.

The work of the musicologist Rudoph Schenker is also relevant. Using the classical symphony as his main study he analyses music horizontally into hierarchical layers. There is the froth of the details, the ornaments, the top-lines, the tambourines and bells. Underneath this there is a more stable harmonic structure of passing modulations and cadences that define intermediate sections of the piece, whether it be an eight bar phrase or a movement. Underneath this, as the very foundation of the whole edifice, there would be the movement away from the tonic chord, (the key of the piece) to the dominant and back. This journey away from home and back again would provide the tension required to motivate and enliven the music, giving it purpose and meaning and engaging the listener on the scale of the whole symphony. Without this foundational distinction between tonic and dominant, the Yin and Yang of the classical symphony, the music would lose its structural integrity. And even before this there would be the duality of sound and silence. It all begins with and depends on a severing or taking apart of an undifferentiated space.

In his book Stalking the Riemann Hypothesis, mathematician and computer scientist Dan Rockmore writes this about the birth of number.

The great German mathematician Leopold Kronecker (1823-1891) said that “God created the natural numbers.” And it is true that the natural numbers – one, two, three, four; and so on they go – appear to have been present from the beginning, coming into existence with the birth of the universe, part and parcel of the original material from which was knit the ever-expanding continuum of space-time.

The natural numbers are implicit in the journey of life, which is a nesting of cycles imposed upon cycles, wheels within wheels. Two is the breathing in and out of our lungs, or the beat of our hearts. The moon waxes and wanes; the tides ebb and flow. Day follows night, which in turn is followed once again by day. The cycle of sunrise, noon, and sunset gives us three. Four describes the cycle of the seasons.

…These natural numbers help us to make sense of the world by finding order, in this case an order of temporal patterns, that lets us know what to expect and when. We notice the rising and setting of the sun, and that cycle of two is given a more detailed structure as we follow the sun through the sky in the course of a day. We turn the temporal telescope around and also see day as part of the larger cycle of the phases of the moon, whose steady progress is situated within the cycle of seasons that make up the year. Patterns within patterns within patterns; numbers within numbers within numbers – all working together to create a celestial symphony of time..

We organise, we count, and therefore we are.

We see here the difference between the usual scientific view and the view that is taken in mysticism. The numbers have to be there from the beginning, since the birth of the numbers is the beginning of existence. Not one thing cannot exist prior to the number one. But they cannot be there from before the beginning. They had to come into existence.

The scientist will usually assume that this is a problem, for it would imply ex nihilo creation. They forget that zero is a number and that it would be a logical error to axiomatise the numbers on a number. If we do this we are left with nothing but cycles on cycles, numbers on numbers, patterns on patterns, fields on fields, turtles on turtles, sets on sets, and no fundamental phenomenon cabable of explaining why there is not just nothing at all.

In metaphysics, even if not in the foundation of mathematics, it would not be enough to say that the natural numbers are there from the beginning. This is not a fundamental theory or an axiomisation. It would be a nonreductive view. It would not explain the birth of the universe or the numbers. To explain this we would have to hypothesise a phenomenon that is prior to number, that cannot be categorised as a number.

A little though shows that there could not be two such phenomena. Like a unity, a numberless phenomenon could not be said to be one or two. Mysticism’s claim that universe is a unity, therefore, is the claim that there is more to the universe than numbers on numbers, cycles on cycles and fields on fields. As well as this there would be the ultimate phenomenon that we have to thank for all these emergent epiphenomena. This is all that would be truly real.

If we deny this real phenomenon then we are left with Russell’s paradox, the problem of how to axiomatise the categories of thought, be they thoughts of sets, numbers, fields, cycles or substances. It cannot be done without the concept of Tao.

 

*For an expert discussion of different conceptions of the void cf.. ‘Hermann Weyl on intuition and the continuum’ by John L. Bell (http://publish.uwo.ca/~jbell/Hermann%20Weyl.pdf)]

Extracts from the Tao Te Ching are from Lao Tsu, Tao Te Ching, Trans. Gia-Fu Feng and Jane English, Wildwood House Ltd (1973)

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