Readers may remember a recent article In which I wrote about how when one considers both the repeated cycles of the liturgy and that of the scales and harmonic intervals in music one sees in both the numbers seven and eight governing the threaded path of a helix (Aquinas, Augustine and Benedict on Seven and Eight in the Psalms and the Liturgy). This described how as we move forward in sacred time, we also repeat cycles (every eighth day for example) so that although every day is distinct from every other, it also shares common aspects with others, for example if it is a Thursday it is like other Thursdays in this respect. Or, the other way around: although I might sing Vespers of the Week 12 of the year using exactly the same form as last year and as I expect to do it next year, it is also a unique moment in time.
One reader, Alexey in Kansas contacted me and made an intriguing suggestion based upon what he read. Here is part of what he wrote to me: "Time then is more than one dimension. Just like, when traveling through space, it is not enough to say “I am at 50 degrees latitude”, — the longitude must be specified as well, so it is not enough to say “50 days passed”, one has to add “it is Thursday".
What I fascinating idea. I know I have heard of multi-dimensional space, but not three-dimensional time. This raises the question as to what it is that brings the helix back to the start again after a 360 degree turn in our original picture of music and the liturgy.
The traditional idea of number considers it to be something that designates both quality and quantity. We are used to the idea of number designating quantity, for this is how we tend to think of it today - 9 is always greater than 8 in magnitude for example. But the idea of numbers possessing different qualities is unusual now. For the ancients, however, this was just as important. So the number eight, as we discussed quantifies, but it also 'qualifies' - in the sense that it assigns a particular quality to something. It associates that count of eight with Incarnation. This means that as we go forward in time, we might say that we have moved forward 50 days in time, this considers only the quantity of time. However, we would also say that this day is a Sunday, and that it is the eighth Sunday in the cycle and so it is a special sort of day, connected to previous and future Sundays. In order to carve out this helical path that allows it to move forward yet return to the same quality, we imagine a three dimension picture of it threading through space, hence Alexey's three dimensions in time. This means that when we move forward it is important to consider not only the magnitude that we count, but also direction. The word for something that has both magnitude and direction, many will know, is a 'vector'. So in other words, Alexey is suggesting that time is vector quantity.
What does this have to do with science? Well this is what strikes me. In the scientific method as I understand it, the scientist observes the data, and proposes a hypothesis to explain it which predicts some new, previously unobserved, effect. Then he tests the hypothesis by carrying out repeated experiments to see if the predictions are realized in practice. Then the hypothesis is considered to have a greater degree of certainty and it is now called a 'theorem'.
The creative part of this process is the development of the hypothesis. The first idea of it in the mind of the scientist is as much the product of an inspired guess as it is reason. The scientist just sees the solution that completes the pattern of data. As soon as the idea occurs, he then uses reason consciously and methodically to check it out. Underlying this process is the assumption that the world is naturally made to be patterned ie it is ordered, symmetrical. This being so it the natural sense of the beautiful that allows the scientist to see the completed pattern in his imagination and propose the hypothesis. His intuition guides him to what to what he thinks it ought to be and then he tests it with reason. There is one famous example of this that one can use to illustrate. In the early Sixties, a tenth sub-atomic particle was discovered when a scientist, Murray Gell-Mann, saw nine points on a graph in the pattern of the Pythagorean tectractys but with the tenth point, which would have been the apex of a triangular arrangement of them, missing. Assuming that the natural order was symmetrical and beautiful, he set out to discover this particle and found the tenth just where he thought it would be. You can read about this in more detail here (Liturgical Science?).
This story illustrates a point, I think. That the greater the intuitive sense of the scientist is of the symmetry and order of creation, that is its beauty, the more likely he is to be able to spot the answer that is consistent with the pattern of nature. So, I would contend, you would make a better research scientist out of the man who has receives the traditional education in traditional beauty. At the core of this is the life lived liturgically. Through our active and ordered participation in the liturgy, the rhythms and patterns of the cosmos are impressed deeply upon our hearts.
There are parallels between artist and scientist in their observation and description of the natural world. The artist that draws well uses combinations of the parabolas and elipses to create the natural looking graceful curves that describe his forms (although he very likely knows these only by a recognition of their graceful shape and would not be able to describe them as a mathematician would). Therefore, the regular exposure to good sacred art, especially in harmony with the liturgy, will enhance the ability of the person to see the world around him in these forms. In the diagram below you can see my version of the plot of the properties of the known particles as it would have appeared to Gell-Mann so that he was inspired to look for the missing apex at the bottom. We can compare this with the pythagorean tectractys, which is a diagramatical representation of the harmonies of music and as shown above in a detail from Raphael's School of Athens.
This goes further.
Pondering over this remark of Alexey's and how even time might be a vector (with magnitude and direction) and not just a scalar (which has only magnitude), reminded me of someone I met years ago in Mountain View, California called Irwin Wunderman. His son was a friend of mine from my time studying metallurgy at Michigan Tech. Irwin was a brilliant man (he was in his seventies, I think, when I met him and he has since died). He was a PhD from Stamford where, he told me, his thesis was so advanced that even in awarding it his advisor told him that they weren't sure that they fully understood it. After leaving Stamford he went on, still in the 1960s, to invent a desk calculator in his garage, which he had patented and then marketed (you can read about this here). He was also an entertaining character who loved to give tours of his house which had been a speakeasy and bordello in the 1920s and had even been raided by the Untouchables. Here he is in his house!
When I met him he had just written a book in which he described a number system he had developed in which he suggested that numbers do not progress linearly (as we normally imagine them) but in fact counting from one to two is a vector operation (even in the absract world of mathematics). In moving from one to two, the vector of the transition is almost linear, but not quite. It moves slightly off in two other dimensions as well. This means that the process of counting follows not a linear scale but a very broad helical path. At the beginning of the conversation he had immediately launched into a complicated description of how his theories worked. I have a degree in materials science (which is the physics of solids) from Oxford University and a Masters in metallurgy from Michigan Technological University. I was never a star student, but it does mean I have more than the average grasp of maths and science. Nevertheless, Irwin lost me in about three sentences. So I stopped him and said: 'Don't tell me how this works. Tell me instead what the important consequences of this are.' Then he told me that if you used his number system, rather than the conventional one, there were no irrational numbers and you could, for example, calculate precisely the area of a circle without having to use an approximate value for 'pi' (ratio of the length of the circumference of the circle to its diameter). Also, he said, through this he had come up with his own unified wave theory in which there was no wave-particle duality in the behaviour of photons, for example. I thought that this was staggering. If he really had done this then it could turn science upside down.
However, Irwin couldn't find anyone to take any notice of him who was capable of understanding his mathematical theories, because he was not associated with any university. He was a complete amateur who had developed this at home. This was so complicated that even most university mathematicians wouldn't understand him. Eventually he had managed to find someone to read and understand it who had some authority and his book was published. But even then, its publication passed largely unnoticed. You can find it on Amazon here. after this I tried to show his book any scientists I knew, but I couldn't get anyone to take me seriously and as soon as anyone started to push me with further questions I couldn't answer them; and again, because Irwin was an amateur they were inclined not believe that it could possibly be true.
Until now, I had not thought about the comparison with the progression of time and the liturgy in a helix, but it is a striking parallel. Perhaps it means that anything that has magnitude (and not just space and time) is three dimensional; because that magnitude is counted by numbers and the number system itself is three dimensional. The question then would be what is else is changing when we count other than conventional modern pure number so that there can be magnitude and direction. I am thinking that perhaps it might be that inherent in number is its quality, as I described it above, which the ancients were aware of and described when they attributed a symbolism to number.
There is another interesting parallel here. When Irwin was describing this to me he said that when the conventional numbers system, in a linear scale remember, was used the error is very small and only becomes significant when you do a large number of counting operations from some fixed point. There are parallels here with music (h/t Fr Z who asked me about this in connection with the topic of liturgy and music). Before the tuning of modern musical instruments to a system of an equal temperament, all notes were sung relative to a pitch decided arbitrarily, say by a leader in a choir. When all notes are sung relative to each other, according to the ear of each singer the intervals and harmonies place each note in a slightly different place, relative to an absolute scale, depending on the route you take to the note. In a choir, this is fine because what happens is that the human ear recognises this and the singer naturally modifies the notes slightly according to the situation and the harmonies are made purer (depending on how good the ear of the singer of course!).
This could work as well in instrumental music where the player has the scope to modify the pitch subtly in response to what is happening around him, for example a stringed instrument without frets, such as a cello. As I understand this, around the time of Bach, they realised that when the player does not have the scope to modify the pitch of the note subtly (such as in a harpsichord, for example, or a flute) in order for instruments to be able to play together in ensemble, they all need to be tuned relative to a single fixed point. The designation of 'equal temperament' refers to the fact that each single interval of pitch is set relative to a single point and is of equal size. This is a compromise and it means that in most situations everything is very slightly out of tune, musical composition was affected by this (I have been told) from this point to take this into account. Here's the point: this being the case, one would expect that just as with the Wunderman mathematical system, the error would become greater of the more iterations there are. In fact this seems to be what we find. It was recognised by those who were working out the new tuning that at a certain point the error becomes large enough to be very clearly audible and so standard modification was introduced, called a 'Pythagorean comma'. This is the difference in pitch that appears between 12 perfect fifths and the tempered seven octaves. In the ideal there should be no difference between the two, however you establish the scale, but there is due to the imposition of equal temperament in the latter case.
Everything that I have written here is highly speculative and if any physicists, mathematicians or experts in musical theory are reading this and tut-tutting and shaking your heads then you are probably right - I am getting out my depth. But I am thinking that there may be enough here for some who really know about these things to be motivated to read Irwin's book and see whether there is anything to it. I would love to think there might be. Maybe this is unifying even more than waves and particles? We might have a bridge between the physical and the metaphysical.
The broader point here in my writing this is to illustrate a conviction that I have that the more we have a deep sense of the beauty of the cosmos (the greatest teacher of this is the liturgy), the greater our creativity in all areas of life for the good of man. I will close with a comment from called Wil Roese, who on reading the first article made this observation: Christ is the Light of the World is simultaneously fully man and fully God, just as light is wave and particle. What an interesting thought!
Below the garage in which Irwin invented his desk calculator; and below that his invention as produced in the 1960s.
What I fascinating idea. I know I have heard of multi-dimensional space, but not three-dimensional time. This raises the question as to what it is that brings the helix back to the start again after a 360 degree turn in our original picture of music and the liturgy.
The traditional idea of number considers it to be something that designates both quality and quantity. We are used to the idea of number designating quantity, for this is how we tend to think of it today - 9 is always greater than 8 in magnitude for example. But the idea of numbers possessing different qualities is unusual now. For the ancients, however, this was just as important. So the number eight, as we discussed quantifies, but it also 'qualifies' - in the sense that it assigns a particular quality to something. It associates that count of eight with Incarnation. This means that as we go forward in time, we might say that we have moved forward 50 days in time, this considers only the quantity of time. However, we would also say that this day is a Sunday, and that it is the eighth Sunday in the cycle and so it is a special sort of day, connected to previous and future Sundays. In order to carve out this helical path that allows it to move forward yet return to the same quality, we imagine a three dimension picture of it threading through space, hence Alexey's three dimensions in time. This means that when we move forward it is important to consider not only the magnitude that we count, but also direction. The word for something that has both magnitude and direction, many will know, is a 'vector'. So in other words, Alexey is suggesting that time is vector quantity.
What does this have to do with science? Well this is what strikes me. In the scientific method as I understand it, the scientist observes the data, and proposes a hypothesis to explain it which predicts some new, previously unobserved, effect. Then he tests the hypothesis by carrying out repeated experiments to see if the predictions are realized in practice. Then the hypothesis is considered to have a greater degree of certainty and it is now called a 'theorem'.
This story illustrates a point, I think. That the greater the intuitive sense of the scientist is of the symmetry and order of creation, that is its beauty, the more likely he is to be able to spot the answer that is consistent with the pattern of nature. So, I would contend, you would make a better research scientist out of the man who has receives the traditional education in traditional beauty. At the core of this is the life lived liturgically. Through our active and ordered participation in the liturgy, the rhythms and patterns of the cosmos are impressed deeply upon our hearts.
There are parallels between artist and scientist in their observation and description of the natural world. The artist that draws well uses combinations of the parabolas and elipses to create the natural looking graceful curves that describe his forms (although he very likely knows these only by a recognition of their graceful shape and would not be able to describe them as a mathematician would). Therefore, the regular exposure to good sacred art, especially in harmony with the liturgy, will enhance the ability of the person to see the world around him in these forms. In the diagram below you can see my version of the plot of the properties of the known particles as it would have appeared to Gell-Mann so that he was inspired to look for the missing apex at the bottom. We can compare this with the pythagorean tectractys, which is a diagramatical representation of the harmonies of music and as shown above in a detail from Raphael's School of Athens.
This goes further.
Pondering over this remark of Alexey's and how even time might be a vector (with magnitude and direction) and not just a scalar (which has only magnitude), reminded me of someone I met years ago in Mountain View, California called Irwin Wunderman. His son was a friend of mine from my time studying metallurgy at Michigan Tech. Irwin was a brilliant man (he was in his seventies, I think, when I met him and he has since died). He was a PhD from Stamford where, he told me, his thesis was so advanced that even in awarding it his advisor told him that they weren't sure that they fully understood it. After leaving Stamford he went on, still in the 1960s, to invent a desk calculator in his garage, which he had patented and then marketed (you can read about this here). He was also an entertaining character who loved to give tours of his house which had been a speakeasy and bordello in the 1920s and had even been raided by the Untouchables. Here he is in his house!
When I met him he had just written a book in which he described a number system he had developed in which he suggested that numbers do not progress linearly (as we normally imagine them) but in fact counting from one to two is a vector operation (even in the absract world of mathematics). In moving from one to two, the vector of the transition is almost linear, but not quite. It moves slightly off in two other dimensions as well. This means that the process of counting follows not a linear scale but a very broad helical path. At the beginning of the conversation he had immediately launched into a complicated description of how his theories worked. I have a degree in materials science (which is the physics of solids) from Oxford University and a Masters in metallurgy from Michigan Technological University. I was never a star student, but it does mean I have more than the average grasp of maths and science. Nevertheless, Irwin lost me in about three sentences. So I stopped him and said: 'Don't tell me how this works. Tell me instead what the important consequences of this are.' Then he told me that if you used his number system, rather than the conventional one, there were no irrational numbers and you could, for example, calculate precisely the area of a circle without having to use an approximate value for 'pi' (ratio of the length of the circumference of the circle to its diameter). Also, he said, through this he had come up with his own unified wave theory in which there was no wave-particle duality in the behaviour of photons, for example. I thought that this was staggering. If he really had done this then it could turn science upside down.
Until now, I had not thought about the comparison with the progression of time and the liturgy in a helix, but it is a striking parallel. Perhaps it means that anything that has magnitude (and not just space and time) is three dimensional; because that magnitude is counted by numbers and the number system itself is three dimensional. The question then would be what is else is changing when we count other than conventional modern pure number so that there can be magnitude and direction. I am thinking that perhaps it might be that inherent in number is its quality, as I described it above, which the ancients were aware of and described when they attributed a symbolism to number.
There is another interesting parallel here. When Irwin was describing this to me he said that when the conventional numbers system, in a linear scale remember, was used the error is very small and only becomes significant when you do a large number of counting operations from some fixed point. There are parallels here with music (h/t Fr Z who asked me about this in connection with the topic of liturgy and music). Before the tuning of modern musical instruments to a system of an equal temperament, all notes were sung relative to a pitch decided arbitrarily, say by a leader in a choir. When all notes are sung relative to each other, according to the ear of each singer the intervals and harmonies place each note in a slightly different place, relative to an absolute scale, depending on the route you take to the note. In a choir, this is fine because what happens is that the human ear recognises this and the singer naturally modifies the notes slightly according to the situation and the harmonies are made purer (depending on how good the ear of the singer of course!).
This could work as well in instrumental music where the player has the scope to modify the pitch subtly in response to what is happening around him, for example a stringed instrument without frets, such as a cello. As I understand this, around the time of Bach, they realised that when the player does not have the scope to modify the pitch of the note subtly (such as in a harpsichord, for example, or a flute) in order for instruments to be able to play together in ensemble, they all need to be tuned relative to a single fixed point. The designation of 'equal temperament' refers to the fact that each single interval of pitch is set relative to a single point and is of equal size. This is a compromise and it means that in most situations everything is very slightly out of tune, musical composition was affected by this (I have been told) from this point to take this into account. Here's the point: this being the case, one would expect that just as with the Wunderman mathematical system, the error would become greater of the more iterations there are. In fact this seems to be what we find. It was recognised by those who were working out the new tuning that at a certain point the error becomes large enough to be very clearly audible and so standard modification was introduced, called a 'Pythagorean comma'. This is the difference in pitch that appears between 12 perfect fifths and the tempered seven octaves. In the ideal there should be no difference between the two, however you establish the scale, but there is due to the imposition of equal temperament in the latter case.
Everything that I have written here is highly speculative and if any physicists, mathematicians or experts in musical theory are reading this and tut-tutting and shaking your heads then you are probably right - I am getting out my depth. But I am thinking that there may be enough here for some who really know about these things to be motivated to read Irwin's book and see whether there is anything to it. I would love to think there might be. Maybe this is unifying even more than waves and particles? We might have a bridge between the physical and the metaphysical.
The broader point here in my writing this is to illustrate a conviction that I have that the more we have a deep sense of the beauty of the cosmos (the greatest teacher of this is the liturgy), the greater our creativity in all areas of life for the good of man. I will close with a comment from called Wil Roese, who on reading the first article made this observation: Christ is the Light of the World is simultaneously fully man and fully God, just as light is wave and particle. What an interesting thought!
Below the garage in which Irwin invented his desk calculator; and below that his invention as produced in the 1960s.
