Though Gödel’s discoveries may have undermined some forms of faith in mathematics, in a manner that seemed to resemble the Pythagorean discovery of incommensurability, Gödel’s view of mathematics was, in fact, Pythagorean. He believed that mathematical truth is something that actually exists apart from any invention by human minds – that his theorems were ‘discoveries’ about objective truth, not his own creations.
This was not a popular idea in the 1930s. Many mathematicians disagreed. In fact, the concept of anything existing in an objective sense – waiting out there to be discovered and not in any way influenced by the actions of the investigator – had been called into question by a development in physics. A far more dramatic and far-reaching crisis than the one caused by Gödel’s incompleteness theorem had occurred in the 1920s and was having a profound effect on the way scientists and others viewed the world. It was the discovery of the uncertainty principle of quantum mechanics.
The way cause and effect work had long seemed good evidence that the universe is rational. It also seemed that if cause and effect operate as they do on levels humans can perceive, they surely must operate with equal dependability in regions of the universe, or at levels of the universe, that are more difficult – or even impossible – to observe directly. Cause and effect could be used as a guide in deciding what happened in the very early universe and what conditions will be like in the far distant future. No one was thinking of belief in cause and effect as a ‘belief’ at all, though, in fact, there was nothing to prove that cause and effect would not cease to operate in an hour or so, or somewhere else in the universe. Then, in the 1920s, came developments that required reconsideration of the assumption that every event has an unbroken history of cause and effect leading up to it.
The quantum level of the universe is the level of the very small: molecules, atoms, and elementary particles. It is on that level that a commonsense description breaks down. Here there are uncaused events, happenings without a history of the sort it is normally assumed any event must have. Atoms are not miniature solar systems. You cannot observe the position of an electron orbiting the nucleus and predict where it will be at a later given moment and what path it will take to get there or say where it was an hour ago – as you could with fair accuracy for the planet Mars in the solar system. An electron never has a definite position and a definite momentum at the same time. If you measure precisely the position of a particle, you cannot at the same time measure its momentum precisely. The reverse is also true. It is as though the two measurements – position and momentum – are sitting at opposite ends of a seesaw. The more precisely you pin down one, the more up-in-the-air and imprecise the other becomes. This is the Heisenberg uncertainty principle of quantum physics – the twentieth century’s ‘incommensurability’. It was first articulated by Werner Heisenberg in 1927. Not only did it undermine faith in a rational universe, it also seemed to undermine the notion that truth was something objective, something waiting out there to be discovered. On the quantum level, your measurement affects what you find.
On the other hand, the existence of quantum uncertainty itself was apparently a very unwelcome piece of objective truth waiting out there that no physicist could change, as much as he or she might wish to, no matter what observational methods he or she used. Einstein in particular rebelled at the notion that no future advance in science and no improvement in measuring equipment was ever going to resolve this uncertainty. Until his death, he went on trying to devise thought experiments to get around it. He never succeeded, nor has he succeeded posthumously as others have found ways to carry out experiments he invented in his head. ‘God does not play dice!’ Einstein wrote on one occasion to Niels Bohr, who was far more ready to accept quantum uncertainty than Einstein. ‘Albert, don’t tell God what he can do!’ Bohr answered. The Bohr-Einstein debate about how to interpret the quantum level of the universe continued and became famous.
It is easy to sympathise with Einstein. The quantum world and the paradoxes implicit in it did not seem to be the work of a rational mind. Einstein might have rephrased the complaint Kepler registered when faced with a similar problem: ‘Heretofore we have not found such an ungeometrical conception in His other works!’ How could what happened to one particle affect another over time and space with no link between them? How could a cat be both dead and alive at the same time – as one had to accept in the famous example of ‘Schrödinger’s cat’? How could something be a wave some times and a particle at others, depending on the experimental situation? It was a Through the Looking Glass world – and still is, in spite of the reassurance that it is possible to predict things on the quantum level of the universe, if one can be satisfied with probabilities. It does seem that the staircase to knowledge about the universe can have a firm footing on the quantum level, with probabilities forming a sort of superstructure above the quagmire. All is far from lost for the Pythagorean climb.
The dawning awareness of a new aspect of the universe, in chaos and complexity theories developed later in the twentieth century, was not nearly so great a shock as quantum uncertainty. However, it did seem to hint that science had been discovering one orderly, predictable system after another only because it was impossible or at least terribly discouraging to try to study any other kind of system in a meaningful fashion. The relatively easy to study predictable systems actually turned out to be the exception rather than the rule. But for those of a Pythagorean cast of mind, it was the discoveries of the repeating patterns in chaos – the pictures deep in the Mandelbrot and Julia sets, and also in nature itself – that gloriously seemed to uphold, as never before, the ancient conviction that beauty and harmony are hidden everywhere in the universe and have nothing to do with any invention of humans. Less immediately mind-boggling, but no less impressive, was the realisation in the study of chaos and complexity that there seem to be mysterious organising principles at work. There are probabilities, but by some calculations they are vanishingly low, that the universe would have organised itself into galaxies, stars, and planets; that life on this earth would have been organised into ecosystems and animal and human societies. Yet that is what has happened. Thus, as with the other challenges to faith in the Pythagorean assumptions underlying science, when scientists began to get a handle on chaos and complexity, the theories having to do with them became not threats but new avenues in the search for better understanding of nature and the universe.
Twentieth-century ‘postmodern’ thinking, combined with suspicions raised by the discovery of quantum uncertainty and our inability to examine the quantum world without affecting it, led to fresh doubts about other Pythagorean pillars of science. Is there really such a thing as objective reality? Is anything real, waiting to be discovered? Does the fact that science continues to discover things that make sense, and suspects or dismisses anything that does not, mean that we are finding out more and more about a rational universe . . . or only that we are selecting the information and discoveries that fit our very Pythagorean expectations?
The assumption of rationality lies at the root of modern arguments about ‘intelligent design’. It is true that the world’s design, as the Pythagoreans found out, is intelligent to a degree that would send any discoverer of a new manifestation to his or her knees – but before what, or whom? Does discovering rationality necessarily mean one has glimpsed the Mind of God? On the other hand, does a good scientist have to repress the strong impression that it does? Those who attack belief in God do so from several directions. One is rather old-fashioned now, but still heard: Everything is so perfectly laid out, in so tight and orderly a design, that there is no room for God to act at any point. It all goes like clockwork. Or, a newer argument: Everything happens – and has always happened – entirely by chance. The impression of any underlying rationality in nature is an illusion. The ‘anthropic principle’ says that if things had not fallen out just the way they have, we could not be here to observe them – and that is the on
ly reason we find a universe that is amenable to our existence. Or . . . our entire picture of the universe is created, by us, in the self-centred image of our own minds, and we are discovering something not far different from the ten heavenly bodies of the Pythagoreans. Plato might have enjoyed the late-twentieth-century discussions about whether mathematical rationality might be powerful enough to create the universe, without any need for God. Quantum theory made possible the suggestion that ‘nothingness’ might have been unstable in a way that made it statistically probable that ‘nothingness’ would decay into ‘something’.
Pythagorean principles and issues also showed up in other ways in twentieth-century culture. Peter Shaffer’s trilogy of plays The Royal Hunt of the Sun, Equus, and Amadeus were all profound explorations of the theme of rationality and irrationality and reflected the sort of love/hate humanity has for both: Is there a Mind behind the universe? Is that Mind sane or mad? Tennessee Williams dubbed the so-called ‘rationality’ of God the rationality of a senile delinquent. In music, ‘twelve tone’ compositions were the most mathematically bound compositions ever written, but this form of music was also clear evidence that the Pythagorean insight had been correct that certain combinations of tones – and only certain combinations – have a deep link with what the human ear recognises as harmonious and beautiful. On Sesame Street, numbers came to life and danced and sang in a way that probably would have delighted the Pythagoreans – if they did not find it irreverent – but probably would have annoyed Aristotle.
The music of the spheres remained a popular metaphor, but in the second half of the century it moved beyond the ‘spheres’.6 As Richard Kerr has put it, ‘the idea of heavenly harmonics is now making a comeback among astronomers. Instead of listening to the revolutions of the spheres, modern astronomers are tuning in to the vibrations within stars’.7
In 1962, astronomers studying the Sun discovered that sound waves travelling through the Sun cause a bubbling of its visible surface, the photosphere.8 They described it as a ‘solar symphony’ that is somewhat like a ‘quivering gong’, or ‘a large spherical organ pipe’, or a ‘ringing bell’, for the Sun has millions of different overtones.9 Ours is not, of course, the only star that vibrates in this way. The giant star XiHydrae is a ‘sub-ultra-bass instrument’, with oscillations of several hours.
In a book entitled Einstein’s Unfinished Symphony: Listening to the Sounds of Space-time, Marcia Bartusiak described the possibility of detecting a black hole ‘by the melody of its gravity wave “song”’.10 Black holes have now indeed joined the heavenly choir. When material falls towards a supermassive black hole, that produces a jet of high-energy particles that blasts away from the black hole at nearly the speed of light. This jet plows into the gas around the black hole, creating a magnetised bubble of high-energy particles. An intense sound wave rushes ahead of the expanding bubble.11 The NASA satellite Chandra, named for Subrahmanyan Chandrasekhar, the first scientist to see that, given Einstein’s theories, black holes were inevitable, has found evidence of acoustic waves like this in the gaseous regions around two super-massive black holes. One of them, at the centre of the Perseus galaxy cluster, plays the deepest note discovered so far in the universe, B flat fifty-seven octaves below middle C.12
Mark Whittle of the University of Virginia has produced a tape of ‘Sounds from the Infant Universe’ which reproduces the power spectrum of the Cosmic Background Radiation – radiation that is still reaching us from the early universe – as an audible sound, covering the first million years of the cosmos in ten seconds.13 In order to make the acoustic waves hearable by the human ear, he had to shift them upward approximately fifty octaves. The tape begins in silence, as the universe did, because there were no acoustic waves as long as the infant universe was symmetrical. Eventually there arose acoustic waves of deeper and deeper tone. The expansion of the universe stretched the wavelengths, making for an overall drop in pitch as the tape continues. The largest variations compare to ‘rock concert volume’.14
The prediction was that a ‘ripple’ in the distribution of galaxies in the universe would reflect the acoustic waves in the Cosmic Background Radiation. At the January 2005 meeting of the American Astronomical Society, the report came that this evidence had been found.15 Those who announced it likened the discovery to ‘detecting the surviving notes of a cosmic symphony’ and the difficulties of the observations to trying to hear the ‘last ring’ of a bell that ‘gets forever quieter and deeper in tone as the Universe expands’.16 One cannot help thinking that Kepler would have been intensely interested in projects like these.
Kent Cullers, who works at SETI, the Search for Extraterrestrial Intelligence, and on whom Carl Sagan based one of his characters in the novel and film Contact, is blind and claims this is an advantage as he listens to signals from outer space. ‘When I hear signals from distance regions, my mind goes out there. I try to ride those waves, extend my senses to a realm where they’ve never been, hear songs from a cloud of gas.’17 In the 1970s, it was proposed that the Pythagorean theorem, or ‘Pythagorean triples’ of numbers that make right triangles, be beamed as messages into space, in the hope that rational life in other star systems might receive the signals and realise that there was rational life on Earth. It is a signal like that that Cullers is hoping to hear, coming to us from deep space – evidence of how truly primordial this knowledge is.
EPILOGUE
Music or Silence
Generation after generation, men and women have recognised the essential truth of the ancient insight that rationality and order underlie the variety and confusion of nature. The image of Pythagoras himself has shifted and occasionally become distorted, but through all the centuries and all the paradigm shifts, this Pythagorean vision has never been extinguished or forgotten, and it has almost always been cherished. He and his first followers could not begin to conceive how vast a landscape lay beyond the door they opened. From unimaginably tiny flickering wisps of uncertainty to the uncountable galaxies, into multiple dimensions, and maybe even to an infinity of other universes. Yet numbers and number relationships seem to have guided the way through this labyrinth of the physical universe as effectively as Pythagoras himself could ever have hoped.
If civilisation as we know it were wiped out and only a remnant were left to start over, would someone make that same discovery? Break the code again? Surely they would! Is it not basic truth? Or . . . maybe they wouldn’t. Maybe the Pythagoreans got it wrong, and we have been living in a dream. Maybe the world really never got beyond a formless ‘unlimited’, and we are only imagining the pattern, or creating it ourselves. The human soul has not proved so easy to map with numbers . . . and yet we are the ‘rational beings’ on the Earth, presumably reflecting the rationality of the universe. How can it be that we are the most difficult of all territory? We do not yet know. Meanwhile most of us are too intoxicated by the music of Pythagoras to suffer a crisis of faith.
We send our tiny beeps into the far distant reaches of space, certain that any intelligent beings out there, no matter how ‘other’ they may be in some respects, could not have failed to discover what our world did . . . sure that our little signalled evidence of rationality will look familiar to them. In spite of the still unsolved mysteries – and the possibility that they may never be solved – our Pythagorean ideal of the unity and kinship of all being tells us this must be so.
Pythagoras . . . are you there?
Appendix
The proof for the Pythagorean theorem that Jacob Bronowski thought may have been used by Pythagoras.1
Start with a right triangle.
Create a square using four triangles identical to that one, but rotated, so that the ‘leading points’ of the triangles point to the four points of the compass (north, south, east, and west), and the long side of each triangle ends at the leading point of its neighbour:
What you now have is a square base
d on the long side of the original triangle – the ‘square on the hypotenuse’. It is this total area that must equal the sums of the squares of the other two sides, if the Pythagorean theorem is correct. As you proceed, remember that however you rearrange these five shapes, their total area stays the same. So, rearrange them into the following shape. Place a rod across your design and look at it carefully. You will see that you have two squares, and they are the squares on the other two sides of the triangle. Using no numbers, you have proved the Pythagorean theorem.
Notes
Chapter 1: The Long-haired Samian
Iamblichus’ Pythagorean Life or Life of Pythagoras is available in translation by Thomas Taylor: Iamblichus Life of Pythagoras (Rochester, Vt.: Inner Traditions International, 1986). It and the biographical treatments by Porphyry and Diogenes Laertius are available in translation by Kenneth Sylvan Guthrie: The Pythagorean Sourcebook and Library: An Anthology of Ancient Writings Which Relate to Pythagoras and Pythagorean Philosophy (Grand Rapids: Phanes Press, 1987). The Guthrie anthology also contains some of the pseudo-Pythagorean works.
Diogenes Laertius’ and Porphyry’s ‘lives’ of Pythagoras are reprinted in K. S. Guthrie.
Jacob Bronowski, The Ascent of Man (Boston: Little, Brown, 1973), p. 156.
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