The speaker alias myself commenced by saying he aimed to give a rapid overview of the subject 1 as a mathematical and physical concept 2 its connection to religion and mysticism and 3 possible social and technological consequences of the elimination of the concept from science and mathematics.
Definition I defined ‘infinity’ as “a process that can be started but never concluded”. Usually the process involves making something ‘bigger and bigger’ or ‘smaller and smaller’ as in Calculus. “Infinity is a process or procedure, not a quantity”. I should have added that I was well aware that are more precise, also more sophistical, mathematical definitions but at the end of the day we come down to this that infinity is a procedure or activity that never terminates and never can.
Is the concept attractive ? “Not to me,” the speaker said. “My dislike of infinity dates back to my childhood when a familiar sight on the breakfast table was a brand of honey which had a bear on the label. The bear was holding a jar of honey with a bear on it holding a jar of honey and so on….This used to torture me at night until I fell asleep with exhaustion.
Necessary as a concept? The Greeks got on very well indeed without it though, arguably, their finitism stopped them developing the science of dynamics. It would have been possible in my view to have developed the calculus without dragging in the infinite but the Greeks just stopped short of doing this though Archimedes came near. Fast forward to the Renaissance. This was a period when the West was liberated from the medieval obsession with ‘infinite time’ (eternity), and the new optic gave rise to the exploration of the physical world by navigation and the loving depiction of the human body in painting and sculpture. “However, the concept of infinity made its fateful appearance with Galileo and others leading eventually to the ‘Infinitesimal Calculus’ (as it was called until very recently) though Newton seems to have had some doubts about the validity of his great invention.” The grip of infinity finally began to loosen at the end of the pragmatic nineteenth century but then mathematics plunged ever deeper into the mire of infinity with Cantor’s theory of the Transfinite, infinity gone mad (and Cantor himself did) . Adapting a simile from Nietzsche, I said that the concept of infinity was like the gigantic statue of a dead god whose baleful shadow lay across the valley below terrifying the inhabitants and stopping them going about their daily business.
The Revealing Case of Pascal “Le silence absolu des espaces infinis m’effraient” (The absolute silence of infinite space terrifies me). Pascal wrote this.
“It is interesting that Pascal, the man who discovered the Law of Uniform Pressure for Gases, built the first working calculator and contributed to the Calculus, had a mystical experience one night around this time of year and from then on abandoned the ‘sterile infinity of mathematics’ for the warmth of a personal relationship with God.”
“But,” I added, “if Pascal had been alive today he would perhaps not have needed to abandon the world and science. For we now know that the universe is not finite ─ we can even judge its extent ─ and it is not silent since we hear it if not with our ears at least with radio telescopes. The universe is no longer forbidding and distant; we know, or think we know, the constituents of the stars and galaxies, however huge, are like grains of sand scattered across the ground. We are part of the universe since, as this gentleman will tell you, [a chemist in the audience] the carbon and other elements in our bodies comes from exploded stars.”
Everything physical is finite At one time almost everything was thought to be ‘infinite,’ ‘eternal’. But we now know, for example, that the speed of light is not infinite, that the universe itself had a beginning in time and has a specific size. Energy is not continuous but can only be distributed in definite quantities (the famous quanta of Quantum Mechanics); molecules and atoms can even be ‘seen’ by electron microscopes.
The Differential Calculus is basically the study of how two sets of quantities change with respect to each other, one variable ‘depending’ on the other. In mathematics the independent variable can be made arbitrarily small. But if you reduce the input of a mechanical system beyond a certain point, this input is unable to overcome internal friction and there is no output whatsoever. And this limit is miles away far from the mathematical one. “Touch the person next to you as lightly as possible. Then lighter still. You will soon get to the point when this person does not recognize the pressure of your hand. Everything is like this, there is always a smallest and largest possible amount in real life. Calculus models an ideal world, not the real one.”
Today there is some talk of there being a finite ‘smallest length’, the Planck scale (10 (exp) –34), but very rarely talk of there being a smallest interval of time, what I call the ksana (from Sanscrit for ‘instant’). Time is actually the most important dimension since, as Pearce wrote, “one can imagine a world without space but not a world without time”. Although the ‘space’ of our dreams is completely distorted, this does not happen with time : one event leads to another just as in real life. In dreams as in real life you never get stuck in a vicious circle going round and round for ever: there is a ceaseless drive onwards and in a single direction. Time is cvery different from space since it only has one ‘dimension’ and it is dislocated from the three spatial dimensions, “the spatial three-dimensional reality must disappear when time is introduced since otherwise there would be no difference from what exists at one moment and the next”.
The Infinite compared with the ‘Non-finite’
Is the universe self-sufficient and self-explanatory? It would seem not since even science is now seriously talking about it coming from something that was there before, and which will perhaps give rise to other, different, universes. This deeper reality, the ‘Origin’, Ain Soph, call it what you will, has (so I would claim) nothing in common with the mathematical concept of infinity – the Buddha is credited with the just observation that “nirvana is neither finite nor infinite”.
The speaker said he envisaged ‘reality’ as made up of two regions with a veil separating them (the veil of Isis). Mystics have lifted a corner of this veil and have sometimes described what they have seen on the other side. The Beyond is so completely different from everything in the physical universe that mystics, quite rightly describe it in contradictory or negative terms. On the other side there is no number, no shape, no name, no elementary particles, no difference between the part and the whole “All is One”.
However, we live on this side of the veil, in the world of separation, the world of extension and number and mathematics and physics should confine themselves to what is measurable and/or deducible from our (ordinary) sense impressions. Above all we should not bring into science and mathematics any knowledge (or delusory imaginings) concerning the ‘non-finite’ domain of reality.
“The Tao that can be named is not the origin al Tao” – the first line of the Tao Te Ching. In Lao Tse’s time, language was the most accurate analytic tool known to mankind : if Lao Tse were alive today he would have written “The Tao that can be numbered or mathematized is not the original Tao”.
I have found this stratagem of separating reality into two, and only two, incompatible regions, one finite, specific, measurable, the other non-finite and immeasurable very useful indeed (Note 1).
Strangely enough, the bridge, inasmuch as there is one, between the two realms is not to be found by reaching out into the vastness with bigger telescopes and torturing oneself with the concept of the infinite, but on the contraryby focussing on the present moment, any moment, this moment. I pointed to the sunlight falling on the grass alongside where we were standing.
Is it possible to show that there is a ‘smallest interval of time’? Is the hypothesis testable?
I made the prediction in one of the early posts on this site, that “during this century science will be able to determine the ratio of the smallest interval of distance to that of the smallest interval of time”. I added that I thought this would not happen in my lifetime. But to my astonishment, someone (Craig Hogan) is currently building a machine he calls an Interferometer in Chicago precisely to show that, as he conceives things, “Space/Time is grainy” or in the current jargon “At a certain level the universe is digital.” Hogan is looking for a basic ‘static’ that goes deeper even than fluctuations of the quantum vacuum and which he sees as the “froth of Space/Time” (Note 2).
There are, incidentally, several thinkers today who view the universe as a giant computer and this came up in the discussion later. “What I note is that a digital computer is made up of a finite collection of bits, carries out finite series of operations sequentially and has two and only two ‘states’, ‘on’ and ‘off’.” In life terms, ‘on’ is ‘existent’ and ‘off’ is ‘inexistent’ and reality is flickering on and off perpetually. “We are bits”, and the person who raised issue, somewhat to my surprise, did not take this as an insult but nodded in agreement.
I also noted that the universe is expanding faster and it does not look like it will ever contract again now. Everything has its time, “You will die, I will die and the universe will die”, the speaker said somewhat melodramatically.
Social and technological aspects of ‘infinity’
The speaker, i.e. myself, did not have time to say too much about social and technological matters because of the cold. He would have liked to say more about how science and mathematics, now everywhere triumphant, have made the world and life almost totally incomprehensible (hence the heasdlong flight towards religious fundamentalism). Theoretical scientists and mathematicians seem to be engaged in a sort of competition, on the one hand they say “See all the improbable or impossible things I can believe in and you can’t!” and on the other “See all the stipid things you believe in and I don’t”. (Things like free will and that you have the ability to change your life overnight if you really want to.) We are moving at an alarming rate towards scientific totalitarianism: science has ceased to be a free enquiry but a matter of signing up to a credo and watch your step if you disagree with Richard Dawkins & co. on any point for you’ll live to regret it — if you’re a professional scientist that is, I can think what I want.
I believe all knowledge is based on sense impressions and this is the point where we should start. No exception should be made for mathematics and speculative science. What is dismissed by science as ‘anecdotal’ is actually in a way more genuine and more real than what is carried out in the artificial environment of laboratories. A practising chemist at a family gathering discussingthese sort of issues, said like a bolt from the blue to everyone;’s astonishment, “Only the experiment is real, all the rest is theory”. I’m not sure that I wouldn’t go one step further and say “Only the experience is real”.
On the social level, he/I referred to a book popular for as a while in the Sixties but now forgotten, Cain’s Book by Alexander Trocchi. In this book, the central character lives on a boat moored near New York. He is paid to be there by the owner and does not have to do anything much except potter around so he has plenty of time on his hands. He sees in the distance the vast city that he calls “the city of outrageous purpose” (an excellent phrase) but rarely ventures into it. He spends his time desultorily (but on the whole enjoyably) looking at the water and occasionally meeting one or two drop outs. He contrasts “the city of outrageous purpose” (spatial) with “the meaningless texture of the present moment” (temporal). He prefers the second to the first obviously – though unfortunately Trocchi’s interest in the ‘texture of the present moment’ took him into hard drugs, an unnecessary and counter-productive move.
Spatial and Temporal Cultures
We live in a spatial civilisation which prizes ‘things’ above sensations. We have an ‘object-orientated outlook’which ultimately goes back to the Greeks whose greatest achievements were strictly spatial (geometry and sculpture). Democritus supposedly said “Nothing exists except atoms and void” and his atoms, like Newton’s, were indestructible and eternal. This view of the world, duly extended by Galileo and Newton, has taken us to where we are now and I certainly don’t want to disparage the fantastic achievements dependent upon it.
But at around the same time as Democritus was active (VIth cetury BC) a homeless wanderer came to exactly the opposite conclusion, namely that “Everything is ephemeral, a ceaseless succession of point-like instants in a state of commotion”. This is the great thought of a timelike civilization and, strangely, though it has given rise to great art and poetry, it never gave rise to a form of science and technology like the spatial take on reality (Note 3). The speaker stated cryptically that the concept of ‘the moment’ will soon give rise to a different kind of science and even a new technology. (My ponderings on this theme will be the subject of a subsequent post.)
Conclusion
I concluded by saying that it was completely appropriate that this discussion was taking place in (or rather just outside) an edifice built in honour of the Buddha. The briefest summary of (Hinayana) Buddhism is the following credo
“The Great Recluse identified the elements of existence (dharma), their causal interconnection (karma) and their ultimate extinction (nirvana)”.
Finally – and this was completely unplanned and a surprise even to me – I said “My message to you is ‘Hold fast to the moment’, ‘Seize the moment’ ”.
Intelligent discussion followed from the audience but we had to call it a day because of the weather.
Postscript Subsequently, I formed the project of giving a series of talks on related subjects in the open air at the Pagoda, probably on the last Sunday of each month (watch this space). If no one turns up it doesn’t really matter as it is a good place to be. SH 15/12/12
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Note 1 This principle of the ‘Seaparation of the Spheres” enables me to dismiss at one fell swoop the Theory of the Transfinite and all the Set Theory that depends on it as nonsense which indeed is how it appears to the ordinary person (if such still exist). I must admit to having some trouble deciding how to fit the ‘reality’, if it be reality, of what is described by the wave function in Quantum Mechanics into my schema — does the Schrodinger equation describe anything that really exists or not? But I’m in good company here since debate on the subject still rages unabated. .
Note 2. See article Scientific American, February 2012 “Craig Hogan believes that the world is fuzzy…… [he] thinks that if we were to peer down at the tiniest subdivisions of space and time, we would find a universe filled with an intrinsic jitter, the busy hum of static. This hum does not come from particles bouncing in and out of being or other kinds of quantum froth that the physicists have argued about in the past. Rather Hogan’s noise would come about if space was not, as we have long assumed, smooth and continuous, a glassy backdrop to the dance of fields and particles. Hogan’s noise arises if space is made of chunks. Blocks. Bits. Hogan’s noise would imply that the universe is digital.
He has devised an experiment to explore the buzzing at the universe’s most fundamental scales.” Scientific American, February 2012
As I see it, if Hogan picks up an irreducible ‘static’ that is regular, this may well be caused by the spatial shift from one ksana to another. If, however, as I would expect, the noise is random, it would not come from ‘Space/Time’ (what I call the Locality) but from stray ‘ultimate events’ springing into existence and then disappearing without being able to form stable event-chains. There are, I suspect, very many more (I nearly said an ‘infinite number’ of) ultimate events that ‘do not make it’ and merely disappear for ever — just as there are many many more elementary particles than the ones that form themselves into stable atoms.
Note 3 The men who elaborated the ‘dharma theory’ certainly had the clarity and intelligence to initiate a scientific revolution but their principal or exclusive concern was ‘soteriological’ : to provide a cure for mankind’s unhappiness. There was no point in delving deeper into the mechanisms underlying the physical (pseudo)world, the world of maya, and so, although the developed a system of logic and psychology (to help people towards enlioghtenment), they never developed a systematic physics. SH
Dharma 