Tag: mathematics

Stuck? 1 Education of a Materialist

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Twenty years ago, early in my retirement, I wrote an essay on ‘Science and Spirituality: Complementary or Contradictory?’ as a submission for a Resurgence Magazine Essay Competition. It expressed my understanding of the dominant thinking of the time, and its limitations, through the lens of science and spirituality. It didn’t win the prize and the essay was just filed away on the computer.

Recently, I came across it, and thought that it would be interesting to reflect on my views as expressed all those years ago. Has anything changed? Am I any wiser than I was then? Are we collectively any wiser than we were then? Did anything change, or am I/we just stuck in my/our thinking? To attempt to answer these questions, I am serialising that essay over a number of blog posts, with any commentary that seems appropriate, in italics. You are welcome to join me on the journey.

Preamble

Science and spirituality are part of our collective experience. That they could be contradictory now seems strange to me, and yet once they seemed so. This series of posts draws parallels between my own personal experience of growth and the corresponding growth of humanity; reconciliation of a complementary science and spirituality is a fundamental part of this process.

Education of a materialist

My school years centred around the 1950s in Lincoln, England. Science was king. I well remember the reverence accorded to white-coated boffins on the television (when we eventually got one in 1953). What they said was treated as gospel. The pressure from teachers was for the sciences. This was the future, what the country needed. Humanities were second best, for those with no aptitude for science.

When we were kids, religion was singing in morning assembly, and being sent to the Methodist chapel on Sundays. The minister told bible stories and warned us of the dangers of alcohol, while parents kept away and did the gardening. Yet we loved the occasional lay preacher who came with song and speeches that stirred our soul with their passion. Except we had no concept of soul.

Spirituality was something we secretly found out about through reading library books. It seemed to be all to do with séances, ouija ouija boards and magic. It was not talked about in polite society, and definitely not recognised as valid by science.

So I emerged from the education system with an essentially materialistic scientific viewpoint, deeply sceptical of religion, and uncomprehending of spirituality. After studying mathematics, I took up what was then called computer science, and soon became information systems engineering. I joined the everyday world of industry, married and eventually we started a family.

That was the preamble, still valid today. To be continued in Stuck? 2 Early doubts.

Featured image of Lego city by Lamiot,
CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0,
via Wikimedia Commons

Metaphor, Map and Model

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Metaphor

1. a figure of speech in which a term or phrase is applied to something to which it is not literally applicable in order to suggest a resemblance…
2. something used, or regarded as being used, to represent something else; emblem; symbol.

dictionary.com

Metaphor is the basis of language and related creativity. While this has always been apparent in the arts and literature, it is perhaps not so readily associated with other fields.

Just consider the two domains of thought that have dominated Western cultures for thousands of years: religion and science.

Religious texts are full of metaphor pointing towards the great religious and spiritual truths that can never be precisely expressed in language. Religions become problematic for human society when these texts are interpreted literally, rather than metaphorically. Then fundamentalism becomes a big problem, as it was for centuries in Europe and still is in many parts of the world. In the terms of Iain McGilchrist’s book The Master and His Emissary, the Left Brain Emissary has usurped the function of the Right Brain Master.

But surely science is different, you exclaim – it’s objective. Piffle! In essence, science makes mathematical models of the real world. And what are these models but metaphors that reach towards the underlying reality. Scientific fundamentalism becomes a problem when the scientist believes that the model accurately describes the real world, rather than being a metaphor, leading to losing touch with reality itself. The map is not the territory (another metaphor).

Of course, science’s handmaidens technology and modern capitalism have this problem in spades. It is not a huge leap to suggest that this Left Brain dominance has significantly contributed to today’s ecological and climate problems, and to the mealy mouthed response to these problems so far.

It’s all metaphor really!

Inspired by Iain McGilchrist’s magnum opus The Matter with Things.
Featured image includes a quote from Genesis I, King James version.

Taking Appearance Seriously

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The Dynamic Way of Seeing in Goethe and European Thought

by Henri Bortoft

taking appearance seriouslyThis challenging book explains where Western thinking went wrong, and points the way towards the revolution in thinking that is needed to get back on track.

I read it on Kindle some time ago, probably not wise for such an erudite work, but it did make it easy to recall a lot of key points by downloading my highlights.

Almost by definition, this is difficult reading, because it does not ‘come from’ the place where Western thinking habitually does these days.

Henri Bortoft has a good shot at making this understandable to such as myself, with an interest in philosophy but no great training or professional expertise. It is of course inspired by the thinking of Goethe, one of the giants of our intellectual history.

I’ve included my edited notes in the following, which may help to give an appreciation of the staggering scope of this book and of Goethe’s thinking. But there is no escape from the effort of reading the book itself if you want to understand its quite revolutionary message.Read More »

All about the numbers

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When a particular subject lights something up inside you, it’s worth taking notice. For me, one of those is the numbers – specifically the whole numbers, or integers. Thus was I from childhood drawn to mathematics, and later to Greek philosophy via Pythagoras. The former gave the outer mechanics of numbers, the latter suggested that numbers had a more mystical and imprecise meaning, leading to later interests in subjects such as numerology, and to astrology, where the numbers lurk in the background.

So I was a sucker for these two books which approach the numbers in completely different ways:

  • Music by the Numbers by Eli Maor
  • The Archetype of Number and its Reflections in Contemporary Cosmology, by Alain Negre

music by the numbersFor people such as me, Eli Maor has written an engaging book about the relationship between music and mathematics. The development of musical scales from Pythagoras to the early 20th century is an interesting story, reasonably well explained, from Pythagoras’s whole number ratios through the equal tempered scale exemplified in the work of JS Bach to the experiments of Stravinsky and Schoenberg.

The fascination still seems to lie in those magical simple ratios of musical resonance: the octave 2:1, the fifth 3:2 and the fourth 4:3, from which are derived the Pythagorean Scale, which is nearly ‘right’, but in the end not adequate for use in orchestras with different sort of instruments, as Maor explains. Always the whole numbers are beautifully simple, but prove too limited to describe the real world, hence the subsequent invention of all the panoply of mathematics, irrational numbers, imaginary numbers, the calculus and on and on.

And in the end, always and tantalisingly, the maths cannot fully describe the real world, which we know thanks to the insights of Kurt Gödel.

archetype of the numberAlain Negre’s book is about number as archetype – the qualitative aspect of number, which was revived in the 20th century by psychologist Carl Jung and physicist Wolfgang Pauli. All begins with 1,2,3, and 4 – just as with the Pythagorean scale. The qualities of these 4 basic numbers are explored and particularly related to the work of Jung, and to the triplicities and quadruplicities of astrology.

There are rather incomprehensible (to me) chapters relating the numbers 3 and 4 to current theories on the evolution of the cosmos – rather speculative, I think. Negre goes on to suggest that the astrological zodiac with the 12 signs is another projection of these number archetypes, including discussion of the axis crosses and the oppositional polarities in a chart of the 12 signs.

So the book is both familiar to me, in an astrological sense, and almost incomprehensible when relating to modern cosmology, which must be partly due to my own failure to keep up with this field. In fact, I had a similar reaction to an earlier work some years ago Number and Time by Marie-Louise von Franz. It feels like there is something important there, but the author has not quite managed to express it in a way that is easily comprehensible to me (of course this may be a commentary on me, rather than on the author’s work).

So yes, number still has that magical pull, but these books didn’t greatly enlightened me. Nor did they blunt that fascination with the numbers.

Music by the Numbers is much the more readable.

Gödel, Maths and Physics

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Edmund M. Law has some fascinating posts on his blog. A recent one had the following quote from Freeman Dyson.

Fifty years ago, Kurt Gödel, who afterwards became one of Einstein’s closest friends, proved that the world of pure mathematics is inexhaustible. No finite set of axioms and rules of inference can ever encompass the whole of mathematics. Given any finite set of axioms, we can find meaningful mathematical questions which the axioms leave unanswered. This discovery of Gödel came at first as an unwelcome shock to many mathematicians. It destroyed once and for all the hope that they could solve the problem of deciding by a systematic procedure the truth or falsehood of any mathematical statement. {53} After the initial shock was over, the mathematicians realized that Gödel’s theorem, in denying them the possibility of a universal algorithm to settle all questions, gave them instead a guarantee that mathematics can never die. No matter how far mathematics progresses and no matter how many problems are solved, there will always be, thanks to Gödel, fresh questions to ask and fresh ideas to discover.

It is my hope that we may be able to prove the world of physics as inexhaustible as the world of mathematics. Some of our colleagues in particle physics think that they are coming close to a complete understanding of the basic laws of nature. They have indeed made wonderful progress in the last ten years. But I hope that the notion of a final statement of the laws of physics will prove as illusory as the notion of a formal decision process for all of mathematics. If it should turn out that the whole of physical reality can be described by a finite set of equations, I would be disappointed.

— Freeman J. Dyson, Infinite in all Directions, 1985

Law presents this under the heading ‘Inexhaustible Mysteries’. To me, it’s just important to be reminded of Gödel’s Theorem from time to time. Mathematics is inherently open-ended, and I believe the implication is also that physics is also open ended. We can never have a model that fully describes reality. There will always be more for mathematicians and physicists to do.

Equally, we will never have a perfect economic system. There will always be space for economists and politicians. And those who seek single solutions to complex problems (e.g. ‘free markets’) are inherently misguided.

See also my post on Godel’s Theorem.

Picture of the tomb of Kurt Godel in the Princeton, New Jersey, cemetery by Antonio G Colombo, from Wikimedia Commons. What a legacy!

Fibonacci Grape Pips?

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2,1,2,2,3,3,3,2,2,5,1,2,2,2,3,5,2…

I was idly counting the pips in each grape off a bunch from E Leclerc (cf Tesco, Kroger). (It seems that France has not really caught on to the fashion for seedless grapes; most on sale had pips. Yes, they were more tasty.) My idle counting had spotted a potential ‘pattern’ – so far these are all Fibonacci numbers, and it is well known that Fibonacci numbers appear frequently in nature. Could it be…?? Then came the next sequence:

4,3,3,1,2,2,5,2,2,6

Now FOUR is not a Fibonacci number, so appears to be anomalous. Well, science does allow for anomalous results that don’t fit the current theory. Then comes the SIX. But here I notice two tiny black dots in the grape – putative pips that did not develop – which makes 8, another Fibonacci number. Maybe I’d missed a black dot with the 4?

So I can hang on to my theory for a while, until more anomalous data emerges. A rather trivial example of the scientific method in action? Of course, there are far too few results to draw conclusions…

Featured image by Thamizhpparithi Maari, via Wikimedia Commons

Reality Is Not What It Seems

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reality coverI recently discovered Carlo Rovelli’s book Reality Is Not What It Seems: The Journey to Quantum Gravity. This was among the big sellers at Waterstones, and I soon discovered why. Rovelli is very good at communicating ‘difficult’ scientific ideas. His subject matter is physics, that most basic of sciences, and this book gives a good overview of the implications of the current thinking of some physicists.

His story begins with the Ancient Greeks and particularly Democritus and his atomism, the essential granular quality of the universe. Although no work of Democritus survived, the essential ideas were rediscovered at the time of the Renaissance, ultimately inspiring Isaac Newton and his model of the existence of particles in space and time, and of forces between them, action at a distance, what became known as gravity.

The next great step was the ‘discovery’ of electric and magnetic fields between particles by Faraday and Maxwell.

With his special theory of relativity in 1905, Einstein brought together space and time, into space-time, and in 1915 his general relativity further integrated spacetime with fields, as covariant fields.

The amazing story of quantum mechanics, developed by many collaborators including Planck, Heisenberg and Dirac, then simplified the physicists’ model of the universe to two things: Spacetime and Quantum fields. And then the ultimate aim of Quantum gravity is to reduce this to just one building block of the whole universe – Covariant quantum fields.

Yes it’s a great story and well worth reading for insight into where the physicists are at, but without the incomprehensible (to most people) maths that lies behind it.

But always remember this. It’s only a model; it’s not reality. And the model doesn’t really understand the interiority of things, life, consciousness, the mystery of existence… i.e. most of what’s important.

 

Mapping the Universe

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I love Mekhi and Joe’s posts on physics on the blog Rationalising the Universe, which brings me more up to date on the enthusiasms for mathematics, physics and cosmology of my youth. But I had to take issue with the conclusion of the recent interesting post on What is a Field, which ended with the following statement:

There we have it, space is no longer a separate entity, space is a field and the universe now consists of fields and particles alone.

That’s exciting. Newton set the ball rolling on mathematical models of the universe, and the current mathematical model of the universe has now simplified to just fields and particles.

But look at the statement again. It says “the universe now consists of…”. Well actually it doesn’t, and I suggest that we still have little idea of ‘what the universe consists of’. But we do have a great model that explains what we see and can measure in a reasonably consistent manner.

The point is

“The map is not the territory”

Alfred Korzybski, 1931

Featured image from the blog Rationalising the Universe

What Newton really thought

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Alert readers of this blog may have realised that I am reading Henri Bortoft’s book Taking Appearance Seriously: The Dynamic Way of Seeing in Goethe and European Thought. Bortoft throws interesting insight into the role of Isaac Newton in creating the modern scientific world, confirming Edi Bilimoria’s article mentioned in an earlier post.

Isaac Newton basically invented modern mathematical physics in his masterwork, Principia Mathematica (1687). To the theory of atomism and mechanical philosophy he added the notion of forces which act between bodies that are not in contact.

Bortoft suggests that from the eighteenth century onwards,  gravity began to be thought of as a ‘property of matter’, as if it were an attractive force inherent to matter. This is not what Newton thought. He did not believe in attraction as a real, physical, force.

For example, in a letter Newton said:

Pray do not ascribe that notion to me, for the cause of gravity is what I do not pretend to know and therefore would take more time to consider of it… Gravity must be caused by an agent acting constantly according to certain laws, but whether this agent be material or immaterial I have left to the consideration of my readers.

So Newton’s major discovery was to the effect that we could create mathematical models of the real world, what we now call ‘physics’. Subsequent founders of modern science were dedicated to the mathematical approach to nature, but ultimately the ascendancy of the mathematical was accompanied by the downgrading of the sensory and increasingly seeing the world as a mathematical abstraction. To many scientists the world became de-spiritualised and dead.

This was not Newton’s intention, although his name is often invoked as the originator of such a viewpoint.

Isaac Newton, Mystic

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Isaac Newton is generally seen as a key founder of modern science, via his major work Principia Mathematica and theory of gravity – which led on to the theory of the ‘clockwork universe’ and much of the modern materialist/atheistic world view.

Newton was indeed a great polymath. What is less known is that his work was inspired by his studies of religion and mysticism, which were at least as important to him as the natural sciences. The idea of a clockwork universe would have been anathema to Newton, as would the idea of atheism.

This is all explained in Edi Bilimoria’s well-researched article ‘Newton’ in the current issue of Paradigm Explorer, magazine of the Scientific and Medical Network.

Interestingly, Newton’s gravity and its attraction were ‘a purely mathematical concept involving no consideration of real and primary physical or mechanical causes’ – which is why his book is about ‘mathematics’ and not ‘mechanics’.

As Edi explains, Newton’s religious ideas were well developed and have little in common with the Christianity of the time, being more related to the view that God is everywhere immanent and transcendent. Quoting Newton himself:

[God] endures forever , and is everywhere present; and by existing always and everywhere, he constitutes duration and space. In him are all things contained and moved…

Of course, many modern scientists have come to a similar viewpoint on the importance of religion. For example, that more modern polymath Albert Einstein:

“Science without religion is lame; religion without science is blind.”

Edi’s article is well worth reading.

Gödel’s theorem

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The excellent recent post Life is so incomplete, by the ‘rationalising the universe’ team, rekindled my interest in Gödel’s Theorem, briefly mentioned in my earlier post on Science, Religion and the New Age:

“Gödel’s theorem tells us that in any model that we construct there will be things that we can neither prove nor disprove – they are outside the scope of the model. A model of everything is impossible.”

Kurt Gödel, perhaps the leading mathematician of this age, published his two incompleteness theorems of mathematical logic in 1931, and these are outlined in the Life is so incomplete.

The point cannot be over-stressed. Objective reductionist science essentially creates mathematical models of the real world. These models can be seductively beautiful and accurate in their predictions of the real world. Yet mathematics itself throws up this wobbly that there are things that any model cannot tell us about, and the model itself may not be provably consistent.

A model of everything is an illusion, a chimera. It is not possible.

And that is before we get to any discussion of the inner and outer of things, subject and object – only the latter of which is really the realm of science.

Featured image by Kedumuc10, via Wikimedia Commons