Gödel, Maths and Physics

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!

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Fibonacci Grape Pips?

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

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

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

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

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

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