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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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.