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!

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