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Gödel Proves Whatever You Like

A theorem about arithmetic, cited as an authority on Lenin’s tomb, God, poetry and the end of physics.

The theorem has small print

The other Gödel page explains the machinery. For this one you need only the statement, and the conditions attached to it.

Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F. — Stanford Encyclopedia of Philosophy, “Gödel’s Incompleteness Theorems”

There are three conditions. F must be a formal system: a precisely specified language, axioms a program could list, and rules of inference a program could apply. F must be consistent. And F must be able to do a certain amount of arithmetic. The statements F fails to settle are statements of arithmetic.

So the theorem applies to Peano arithmetic and to set theory. It does not apply to France, the Bible, a bicycle, your marriage or the universe. None of those is a formal system, and nobody was expecting them to settle questions in number theory.

Exhibit A: Lenin’s tomb

Régis Debray, French philosopher and former comrade of Che Guevara, in Le Scribe (1980):

Ever since Gödel showed that there does not exist a proof of the consistency of Peano’s arithmetic that is formalizable within this theory (1931), political scientists had the means for understanding why it was necessary to mummify Lenin and display him to the ‘accidental’ comrades in a mausoleum, at the Centre of the National Community. — Debray (1980), as quoted by Sokal and Bricmont

He promoted this to a law of politics:

The secret takes the form of a logical law, an extension of Gödel’s theorem: there can be no organized system without closure and no system can be closed by elements internal to that system alone. — Debray, Critique of Political Reason (1983), as quoted by Sokal and Bricmont

From which he deduced that government of the people by the people is “a logically contradictory operation”. Michel Serres, of the Académie française, was impressed:

In applying Gödel’s theorem to questions of the closed and the open, as they relate to sociology then, with one gesture Regis Debray recapitulates and concludes the history and the work of the previous 200 years. — Serres (1995), as quoted by Sokal and Bricmont

Serres calls the result the “Gödel–Debray principle”. One of the two named parties proved something.

Sokal and Bricmont, two physicists, replied in Intellectual Impostures:

There is quite simply no logical relationship between this theorem and questions of sociology. — Sokal and Bricmont (1998)

Gödel’s theorem will still be true in ten thousand years or a million years, but no one can say what human society will look like so far in the future. — Sokal and Bricmont (1998)

Debray’s response

He partly came round. In a 1996 lecture, as Sokal and Bricmont report it, he admitted that “Gödelitis is a widespread disease” and that “extrapolating a scientific result, and generalizing it outside of its specific field of relevance, can lead … to gross errors”. His own use, he said, was “simply metaphorical or isomorphic”.

A metaphor is fine. A metaphor does not get to be called “a logical law”, and you cannot deduce the necessity of embalming from it.

Exhibit B: God, by way of a bicycle

Perry Marshall is a marketing consultant whose article calls incompleteness “the #1 mathematical discovery of the 20th century”. He tells us what the theorem says, in quotation marks:

Gödel’s Incompleteness Theorem says: “Anything you can draw a circle around cannot explain itself without referring to something outside the circle – something you have to assume but cannot prove.” — Marshall, “Gödel’s Incompleteness Theorem”

It says nothing of the kind. The quoted sentence is Marshall’s. He goes on:

You can draw a circle around a bicycle but the existence of that bicycle relies on a factory that is outside that circle. The bicycle cannot explain itself. — Marshall

If we draw a circle around all matter, energy, space and time and apply Gödel’s theorem, then we know what is outside that circle is not matter, is not energy, is not space and is not time. It’s immaterial. … Therefore whatever is outside the largest circle is a conscious being. — Marshall

A bicycle is not a consistent, effectively axiomatised theory of arithmetic. Neither is the universe. “Apply Gödel’s theorem” to a bicycle and nothing happens, for the same reason nothing happens when you apply the quadratic formula to a sandwich.

The logician Panu Raatikainen, more politely:

Perhaps a person who is inclined to see evidence for God’s existence everywhere can also see it in Gödel’s theorems, but in themselves, these results have no such implications. — Raatikainen (2005)

Exhibit C: Chopra gets it backwards

Deepak Chopra, explaining in 2016 why artificial intelligence will never rival the mind:

His first theorem says that incompleteness is the fate of any logical system; there will never be a system that explains everything. — Chopra, Huffington Post (2016)

We are trapped by the implications of Gödel’s second theorem, which holds that a logical system cannot reveal its inconsistencies. Blindness is built in. I know that I am humanizing mathematics, which marks me as a total outsider, but systems engulf us at every turn — systems of politics, religion, morality, gender, economics and, above all, materialism. — Chopra (2016)

The first theorem does not cover “any logical system”. Plenty of logical systems are complete, as we will see. And the second theorem says the opposite of what Chopra says. A consistent system cannot prove its own consistency. An inconsistent system reveals its inconsistencies without any trouble, because it proves everything.

He is right about one thing. It does mark him as a total outsider.

Exhibit D: the end of physics

The physicists are a harder case, because they are clever and they hedge. Stanley Jaki, a priest and physicist, did not hedge:

Then it follows that there can be no final physical theory which would be necessarily true at least in its mathematical part. This should seem elementary and obvious. One may feel that one merely has to put two and two together. — Jaki, “A Late Awakening to Gödel in Physics”

He recalls putting the point to Murray Gell-Mann on a panel at the 1976 Nobel Conference:

He shouted back rather angrily. “Why not?” “Because of Gödel’s theorem,” I replied. “Whose theorem?” he asked again. I said again “Gödel’s theorem.” Then I had to spell out Gödel’s name which Gell-Mann apparently had not heard before. — Jaki

I would like to suggest another reading of “Whose theorem?”

Stephen Hawking gave a 2002 lecture titled “Gödel and the End of Physics”. He says “reminiscent” and “analogy” throughout. The one place he slips is where he says a physical theory is self-referencing, like Gödel’s sentence, so that “one might therefore expect it to be either inconsistent, or incomplete.”

Freeman Dyson went the whole way, in the New York Review of Books in 2004:

Now I claim that because of Gödel’s theorem, physics is inexhaustible too. The laws of physics are a finite set of rules, and include the rules for doing mathematics, so that Gödel’s theorem applies to them. — Dyson (2004)

The logician Solomon Feferman wrote in. If the laws of physics are formalised in a system S, then:

… if S is consistent, then there are propositions of higher arithmetic which are undecidable by S. But this tells us nothing about the specifically physical laws encapsulated in S, which could conceivably be complete as such. — Feferman, letter to the NYRB (2004)

The undecided statements are arithmetic. The physics could be finished on a Tuesday and Gödel would have no objection.

Dyson’s response

First come the responses from nonexpert readers who write to tell me how much they like the review. Second come the responses from expert readers who write to correct my mistakes. I am grateful for both categories of response, but I learn much more from the second category. … I am grateful to Solomon Feferman for explaining why we do not need Gödel’s theorem to convince us that science is inexhaustible. — Dyson (2004)

Make a mistake in mathematics and you start again. That is what it looks like. It took him two months.

Exhibit E: poetry

Paul Virilio, on where modern science is heading:

… the promised surrealist peaks of scientific theory, peaks that culminate in Gödel’s theorem: the existential proof, a method that mathematically proves the existence of an object without producing that object. — Virilio (1991), as quoted by Sokal and Bricmont

That is not Gödel’s theorem, or any theorem. Julia Kristeva, meanwhile, applied the axiom of choice to poetic language, found the result “close to Gödel’s observation”, and added that the poet Lautréamont “was one of the first consciously to practice this theorem.” Sokal and Bricmont:

It is rather improbable that Lautréamont (1846–1870) could have ‘consciously practiced’ a theorem of Gödel–Bernays set theory (developed between 1937 and 1940) or even of set theory tout court (developed after 1870 by Cantor and others). — Sokal and Bricmont (1998)

They add that Gödel “proved exactly the opposite of what Kristeva claims”.

The Bible, the Constitution and Ayn Rand

The late Torkel Franzén wrote a whole book on this, having spent years correcting people on the internet. His collection includes claims that Gödel shows the Bible, the US Constitution and Objectivism must each be incomplete or inconsistent. His reply:

Nobody expects the Bible, the laws of the land, or the philosophy of Ayn Rand to settle every question in arithmetic. — Franzén (2006)

Of course the Constitution is incomplete in the everyday sense. As he puts it in the book, “it does not tell us whether or not wearing a polka-dot suit is allowed in Congress.” Nobody needed a theorem for that.

Four facts that would have prevented all of this

First, these things are not formal systems. Franzén:

There is no formally specified language, and there are no formal rules of inference in the logical sense associated with quantum mechanics, the Bible, a system of laws, and so on. — Franzén (2006)

Second, incompleteness is not what happens to anything “sufficiently complex”. Substantial mathematical theories are complete:

The theory of only addition of natural numbers but without multiplication (often called “Presburger arithmetic”), for example, is complete (and decidable) … the so-called theory of real closed fields (RCF), is both complete and decidable, as was shown by Tarski (1948); he also demonstrated that the first-order theory of Euclidean geometry is complete and decidable. — Stanford Encyclopedia of Philosophy

Euclidean geometry is complete. So much for “no system can explain everything”.

Third, “unprovable in F” does not mean “unprovable”. The encyclopedia again: “For any statement A unprovable in a particular formal system F, there are, trivially, other formal systems in which A is provable (take A as an axiom).” That is a syntactic move, not an epistemic one: adding A makes it provable rather than known, and the larger theory may be unsound.

The limitation that does survive is uniform and unmysterious. No single consistent, effectively axiomatised theory strong enough for arithmetic settles every arithmetic sentence. That is a long way from a truth lying beyond the reach of reason.

Fourth, Gödel drew the opposite moral. He thought his results showed the reach of mathematical insight. Of the sceptical reading he said that one gives up the belief that reason can answer the clear questions it poses

… not because the mathematical results achieved compel one to do so but because that is the only possible way, despite these results, to remain in agreement with the prevailing philosophy. — Gödel (1961)

In their favour

Analogy is allowed. A theorem can inspire a thought about politics or physics, and Hawking was careful to call his an analogy. The offence is the word “therefore”.

And three of the defendants improved. Debray diagnosed Gödelitis in himself. Kristeva, Sokal and Bricmont note, long ago abandoned this approach. Dyson retracted in print within two months and thanked the man who corrected him.

The others are still drawing circles around bicycles.

References

• Sokal, A., and Bricmont, J. (1998). Intellectual Impostures (US title Fashionable Nonsense). Profile Books. Chapters 3, 10 and 11. The Debray, Serres, Kristeva and Virilio passages are quoted from here, in their translations.
• Franzén, T. (2005). Gödel’s Theorem: An Incomplete Guide to Its Use and Abuse. A K Peters. Franzén, T. (2006). “The Popular Impact of Gödel’s Incompleteness Theorem.” Notices of the AMS 53(4), 440–443.
• Raatikainen, P. “Gödel’s Incompleteness Theorems.” Stanford Encyclopedia of Philosophy. Raatikainen, P. (2005). “On the Philosophical Relevance of Gödel’s Incompleteness Theorems.” Revue Internationale de Philosophie 59.
• Dyson, F. (2004). “The World on a String.” New York Review of Books, 13 May. Feferman’s letter and Dyson’s reply: 15 July 2004.
• Hawking, S. (2002). “Gödel and the End of Physics.” Dirac Centennial lecture, Cambridge.
• Jaki, S. L. (c. 2004). “A Late Awakening to Gödel in Physics.”
• Marshall, P. “Gödel’s Incompleteness Theorem: The #1 Mathematical Discovery of the 20th Century.” perrymarshall.com.
• Chopra, D. (2016). “AI Will Never Rival the Deep Complexity of the Human Mind.” Huffington Post.
• Gödel, K. (1961). “The Modern Development of the Foundations of Mathematics in the Light of Philosophy.” In Collected Works, vol. III, Oxford University Press, 1995.