What does Gödel's theorem mean ?

Gödel's theorem is a result in mathematical logic, which is often stated as showing that « there are true things that cannot proved ». It is sometimes used to comment on the limits of science, or the superiority of human intuition. Here I want to clarify what this theorem means and what the epistemological implications are.

First, this phrasing is rather misleading. It makes the result sound almost mystical. If you phrase the result differently, by avoiding the potentially confusing reference to truth, the result is not that mystical anymore. Here is how I would phrase it : you can always add an independent axiom to a finite system of axioms. This is not an obvious mathematical result, but I wouldn't think it defies intuition.

Why is this equivalent to the first phrasing ? If the additional axiom is independent of the set of axioms, then it cannot be proved from them (by definition). Yet as a logical proposition it has to be either true or not true. So it is true, or its negation is true, but it cannot be proved. What is misleading in the first phrasing is that the statement « there are true things » is contextual. I can start from a set of axioms and add one, and that new one will be true (since it's an axiom). Instead I could add its negation, and then that one will be true. That the proposition is true is not a universal truth, as it would seem with the phrasing « there are true things ». It is true in a particular mathematical world, and you can consider another one where it is not true. Famous examples are Euclidean and non-Euclidean geometries, which are mutually inconsistent sets of axioms.

So, what Gödel's theorem says is simply that no finite system of axioms is complete, in the sense that you can always add one without making the system inconsistent.

What are the epistemological implications ? It does not mean that there are things that science cannot prove. Laws of physics are not proved by deduction anyway. They are hypothesized and empirically tested, and all laws are provisory. Nevertheless, it does raise some deep philosophical questions, which have to do with reductionism. I am generally critical of reductionism, but more specifically of methodological reductionism, the idea that a system can be understood by understanding the elements that compose it. For example : understand neurons and you will understand the brain. I think this view is wrong, because it is the relations between neurons, at the scale of the organism, which make a brain. The right approach is systemic rather than reductionist. Many scientists frown at criticisms of reductionism, but this is only because they confuse methodological and ontological reductionism. Ontological reductionism means that reality can be reduced to a small number of types of things (eg atoms) and laws, and everything can be understood in these terms. For example, the mind can in principle be understood in terms of interactions of atoms that constitute the brain. Most scientists seem to believe in ontological reductionism.

Let us go back now to Gödel's theorem. An interesting remark made by theoretical biologist Robert Rosen is that Gödel's theorem makes ontological reductionism implausible to him. Why ? The theorem says that, whatever system of axioms you choose, it will always be possible to add one which is independent. Let us say we have agreed on a small set of fundamental physical laws, with strong empirical support. To establish each law, we postulate it and test it empirically. At a macroscopic level, scientists postulate and test all sorts of laws. How can we claim that any macroscopic law necessarily derives from the small set of fundamental laws ? Gödel's theorem says that there are laws that you can express but that are independent of the fundamental laws. This means that there are laws that can only be established empirically, not formally, in fact just like the set of fundamental laws. Of course it could be the case that most of what matters to us is captured by a small of set of laws. But maybe not.

2 réflexions au sujet de « What does Gödel's theorem mean ? »

  1. The Godel sentence is "true" in a somewhat more meaningful way. Godel's theorem only applies to systems that are rich enough to represent arithmetic, and the way the construction works is to find an equivalence between statements in the system, and statements about arithmetic. In particular, the construction gives a statement that more or less means "this statement cannot be proved" which is equivalent to a statement about arithmetic that says there is a certain polynomial of multiple variables that has no zeros. The statement must be "true" because if it were false, you could find zeros of the polynomial (and also intuitively, if it were false it would say that it could be proved which would be a contradiction). It is possible to find a model of the system in which the statement is false, but it would have to be one in which arithmetic behaves weirdly.

    I don't really think any of this bears on reductionism because the Godel sentences are highly contrived and practically irrelevant. There have been shown to be interesting statements that are independent of standard axiomatisations of mathematics. Cohen showed the independence of the axiom of choice and the continuum hypothesis from ZF set theory. There's even a whole branch of mathematics dedicated to investigating how to do this sort of thing (the "reverse mathematics" of Harvey Friedman). However, that's all highly specific to mathematical statements. The "Godel sentence" of an axiomatisation of a physical system like the brain would be very likely to be highly contrived and uninteresting, and there's no guarantee that an interesting and independent statement could be found (and in my opinion, it's unlikely).

    I think the real objection to reductionism is that pratically speaking it's unlikely to help.

  2. The statement that one constructs to make the general proof of the theorem is highly contrived, sure. But that's to be expected because there is no general recipe to produce new "interesting" axioms. That doesn't mean there aren't any - especially as "interesting" seems rather subjective. Anyway, that's more or less what I said in my concluding sentence.

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