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Godel's Incompleteness Theorems seem to get bandied about all the time and people claim all sorts of philosophical conclusions from them. Personally, I think that the meaning of the theorems is unclear, though many would disagree. However, I do understand what they actually say and I've noticed that discussions about Godel tend to be riddled with misconceptions. So I'd like to spark off a debate by describing the results in what I hope is fairly accessible language. I apologise if the tone is patronising. I should also make clear that I am not an expert on this.
I think it is fair to say that up until the nineteenth century, mathematicians believed mathematics to possess an independent existence. This is
realism (or Platonism, though that is misleading) and many present day mathematicians hold this view, at least superficially.
So when Euclid described his
geometry, many believed it to be the one true geometry. It seemed "obvious", after all. But the arrival of
non-Euclidean geometry, which was much contested for some time, seemed to undermine this. A modern realist would say that what mathematicians hadn't realised is that, as well as Euclidean geometry, there also exists non-Euclidean geometry.
A formalist would say something else. She might say that mathematics only exists insofar as the writing one uses to describe it. Mathematics, by this view, is a game where one specifies the rules and plays. So by a certain definition of geometry (specified by the first four Euclidean postulates) there are three geometries. None of them is "true", they are simply consequences of the geometrical game one chooses to play.
One of the first major proponents of formalism was David Hilbert (probably the greatest mathematician of his day and the most influential figure in 20th century maths). Hilbert, along with others, most notably Russell and Whitehead, wanted to place maths on a sure formalist footing. That is, for each area of maths, write down the rules of the particular game and show that everything follows from those rules.
The way to do this was to use formal logic. This is a very cumbersome type of language, that no mathematician could really work or think in, but which contains sufficient expressive power to avoid the Euclidean trap. Moreover, it is extremely precise so one can prove things about the manipulation of this language - metamathematics. Briefly, a logical system contains axioms and rules of inference. For a formalist, the axioms are arbitrary assertions in logic that one decides are "true" for this particular game. The rules of inference are "moves" that allow one to pass from one "true" statement to another. As it stands, a logical system has no intrinsic meaning. However, in cases of interest one chooses the axioms to capture the essence of some area of maths, say arithmetic. So one usually has an interpretation in mind.
Hilbert wanted to cut the uncertainty out of mathematics and put it on a sound footing, as had been done for the previously suspect calculus of
Newton and
Leibniz. I think that Hilbert envisaged acquiring an iron clad certainty in his anti realism. Namely, he wanted to be able to take an area of maths and pin it down with some axioms. This would involve completely identifying the maths with the logical system so that anything that might be said about the maths could be obtained by playing the axiom game.
Here I should introduce some terminology about logical systems. A proof is a sequence of moves in the axiom game. The end of the sequence will be a logical statement that has been proved. You should think of this as a formalist notion of "truth", which of course needs to be defined as nothing is given.
A logical system is said to be complete if every statement in the system either has a proof or its converse has a proof. Intuitively, a system is complete if every statement is either formally true or false. This is exactly what you might expect if you wanted to capture everything about arithmetic, for instance.
A logical system is said to be consistent if you can't prove any logical contradictions (things like, "This number is both even and not even"). Contradictions are bad if you want to model maths. In fact, it is much worse than you think since in an inconsistent system everything has a proof. In other words, if you have inconsistency then everything is both formally true and formally false.
Now along comes Kurt
Godel (more properly, Goedel), who was himself a realist, to demonstrate in a fairly simple manner that Hilbert's highest hopes for formalism are in vain. (Formalism isn't completely discredited by Godel, however, just limited.) Using self referentiality and a version of the Liar paradox, Godel proved the following two Incompleteness Theorems:
Consider any consistent logical system, T, which is able, under some interpretation, to describe arithmetic.
1. T is always incomplete.
2. You can't determine if T is consistent by playing the axiom game in T.
So 1 says that there are always things that are neither formally true nor formally false. Most people interpret this to mean that you can never use logical systems to capture all of maths. There are more radical interpretations, however.
The second says that if you play the axiom game in T, you can never be sure that the whole thing won't collapse solely by playing this game. Some of you will no doubt realise that you could think of T from a different, larger perspective and there is a lot of maths in the vein. However, for the uncompromising formalist, to think of T in a different way means placing it "inside" another axiom game which you equally cannot be sure won't collapse. And if the large system collapses it brings the smaller one down with it. Hence the ultra strict formalist can never be sure that all of maths doesn't collapse.
Note that if all of maths "collapses" then 2+2=5 and also 2+2=10 and 2+2=129, etc, etc. To the realist this is absurd. The realist would interpret Godel's Theorems as saying that mathematical truth is not the same as axiom games and that sometimes, you can only see things are true by thinking "outside the box". This all makes sense for the realist since truth is for him a property of real mathematical objects. |
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