[FOM] Is Godel's Theorem surprising?

Mark van Atten Mark.vanAtten at univ-paris1.fr
Sat Dec 9 05:01:11 EST 2006

(The following is an extract from the chapter 'Gödel's logic', by 
Juliette Kennedy and me, for Elsevier's upcoming Handbook fo the History 
of Logic)

Brouwer must have realized already around 1907 that one can diagonalize 
out of formal systems. In his dissertation of that year he noted that 
the totality of all possible mathematical constructions is `denumerably 
unfinished'; by this he meant that `we can never construct in a 
well-defined way more than a denumerable subset of it, but when we have 
constructed such a subset, we can immediately deduce from it, following 
some previously defined mathematical process, new elements which are 
counted to the original set'. In one of the notebooks leading up to his 
dissertation, Brouwer stated that `The totality of mathematical theorems 
is, among other things, also a set which is denumerable but never finished'.

In 1928, Brouwer gave two lectures in Vienna: on March 10, on general
philosophy and intuitionistic foundations of mathematics; and on March 
14, on the intuitionistic theory of the continuum. According to Hao 
Wang, `it appears certain that Gödel must have heard the two lectures'; 
indeed, Gödel wrote to Menger on April 20, 1972 that `I only saw 
[Wittgenstein] once in my life when he attended a lecture in Vienna. I 
think it was Brouwer's'. An entry in Carnap's diary for December 12, 
1929, states that Gödel talked to him that day `about the 
inexhaustibility of mathematics (see separate sheet) He was stimulated 
to this idea by Brouwer's Vienna lecture. Mathematics is not completely 
formalizable. He appears to be right'. On the `seperate sheet', Carnap 
wrote down what Gödel had told him:

`We admit as legitimate mathematics certain reflections on the grammar 
of a language that concerns the empirical. If one seeks to formalize 
such a mathematics, then with each formalization there are problems, 
which one can understand and express in ordinary language, but cannot 
express in the given formalized language. It follows (Brouwer) that 
mathematics is inexhaustible: one must always again draw afresh from the 
``fountain of intuition''. There is, therefore, no characteristica 
universalis for the whole mathematics, and no decision procedure for the 
whole mathematics. In each and every closed language there are only 
countably many expressions. The continuum appears only in ``the whole of 
mathematics'' ... If we have only one language, and can only make 
``elucidations'' about it, then these elucidations are inexhaustible, 
they always require some new intuition again.'

This record contains in particular elements from the second of Brouwer's
lectures, in which one finds the argument that Gödel refers to: on the 
one hand, the full continuum is given in a priori intuition, while on 
the other hand, it cannot be exhausted by a closed language with 
countably many expressions.

This explains why Brouwer could say (e.g., to Freudenthal and to Wang) 
that he was not surprised by Gödel's incompleteness theorems.

But of course, in the theorems that Gödel, after having been inspired by
Brouwer's lectures, eventually arrived at, he went considerably beyond 
Brouwer. As Gödel stressed (without reference to Brouwer's lectures) in 
his letter to Zermelo of October 12, 1931: "I would still like to remark 
that I see the essential point of my result not in that one can somehow 
go outside any formal system (that follows already according to the 
diagonal procedure), but that for every formal system of metamathematics 
there are statements which are expressible within the system but which 
may not be decided from the axioms of that system, and that those 
statements are even of a relatively simple kind, namely, belonging to 
the theory of the positive whole numbers."

Mark van Atten.

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