Gödel was a member of the Vienna Circle, whose philosophical position as a group was Logical Positivism, or Logical Empiricism.

The SEP article on him states that among his philosophical views were realism, rationalism, and Platonism. These all seem to be in radical contradiction with the Logical Positivists anti-realism, empiricism and general disregard for metaphysics.

How is it that Godel could hold these views while being a member of the Vienna Circle?

  • Was he a member of the circle only "socially", while dissenting with the group's (presumably majority) philosophical views?
  • Did he adopt these views only later in his life, after leaving the circle?
  • Are these views actually compatible with the Vienna Circle's, and I am missing something? How could his views ever be reconciled with any empiricist world view, let alone the Logical Positivist view?

Gödel was a young man in search of a place to belong, many young intellectuals were attracted to the Vienna Circle for its pluralism and tolerance. But it wasn't purely social. Gödel was clearly interested in mathematics, in 1925-26 Schlick, the circle's founder, gave lectures on philosophy of mathematics which Gödel attended. But it was Carnap, who really pulled Gödel in and turned his sights towards logic. Goldfarb in On Gödel's Way In describes Gödel's path towards the incompleteness theorem that begins with Carnap's 1928 lecture, and his pseudo-completeness theorem based on a question begging definition of logical consequence.

It is hard to say if Gödel's later philosophy of mathematics was determined by his entry jewel, the incompleteness theorem (it certainly played a key role in his arguments), or if it only confirmed his pre-existent disposition. In any case, in the early years he was clearly more interested in logical and linguistic aspects of logical positivism than in its epistemology. By the way, although Gödel's mature philosophy is often glossed as Platonism, it is closer to Aristotelian realism, where ideal objects exist "beside" material ones as their forms, not in a separate realm. In particular, Gödel was self-admittedly influenced by Husserl's modernization of Aristotelian realism.

By the way, Quine was also closely associated with the circle in 1930s, and boy did he and Gödel break logical positivism.

  • Those two, Goedel and Quine? "Logical positivism is dead. . . . Who has done it? ... I fear that I must admit responsibility." - Popper wrote this in his autobiography... – sand1 Jan 16 '16 at 14:37
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    @sand1 I think it's fair to say that, despite what Popper thought, "Two Dogmas..." and "Truth by Convention" are devastating to the positivist program -- I'm inclined to think more so than any of Popper's work. While we're at it we can toss in Putnam's "What Theories Are Not", which many consider to be the final deathblow, completely demolishing any hope for a legitimate characterization of the positivists so-called "observation language" which played such a large role in positivist philosophy. – Dennis Jan 17 '16 at 1:07

By all accounts, Gödel rarely uttered a word when he attended the logical positivist's meetings in Vienna. So it is not clear if Gödel was originally opposed to logical positivism even if his opposition became absolute as time passed.

The refutation of logical positivism began when Russell pulled the rug out from under Frege with his famous paradox. We still have the polite but lethal letter Russell sent to Frege in 1902 :

Dear Colleague, ... I find myself in complete agreement with you in all essentials... I find in your work discussions and distinctions... one seeks in vain in the work of other logicians. There is just one point in which I have discovered a difficulty. ...

Hilbert's response was to introduce formalism in an attempt to bypass Russell's paradox. Gödel took up Hilbert's monumental programme by attempting to see whether one could prove the consistency and completeness of a formal axiomatic system for mathematical analysis. Of course, he discovered that neither was possible.

What Gödel first discovered was that mathematical truth cannot in principle be confined to a formal system - i.e., truth is not reducible to proof; syntax cannot supplant semantics; intuition cannot be dispensed with in mathematics, indeed, even in arithmetic. This was first metaphorical nail in the logical positivist coffin.

The second nail in the coffin of logical positivism came when Gödel demonstrated that if a given system of axioms for arithmetic were in fact consistent, then it could not be proved consistent by the system itself. In other words, only an inconsistent formal system can prove its own consistency.

The logical positivists did not go down without a fight. By all accounts, Hilbert's first response to Gödel's results was anger followed by denial. He became the first, but by no means the last, to propose an anti-Gödel principle; an ad hoc principle to be appended to formal mathematics simply to block the application of Gödel's theorems. Gödel was apparently genuinely irritated by this. Hilbert's idea was to append a new rule of deductive inference that would allow for the employment of infinitely many premises. Gödel was quick to point out that this would violate the very idea of a formal system. The cure would kill the patient.

Few of the logical positivists, or philosophers in general, appear to have initially fully understood Gödel's results - the exception being von Neumann. Gödel simply stopped responding to Zermelo's frequent letters. Carnap too, the first to hear Gödel's result, appears to have failed to comprehend its implications for some time. Gödel famously rubbished Wittgenstein's criticisms as trivial. And a host of others.

So although it is unclear whether Gödel initially opposed logical positivism, it is clear that his results spelt its demise and that he well understood it.

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