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In a lecture on philosophy of science I recently listened to, it was stated that Quine was the one who decisively refuted the logical positivist program. I've also read that Quine and Popper were significant challengers, but that it was Kuhn who effectively ended the logical positivist movement.

It seems to me that before any of those challengers, Gödel's incompleteness theorem would have dealt a major blow to logical positivism. If theories based on logic are inevitably incomplete or inconsistent, wouldn't it follow that stronger theories based on logic in conjunction with empirical evidence are inevitably incomplete or inconsistent as well?

On the other hand Gödel himself was a member of the Vienna Circle, and was surely sympathetic to the logical positivist's views. They couldn't have just ignored him.

My questions:

  1. Is it fair to say that Gödel's result seriously challenges the logical positivist program?
  2. How did the logical positivists respond to Gödel's incompleteness theorem?
Alexander S King
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    It's not logic that is incomplete but arithmetic. First order logic is complete. – Quentin Ruyant May 17 '15 at 13:58
  • Editing to take into account your comments @quen_tin and WillO . – Alexander S King May 17 '15 at 14:02
  • Interesting.... –  May 17 '15 at 16:02
  • As far as I know and can infer from its Wikipedia article, positivism is a sociological approach, unrelated to mathematics such as the the incompleteness theorems (yes, plural). Presumably, then, you mean something else when you refer to "logical positivists". Exactly what? Oh, wait, no matter, I found it. – Cheers and hth. - Alf May 17 '15 at 22:39
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    There has been some discussion regarding the domain in which Gödel's incompleteness theorems apply. They apply within any formal system as or more powerful than the Peano axioms. They do not apply within Presburger arithmetic. – Tuukka Virtaperko May 17 '15 at 17:04
  • Gödel's 1931 incompleteness theorem was proved in the context of Russel and Whitehead's Principia Mathematica, which is not exactly all "theories based on logic". It took time for generality and implications to be understood and to sink in. Many didn't until 1950s. And the positivists' core analytic-synthetic distinction, between "factual" and "metaphysical", doesn't require complete theories or reduction of mathematics to logic. It is that that Quine later dismantled. – Conifold May 18 '15 at 03:07

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There is boundless faith in second-order logic, because people do not realize that testable second-order theories are almost always really sorted first-order theories in disguise. So science that is not 'spooky' is basically first-order.

People in the 1980's were still trying to prove that some second-order theory of the reals was consistent and complete. (I was a (largely uninterested) student of Gaisi Takeuti (of second-order cut elimination), and Lou van den Dries (of Real Closed Field theory), both of whose work even at that time could be seen as advancing different paradigms trying to converge in that direction.)

Once you leverage logic out of the first order, you get infinite computation and tons of infinite regresses trying to relativize Godel. So folks had a reason not to attach to it strongly.

It may be the first post-modern fact, but it also the easiest to dismiss. Reductionists will take the out to wish it away, even though dismissing it burdens one with all sorts of infinities that would otherwise just annoy them. At the same time, most people of the contrasting bent would really like all that stuff that the theorem does not apply to, to be real, maybe even more real than the science. So it kind of loses either way.

  • I need to learn more about 2nd order logic. – Alexander S King May 17 '15 at 16:49
  • Gödel's theorem applies to any order logic, correct? Why were they even trying? – Alexander S King May 17 '15 at 16:59
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    No, it does not. You have to be able to unwind every formula into a countable succession of propositions, each stated with a finite number of symbols in order to perform the diagonalization. If you allow self-references that each must be kept separate for lack of ambiguity, then unwinding the propositions can call for infinitely many symbols that might refer to different sets of things. So you cannot name every symbol with a number in a deterministic way, and the proof cannot proceed. –  May 17 '15 at 17:21
  • That second paragraph is meant to indicate that without the technobabble. If Godel really had totally popped the chance of getting some total theory via another order of logic, high-powered academic logicians at institutions like UIUC would have stopped obsessing over total second-order or nonstandard theories well before the late 1980's. –  May 17 '15 at 17:47
  • ('self-references' is a typo, just set references in general will do. You can't easily conflate the variables like you can when all quantifiers range over everything.) –  May 17 '15 at 21:11
  • I think I can prove outright that Godel must fall in second order logic. I was looking at a Godel breaker and had to be told what I was working with was second order logic as I didn't even know it existed. Incidentally, Turing's incompleteness theorem will fall in at least one second order logic. – Joshua May 17 '15 at 21:22
  • Turing has an incompleteness theorem? or are you just talking about the halting problem? – Alexander S King May 18 '15 at 01:21
  • None of this is relevant because none of it was known in 1930s. It wasn't even clear then that Gödel's theorems apply to all first order theories with arithmetic. – Conifold May 18 '15 at 03:14
  • @Conifold The logical positivist program extends well beyond the Vienna circle and is still with us today. Whether it directly influenced those people, it figures in the ongoing struggle between the assumption that we can clarify logic indefinitely, and the insistence that meaning is more complex than computation. –  May 18 '15 at 14:59
  • @jobermark The responses to this post http://philosophy.stackexchange.com/questions/7239/is-g%C3%B6dels-incompleteness-theorem-still-valid-if-one-uses-a-higher-order-logic?rq=1 seem to contradict your answer w/r to 2nd order logic. Is there something I am missing? – Alexander S King May 18 '15 at 15:35
  • In some sense, second-order logic on the reals is "not even" incomplete. There is no systematic proof theory of the second order logic that does not fall prey to the halting problem, and without one, it remains possible to have consistency and completeness, but lack what is called 'effectiveness'. If you generalize the first-order deduction in a way that rules out recursion in proofs (transfinite induction, e.g.) you maintain Godel. If you don't, you never quite know whether the system is consistent and complete, but it still might be. This is only endless hope, not a reasonable position. –  May 19 '15 at 13:41
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Quoting Wikipedia,

” Carnap envisioned a universal language that could reconstruct mathematics and thereby encode physics.[9] Yet Kurt Gödel's incompleteness theorem showed this impossible except in trivial cases, and Alfred Tarski's undefinability theorem shattered all hopes of reducing mathematics to logic.[9] Thus, a universal language failed to stem from Carnap's 1934 work Logische Syntax der Sprache (Logical Syntax of Language).[9] Still, some logical positivists, including Carl Hempel, continued support of logicism.

where reference [9] is

” Jaako Hintikka, "Logicism", in Andrew D Irvine, ed, Philosophy of Mathematics (Burlington MA: North Holland, 2009), pp 283–84

So, regarding your first question

” Is it fair to say that Gödel's result seriously challenges the logical positivist program?

if it is rephrased as “challenged” then the Wikipedia quote above applies and the answer is a clear yes.

However, if you mean whether there is still a challange going on, literally “challenges”, then that depends on what exactly you mean by logical positivism for the present, and I have no answer.

Presumably, though, a continued challenge based on some evolution of the concept of logical positivism is not an issue, or else it would surely have been mentioned.

Regarding your second question,

” How did the logical positivists respond to Gödel's incompleteness theorem?

according to the Wikipedia quote above it was a mixed response. Some people, including Carl Hempel, “continued support of logicism”. Assuming that they didn't fail to understand the import of Gödel's work, one must conclude that these people did not see Carnap's “universal language” for reducing mathematics to logic, as essential to the logical positivism, whatever they then defined it as.

Cheers and hth. - Alf
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    Logicism, reducing mathematics to logic, for Carnap and others was a means to an end, justifying the distinction between empirical and metaphysical, the analytic synthetic distinction, and verificationism based on it. As it turned out, they could easily give up logicism, while preserving the rest. E.g. in 1936-1937 Carnap replaced "verification" of universal laws with mere "confirmation", coming in probabilistic degrees. What they could not survive was Quine's demolition of the analytic synthetic distinction in Two Dogmas of Empiricism (1951). – Conifold May 18 '15 at 04:22
  • @Conifold that's helpful. Do you want to elaborate on it and present it as an answer? – Alexander S King May 18 '15 at 15:30
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According to this SEP article Carnap responded to Gödel's incompleteness theorem by appealing, in The Logical Syntax of Language, to an infinite hierarchy of languages, and to infinitely long proofs. Gödel's theorem (as to the limits of formal syntax) is also at least part of the reason for Carnap's later return from Syntax to Semantics.

Tarski also shared Quine's misgivings about analyticity when they discussed these issues with Carnap at Harvard . . . Their scepticism found its target in Carnap's ingenious measures in Logical Syntax taken to preserve the thesis that mathematics is analytic from the ravages of Gödel's incompleteness theorems . . . Commonly, Gödel's proof is taken to have undermined the thesis of the analyticity of arithmetic . . . Carnap responded by stating that arithmetic demands an infinite sequence of ever richer languages and by declaring analytic statements to be provable by non-finite reasoning . . . Carnap's move highlights the tension within Logical Syntax between formal and crypto-semantic reasoning. It thus points ahead to his acceptance of semantics in 1935— only one year after the publication of Logical Syntax and contrary to his opposition against it expressed in that book.

So yes, Gödel's incompleteness theorem was a serious problem for the Logical Positivists. It was one of several nails in the coffin that gradually closed upon LP..

Ram Tobolski
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  • "to be provable by non-finite reasoning " , does anybody take that seriously?!? – Alexander S King May 19 '15 at 20:15
  • @Alexander S King Gentzen proved consistency of arithmetic by "non-finite reasoning", it's pretty convincing. http://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof By the way, it was Carnap who introduced Gödel to the ideas that led to the incompleteness theorem http://dash.harvard.edu/bitstream/handle/1/3153305/Goldfarb_OnGodelsWay.pdf?sequence=2 And Gödel's incompleteness was not a problem, let alone "nail in the coffin", for Carnap, his main work Logical Syntax (1934) is based on it, see Michael Friedman's paper. http://www.mcps.umn.edu/philosophy/11_3friedman.pdf – Conifold May 19 '15 at 22:32
  • "No attempt is made to give a finitary consistency or conservativeness proof for classical mathematics. Carnap takes Gödel's results to show that the possibility of such a proof is "at best very doubtful"… most original and fundamental philosophical move: we are to give up the "absolutist " conception of logical truth and analyticity common to Frege and the Tractatus. For Carnap, there is no such thing as the logical framework governing all rational thought. Many such frameworks, many such systems of what Carnap calls L-rules are possible: and all have an equal claim to "correctness"." – Conifold May 19 '15 at 22:43
  • @Conifold Friedman adds "Alas, however, it was not meant to be. For Gödel's results decisively undermine Carnap's program after all" – Ram Tobolski May 19 '15 at 23:17
  • And then he writes:"But why should this circumstance cause any problems for Carnap? After all, he himself is quite clear about the technical situation; yet he nevertheless sees no difficulty whatever for his logicist program." This is Friedman's hindsight assessment, not Carnap's. "What the logicist wishes to maintain is not a reduction or justification of classical mathematics... but simply that classical mathematics is analytic". And "to appreciate the full impact of Gödel's results" he cites Quine's criticisms of analyticity in 1951, which apply regardless of incompleteness. – Conifold May 20 '15 at 01:10
  • @Conifold Friedman further writes: "Here is where Gödel's Theorem strikes a fatal blow . . . In this sense, Gödel's results knock away the last slender reed on which Carnap's logicism (and antipsychologism) rests" – Ram Tobolski May 20 '15 at 09:13
  • @Conifold Friedman's thesis is that the Two Dogmas's arguments applies to notions such as non-combinatorial synonymy, which are not involved in Carnap's Logical Syntax. It was Gödel who refuted Logical Syntax. When Carnap was made to realize that, he moved to Introduction to Semantics. And it was that later work that was refuted by Quine. Two different nails. – Ram Tobolski May 20 '15 at 09:48