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Do Gödel's results have any impact on the theory of knowledge: realism vs anti-realism, the existence of the noumenal, the existence of synthetic a priori truths, etc... or are they just relevant to math and logic, and finding any epistemological consequences to them is typical "reading too much into Gödel"?

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Gödel's theorems only apply to specific theories. In particular, they must be capable of proving all the provably true statements of arithmetic.

Gödel's theorems have very strong implications if you presume the universe you live in is capable of proving all true statements of arithmetic. This creates neat tail-chasing loops if you believe that the mathematics you have in your head can prove such statements.

However, if you do not start from the presumption of being able to prove all true statements of arithmetic, Gödel's incompleteness theorems do not affect your epistemology. If anything, it states that if you wish to believe the world has several very important and desirable features (such as provability, completeness, etc.), the world cannot be fully defined within a mathematical theory that includes the rules of arithmetic. This means that if you wish to believe the world has those several desirable features, the rules of arithmetic must be thought of as nothing more than an abstract notion which is a useful tool for making sense of the world around you, rather than a fundamental notion which defines that world. Alternatively, you may give up one of those features (provability happens to be popular amongst religions).

EDIT: Conifold's question pointed out that I had not tied this to epistemology. This approach identifies some limits as to what knowledge are possible. It is not possible to know that the universe is constructed in a way which can prove all statements in Peano Arithmetic without knowing, by correlary that the universe is incomplete, inconsistent, unprovable, intractable (recursively-enumerable set of axioms), or illogical (doesn't follow the traditionally recognized definitions of logic, truth, or falsehood).* Knowing that your universe can prove all statements in arithmetic and knowing your universe does not have any of those undesirable traits are logically inconsistent with themselves, thus you do not have to concern yourself with the possibility of knowing thus.

*Disclaimer: There are a few other ways to sidestep the Incompleteness Theorems, but these do a good job of capturing the main argument. Gödel states that any omega-consistent system must be at least one of: incomplete (at least one true statement that cannot be proven, i.e. an axiom), inconsistent (at least one statement can be proven to be both true and false), or unprovable (at least one true statement cannot be proven). Intractable arises from the fact that Gödel's proofs require the assignment of a Gödel number, which cannot be done if the number of axioms in the system is not recursively enumerable. Illogical points out that all of Gödel's proofs presumed some standard assumptions about math. There's also a few other escapes, such as not having a multiplication function that is total, but people usually get lost when trying to describe that one.

  • Shock horror! You mean the world isn't made of mathematics? Oh no! That's worth a brownie point. – John Duffield Dec 2 '15 at 20:05
  • @JohnDuffield It at least puts meaningful constraints on what "made of mathematics" could actually mean ;-) – Cort Ammon Dec 2 '15 at 20:06
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    How can universe prove statements of arithmetic, I mean what does that mean? – Conifold Dec 3 '15 at 0:53
  • @Conifold A universe that contains a set of statements which prove all true statements in arithmetic. For example, if you believe that there exists a book which contains all truths, including the statement that all statements in the book are true, then the world you believe to live in is powerful enough to generate the set theory construction for Peano axioms using that book. Since all statements in the book are true (the assumption made when defining this particular book), and the statements can be used to define Peano arithmetic, that universe proves all of arithmetic. – Cort Ammon Dec 3 '15 at 1:04
  • But a book needs a reader, statements come from a reading, otherwise they are scribbles on paper. Does universe come with a resident reader, or does it mean something like powered by a Turing machine with infinite tape? First I thought you meant something like "contains a structure isomorphic to the standard model of arithmetic", but thenit can also contain non-standard models, in which case we may have reasons for adopting both Godel sentences and their negations, so no "all truths of arithmetic". Certainly if it contains a standard model of set theory it will also contain non-standard ones. – Conifold Dec 3 '15 at 1:31
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Do Godel's results have any impact on the theory of knowledge: realism vs anti-realism, the existence of the noumenal, the existence of synthetic apriori truths, etc... or are they just relevant to math and logic, and finding any epistemological consequences to them is typical "reading too much into Godel"?

The latter. Gödel's incompleteness theorems are to do with mathematical logic. Knowledge involves evidence and experiment too. As for noumena, that's an issue in its own right. Can you show me some? The synthetic apriori truths are an issue too. They're rather like "the laws of physics". They don't have any ontological reality. It's like the Terry Pratchett quote: "Take the universe and grind it down to the finest powder and seive it through the finest seive, and then show me one atom of justice, one molecule of mercy". What exists is space and waves and things. They do what they do because of the way they are, not because of truth or law.

Let's take a look at the theorems. I think the Wiki articles are pretty good:

"The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (i.e., any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic)".

That's just to do with the natural numbers, which are merely a concept. You cannot point up to the clear night sky and say "look, there's a seven". It doesn't have a huge amount to do with the real world of science and observation and experiment. The other one is more to do with science, but it still deals with theories and arithmetic truths, which are abstract things:

"For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent."

You cannot point up to the clear night sky and say "look, there's a theory" either. A theory is just a concept too. It describes how things are and it makes predictions, which we test with experiment. All Gödel is really saying is that a theory can't be introspective, it can't prove itself. So whilst this stuff is relevant to maths and logic and theory, you shouldn't read to much into it. But don't think Gödel is a waste of time. See A World Without Time: The Forgotten Legacy of Godel and Einstein. It's by philosopher Palle Yourgrau, and IMHO it's of crucial importance to understanding space and waves and things. To understanding the world and the reality that's out there. I can hold my hands up and show you a space, I can waggle my hands and show you motion. But can you show me time?

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This isn't probably what you're looking for, being continental and left-field; however Lacan uses Godel in his re-reading of psychoanalytical theory; he uses his theorem analogically ie by analogy or metaphor (which Iris Murdoch, takes to be an important form of thinking) ie as in 'there is a lack in the other, the other is inconsistent and incomplete'.

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The consequences of Godel's theorems have been somewhat exaggerated. In philosophical terms one could explain them informally by saying that with restricted means only modest goals are attainable. Gentzen's consistency proof from 1936 shows Peano's arithmentic to be indeed consistent. But it uses something unusual, transfinite induction, which looks less certain that what it purposes to ascertain. When Godel announced his first result nobody was surprised: people knew that something like this could turn up, but expected that it would be patched somehow. Hilbert and Bernays published a second edition of their Grundlagen just by noting a difficulty which they expected to be soon overcome. Only later historians began to describe it as a major blow, even if in fact it is mostly a nagging problem.

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    "When Godel announced his first result nobody was surprised ..." Really ???? See Paolo Mancosu, Between Vienna and Berlin : The Immediate Reception of Godel's Incompletenss Theorems (1998) : 1/2 – Mauro ALLEGRANZA Dec 3 '15 at 7:55
  • ... "`What is going on with Mr Godel? I hear all kinds of exciting things but I cannot make outwhat this is all about’ ." So wrote Heinrich Scholz to Rudolf Carnap on April 16, 1931. The rumour of Godel’s groundbreaking discoveries was spreading quickly and Godel’ s work was immediately perceived as having tremendous importance for the debate on the foundations of mathematics." 2/2 – Mauro ALLEGRANZA Dec 3 '15 at 7:55
  • Godel's results were not unexpected and it took 10 years to convince the community that they are valid. Veblen (1904), Frenkel (1928), Skolem and Finsler (1926) had looked into the idea and Finsler later claimed priority. Hilbert had doubts about the validity, Carnap in Vienna was also reserved. Quine was in Vienna 1932-3 and learnt about the result back in America but did not mention it in his 1934 book. Godel's achievements are impressive against Hilbert's program: Brouwer is reported to have said "I am not surprised at all. It was to be expected" (see van Dalen). – sand1 Dec 3 '15 at 10:37
  • G's result was so "unsurprising" that Hilbert's student : Ackermann, devoted his doctoral dissertation (1924) to the consistency proof of arithmetic, based on Hilbert's Epsilon Substitution Method : 1/2 – Mauro ALLEGRANZA Dec 3 '15 at 11:00
  • ... "Ackermann's procedure applied to a system of 2nd-order arithmetic [...] Further difficulties with 2nd-order epsilon terms surfaced, and it quickly became apparent that the proof as it stood was fallacious. However, no one in Hilbert's school realized the extent of the difficulty until 1930, when Gödel announced his incompleteness results. Until then, it was believed that the proof (at least with some modifications introduced by Ackermann, some of which involved ideas from von Neumann's version of the epsilon substitution method) would go through at least for the first-order part." 2/2 – Mauro ALLEGRANZA Dec 3 '15 at 11:02
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I do not see any impact of Goedel's result to epistemology.

Broadly speaking, epistemology deals with the question how we recognize objects and facts from reality. It refers to the relation (issue 1) between the world (issue 2) and our model (issue 3) of the world.

Goedel's result refers to the power of axiomatized theories. This issue neither refers to the world (issue 2) nor to the relation (issue 1) between world and model. At most, it refers to properties of our world models (issue 3).

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