I was eager to start this book, but it does not seem to be able to address my questions.

A very engrossing book on Kurt Gödel. It covers his life and the Incompleteness Proof that he formulated, making it understandable. I have read Kurt Gödel's proof, but as I said in my review of that book, I did not really understand it. Maybe now I will return to that book.

Anyway, there are a number of things I did know about Gödel, but some things I did not. I remembered the Einstein-Gödel friendship, but I did not realize that his Incompleteness Proofs were taken the wrong way: that is, they were not intended to turn the world of mathematics upside-down per se, but rather to be an ironclad answer to the members of the Vienna Circle that supported Positivist Views.

In any case, I would really like to find more books in this Great Discoveries series. Hopefully, they are all as good as this one. ( )

A good, not great, focused biography on one of the more misunderstood mathematician/logicians of the 20th cent. Inefficient exposition drags down an otherwise effective book. Worth it for a generally readable explanation of the theorems and the (lonesome) end. ( )

“It is really not so surprising that Wittgenstein would dismiss Gödel’s result with a belittling description like ‘logische Kunstücke,’ logical conjuring tricks, patently devoid of the large metamathematical import that Gödel and other mathematicians presumed his theorems had. Gödel’s proof, the very possibility of a proof of its kind, is forbidden on the grounds of Wittgensteinian tenets that remained constant through the transformation from ‘early’ to ‘later’ Wittgenstein, where early Wittgenstein had a monolithic view of language and its rules and later Wittgenstein fractured language into self-contained language-games, each functioning according to its own set of rules. He was adamant on the impossibility of being able to speak about a formal language in the way that Gödel’s proof does.”

In “Incompleness - The proof and Paradox of Kurt Gödel” by Rebecca Goldstein

Wittgenstein: “Hi Kurt, as you appear to be a professional mathematician working in the field, and after having written my “Tractatus Logico-Philosophicus”, I wonder if you can confirm whether these points are true, points I always wondered about whenever I read on articles connected to the work of Cantor, you and Cohen:
1) Primary school arithmetic has never been proved to be consistent, so theoretically a snotty kid could one day do correct arithmetic manipulations which lead to the result 0=1 i.e. Maths cannot currently prove this won't happen?
2) Your 2nd incompleteness theorem states roughly that a proof of the consistency of a consistent system which includes arithmetic does not exist in the language of that system. Now maybe there exists a proof of the consistency of system A (CON(A)) in the language of system B, but if system B again includes arithmetic you don't know if CON(B) is true and therefore cannot trust the proof of CON(A) in system B, and so on. But is it possible that CON(A) could be proved to be true without any dependencies by some wholly other method?”

Gödel: ”From 1) Commutative law:
For addition: a + b = b + a, ergo unless one creates a new or addendum to this law, one unit will never equal zero. Remember when Euclid's parallel line axiom was changed and the math, based on these new axioms, was useful for spheres and hyperspheres? Same thing “me “thinks. I'm a Computer Scientist with a minor in Physics though. What the hell do I know?"

Wittgenstein: “But you are assuming the axioms are consistent. Gödel said that a system which is powerful enough to include arithmetic cannot prove its own consistency. Given, then, we can't prove the consistency of the axioms, we may end up with a contradiction, e.g. 1=0. (In the case of Geometry, Euclid's axioms have been shown to be consistent, so the situation is different to that of arithmetic.)

Gödel: “A system cannot prove itself GIVEN the axioms in the system. It would need new axioms to prove these old axioms. But the new axioms would need newer axioms. And so on. Isn't this what you proved? We are always at least an axiom away from a house built on rock. Euclid's axioms have the same problem. Or maybe I'm wrong.”

Wittgenstein: “The question is about consistency of a set of axioms. If system A can handle arithmetic then a proof of system A's consistency cannot be provided from system A's axioms. Maybe a proof of A's consistency can be provided in system B, but then the question becomes can we trust that proof given we can't prove the consistency of B from its own axioms, and so on. All of which means that the consistency of elementary school arithmetic has never been proven, and so the appearance of a contradiction has not been ruled out mathematically. I don't think this applies in the case of Euclid's axioms. I believe they have been shown to be consistent.”

Gödel: “What the hell Wittgenstein????”

NB: This conversation took place in German. This dialectic is presented in translation, because my blog friends would complain about it.

The trouble with philosophy is that it is the residue of thought that cannot be answered elsewhere. Thales postulating that everything is made of water is physics as much as metaphysics (albeit a physics founded on pure speculation). Eventually though physics got its act together, developed its own rules and methodologies and never looked back. All the better for physics but it left philosophy somewhat diminished. Every discipline, pretty much, has its origins in philosophy. Philosophy is, in a sense, just science that hasn't got its act together yet. That is why the history of philosophy is so fascinating and so much modern philosophy (including Wittgenstein; "Philosophical Investigations" disproves the Tractatus and dissolves philosophy completely) is pretty sterile stuff.

It sounds rather Wittgenstein was possessed was a very strange case of the Kierkegaardian Malady in its distinction between not believing in Gödel's theorems and having faith in Math (in a Kierkegaardian sense the same as believing in God and having Faith in God at the same time; but then Kierkgaard knocks the socks off Wittgenstein any time of the day).

Bottom-line: Goldstein’s take both on Gödel and Wittgenstein’s opposing views is one of the best I’ve ever read. Her explanation on the concrete way Gödel went about proving both theorems is much better than Newman’s and Nagel’s book . ( )

Pretty good discussion of Godel and his incompleteness theorem with a not so clear explanation of it at the end. ( )