Nicht-Logische-Kunstücke: “Incompleness - The proof and Paradox of Kurt Gödel” by Rebecca Goldstein

“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 .

Calabi-Yau Spaces 2: "The Shape of a Life: One Mathematician's Search for the Universe's Hidden Geometry" by Shing-Tung Yau, Steve Nadis

“My proof, I told them [Andrew Strominger and Edward Witten], was motivated by physics, specifically the notion that even in a vacuum, a space with no matter, gravity could still exist. I felt certain that this must be important for physics, though I was not sure of the exact ramification.“

In “The Shape of a Life - One Mathematician's Search for the Universe's Hidden Geometry” by Shing-Tung Yau, Steve Nadis

“String theory further postulated that we inhabit a ten-dimensional universe consisting of the three familiar (and infinitely large) spatial dimensions, one dimension of time, and six  additional miniature dimensions that are wound up into a tight coil and thereby hidden from view.  The question that Candelas and Strominger, among others, were grappling with concerned the geometry of the six shrunken, or ‘compactified’ dimensions. What, exactly, is the shape into which these extra dimensions are confined? Strominger knew they needed a manifold, or space, with well-defined properties, including a special kind of symmetry called ‘supersymmetry,’ which turns out to be an intrinsic feature of the manifolds, of the variety called Kähler, whose existence I had proved. Supersymmetry is also a requisite feature of many versions of string theory, which is why it’s sometimes called ‘superstring theory’ instead.”

In “The Shape of a Life - One Mathematician's Search for the Universe's Hidden Geometry” by Shing-Tung Yau, Steve Nadis

I wear a giant panda suit outside a Panda Burger giving out promotional leaflets. As this job is a bit easy and I can do it without too much conscious effort... the only thing I have to watch out for is farting as it is unpleasant trapped in that panda suit... anyhow I digress ... this gives me a LOT of time to think about serious issues such as time and the merits of thinking about Physics in a Panda suit in scorching hot Lisbon in the Summer... 

Mark Twain said that scientific facts give rise to speculations, which of course are tested if possible. For the most part, math is not about "numbers" but largely about properties of, and relationships among highly abstract objects. Indeed, mathematics as a profession is a risk and self-sacrifice. One has to devote time and effort to one's field before one gets to appreciate it and produce results worth of publication. But there is always a risk that, even if one gains an understanding - which in itself is rare and precious - it will not be followed by original results, stalling one's academic career. This stalling of career due to the lack of originality is normally a direct result of being risk averse and not pushing yourself hard enough. Mathematics is an essentially creative activity: you are bound to achieve something if you are genuinely interested...Tricky thing defining maths. Even if the definition is true, it never looks very interesting. Certainly not as interesting as mathematics itself. It's certainly made a wee bit of progress from counting. Over the last few thousand years... There was that Archimedes and that other Euclid guy. And that Al Khwarizmi dude. Some Newton bloke. Euler, Gauss, a whole truckload of Bernoullis, Fourier, Cauchy, Poincare, Riemann, Noether, Cantor, Goedel, Brouwer... feel like I've forgotten a few hundred really big names but I just can't put my fingers on them...Reducing maths to numbers is kind of like saying all cooking is really just a matter of making 2 minute noodles.

My querky moment while learning mathematics was during a moment of boredom when I took the differences of successive calculated polynomial values and continued taking differences of the results. It turns out this is the basis of the difference engine that Babbage designed, and how mathematical tables were created before the advent of electronic calculators and computers. Probably unsurprisingly I took up Engineering which makes use of a myriad of mathematical techniques and valid short cuts, many of which are never taught to scientists and mathematicians in my experience.

There's something sublime, mystical and ineffable about such problems. You'd think maths would be easy, just counting, but hidden within those ostensibly basic concepts are such convolutions and crenelations and complications. It's amazing that 1+1 can get to such things like Fermat's Last Theorem and imaginary numbers or that Calabi-Yau Manifolds can be applied to Physics, namely String Theory and General Relativity. Let alone whatever these things are on about.

I just wish Yau had written a more math-oriented biography. We don't really get math insights on how he got to prove some of the things important to Physics, namely the Calabi-Yau conjecture. It's all very vague... If you want that to dig deeper into the math part of some of these topics, you should read “The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions” by the same authors.

Coda: No-one uses Calabi-Yau in a sentence (apart from Woody Allen in a New Yorker piece). It inspired me...

I wish my house was a Calabi-Yau Space,

a place where I could tell fiction from fact

I'd invite politicians to sit in the middle

Then I'd focus the heat so it's hot as a griddle

I'd make then elucidate policies at length

And keeping them talking to sap all their strength

And right at the end I would shout and declare

"Your lies and deceit are now totally clear

My house has deciphered your thoughts and your words

And showed them as nothing but bright polished turds

I'm leaving you now and I'll never come back

This part of my house is now fading to black....

NB: It was kind of interesting to read about Yau’s take on the feud between Yau and Chern and also his attempt at explaining what happened with the Poincaré Conjecture (he was accused of “stealing” Perelman’s discovery by having some of his students develop a more rigorous proof of Perelman’s demonstration).