Friday, May 20, 2005

First thoughts on Rebecca Goldstein’s, Incompleteness: The proof and paradox of Kurt Gödel

I bought this book despite myself. I’ve carefully studied Gödel’s Incompleteness Theorems and expected Goldstein to give a soft, non rigorous, largely biographical treatment which wouldn’t teach me anything new. I bought the book almost out of duty – it is after all in the subject I care most deeply about – I should read it just in case. I am glad I got took the chance. This is a great book, and I am not one to use the term loosely.

The power of the book doesn’t come from its treatment of the theorem itself (she does an adequate job, but others have done better. See for example Nagel and Newman’s classic, Gödel’s Proof for a fine non-technical treatment); rather the books achievement is that it puts Gödel’s work in context. Goldstein successfully (and finally) gives Gödel’s theorems the philosophical interpretation that he himself would have intended.

Before reading Incompleteness I often wondered why Gödel, an avowed Platonist, did most of his work in Mathematical Logic, the most formalist of all mathematical fields. Also, why did he join the Logical Positivists of Vienna who in their way were the most extreme kind of Formalists; and lastly why did Gödel associate himself with a group who revered the teachings of Wittgenstein – the very same Wittgenstein who essentially claimed that all of a mathematics was a mere tautology (a claim that was almost surely quite repulsive to Gödel, and to almost every other mathematician).

Goldstein answered all of this (and more). She gets her answers not from the mathematics, but from the story of Gödel’s life and the philosophical battles that drove him.

In brief, the story is that the Logical Positivists essentially believed that truth lived in the precise, meaning-aware use of language. According tho them, it is only possible to identify a statement as being true or false by proving or disproving it by experience. Logic and mathematics was excluded from this rule; they claimed that mathematics was a branch of logic and was for all intents and and purposes a mere tautology.

Gödel on the other hand was a Platonist; he believed that mathematicians uncovered truths about the universe, and mathematical concepts were merely communicated by—but not contained within—its equations and symbols. Yet, confusingly, Gödel belonged to a Positivist group. He largely stayed silent through their meetings, neither objecting nor agreeing, for that was not his way.

But the internal storm of disagreement that welled within him did lead him to prove that Positivists were wrong. He proved that the structural manipulation of mathematical symbols could not yield all statements that we know to be true. He demonstrated a ‘true’ statement that was not provable—which should have banished Logical Positivism for ever.

Yet it didn’t; for Godel, before Goldstein’s book, was never well understood.

I’ll have a lot more to say about all this in the coming weeks.

Sunday, May 15, 2005

5 good uns

Ah, Why, ye Gods, should two and two make four --Alexander Pope

In mathematics there are no true controversies. --Gauss

Logic is the art of going wrong with confidence --Anonymous

Let us suppose there are things like the truth --Xenphanes

We know truth, not only by reason, but also by the heart --Pascal

--Gaurav Suri

Thursday, May 12, 2005

Infinity in the physical world

Here’s a section from an essay on Infinity by Hector Parr (which has several inaccuracies within it). I bring it up because it brings many of our beliefs about infinity into high relief:

“The world of the Pure Mathematician is far removed from the real world. In the real world there is no difficulty finding the length of the diagonal of a square and expressing this as a decimal, but in the perfect world of Pure Mathematics this cannot be done. In the real world we know the process of counting the natural numbers can never be completed, so that the number of numbers is without meaning, while the mathematician finds it necessary to say that if the process were completed, the number would be found to be Aleph-0. These are harmless follies; what the mathematician gets up to inside his ivory tower need not concern those outside.
But the mathematician's ideal world did impinge on reality at the beginning of the twentieth century when Russell (1872-1970) attempted to reduce all mathematical reasoning to simple logic. Even the natural numbers themselves could be defined in terms of a simpler concept, but to make this possible Russell found it necessary to assume that the number of real objects in the universe is itself infinite. Here again I find it astonishing that he made this assumption so glibly. He called the principle his "Axiom of Infinity". Now an "axiom" is something which is self-evident, unlike a "postulate", which is assumed for convenience even though it is not self-evident. Why did Russell not refer to the principle as his "Postulate of Infinity"? To me it is far from self-evident that there are an infinite number of things in the universe; in fact I cannot see that the statement has any meaning. Infinity is an indispensible concept in Pure Mathematics, but is it not meaningless when applied to the number of real things?”

I disagree with a lot of this selection and I’ve reproduced it here because Parr’s view is a fairly commonly held one. First, he says that in the real world there is no ‘difficulty’ expressing the diagonal of a square as a decimal, but in pure mathematics ‘this cannot be done.’ This statement is either vague or meaningless. Mathematicians have no difficulty assigning a decimal to a diagonal; but they do so as an approximation, not as a precise value. In the ‘real world’ the exact same condition holds – it’s just that for most purposes an approximation suffices. Second, Parr writes that in the real world the process of counting cannot be completed, while the mathematician says that if the process were completed, the number would be Aleph-Zero. This is incorrect. Mathematicians know that counting the Naturals is an unending process. The need for Aleph-Zero arises because Cantor proved that there are several levels of infinity and that he named the beginning level to be Aleph-Zero.

The idea of the ideal world impinging on reality with Russell’s reduction of mathematics to logic is incorrect. None of Russell’s theories required there to be an infinite number of objects in the physical universe. But here finally we are at the real question Parr is attempting to raise: Is there a true infinity in the physical universe? I tend to think not. The universe is likely finite (although unbounded); it likely has a finite life, so time is finite; Some people may argue that God is infinite, but we’re getting into mysticism there. So let’s say that despite being wrong in all the steps of his argument, Parr’s conclusion is correct – that infinity does not exist in the physical world.

Does that somehow make the mathematics of infinity a mere game?

I think not. If one acknowledge 1, 2, 3…, one must acknowledge (countable) infinity and from there it’s a few elegant steps to many levels of infinity. Formalists argue that infinity is meaningless; but it easy for me to be on Godel’s side who argued that if we perceive something about the universe with our mind (e.g. infinity), then it is at least as real the stuff we perceive with our senses (e.g. my 2 year old throwing my phone on the floor)

--Gaurav Suri

Saturday, May 07, 2005

The biological roots of mathematical proof

A great article about the biological roots of music proposes that: “music is merely "auditory cheesecake," or "an evolutionary accident piggy-backing on language," as Daniel J. Levitin at McGill University explained in a recent issue of the journal Cerebrum. But many scientists—Levitin among them—don't agree. "Some researchers are finding that listening to familiar music activates neural structures deep in the ancient primitive regions of the brain, the cerebellar vermis," Levitin writes. "For music so profoundly to affect this gateway to emotion, it must have some ancient and important function.””

I wonder the same thing about the human ability to do proofs. Proof is a recent invention which—if you really examine it—seems to be based on some pretty shady ground. Some professional mathematicians think the emphasis on strict proof is a mystification. G.H. Hardy, one of the most eminent pure mathematicians, commented: "There is strictly no such thing as mathematical proof; proofs are what Littlewood and I call gas, rhetorical flourishes, devices to stimulate the imaginations of pupils". Lets assume for a second that Hardy is right (I believe he is). This would make proof remarkably similar to music: a profound activity, beautiful in its way, but without any obvious function.

What then are the biological roots of proof? Are they some accidental piggy backs on our traditional reasoning abilities? Or are they a direct consequence of language?

--Gaurav Suri

Tuesday, May 03, 2005

Beautiful Equation

Consider the series of the reciprocal of squares. If you have not seen it before, you’ll never believe what it converges to:

1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + 1/49 + … = π2/6 (this is pi squared divided by 6)

Yes, π2/6!! That is the same π that you are familiar with: the ratio of a circle’s circumference and diameter. How did it get into this equation? What could a ratio pertaining to circles have to do with the sum of reciprocals of squares? Is this not truly miraculous?

Sometimes when I look at this equation with fresh eyes, I’m amazed and in awe all over again. Equations like this represent why I fell in love with mathematics. There are so many unexpected connections, so much order when you would expect none, a mostly hidden tapestry into which we get a few limited glimpses through the efforts of our brightest minds.

Who made these connections? Why do they exist?

The proof unfortunately is not as beautiful as the result. It justifies rather than explains

--Gaurav Suri