Dennis Hollingsworth

Incompletely

Chew on this:

It is difficult to overstate the impact of his theorem and the possibilities that opened up from G?del's extraordinary methods, in which he discovered a way for mathematics to talk about itself. (Ms. Goldstein compares it to a painting that could also explain the principles of aesthetics.)

I... don't... think... aesthetic principles can be so discrete... hmmmmm.

Here's more context for this snippet:

Before G?del's incompleteness theorem was published in 1931, it was believed that not only was everything proven by mathematics true, but also that within its conceptual universe everything true could be proven. Mathematics is thus complete: nothing true is beyond its reach. G?del shattered that dream. He showed that there were true statements in certain mathematical systems that could not be proven. And he did this with astonishing sleight of hand, producing a mathematical assertion that was both true and unprovable.

It is difficult to overstate the impact of his theorem and the possibilities that opened up from G?del's extraordinary methods, in which he discovered a way for mathematics to talk about itself. (Ms. Goldstein compares it to a painting that could also explain the principles of aesthetics.)

The theorem has generally been understood negatively because it asserts that there are limits to mathematics' powers. It shows that certain formal systems cannot accomplish what their creators hoped.

But what if the theorem is interpreted to reveal something positive: not proving a limitation but disclosing a possibility? Instead of "You can't prove everything," it would say: "This is what can be done: you can discover other kinds of truths. They may be beyond your mathematical formalisms, but they are nevertheless indubitable."

In this, G?del was elevating the nature of the world, rather than celebrating powers of the mind. There were indeed timeless truths. The mind would discover them not by following the futile methodologies of formal systems, but by taking astonishing leaps, making unusual connections, revealing hidden meanings.

Like Einstein, G?del was, Ms. Goldstein suggests, a Platonist.

Of course, those leaps and connections could go awry. G?del was an intermittent paranoiac, whose twisted visions often left his colleagues in dismay. He spent his later years working on a proof of the existence of God. He even died in the grip of a perverse esotericism. He feared eating, imagined elaborate plots, and literally wasted away. At his death in 1978, he weighed 65 pounds.

But he was no postmodernist. Late in his life G?del said of mathematics: "It is given to us in its entirety and does not change, unlike the Milky Way. That part of it of which we have a perfect view seems beautiful, suggesting harmony." That beauty, he proposed, would be mirrored by the world itself. These are not exactly the views of an acolyte devoted to Relativity, Incompleteness and Uncertainty. And Einstein was his fellow dissenter.

Maybe all of the quantum uncertainties and relativistic floatery in PostModernland is all curled up tight like the other seven or so string theory dimensions inside the crannies of atomic nucleus....

In other words, maybe we've misapplied subatomic ideas to superatomic everyday life? Cool article (book review), harvested like most of the other cool articles in this here blog, at Arts and Letters Daily.