" My approach to understanding the full implications of Gödel's work is mathematically analogous to the ideas of thermodynamics and Boltzmann and statistical mechanics. You might say, not completely seriously, that what I'm proposing is 'thermodynamical epistemology'!"
- G. Chaitin

"Those who cannot cope with mathematics are at best, tolerable sub-humans who have learned to bathe, wear shoes and not make messes in the house."
- Robert Heinlein

A baker knows when a loaf of bread is done and a builder knows when a house is finished. Yogi Berra told us "it ain't over till it's over," which implies that at some point it is over. But in mathematics things aren't so simple. Increasingly, mathematicians are confronting problems wherein it is not clear whether it will ever be over.

People are now claiming proofs for two of the most famous problems in mathematics — the Riemann Hypothesis and the Poincare Conjecture — yet it is far from easy to tell whether either claim is valid. In the first case the purported proof is so long and the mathematics so obscure no one wants to spend the time checking through its hundreds of pages for fear they may be wasting their time. In the second case, a small army of experts has spent the last two years poring over the equations and still doesn't know whether they add up.

Consider all English phrases that uniquely identify a real number, such as "that positive real number whose square is two", or "the ratio between a circle's diameter and its circumference. For any given positive integer n, there are a finite number of such phrases of length n (this might be zero if there are no meaningful phrases of length n, such as when n is 1 or 2). By listing first all phrases of length 3 in some order (say, alphabetical order), then all phrases of length 4, and so on, we generate an infinite ordered list of all such phrases. Call this the "Richard phrase list".

Now define a real number as follows:

Abstract
We discuss mathematical and physical arguments against continuity and in favor of discreteness, with particular emphasis on the ideas of Emile Borel (1871–1956).

Introduction
Experimental physicists know how difficult accurate measurements are. No physical quantity has ever been measured with more than 15 digits or so of accuracy. Mathematicians, however, freely fantasize with infinite-precision real numbers. Nevertheless within pure math the notion of a real number is extremely problematic.

We’ll compare and contrast two parallel historical episodes:
1. the diagonal and probabilistic proofs that reals are uncountable, and
2. the diagonal and probabilistic proofs that there are uncomputable reals.

Both case histories open chasms beneath the feet of mathematicians. In the first case these are the famous Jules Richard paradox (1905), Emile Borel’s know-it-all real (1927), and the fact that most reals are unnameable, which was the subject of Borel, 1952, his last book, published when Borel was 81 years old James, 2002. In the second case the frightening features are the unsolvability of the halting problem (Turing, 1936), the fact that most reals are uncomputable, and last but not least, the halting probability , which is irreducibly complex (algorithmically random), maximally unknowable, and dramatically illustrates the limits of reason Chaitin, 2005.

In addition to this mathematical soul-searching regarding real numbers, some physicists are beginning to suspect that the physical universe is actually discrete Smolin, 2000 and perhaps even a giant computer Fredkin, 2004, Wolfram, 2002. It will be interesting to see how far this so-called “digital philosophy,” “digital physics” viewpoint can be taken.

Gregory Chaitin has devoted his life to the attempt to understand what mathematics can and cannot achieve, and is a member of the digital philosophy/digital physics movement. Its members believe that the world is built out of digital information, out of 0 and 1 bits, and they view the universe as a giant information-processing machine, a giant digital computer. In this book on the history of ideas, Chaitin traces digital philosophy back to the nearly-forgotten 17th century genius Leibniz. He also tells us how he discovered the celebrated Omega number, which marks the current boundary of what mathematics can achieve. This book is an opportunity to get inside the head of a creative mathematician and see what makes him tick, and opens a window for its readers onto a glittering world of high-altitude thought that few intellectual mountain climbers can ever glimpse.

Elegance is more than just a frill in life; it is one of the driving criteria behind survival.
- Douglas Hofstadter

Researchers need enormous computer power to forecast changes in the Earth's climate, but they can predict the speed of a ball rolling down a ramp with pencil and paper. Stephen Wolfram claimed in his 2002 best seller, A New Kind of Science, that there is a clear dividing line between complex problems that require computer crunching and those for which equations alone will do. He argued that many important problems are more like the climate than the ball. But according to the 20 February PRL, his definition of complexity is imperfect because many of the problems he classified as complex are easily solved, as long as you can accept approximate answers. The results suggest that the traditional approach of physics--the equivalent of pencil and paper--is more widely applicable than Wolfram's analysis implies.

The physicist rightly dreads precise argument, since an argument which is only convincing if precise loses all its force if the assumptions upon which it is based are slightly changed, while an argument which is convincing though imprecise may well be stable under small perturbations of its underlying axioms.
— Jacob Schwartz

There are two kinds of ways of looking at mathematics... the Babylonian tradition and the Greek tradition... Euclid discovered that there was a way in which all the theorems of geometry could be ordered from a set of axioms that were particularly simple... The Babylonian attitude... is that you know all of the various theorems and many of the connections in between, but you have never fully realized that it could all come up from a bunch of axioms... Even in mathematics you can start in different places... In physics we need the Babylonian method, and not the Euclidian or Greek method.
— Richard Feynman

At New York's Kennedy airport today, an individual later discovered to be a public school teacher was arrested trying to board a flight while in possession of a ruler, a protractor, a setsquare, a slide rule, and a calculator. At a morning press conference, Attorney General John Ashcroft said he believes the man is a member of the notorious al-gebra movement. He is being charged by the FBI with carrying weapons of math instruction.

"To how many places does nature carry out PI when she makes each successive bubble in the white-cresting surf of each successive wave before nature finds out that PI can never be resolved?... And at what moment in the making of each separate bubble in Universe does nature decide to terminate her eternally frustrated calculating and instead turn out a fake sphere? I answered myself that I don't think nature is using PI or any of the irrational fraction constants of physics."
-Buckminster Fuller(Synergetics II, p. 233).

The Goldbach conjecture states that all positive even integers greater than or equal to four can be expressed as the sum of two odd prime numbers.

Cristian Goldbach (1690-1764) submitted his conjecture to Euler who dismissed it as trivial, however as of today (2003) it has not yet been proved.

See this page for more details.