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Chapter 2 on Incompleteness
Extract - first two pages
In the previous chapter we saw how nature limits the certainty we can expect from the material world. Nature allows us to probe only so far into the mystery of reality, beyond this we are in danger of becoming lost in paradox and confusion. Does this mean that we have lost forever the hope of certainty?
If, through our acts of participation in nature, limits are placed on the extent of our knowing, then at least we should be able to find certainty in the abstract products of our own minds. Above all, shouldn't we be able to discover certainty within the world of mathematics? This is exactly what the philosopher Bertrand Russell believed as, in the year 1900, he listened to the Giusseppe Peano speak with great clarity about the foundations of mathematics and decided to devote himself to proving their absolute rigor..
The power and beauty of mathematics
The dream of structuring the world according to mathematical principles began long before the rise of modern science. The Pythagorean brotherhood of ancient Greece believed that "all is number." In the Middle Ages, mathematical harmonies were the key to both music and the architecture of great buildings. The paintings of Piero della Francesca (1420?-1492) take us into a world of deep mathematical order and balance. The same sense of harmony and proportion is found in the music of J.S. Bach, and, thanks to the researches of the cellist Hans-Eberhard Dentler, we now know that Bach's Art of Fugue was influenced by Pythagorean number symbolism. (Hans-Eberhard Dentler "L'Arte della Fuga di Johann Sebastian Bach: un'opera pitagorica e la sua realizzazion", Skira, Milan, 2000.)
Where we find certainty and truth in mathematics we also find beauty. Great mathematics is characterized by its aesthetics. Mathematicians delight in the elegance, economy of means, and logical inevitability of proof. It is as if the great mathematical truths just have to be that way and no other. This light of logic is also reflected back to us in the underlying structures of the physical world through the mathematics of theoretical physics.
In The Study of Mathematics , Bertrand Russell, put it this way: "mathematics takes us into the region of absolute necessity, to which not only the actual world, but every possible world, must conform." For the philosopher, "mathematics is an ideal world and an eternal edifice of truth. …in the contemplation of its serene beauty man can find refuge from the world full of evil and suffering."( Frederick Copleston paraphrasing Russell "The History of Philosophy Vol. VIII Bentham to Russell" Bantam, Doubleday Bell, NY, 1985) ) For the astronomer James Jeans(1877-1946), "God is a mathematician". And there is a saying amongst mathematicians that "God made the numbers. All the rest is made by humans."
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