David Peat

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TwoON INCOMPLETENESSIn the previous chapter we saw howNature limits the certainty we can expect from the material world andallows us to probe only so far into the mystery of reality; beyond thiswe are in danger of becoming lost in paradox and confusion. Does thismean that we have lost forever the hope of certainty?If, through our acts of participation in nature, limits are placed onthe extent of our knowing, then at least we should be able to find certaintyin the abstract products of our own minds. Above all, shouldn’twe be able to discover certainty within the world of mathematics? Thisis exactly what the philosopher Bertrand Russell believed as, in theyear 1900, he listened to Giuseppe Peano speak with great clarity aboutthe foundations of mathematics and decided to devote himself to provingtheir absolute rigor.27
28 From Certainty to UncertaintyThe Power and Beauty of MathematicsThe dream of structuring the world according to mathematical principlesbegan long before the rise of modern science. The Pythagoreanbrotherhood of ancient Greece believed that “all is number.” In theMiddle Ages, mathematical harmonies were the key to both music andthe architecture of great buildings. The paintings of Piero dellaFrancesca (1420?–1492) take us into a world of deep mathematical orderand balance. The same sense of harmony and proportion is foundin the music of J. S. Bach, and, thanks to the research of the cellistHans-Eberhard Dentler, we now know that Bach’s Art of the Fugue wasinfluenced by Pythagorean number symbolism. 1Where we find certainty and truth in mathematics we also findbeauty. Great mathematics is characterized by its aesthetics. Mathematiciansdelight in the elegance, economy of means, and logical inevitabilityof proof. It is as if the great mathematical truths can be no otherway. This light of logic is also reflected back to us in the underlyingstructures of the physical world through the mathematics of theoreticalphysics.In The Study of Mathematics, Bertrand Russell put it this way:“[M]athematics takes us into the region of absolute necessity, to whichnot only the actual world, but every possible world, must conform.”For the philosopher, “mathematics is an ideal world and an eternaledifice of truth. . . . [I]n the contemplation of its serene beauty mancan find refuge from the world full of evil and suffering.” 2 For the astronomerJames Jeans (1877–1946), “God is a mathematician.” Andthere is a saying amongst mathematicians that “God made the numbers.All the rest is made by humans.” 31Hans-Eberhard Dentler. L’Arte della Fuga di Johann Sebastian Bach: Un’operapitagorica e la sua realizzazione (Milan: Skira, 2000).2Frederick Copleston paraphrasing Russell, The History of Philosophy, vol. 8:Bentham to Russell (New York: Bantam, Doubleday Bell, 1985).3James Jeans, The Mysterious Universe (Cambridge: Cambridge University Press,1930).
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SevenRE-ENVISIONING THE PLANETThe h
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POSTSCRIPTThis book was completed b
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AppendixGÖDEL’S THEOREMEarlier a
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228 IndexPalio of Siena, 195 n.4Pa
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David Peat

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