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The Turing Essays
Uncomputable Numbers
Real numbers we can never know the value of
We all remember learning that the decimals of pi are infinite in number, 3.14159265359… Some of us even recall learning that you can approximate upper and lower bounds on the value of pi to as high of a degree as you want by measuring the sides of polygons. As the number of sides of the polygons approach infinity and the length of their sides approach zero, your approximation gets closer.
Archimedes (c. 287- 212) invented this early ‘polygonal algorithm’, which dominated for over 1000 years as the most efficient way of computing pi to any desired precision. The very primitive and geometric algorithmic function serves the purpose of estimating a real number, pi, whose value must be computed, as it is not expressible as a rational number (fraction/ratio).
The modern study of computability began around 1900 with the announcement of Hilbert’s ‘finitist’ program to axiomatize the foundations of mathematics. This after Hilbert himself had showed in 1899 that the consistency of geometry reduced to that of the real numbers, which in turn, reduced to the arithmetic results of Dedekind (Soare, 2013). Hilbert’s program (1904), hence, was established to try to similarly exploit the finiteness of mathematical proofs to show that contradictions…
