46
)( nT k in
(ii) Feinstein claims to show that, given the formula given above for
1 )( = nT k for a given k and n we
k λ
k ρ
terms of
and
, to show that
)( n
)( n
would need to specify each of the zeros and ones in the sequence (1, 0, 0, 1, 1, 0, 1, 1, ...)
(iii) Feinstein then says that if there were a proof of the Thwaites conjecture, this could be written in a document in a finite number of characters, e.g. typed up in Word, needing a finite number of zeros and ones to store the file in binary on a hard drive of a computer. (iv) Lastly, Feinstein essentially argues from the idea of a random sequence of zeros and ones, as defined in a moment, to the conclusion that the information stored in the computer file would never be enough to include all of the
)( nT k formula in terms
information in the Thwaites conjecture, as shown by the
of zeros and ones.
For anyone used to ‘standard’ mathematical proofs, the proof offered by Feinstein is bizarre to say the least. The idea of using the length of a possible proof, and how it might be stored on a computer, to show that no proof is possible is totally unlike any other proof encountered in number theory. One may even be led to suspect that Feinstein is a crank (this is hinted at in some internet discussion). However, whether or not Feinstein’s argument is correct, the ideas that he uses are actually rooted in a very rigorous and valid area of mathematics, albeit a fairly recent one, as discussed next. The area of mathematics / computer science called Algorithmic Information Theory is largely the result Gregory Chaitin who has worked on the theory since the 1960s. He has produced several serious textbooks on the theory (published in the Springer-Verlag series, one of the most respected series of mathematics textbooks) along with several more popular expositions of the theory. One of the key ideas, used in Feinstein’s argument is that of ‘randomness’. Chaitin states himself that the word ‘random’ is a poor choice, for there is nothing truly random about the concept as used in the theory. The idea instead is that a random number is one that cannot be stated more succinctly that simply writing it out. A good example of a non-random number is the number 1000000000000000000000000. This can be written using 25 digits as just shown, or can be written much more succinctly as 1 × 10 24 . The reason why we can Algorithmic Information Theory
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