Download Mathematics by Experiment, 2nd Edition: Plausible Reasoning by Jonathan Borwein, David Bailey PDF

By Jonathan Borwein, David Bailey

This revised and up to date moment variation continues the content material and spirit of the 1st version and encompasses a new bankruptcy, "Recent Experiences", that offers examples of experimental arithmetic that experience come to mild because the ebook of the 1st version in 2003. For extra examples and insights, Experimentation in arithmetic: Computational Paths to Discovery is a hugely instructed better half.

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Extra info for Mathematics by Experiment, 2nd Edition: Plausible Reasoning in the 21st Century

Example text

As in science, mathematics can advance by making mistakes and then correcting or even re-correcting them. The “fallibilism” of mathematics is brilliantly argued in Lakatos’ Proofs and Refutations. 3. There are different versions of proof or rigor. Standards of rigor can vary depending on time, place, and other things. The use of computers in formal proofs, exemplified by the computerassisted proof of the four color theorem in 1977, is just one example of an emerging nontraditional standard of rigor.

R(3) = R(3, 3) = 6 and R(4) = R(4, 4) = 18. This is often described as saying that at a six-person dinner party either there are three friends or there are three strangers. Such numbers R(k, 1) are very hard to compute. Indeed, Paul Erd˝ os suggested that if an alien demanded we give it the value of R(5) in order to save the Earth, we should set all mathematicians and computers to calculating the value. But if the alien demanded R(6), then humanity should attempt to destroy it before it destroyed us.

14159245358979323846464338327950278419716939938730582097494182230781640... 14159265358979323846264338327950288419716939937510582097494459230781640... ” But the next 13 digits are correct! Then, following another erroneous digit, the sequence is once again correct for an additional 12 digits. In fact, of the first 46 digits, only four differ from the corresponding decimal digits of π. Further, the “error” digits appear to occur in positions that have a period of 14, as shown above. Such anomalous behavior begs explanation.

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