| From page 12: > Why should we believe in real numbers, if most of them, it turns
out,[^15] are maximally unknowable like Ω? [^16] The footnotes: > [^15]: See the chapter entitled The Labyrinth of the Continuum in [Chaitin, 2005] > [^16]: In spite of the fact that most individual real numbers will forever escape us, the notion of an arbitrary real has beautiful mathematical properties and is a concept that helps us to organize and understand the real world. Individual concepts in a theory do not need to have concrete meaning on their own; it is enough if the theory as a whole can be compared with the results of experiments. --- The reference [Chaitin, 2005] in footnote 15 links to.. Meta Math! The Quest for Omega - http://arxiv.org/abs/math/0404335 > This book presents a personal account of the mathematics and metamathematics of the 20th century leading up to the discovery of the halting probability Omega. The emphasis is on history of ideas and philosophical implications. Irreducible Complexity in Pure Mathematics - http://arxiv.org/abs/math/0411091 > By using ideas on complexity and randomness originally suggested by the mathematician-philosopher Gottfried Leibniz in 1686, the modern theory of algorithmic information is able to show that there can never be a "theory of everything" for all of mathematics. |