There's Something About Gödel by Francesco Berto download in iPad, ePub, pdf
Pattern recognition is very useful for the purpose of inventing new mathematics, but it is not itself formal Mathematics unless it is mappable in a faithful way to machine code. And last but not least, the final chapter with the description of the Wittgenstein's point of view throws on the reader new doubts and fascinating vision. The axiom of choice or transfinite ordinals may be intuitive to others of course, intuition is subjective. If I am allowed, I would like to ask an impertinent question.
By education I am a physicist and work as applied mathematician and engineer. The only philosopher I can relate to is Wittgenstein. It acquires a meaning only when it is proved as a Theorem.
It is also an exploration of the most controversial allegedphilosophical outcomes of the Theorem. In particular, Aristotle was absolutely right distinguishing between potential and actual infinite, saying mathematical discourse should only dwell on the former. All this may seem conservative and old fashioned, but it makes perfect sense to me. Without a proof, a proposition with quantifiers over actually infinite classes lives in the limbo of abstract nonsense. In light of quantum computing this is physically possible, so I think there is hope and I am making good progress at understanding my problem now.
The first half is a gentle but rigorous introduction to the incompleteness theorems for the mathematically uninitiated. Socrates and the inexhaustibility of mathematics. Godel and Artificial Intelligence.
And the last part of the book is as much important as the technical one, because the author clarify well how people in different fields use and misuse Godel's results. To accept real random numbers in a Constructive Mathematics context, one clearly needs to go beyond recursive Arithmetics and accept stochastic recursion as a primitive concept. Grazie a questo libro mi sono chiarito le idee su molti aspetti che altri testi non spiegavano bene, o peggio, sui quali vengono spesso diffuse interpretazioni errate. But the more I try to understand Godel, the Hilbert program and Classical Mathematics, the more I am puzzled by all this. Really well written and the author was able to deeply explain in detail Godel's famous results without getting the reader lost.
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