Tuesday, October 2, 2012

Logicomix - An Interesting Read

My friend lent me the book 'Logicomix - An epic search for truth'. What follows below is my notes from reading the book.

We usually associate comics with lighter subjects, but this book has attempted to present a very serious subject - the doomed search for mathematical certainty, doomed because Goedel put an end to it. What makes the book more interesting is that the authors are also characters in the book, along with Russel and others.

This book presents the story of Russel's quest for mathematical certainty and also covers a surprising amount of information about logic, mathematics and philosophy.

The giants of late 19th and  20th century mathematics and philosophy like Henri Poincare, Gottlob Frege, Kurt Goedel, Dave Hilbert , Von Neumann and Ludwig Wittgenstein along with their works all figure in this book.

I should say that the attempt to present a serious subject through comics has also succeeded because the book is readable, makes the content stick and  i couldn't put it down. I completed it in a train journey, in 5 hours.

For e.g. the reason for Principia Mathematica is provided like this in Russel's words

1. Mathematics must be based on logic
2. Frege creates the right logic (based on sets)
3. I find paradox  (Russel's paradox)
4. Whitehead and I must fix it ( Principia Mathematica)

Russel strived worked hard on #4 above till Goedel proved that it was impossible.

The book says it in Goedel's words as  "There will always be unanswerable questions. What i have proved in essence, is that arithmetic, and thus also any system based on it, is of necessity, incomplete". 

The same written in serious language, at the end of the book is given below.

Goedel shocked the mathematical world by proving, in his famous paper "'On undecidable propositions in the Principia Mathematica and Related systems', that any consistent axiomatic system for arithmetic, in the form developed in the Principia, must of necessity be incomplete. More precisely, the first of the two incompleteness theorems establishes that in a logical axiomatic system rich enough to describe properties of the whole numbers and ordinary arithmetic operations, there will always be propositions that are grammatically correct by the rules of the system, and moreover, true, but cannot be proven within the system. The second incompleteness Theorem states that if such a system were to prove its own consistency it would be inconsistent".

The above shows how much simplified the language used in the book is.This book should be looked at as an introduction to a very serious subject, where it does a very good job, and not an in depth study of the subject.   Makes me wonder why school text books don't come out in this format.

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