Categories: archive |books

Book: Logicomix: An Epic Search For Truth

  • Read: March 2012
  • Rating: 9.0/10

Logicomix: An Epic Search For Truth by Apostolos Doxiadis and Christos Papadimitriou is a graphic novel that presents itself as historical fiction, or fictional history of the logic behind mathematics. The book is fantastic and definitely worth a read by anyone who is a lover of logic, math, graphic novels, programming, computer science, or anything else, really. Apostolos and Christos did an excellent job in introducing a lot of basic concepts and how some of those concepts came to being. Of course, some of the material is fictionalized, but that doesn’t stop the book at being an good resource on getting started to know some of the people who made things happen in the principles of mathematics.

My Notes

I don’t really have a lot of notes. Instead, I’ll list here some of the topics and people in the book.

Topics

  • Algorithms
    • Methodical, step-by-step procedure described in terms of totally unambiguous instructions, which starts at a specified initial condition and terminates with the desired outcome.
  • Axiom
    • Agreed-upon first principles that don’t require proof.
  • Foundations of Mathematics
  • Incompleteness Theorem
    • The two theorems by Kurt Godel which show that:
      • 1) Any consistent axiomatic system for arithmetic, in the form developed in the Principia, must of necessity by incomplete.
      • 2) If such a system were to prove its own consistency it would be inconsistent.
  • Intuitionism
    • Philosophy of mathematics that is based on the belief that intuition and time are fundamental to mathematics.
    • The belief that logic is founded on mathematics, and not the other way around.
  • Logic
    • The study of methodical thinking, deduction, and demonstration.
  • Predicate Calculus
    • Synonym with predicate logic and first-order logic.
  • Principia Mathematica
    • Extremely influential but controversial work by Alfred Whitehead and Bertrand Russell. It attempted to salvage Frege’s work which was decimated by Russell’s Paradox.
  • Proof
    • The process of arriving at the logical verification of a mathematical or logical statement, starting from a set of agreed-upon first principles.
  • Russell’s Paradox
    • A flaw in Cantor’s Set Theory. Using the definition of a set of all sets which don’t contain themselves, it is Russell’s question: “does this set contain itself or not?”
      • If it does contain itself, it follows that it is one of the sets which don’t contain themselves, and thus cannot contain itself.
      • If it doesn’t contain itself, then it does not have the property of not containing itself, and thus does contain itself.
      • Which is to say it is a paradox
    • His paradox deals with self-reference, like Euboulides’ statement “I am now lying to you”
  • Self-reference
    • The quality of a statement of referring to itself.
  • Set theory
    • The study of collections of objects united by a common property.
  • Tractatus Logico-Philosophicus
    • Ludwig Wittgenstein’s solution of “all the problems of philosophy”.

People

  • Aristotle
    • One of Greek’s most influential philosophers.
    • Contributed greatly to systematization and exposition of logic.
  • George Boole
    • Great contributor to logic in mathematics.
    • Developed the idea that logical propositions can be expressed in a purely symbolic language which allows them to be manipulated by operations similaty to the operations of elementary arithmetic.
    • “and”, “or”, and “not” can be traced back to his work.
  • Georg Cantor
    • Made a great deal of progress in set theory.
    • Theorem that set of all real numbers is uncountable, but set of all rational numbers is countable.
  • Euclid
    • Greek mathematician
  • Gottlob Frege
    • Considered the father of modern logic
  • Kurt Godel
    • Proved Completeness Theorem
    • Later proved Incompleteness Theorem
    • Proved that Cantor’s Contiuum Hypothesis is consistent with the axioms of set theory.
  • David Hilbert
    • One of the greatest mathematicians in history.
    • Pioneered many methods of proofs.
    • Famous cries of “In mathematics, there is no ignorabimus” (no “we shall not know”).
  • Gottfried Leibniz
    • Most important logician after Aristotle and before Boole.
    • Discovered calculus ratiocinator.
  • Giuseppe Peano
    • Created a notation for first-order logic and a system of axioms for arithmetic, that is still in use.
  • Henri Poincare
    • The greatest mathematician of his time, and the last universal mathematician.
    • Made many contributions to many different fields in mathematics.
  • Bertrand Russell
    • Main character in the book (or one of).
    • One of the greatest contributors in mathematical logic, along Aristotle, Boole, Frege, and Godel.
  • Alan Turing
    • The father of computer science.
    • Creator of Turing Machines - whether, given a logical system, there is an algorithm for deciding whether a proposition is provable within the system or not (his answer is “no”).
  • John Von Neumann
    • Contributor to many areas of mathematics.
    • Called “the last of the great mathematicians”.
    • One of the first great computer scientists.
  • Ludwig Wittgenstein
    • Writer of Tractatus Logico-Philosophicus.