Irreducible Complexity in Pure Mathematics Gregory Chaitin
My story begins in 1686 with Leibniz's philosophical essay Discours de métaphysique, un English, Discourse on Metaphysics, where Leibniz discusses how we can distinguish between facts that follow a law, and lawless, irregular, chaotic facts. How can we do
Leibniz's idea is very simple and very profound. It's in section VI of the discours. It's the observation that the concept of law becomes vacuous if arbitrarily High mathematical complexity is permitted, for then there is always a law. Conversely, if the law has to be extremely complicated, then the data is irregular, lawless, random, patternless, and also and irreducible. Page 2
Appendixes
What is Gödel's proof?
Let's start with the paradox of the liar: "This statement is false." This statement is true if and only if it's false, and therefore is neither true nor false.
Now let's consider "This statement is unprovable." If it is provable, then we are proving a falsehood which is extremely unpleasant and is generally assumed to be impossible.
The only alternative left is that this statement is unprovable. Therefore it's in fact both true and unprovable, and mathematics is incomplete, because some truths are unprovable.
Gödel's proof constructs self-referential statements indirectly, using their Gödel numbers, which are a way to talk about statements and whether they can be proved by talking about the numerical properties of very large integers that represent mathematical assertions and their proofs.
And Gödel's those proof actually shows that what is incomplete is not mathematics, but individual formal axiomatic mathematical theories that pretend to be theories of everything, but in fact failed to prove the true numerical statement "This statement is unprovable."
The key question left unanswered by Gödel: is this an isolated phenomenon, or are there many important mathematical truths that are unprovable? Pages 9-10
Extensive computer calculations can be extremely persuasive, but do they render proof on necessary?! Yes and no. In fact, they provide a different kind of evidence. And important situations, I would argue that both kinds of evidence are required, as proofs may be flawed, and conversely computer searches may have the bad luck to stop just before encountering a counter- example. Page 13
Love and Math: The Heart of Hidden Reality
Edward Frenkel
(New York: Basic Books, 2014)
We envisioned it as an allegory, showing that a mathematical formula can be beautiful, like a poem, a painting, or a piece of music. The idea was to appeal not to the cerebral but rather to the intuitive and visceral. Let the viewers first feel rather than understand it. We thought emphasizing the human and spiritual elements of mathematics would help inspire viewer’s [sic] curiosity. Mathematics and science in general are often presented as cold and sterile. In truth, the process of creating new mathematics is a passionate pursuit, a deeply personal experience, just like creating art and music. It requires love and dedication. Pages 232-3