 No matter how hard I try my Kurt Gödel jokes always seem incomplete or recursive very much like my Kurt Gödel jokes Kurt Gödel was an Austrian born philosopher and mathematician back when doing philosophy while you were studying for a mathematics degree was totally Trendy and socially acceptable. He made one of the most important and lasting contributions to well both really It's not often that you go up to a chalkboard to do some math and end up proving something that has massive implications for philosophy I mean, I usually just forget how partial fraction decomposition works and end up embarrassing myself But Gödel's proof actually demonstrates something that makes people who know advanced mathematics Uncomfortable a dark secret at the center of the language that we used to describe both the motion of the planets and the operation of computers On some level mathematics is fundamentally broken What I'm talking about is Gödel's incompleteness theorem a sort of black hole at the center of mathematics from which no Calculator can save you around the time that Gödel started thinking about these things There was a movement to define mathematics in the simplest most rigorous terms possible This was mostly because of discoveries of really embarrassing inconsistencies in previous mathematical work Seemingly well constructed proofs that ended in contradictions if you looked hard enough This is why mathematicians are paranoid about stuff like dividing by zero you look the other way for half a second It's suddenly one equals two black as white dogs and cats living together mass hysteria to prevent this sort of thing from happening again Many were working on establishing a rigorous foundation for basic arithmetic most notably Bertrand Russell who published a massive Three-volume work entitled the Principia Mathematica, which is absurdly rigorous how rigorous well It takes until halfway through the second book to prove that one plus one equals two Russell actually put a footnote next to this saying this equation has proven to be useful from time to time But even though this was one of the best foundations for mathematics to date There was still something not quite right about it There were a couple assumptions that Russell had to make in order to get the rules of a system to work the way that they needed to Assumptions that he figured that someone would eventually prove were entirely necessary and valid And then we'd finally prove that math worked all the way down to its most fundamental principles and then Gödel published his incompleteness there. That's what everybody calls it now But he actually called his paper on Formally undecidable propositions in the Principia Mathematica and related systems, which must have stung a bit for Russell It sounds fairly complicated, but unlike a lot of really advanced mathematics. It's fairly intuitive to understand So there are certain statements in English that you can't really say are true or false statements like this sentence is false if you say that it's true then it becomes false if you say that it's false Then it becomes true any attempt to confirm it as true or false ends in failure. So it's really undecidable Undecidability is really the worst possible thing that could happen to a mathematical system It's okay to have things that are outside of the system like 2 plus Scorpio But every statement that you can make inside the system should be either explicitly true or false It should be what mathematicians call complete Gödel proved mathematically that not just Russell's system But any system of sufficient complexity to add numbers together Necessarily contained a statement like this that was undecidable That means that every single system of mathematics that can do basic arithmetic has an embarrassing Undecidable statement like this lurking somewhere inside something that's called the Gödel sentence The Gödel sentence is usually some variant of the phrase this statement is undecidable in this system Just like that statement in English, you can't really say if it's true or false This is kind of depressing for mathematicians because it means that math can't ever be totally vindicated That no useful system of arithmetic can ever be complete that as soon as you can add two numbers together your buckets been kicked But it also has some broader implications like fundamentally a computer functions by adding numbers together So a computer is a mathematical system that can add numbers What does Gödel have to say about that? In fact the incompleteness theorem is intimately linked to an idea in computer science called the halting problem The fact that you can't really calculate whether or not a particular program is ever going to end No computer can process every possible instruction in a finite period of time There's always some input that will just leave it crunching numbers forever think about that for a second No matter what computer you're using Mac PC cell phone, whatever there is a number that will break it All sorts of other stuff functions on mathematical principles as well I've talked before about how your brain is very much like a computer It's been theorized that brains have Gödel sentences to sci-fi author Philip K. Dick once said there exists for everyone a Sentence a series of words that has the power to destroy them. That might be totally accurate There are also systems of ethics that function on algorithms there are ideas that the fundamental nature of the universe is mathematical and Every time you make a list if you allow an individual list item to reference the list as a whole you might be in trouble Or maybe mathematics doesn't need to be rigorously defined in order to do cool things for us Maybe Godel's incompleteness theorem is just an interesting special case. Please leave a comment and let me know what you think Thank you very much for watching. Don't forget to blah blah subscribe blah share and I'll see you next week