And He took the five loaves and the two fish, and looking up to heaven, He blessed and broke and gave the loaves to the disciples; and the disciples gave to the multitudes. So they all ate and were filled, and they took up twelve baskets full of the fragments that remained. – Matthew 14.
The logician Willard Quine defined a paradox as an “absurd” statement backed up by an argument.
The famous result of Banach and Tarski definitely counts as a paradox by this definition. They proved that it is possible to take a unit sphere (a ball with radius 1), divide it into five pieces, then by rotations and translations reassemble it into two unit spheres.
The Liar Paradox is simple enough to explain – is the following statement true or false?
This statement is false.
If it’s true then it’s false, but if it’s false then it’s true … nothing works.
In my not-so-humble opinion, most (maybe all) paradoxes are the last step in a proof by contradiction that some unstated assumption is false.
In this case, the assumption is that the above statement is meaningful – is either true or false. The assumption is false, the statement is meaningless. End of paradox.
In the last post there was plenty about Gödel but not much about Grammar and Go-the-game. Time to pay my debt!
Basically I said that Gödel’s results proved that no fixed set of facts and rules can on their own form the basis of mathematical knowledge. I said that hard-earned experience is indispensable. That mathematics is ultimately an experimental science. (This is not the usual take on Gödel’s work.)
More than eighty five years ago Kurt Gödel proved, roughly speaking, that no fixed set of a formal facts (like 23+14=37) and rules (like x+y = y+x) can establish the truth or falsity of every theorem about arithmetic over the counting numbers.
This result, known as Gödel’s Theorem, has a lot of formal and informal consequences. It means there is no computer program that can infallibly decide whether or not a statement about arithmetic is true or false. It means we will never know everything about arithmetic, though we may know more and more as time goes on. It means, however, that this knowledge will not come about purely as a result of manipulating formal facts and rules. We will have to rely on other sources, including experiment.
Even more interesting is the fact that this situation – the limits of facts and rules – reappears in other domains, including games, natural language, and even psychology.
I’ve already described the relatively simple Monadic Hybrid Calculus that allows you to say simple things formally without bound variables. For example, sG may mean “Socrates is Greek”, [G]M “all Greeks are Mortal”, and the conclusion “Socrates is Mortal”, sM.
The MHC has a big brother, the Hybrid Predicate Calculus (HPC), which (apparently) has the power of full predicate logic. But at a certain point, it gets weird!
A while back I decided that it would easy and useful to video record some lectures. Little did I know!
I like true/false exam questions and through my career have thought up hundreds of them. Every now and then, for comic relief and to inflate the grades, I include some that are ridiculously easy. However, I’ve never found one that is so ridiculous that everyone gets it right. I always had a few takers. Here are some of my favourites.