Today I am going to spill the beans about a huge gap in our understanding of the world and of mathematics. In my view this is probably the biggest stumbling block in our understanding of the philosophy of science and a massive obstruction in the way of advancing mathematics. That stumbling block is probability.
I recently read an article in a lovely little book entitled "What we believe but cannot prove" (ed. John Brockman). The article suggested a simple problem in mathematics that the author (Freeman Dyson) believes we will never resolve. The problem is the following: Consider numbers in the series 1,2,4,8,16,32 (where each time you multiply by 2) and in the series 1,5,25,625 (where each time you multiply by 5). Can you obtain a number in the second series by reversing the order of a number for the first series (i.e 1,2,4,8,61,23)?
Dyson argues that (upon the assumption that the digits in the first series behave kind of randomly) that the answer is probably no (With less than 0.001% chance) but that this must also be unprovable if the assumption is correct. It seems a plausable argument but it got me thinking because we mathematicians do not have a model of probability.
Certainly there are probability theorists and statisticians but they are just doing the best they can of a bad job. I'll do my best to explain the extent of the problem in terms not requiring a mathematical education.
First, there is an assumption that is necessary. A few mavericks do not agree with the assumption but the majority of mathematicians agree with it and use it every day.
If you have a set of sets where each one has something in it and no two share an member then: There is another set containing exactly one thing from each of those sets. I'm hoping that seems obvious to you. Its called the axiom of choice because it allows you to make lots of choices all at once.
Probability theory works quite well when we have a finite number of things that are random. It only gets into trouble when you do something like flip a coin infinitely many times. This matters because in the philosophy of science this sort of thing is going on all the time. It also matters in mathematics because damn near everything we do these days can be linked to such possibilities.
Now suppose I promise to give you £100 if the sequence of coin tosses has a certain property X. How much should you pay to take up the bet? If I said the coin sequence must contain a head, then you'd bet anything up to £100. If I said that the coin sequence must never contain a string of heads longer than any string of tails before it then you shouldn't bet (this is almost impossible). You would be justified in wanting a cut and dried betting strategy for any property X so long as it's properly defined (no abiguity is allowed). Here's the rub: Assuming the axiom of choice there is a property such that there is no sensible amount of money to bet. That doesn't mean you should bet £0 or £100. It means whatever money you'd bet you can't justify it. Standard probability theory cannot deal with these situations. This is the main reason some mathematicians doubt the axiom of choice. But this axiom does really seem obvious.
I can't tell you what the problem really is or how it should be resolved. But I can give you my suspicions. I think our notion of probability doesn't work with our philosophy of science. I think the notion of infinitely many random events is the notion causing trouble. I shall write another post explaining why I think Occham's razor sheds light on this (and taking my old post further at the same time).