"Dedicated to making Likelinesses the entity of prime interest"
Despite being a professional mathematician, I have never felt totally at ease with the theoretical definition of "probability". I know some will view this as being a very unprofessional attitude to take -I should be willing to take definitions as they are, and just argue the consequences and implications in a non-judgemental way- but there we are.
I suppose there are two main reasons for my uneasiness. One is about the terminology, and the other is about the commonest concept of probability: frequentist probabilility.
The problem I have with the terminology is that it is so wide-ranging that it includes things which I simply do not want to call probabilities. For example, if I take a possibly-biased coin and toss it a number of times then I can use, say, the Law of Succession to estimate the probabilities of Heads and Tails. The difficulty is that those estimates are not the probabilities, they are just estimates; yet they meet the technical definition of "probability". Are they "probabilities", or are they not? There is, at the very least, a lack of clarity which ought not exist.
Or take relative frequencies. I doubt whether there is any mathematician alive who would not feel uneasy about calling them "probabilities", yet they do meet the technical definition of being probabilities.
I just wish that those responsible for setting up the modern definitions had used some other term -doing so would not have meant disturbing the theory and the subsequent good works in any way- and then left the term "probability" to be defined in a context-sensitive way.
Let me give you an analogy to what I wish had happened: the concept of distance. When it came to formalising the concept of "distance", that particular word was avoided; instead, the word "metric" was introduced and it was given the abstract definition. This meant that the word "distance" remained free to be used in any context once the metric for that context had been established. Once a metric space has been established then we can, and do, talk about the "distance from x to y" quite freely and without ambiguity to mean d(x,y) where d is the metric.
Large swathes of Applied mathematics are concerned with estimates. Yet I know of no other area where terminological confusion between an entity and an estimate of that entity would be tolerated. Yet there is such confusion when it comes to probabilities.
This confusion permeates not only Applied mathematics but also Pure. If you want examples then I offer two for your consideration. Details lie elsewhere on this site, but let me give you the Perfect Cube Factory and the difficulties Johnson had with his Combination Postulate.
Moving on to my uneasiness with frequentist probabilities, there is a theorem which states that the limit of a convergent sequence cannot be deduced from any finite number of its terms. More than that, that it cannot even be deduced that a sequence is (or is not) convergent from any finite number of terms.
The implications for the concept of frequentist probability are severe: namely, that it is a theoretical construct which cannot in practice be used.
"Really?", I can almost hear you asking. "What have I been using every day of my working life, then?". The answer is that, unless you are a theoretician, you have not been using frequentist probabilities; you have been using estimates of frequentist probabilities.
At this point, I can visualize you shrugging your shoulders and saying "So what?"
The "so what" is that the consequences -for the theory and the practice- are far greater than you might believe.
This site is about sorting out all the above.
Now read the Introduction.