This post is not an attempt to answer the question of the subject, but merely to discuss related topics. The idea dates back to the time when I first learned what is the likelihood function in probability theory.
When one encounters a philosophical problem, it is often the case that the problem is stated clearly enough to cook up a paradox in your mind, but vaguely enough not to see any reasonable way to approach the solution. A famous example is the free will problem: whether or not people have an actual, true free will. The problem is (or seems to be) clearly stated: we all (think that we) know what is meant by free will and so it makes sense to ask whether it exists; also there are two opposite intuitions about the issue: on the other hand we do experience (something that we interpret as) free will, but on the other hand aren’t our brains just a highly deterministic physical system? But this question (stated this way) gives no clue on what is the correct way (if any) to approach the solution.
I often think that many such philosophical questions might be solved by practicing the specific science which deals with the concepts in question. For instance the aspect of free will probably belongs to cognitive science and psychology. (Some people think that it belongs to religion; fine with me.) Thinking that way, I believe that many (but not all) philosophical problems concerning mathematics, can be solved mathematically. Gödel’s theorems provide a good example: they have certainly deeply influenced the philosophy of mathematics, not to mention many other branches of philosophy.
I would like to discuss the so called Hume’s argument given by David Hume. As the free will problem, it is also a philosophical problem stated such that the paradox is evident. How do I know that if I now jump out of the window, then I will fall down? Maybe I will fly up to meet the gods? How do we know that the Sun will not turn into an elephant tomorrow? Or rhinoceros? How do we know that all the rhinoceroses in the world will not turn into elephants tomorrow? We could refer to our scientific theories, but how do we know that they (will) hold (tomorrow)?
I’ll give two reasons why the Hume’s argument is not worth taking seriously (in some sense) and one of the arguments is mathematical and other is philosophical (I guess). The “in some sense” above takes place because of course, Hume’s right. We cannot know for sure. But since we do not know for sure, we should attach probabilities to the various possible events, do not we? By Hume, no! Because the probability that the Sun turns into an elephant any second might be 0.999! It might be the case that we have just been unbelievably lucky in that the Sun is still the Sun! It is like flipping a coin: even if you get 10000 heads in a row, the probability of getting a tail on the next move is still one half, provided the coin is fair…wait!…how did we know that the coin is fair? Did we flip it before to measure that?
Fortunately, there is the concept of likelihood. Suppose you have a coin and you do not know whether it is fair or not, and you do not know what is the probability of getting tails or heads. After you get 10000 heads in a row without getting any tails, you can ask: what is the likelihood that the probability of tails is 0.5? Whatever it is, the likelihood must be less than the same for 0.25 and so on.
Mathematically, writing X for the set of observed data and Θ for the set of parameter values, the expression P(X | Θ), the probability of X given Θ, can be interpreted as the expression L(Θ | X), the likelihood of Θ given X. The interpretation of L(Θ | X) as a function of Θ is especially obvious when X is fixed and Θ is allowed to vary. (I copypasted this paragraph from wikipedia, forgive me.)
When I learned about this, I immediately saw a solution to Hume’s argument. Thus, as we observe the Sun rising every day, the likelihood for the “parameters” (the real and possibly absurd laws of nature) to be such that the probability of the Sun rising is 1 is approaching 1. Similarly the likelihood for it to turn into an elephant with probability 1/2 approaches zero.
The difference of this to the response “the probability of sun rising is approaching 1″ is that we cannot know the “real” probability, but we can calculate the likelihoods.
For the sake of completeness I would also like to give the obvious philosophical (pragmatic) response to the Hume’s argument. Suppose the world is a totally arbitrary place and tomorrow something very weird could happen. Thus there are two options: either everything will be as we normally predict, or not. So we would like to prepare ourselves to both events. Next we notice that we can be prepared for the “normal tomorrow”, but we cannot be prepared for the “absurd tomorrow”, simply because we have no idea what it could be. So the only reasonable thing to do is to prepare ourselves for the “normal tomorrow” and hope that it will happen. Thus, believe we in the laws of physics (or whatever) or not, we should act as we believed them to be true. Sounds like a pragmatists argument, that is why I put “pragmatic” into brackets above. But I claim further that it is not only a pragmatic argument (that’s why brackets!). Namely if you see yourself acting according to certain believes, shouldn’t you consider believing them already?
Probably the following argument is fallacial, but you can see what I mean. If you are say a platonist and do not want to base your believes on “what is the most rational way to act” -paradigm, you can use Occam’s razor to infer that “if we know anything to be true at all, then we know the laws of physics are not a matter of chance”, whatever “true” means for you. Then apply Occam’s razor and infer “We do know anything at all” and then apply modus ponens to infer that the laws of physics are not a matter of chance…
I still prefer the mathematical argument, although it says the same, I think.