The issue is too involved to give a full justification of induction here, but I will try to give a very general idea. (This was on my mind a while back as I got asked about it in an interview.)
Even if we don’t assume that we can apply statistics in the sense of using past observations to tell us about future observations, or observations about some of the members of a group to tell us about other members of a group, I suggest we are justified in doing the following.
Given a reference class of possible worlds in which we could be, in the absence of any reason for thinking otherwise, we are justified in thinking that any world from the reference class is as likely as any other to be our world. (Now, this may seem an attempt to sneak statistics in—but, really, all I said was that if we have a list of possible worlds that we could be in, and we don’t know, then we our views on probability merely indicate that we don’t know.)
The next issue is how this reference class is constructed—more specifically, how each member of the reference class is constructed. It may seem to make sense to construct each world by “sticking bits of space-time together”, but I suggest that this itself implies an assumption. After all, many things in a world can be abstract entities: How do we know what appear to be basic things aren’t? Furthermore, why build the reference class like that? What is the justification? It also forces a particular view of physics onto us. What about views of physics were space-time may not be fundamental? They would be eliminated from the referenc class.
The only justifiable way of building the reference class is to say that the world is an object, and that the reference class of worlds is “Every formal description of a world”. Rather than make assumptions about what space is, what time is, etc, we should insist that the description merely describes the world, including its history as an object. Such a description is out situation at any time. At any time, we live in a world which has some description, and all I am saying is that the reference class is all possible descriptions. Now, it may seem that I am trying to sneak laws of nature and regular behavior in by the backdoor here, but I am not: If we can’t demand that a world be formally describable we are being incoherent. If we can’t demand that the reference class contains every such formal description, surely the most general idea we could have of building a reference class, we are imposing something more specific, with all kinds of ontological assumptions, on it.
Now, if we see regular patterns in a world, this justifies expecting those patterns to continue. For a pattern to be made by the description specifiying each element individually will take a lot of information. Therefore, the description must be highly specific and only a small proportion of possible world-descriptions in the reference class will comply. On the other hand, if the pattern is made by a small amount of information in the world-description, which describes the entire pattern, this is much less specific and a greater proportion of possible worlds will comply: We are demanding less specific information content in a possible world for it to be ours. Therefore, if we see a regular pattern, it is much more likely that our world is one of the large proportion of worlds where that pattern results from a small amount of information in the description that one of the much smaller proportion of worlds where it results from a much greater amount of information in the description.
A pattern which results from a small amount of information in the world description should be expected to be continued, because that is the very idea of a pattern generated by a small amount of information. For example, if you find yourself living in a world which looks like part of the Mandelbrot set, you should think it more likely that you live in a world where the Mandelbrot rule is part of the description of that world and expect to see more Mandelbrot pattern in every places.
Therefore, patterns should be expected to be continued.
I also suggest that Hume’s problem of induction only appears in the first place because people have the misplaced idea that the reference class should be built up second by second, from the point of view of a being inside time, when it should ideally be built from the point of view of an observer not restricted in that way.
I’ll give a reworded version of this, to take it out of the context of a belief system with which we are familiar. I’m not intending any mockery by this: It is to make a point about the claims and the evidence:
“Let us stipulate that, on Paris Hilton’s birthday, a prominent Paris Hilton admirer claims to have suddenly become a prophet. They go on television and answer questions on all topics. All verifiable answers they give, including those to NP-complete questions submitted for experimental purposes, turn out to be true. The new prophet asserts that Paris Hilton is a super-powerful being sent here from another world, co-existing in space with ours but at a different vibrational something or whatever. Paris Hilton has come to show us that celebrity can be fun. The entire universe is built on celebrity power. Madonna tried to teach us this when she showed us how to Vogue but we did not listen and the burden of non-celebrity energy threatens to weigh us down into the valley of mediocrity when we die instead of ascending to a higher plane where each of us gets his/her own talkshow with an army of smurfs to do our bidding. Oh, and Sesame Street is being used by the dark energy force to send evil messages into children’s feet. (The brain only appears to be the source of consciousness: Really it is the feet. Except for people with no feet. (Ah! I bet you thought I didn’t think of that.) Today’s lucky food: custard.”
There is a website where you can suggest questions to put to the new prophet. Not all submitted questions get answered, due to time constraints, but interesting ones do get in reasonably often. Are there any questions you’d like to ask?”
The point I am making here is that the above narrative is absurd, and even if he can demonstrate some unusual ability with predictions or NP problems (and I admit the NP problems would really impress me), there is nothing that makes that explanation more sensible than any number of other stupid explanations. Nor does he have an automatic right to be believed: His explanation is just too stupid.