Causality, Randomness and Induction

And having inquired into the psychological nature of our use of induction, what should our conclusion be? Is it rational to continue using induction while believing that it has no adequate justification?

"Rational" may simply mean that we find inductive inference useful so far, even if we have no good reason to think that the next time it will work. Especially since there is nothing better. If I need something to bang a nail into the wood, and if I don't have a hammer, but I have a stone, it is rational to use the stone. It would be better, perhaps to use a hammer, but since I don't have a hammer, such is life. "The Hindoo does the best he candoo". Of course, the question remains, what is supposed to be the matter with induction that it needs justification? We don't feel that way about deduction. Why induction? What is it that induction needs that deduction apparently has?

By the way, Hume not only thought that we had no choice but to use induction in the sense that there was no better choice, he seems also to have thought that we literally could not choose not to use inductive inference since it was (to be anacronistic) hard-wired into us. So that is why we use induction (in the psychological sense of "why?" which asks for an explanation, and not a justification of our practice). And, as I said before, it was only the answer to that sense of "why?" that we could expect to get. The justification of induction, if there is one, was beyond the limits of human knowledge, just as Hume thought that it went beyond the limits of human reason to expect an answer to the question, do we know there is an external world. The justification of our belief in induction and the externiality of the world are the two mysteries which we can never expect to crack.

The justification problem isn't specific to induction, it's general. All justifications which are neither infinite chains nor circularly dependent, have unjustified bases.

Well, that really depends on whether you hold that justification can be be only foundational (a la Descartes) or whether you think that justification can consist in coherence (a la Quine). See the classic paper by Earnest Sosa detailing and contrasting these two basic view of justification. The Raft and the Pyramid.

http://www3.interscience.wiley.com/journal/120156965/abstract?CRETRY=1&SRETRY=0

Quine, for instance, famously said that in regards to truth, he advocated the correspondence theory. But in justification, he advocated coherence. An admirable application of the coherence view of justification is to be found in the little book by Quine and Joseph Ullian called, The Web of Belief.Your view of justification strictly pertains to the foundational theory of justification, and is one of the motives for turning to a coherence view of justification which is not subject to your objections (if that is what they are).

But I wonder whether even if we accept the foundation theory of justification, when deductive justification (not inductive justification) is vulnerable to your objections. After all, apparently the reason that induction is believed to require justification is only that is lacks what deductive inference is suppose to have. Namely, deductive inference is self-justifying. Deductive inference is truth-preserving, and inductive inference is not. So that inductive inference seems to need justification just because it is not deductive inference. The interesting question is whether that is a good reason to think that inductive inference needs justification. If not, then the problem of induction seems bogus.

What is meant by this idea of a random universe?
A random universe would lack any sort of pattern or order, since everything that happens is simply a random outcome. No life (As we understand it) could ever exist in this universe. Literally nothing would make sense. There are of course studies in quantum mechanics that may suggest randomness (or probability based outcomes), but the general structure of our universe from micro to macro follows the same repeating patterns of interaction, hence allowing us to make sense of it (To a degree).

I would say that the universe is simply meaningless, requiring human culture to give it meaning. Meaninglessness is not the same as randomness; the latter is meaningful, a human conception.
On the other hand, I also think that the meaninglessness experienced--and culturally resisted--by humans is ultimately transcended in a grand mystical sense by notions such as Brahma, the ultimate ground of all being/reality.