It is not immediately obvious why there should be a paradox here. Not to encourage any invidious comparisons, but I am a contingent being who has an ongoing involvement with the world. To say I am contingent is to to say that I might not have existed; but that is no problem so far as my involvement, since fortunately for me I do exist in actuality. Suppose that there is a God and He has ongoing involvement with the world and indeed, unlike me, he created it too. The additional supposition that, again unlike me, he could not have not existed, seems to pose no additional barrier to His involvement. To express the matter in the possible world talk that philosophers like, to be involved with the actual world you must be in the actual world; but your additional presence in other worlds does not prevent this involvement.

I'm not going to comment on infinity, but on the general point. First of all, which existence is in question: a concept or a thing? If a thing, it would seem possible that there could be something that exists but that we cannot conceptualise. So here there would be no question of a definition, yet the thing exists. If a concept, well, one in fact argue that some concepts must be undefined, or else the chain of definitions would have to go on for ever. (Though circles might be another possibility.) But if it is OK, indeed obligatory, that some concepts are undefined, then maybe it is OK for a concept to be indefinable. There is a great deal of philosophical excitement over the question of what it is to grasp a concept, but I think that it is widely agreed that definition is not the only way in.

I think that almost all of the central figure in the western philosophical canon would have considered themselves philosophers, though not necessarily academic. Some of them were also notable for other things -- for example Aristotle for biology and Descartes and Leibniz for mathematics. There have also been great scientists who might not have considered themselves philosophers but did some significant philosophical writing -- for example, Galileo, Boyle, Newton and Einstein. But perhaps one shouldn't put too much emphasis on disciplinary affiliation. And even if one is in the first instance a philosopher, it is a real plus if philosophy is not the only thing you know about since (among other things), so much philosophy is Philosophy of X, where X is some other discipline.

Knowledge often has practical value, and I would go along with the idea that it sometimes has intrinsic value too: it is good in itself. But I'm not convinced that all knowledge has intrinsic value. For example, I'm not sure that there is even a little intrinsic value to knowing exactly how many hairs I have on my head. So there might be some bits of self-knowledge that are not intrinstically valuable. And I agree with you that there may be some bits of self-knowledge that have negative practical value. For example, knowing about some of one's faults might make them worse. (Knowing how often one tends to stammer might make one stammer even more often.) So it looks like there could be cases of self-knowledge with no instrinsic value and negative practical value.

This is a classic pr0blem in philosophy, the problem of 'spectrum inversion'. Even if you see blue like I see red, and vice versa, it is very difficult to see how we could ever tell. I cannot see your experiences, and you would use the words 'blue' and 'red' the same way I do, since you were taught to say 'blue' when you saw blue objects and 'red' when you saw red objects, even if your experiences were different. It's interested that spectrum inversion is different from color-blindness. There we can tell, because color blind people can make fewer discriminations.

In philosophy it's often a good idea to begin with a clear case where we are particularly confident about what the right answer is, then move to subtler cases. If the question is whether water ever floats, the clear case is ice.

What makes you the same person over time may not be that there is some essence that you have at each time, but rather that your different 'slices' are related to each other in the right way. Think of climbing rope. Modern synthetic ropes do have a single filament running their entire length, but the old-fashioned ropes are made of of many relatively short fibers woven together. So there is no fiber that runs the entire length, but still different 'slices' of the rope are all part of the same rope, because of the way they are connected to each other. Admittedly, just what 'related to each other in the right way' comes to in the case of people is a more difficult question. If you want to pursue the topic of personal identity, a good place to start is John Perry's A Dialogue on Personal Identity and Immortality .

Scientists do seem to have a strong preference for simple theories, though the relevant concepts of simplicity are not at all easy to analyze. It is also very difficult fully to justify a simplicity preference, since scientists seem to prefer simpler hypotheses because they think them more likely to be correct, not just because they are more economical, and it is hard to see how we could justify the claim that the universe is more likely to simple without begging the question. But I don't think that a claim that assigns an infinite quantity is automatically simpler than one that assigns a finite quality. 'There are infinitely many rocks' mayh not be relevantly simpler than 'There are 437 rocks.' Of course there many more claims assigning specific finite numbers of rocks than there are claims assigning an infinite number of rocks (infinitely many more, in fact), but I think the relevant comparison is between specific claims.

I'm with you: I don't see how indeterminacy helps with free will. That's what makes free will such a tough nut: we seem to lose either way. If everything is determined, then it looks like we don't have free will, because we could not have done otherwise that what we do. If not everything is determined, then this might mean that some of our actions have a random element, but then these actions hardly seem ours: they certainly don't seem to involve any more free will than under determinism. It's heads or tails; if it's heads, we lose; if it's tails, we lose; so we just lose.

I don't think scientific research is algorithmic, unless you use the term so broadly so that any non-random behavior counts. Research certainly doesn't follow rules that the scientists can articulate, and I think there are reasons to believe that they aren't following anything like a full set of tacit rules either. Thomas Kuhn mounts an extended case for an alternative to a rule-based account of research in his classic The Structure of Scientific Revolutions .