In physics there is the Zeroth Law of Thermodynamics, a transitivity of temperatures. If A and B have the same temperature, and B and C have the same temperature, then A and C have the same temperature. (A could be melting ice, B expanding mercury, and C bourbon at the correct temperature.) The name for the Law was devised after the names for the three laws of thermodynamics had become standard.

How much do you need to know about science to begin learning about the philosophy of science? Some but not a great deal if you are interested in very general metaphysical questions about e.g. the nature of explanation, laws and causation, and in very general methodological questions about how scientific theories are confirmed and refuted. You'll need to know quite a lot more if you are interested in understanding more specific foundational questions about the interpretation of quantum mechanics or are worrying about the nature of natural selection.

Similarly, how much do you need to know about mathematics to begin learning about the philosophy of mathematics? Some but not a great deal if you are interested in very general metaphysical and epistemological questions about e.g. the nature of numbers and the nature of our knowledge of such things (if they are "things"). Quite a lot more if you are bugged by more specific questions about how we are to settle axioms for set theory or to decide whether category theory offers a new kind of foundational framework for mathematics.

Most (but not all) beginners get into the philosophy of science via the pretty general metaphysical and methodological questions -- on which typical introductory books concentrate. Likewise most but not all beginners get into the philosophy of mathematics via the pretty general questions which require little mathematical background to grasp. And you can read a really excellent introduction like Stewart Shapiro's Thinking about Mathematics, which concentrates on the more general questions, without knowing much mathematics at all.

For more on this general theme, see also my answer to Qn 2938.

You also mention Russell. Well, you don't need a great deal of mathematical background to read The Principles of Mathematics -- but you do need a lot of historical background to understand what on earth is going on in some of Russell's pretty puzzling philosophical discussions: it is a confused and confusing great shambles of a book. That's not the place to start! As for Principia that is best admired from a great distance, and is pretty much now a historical curiosity -- a noble failure.

You are right that the points in a 4 inch line segment can be put into one-to-one correspondence with the points in a 2 inch line segment. Think of a line swinging through both line segments, the way a door swings through a shorter path nearer its hinge and a longer path further from the hinge. The swinging line matches any point in one with a point in the other. Therefore, they have the same number of points--an infinite number. However, that is not a strike against the claim that the line segments have different lengths. The points are dimensionless, and the length of a line segment is not a function of the number of its dimensionless points. So the 4 inch line segment is still twice the length of the other.

There are a few kinds of support. One is that one can prove certain special cases of the conjecture that seem inherently unrelated, so one thinks that these special cases must really be true because a certain generalization of them is true---and that's what one conjectures. But conjectures are often based upon a dim and hard to express appreciation for "what is going on", so that it just sort of seems as if the thing ought to be true. One can sometimes give reasons to think things ought to work out that way, but they wouldn't be the kinds of reasons that would count as a proof.

There are two questions here that need to be distinguished: (1) Does the decimal expansion of pi contain a large but finite string of consecutive 7's--say, 1000 consecutive 7's? (2) Does it contain an infinite string of consecutive 7's?

The second question is the easier one. The only way that pi could contain a string of infinitely many consecutive 7's is if all the digits are 7's from some point on. And, as William has pointed out, that can't happen because pi is irrational.

But the first question is harder. The digits of pi "look random." Imagine a number whose digits are generated by some random process--for example, we might roll a 10-sided die repeatedly to generate the digits. One could compute the probability that a string of 1000 consecutive 7's appears in the first n digits of this number. For n =1000, this number would be extremely small--you'd have to roll 1000 consecutive 7's on your first 1000 rolls, and that's very unlikely. But as n increases, the probability increases, and in fact as n approaches infinity the probability approaches 1. Thus, if you generate an infinite string of digits this way, then the probability that you will eventually get 1000 consecutive 7's is 1, although that doesn't mean that you're guaranteed to get 1000 consecutive 7's. (For more on the difference between "probability 1" and "guaranteed to happen," see this question. Well, actually the discussion there is about the difference between "probability 0" and "guaranteed not to happen," but the idea is the same.)

This heuristic argument makes it seem very likely that the decimal expansion of pi contains 1000 consecutive 7's. But it doesn't actually prove anything--after all, the digits of pi are not generated by a random process, they just "look random" to us. If pi is a normal number, then it must contain 1000 consecutive 7's. But proving that pi is normal is an unsolved problem.

It doesn't imply that, because that's false. If pi had an infinite number of the same digits repeated, it would be a rational number. But it's an irrational number. See Wikipedia's article on pi.

You certainly don't need to be able to do original research in maths to be able to work on the philosophy of maths. But you will need to be able to follow whatever maths is particularly relevant to your philosophical interests. How much maths that is, which topics at which levels, will depend on your philosophical projects. For example, compare and contrast the following questions (not exactly a random sample -- they all happen to interest me!):

1. "Is our basic arithmetical knowledge in any sense grounded in intuition?" Evidently, you don't need any special mathematical knowledge to tackle that.
2. "Can a fictionalist about mathematics explain its applicability?" Again, I guess that acquaintance with the sort of high school mathematics that indeed gets applied is probably all you really need to know to discuss this too.
3. "Just what infinitary assumptions are we committed to if we accept applicable mathematics as true?" Here you do need to get more into the maths, and know quite a lot about what can be reconstructed in various weak subsystems of full analysis, and about what infinitary assumptions these subsystems need.
   
4. "Is there a unified justification for all the axioms of the standard set theory ZFC?" You better know a bit of set theory to tackle this -- but you perhaps needn't know much about e.g. large cardinal axioms that take us beyond ZFC!
5. "Do we need axioms that take us beyond ZFC? -- and if so, what?" For this, by contrast, you will need to know more about the fancier reaches of set theory.
6. "Does category theory in any sense provide 'foundations' for mathematics rivalling the set theoretic approach?" Well, plainly, you better know some category theory to tackle this one.
   
7. "Why is the question whether P = NP so hard?" And you'll need to know qutie an amount about complexity theory for this one.
   
8. "How satisfactory is Lakatos's model of the growth of mathematical knowledge?" Here, by contrast, you'd be better equipped if you know a little about a lot of mathematics, rather than a lot about a little, so your discussions aren't to suffer from an impoverished diet of examples.

And so it goes. What maths you need to know will vary a lot with which philosophical questions bug you. And sometimes it is difficult to tell in advance, when you start thinking about a topic, just how embroiled you will need to get in the mathematical nitty-gritty. But that's part of the fun.

You are reasoning correctly--mathematics deals with possibilities and physics with actualities (even though in quantum mechanics these are probabilistic). Theory in physics is often expressed mathematically, but that does not make it mathematical knowledge. Some theoretical advances in physics can come from working in an armchair and extending the mathematical implications of (already accepted and contingently true) theory. The actual status of mathematics (a priori or not) is debatable (Quine etc claiming that mathematics is an empirical theory like any other). But you are correct that physics is not mathematics, and the sort of evidence that confirms physical theory is not (or perhaps, not entirely) the evidence (or other considerations) needed to confirm mathematics.

If I remember correctly, and I may well not, David Lewis explicitly argues that there are uncountably many propositions in Plurality of Worlds and uses this as an argument against any view that would try to reduce propositions to sentences. At the very least, he does consider this issue. So here's an argument that I think I remember from that book that we can consider, anyway. It is based upon the claim that, for any real number x, there ought to be a proposition---a possible content of thought---that I am shorter than x inches tall. Indeed, each such proposition could be expressed by a sentence. All we have to do is give the real number x a name, say, "Fred", and then the proposition will be expressed by the sentence "I am shorter than Fred inches tall". But if so, then there are at least as many propositions as there are reals.

The key to this argument, note, is the observation that the claim "For every proposition p, there could be a sentence S that expressed it" is much weaker than "For every proposition p, there is a sentence S that expresses it". It's the latter that the reductionist needs. But the argument needs only: If there could be a sentence S that meant that p, then there is a proposition that p---assuming that other, uncontroversial conditions required for the proposition to exist are met. E.g., there might have been propositions other than the ones there are, because there might have been people other than the ones there are, and then there would have been propositions about them that there in fact aren't. But that's not what's happening in this case.

That there should be uncountably many propositions for this kind of reason does not, so far as I can see, imply anything bad about the possibility of communicating. Each actual (meaningful) sentence corresponds to---that is, expresses---just one proposition (modulo worries about context). But there are lots of propositions that are not, in fact, expressed by any sentence.

My own preference, for what it is worth, is to refuse to talk of propositions, period. But that's a different matter. If we are going to talk of propositions, then it actually seems quite hard, for the reasons just given, to keep them countable.

If the term 'proposition' is used to mean a sentence -- a string of symbols, be they spoken, written, gesticulated or whatever -- then I suppose there could be uncountably many propositions if we allow there to be propositions of infinite length. In that case, one ought to be able to diagonalise on them just as one does with the infinite decimal expansions of the real numbers. But I think it's reasonable to stipulate that we're only going to countenance finitely long sentences. After all, they wouldn't be much use in communication, if you could literally never get a sentence out.

Alternatively, if the set of symbols is itself uncountable, then that will certainly lead to an uncountable infinity of strings of such symbols. But it seems reasonable to stipulate against that case too. Communication would once again be thwarted, because we don't seem to have the perceptual capacity to discriminate between uncountably many different symbols -- indeed, our discriminatory abilities probably only extend to a finite (though no doubt very large) number.

But 'proposition' has been used in other ways over the years. So what if we take it to mean not the string of symbols itself, but rather the non-linguistic thing that such a sentence is expressing: a thought, or something of that kind? Well, then maybe there could be uncountably many propositions: we'd need to have a great deal more spelt out about the details of this notion before we'd be in a position to assess that. But (leaving aside any eccentric conceptions of sentences and symbols like those just discussed) there could no longer be a one-to-one correspondence between propositions and sentences. Which would mean that sentences wouldn't do a very good job of expressing propositions -- which is, after all, what they're there for. For there to be uncountably many propositions, and yet only countably many sentences, a single sentence would need to correspond to more than one proposition. But then your interlocutor would have no way of judging which of these various propositions you were actually intending to communicate.