I doubt that "because" is as finicky as you seem to be suggesting it is. I think it's perfectly true that Fred belongs to the set because he is only 14, and it's perfectly true that Fred belongs to the set because he is less than 15. I'm not familiar with any explanatory concept according to which one of those facts about Fred, but not the other, explains Fred's membership in the set. In any case, I'm confident that "because" does not stand for any such concept.

If there is a paradox here, I don't think it will have anything to do with a conflict in the conditions for set membership. Let's leave aside that there may be sorites-style paradoxes arising from the vagueness of the predicates "young girl" and even "female human." I suspect that those paradoxes can be solved in the "epistemicist" way (see this link ). One and the same individual can possess various mutually consistent properties: she can be a young girl (at a specified time t ), a female human being (at any time during her existence, including at time t ), and so on. So Miss X can belong to the set of girls who are young at t , the set of female human beings, the set of human beings, the set of mammals, the set of things referred to by you in your question above, etc. She would belong to each of those different sets for different but compatible reasons. I don't see anything paradoxical about that.

I don't see a contradiction here any more than I did back at Question 26649 , which is nearly identical. Yes, the empty set has the property of being empty and is the only set having that property. But the emptiness of the empty set doesn't imply that the empty set has no properties. On the contrary, it has the property of being empty, being a set, being an abstract object, being distinct from Mars, being referred to in this answer, etc. Why would anyone think that the empty set must lack all properties?

Why must either be "more real" than the other? I can't make sense of "more real," anyway, as a comparison. Are shadows less real than the 3D objects that cast them? Shadows are dependent in a way in which 3D objects are not, but I don't see how that makes shadows any less real when they exist. Some philosophers say that the number 2, being an abstract object, exists necessarily (i.e., in all possible circumstances), whereas your two feet exist only contingently (i.e., in some but not all possible circumstances). But that view does not imply that the number 2 is any more real than your two feet. Other philosophers say that the number 2 exists but not your two feet, because they say that "anatomical foot," being a linguistically vague term, fails to denote anything in the world. (I think they're mistaken.) Still other philosophers would say that neither the number 2 nor your two feet exist. But none of that, I think, implies that one is more real than the other. Is Donald Trump more real than the...

I would question your assumption that positions, magnitudes, etc., whose measure is irrational "cannot be precisely described mathematically." Consider a simple-minded example: In a given frame of reference, some point-particle is located exactly pi centimeters away from some other point-particle. I think that counts as a precise mathematical description of the distance between the two particles, even though it uses an irrational (indeed, transcendental) number, pi, to describe the distance. It's true that any physical measurement of that distance -- say, 3.14159 cm -- will be precise to only finitely many decimal places and therefore will be only an approximation of the actual distance. But the description "pi centimeters apart" is itself perfectly precise, despite the irrationality of pi.

I think it does, or at least I think the burden of proof is on anyone who says that "exist" is systematically ambiguous, meaning one thing when applied to numbers and another thing when applied elsewhere. It's widely held that abstract objects such as numbers, if indeed they exist, don't exist in spacetime, whereas concrete objects such dogs clearly do exist in spacetime. But that doesn't affect the meaning of "exist" itself. In particular, it doesn't imply that "exist" means "exist in spacetime." Otherwise, the expression "exist in spacetime" would be redundant and the expression "exist but not in spacetime" would be self-contradictory, neither of which is the case. Analogy: It's a fact that some things exist aerobically and some things exist anaerobically, but that fact doesn't tempt anyone to say that one or the other kind of thing doesn't really exist, or to say that "exist" just means "exist aerobically." So I see no reason not to say that numbers, if they exist, exist nonspatiotemporally,...

Pretty clearly, zero and infinity aren't the same thing. For example, the number of prime numbers is infinite and (therefore) definitely not zero. But I'm not convinced that zero and infinity are opposites either. (I'd be more inclined to say that negative infinity and positive infinity are opposites.) One reason is this: "zero" and "none" are often synonymous, as in "I own zero unicorns; I own none." The opposite of "none" is "all" (whereas the contradictory of "none" is "some"). But "all" and "infinitely many" are not synonymous: for example, even if we collect all the grains of sand in the world, we will collect only finitely many grains.

I agree with you that the sources of truth in mathematics can't be physical. For it seems clear to me that there would be mathematical truths even in a world that contained nothing physical at all (for instance, it would be true that the number of physical things in such a world is zero and therefore not greater than zero, not prime , etc.). So the sources of mathematical truth must be other than physical: if you like, metaphysical. Does this fact mean that all mathematicians are doing metaphysics? I don't think so. Metaphysicians can investigate the sources of truth in mathematics and the ontological status of mathematical truth-makers. But mathematicians themselves can simply make use of those truths without having to delve into what it is that makes those mathematical truths true.

I'm strongly inclined to say yes . Here's an argument. If there's even one technological civilization elsewhere in our unimaginably vast universe, then that civilization must have discovered enough math to produce technology. But we have no reason at all to think that it's a human civilization, given the very different conditions in which it evolved: if it exists, it belongs to a different species from ours. So: If math depends on human consciousness, then we're the only technological civilization in the universe, which seems very unlikely to me. Here's a second argument. Before human beings came on the scene, did the earth orbit the sun in an ellipse, with the sun at one focus? Surely it did. (Indeed, there's every reason to think that the earth traced an elliptical orbit before any life at all emerged on it.) But "orbiting in an ellipse with the sun at one focus" is a precise mathematical description of the earth's behavior, a description that held true long before consciousness emerged here....

You've asked a terrific question! I wish I were more qualified to venture an answer to it. As you suggest, a sound direct proof of a theorem shows that the theorem must be true, in the broadest possible sense of "must." But a sound indirect proof shows the same thing. The difference, if any, seems purely psychological: some people find one proof psychologically more satisfying than the other. My sense is that some philosophers of math take this psychological difference very seriously and propose far-reaching revisions to classical math on the basis of it. You might take a look at the SEP entry on intutionism in the philosophy of math , particularly the discussion of constructive and nonconstructive proofs. The entry includes other helpful links and references too.