As a warm-up exercise, consider the following two infinite ordered sets of numbers. Firstly, take the negative and positive integers in their 'natural' ordering ... -4, -3, -2, -1, 0, 1, 2, 3, 4, ... trailing off unendingly to the left and to the right. Second, take all the negative numbers, in increasing size, followed by zero and all the positive numbers: -1, -2, -3, -4, ... o, 1, 2, 3, ... Now, in both orderings, any positive number is preceded by an infinity of numbers (including all the negative numbers). But there is a very important difference between the two cases -- they have, as mathematicians say, different 'order types'. One big difference is this: there is no first member of the first ordering (i.e. for any given element of the ordered series, there's an earlier one); but there is a first member of the second ordering (namely, -1). To bring out another difference, suppose in each ordering we take one of the negative numbers, and we ask: can we...

Plato famously thought that you should master mathematics before you turn to philosophy. No one would quite think that these days (though it is interesting that, among my faculty colleagues, over a quarter of us in fact have first degrees in mathematics, and turned to philosophy later). But perhaps Plato has a point. For a start, it is good to first hone your skills of sharp accurate reasoning, and to practice intellectual humility in the face of really difficult problems, when working on matters that aren't too conceptually tangled (let alone often bound up with your emotions or with cultural/religious prejudices). Not that doing serious mathematics is the only way of getting in the practice, of course: doing a degree in classics is another well-trodden route! Further, outside moral and political philosophy, a great deal of the best work -- philosophy that isn't just the higher arm-waving waffle -- is closely bound up with science, in the broadest sense. To do serious philosophy of...

Which mathematics manages to describe the physical world? Mathematicians offer us, e.g., Euclidean and non-Euclidean geometries of spaces of various dimensions (and the non-Euclidean geometries come in different brands). They can't all correctly describe the world, since they say different things even about such simple matters as the sum of the angles of a three-dimensional triangle. But we hope that one such geometry does indeed describe the sort of structure exemplified by physical space (or better, physical space-time). That's pretty typical. Mathematicians explore all kinds of different possible structures. Only some of them are physically exemplified. For example, group theories explore patterns of symmetries; some of the patterns are to be found in the world -- but I guess that no one thinks e.g. that the Monster Group is physically instantiated. Mathematical physicists tell us which kinds of structures are to be found in the physical world and then use the appropriate mathematics to...

Take, as an extreme case, the argument "The earth is round; hence the earth is round". The inferential move is trivially deductively valid (there is no possible way the premiss can be true and the conclusion false); and the premiss is true. So the argument is sound . But of course, the argument would be quite useless in an exchange with a latter-day flat-earther! He could rightly complain that, given his views, that argument just begs the question at issue between you. So, in that context, the argument would be sound but question-begging.

There is more relevant discussion in response to Question 2107 , where I remark on the moral differences between early fetuses and newborn infants that we seem to make in our thinking about the natural or accidental death of fetuses as against babies.

Zeno most famous paradoxical argument seems to show that Achilles can never overtake the tortoise. Plainly, the conclusion of Zeno's argument is false: that can be shown by Achilles just walking along, overtaking the tortoise! That's why the argument -- which seems to go from true premisses via plausible reasoning to the palpably false conclusion -- is a paradox . But of course, just re-iterating that the conclusion is false doesn't solve the paradox, if that means explaining just where Zeno's reasoning goes wrong. It's perhaps not helpful, then, to talk about "refuting" the paradox, for that's ambiguous. It could mean showing the conclusion is false , or it could mean explaining where the bug is in Zeno's reasoning . Doing a bit of walking (Achilles overtaking the tortoise yet again) suffices for the first, but not for the second!

I'd start with some more modern books actually written for beginners, before tackling Popper, Nagel et al. Two that in my experience work well with students are Alan Chalmers' What is This Thing Called Science , and Alexander Bird's Philosophy of Science .

It might very well be that you have had feelings that you have interpreted as feelings of the presence of God (you didn't imagine having the feelings that you interpreted that way). But the question, presumably, is whether you were right in so interpreting them. The fact that, given your cultural setting, you found it very natural at the time to interpret the feelings in a certain way may be hardly surprising. But presumably that is in itself not a strong reason for supposing that the interpretation you put on your experiences was actually correct. After all, others brought up differently -- e.g. in a cheerfully atheistic environment -- are no less prone to various patterns of exalted feelings from time to time: but they will no doubt want to interpret the experiences in a very different way. Though actually, things probably go deeper than mere patterns of acculturation. Dan Dennett has argued that our tendency to over-interpret certain experiences as betokening the presence of...

It is not entirely straightforward to come up with a cogent statement of determinism. But perhaps something along the following lines will do: our world is a deterministic one if the laws of nature are such that, given the past and current state of the world, there is only one possible way its future state can evolve. If you prefer that in "possible world" talk, then the idea is that the actual laws are such that any other possible world which shares these same laws, and whose past and present duplicates that of the actual world, will also be a future duplicate. Thus understood, the claim that our world is a deterministic one is a claim about the shape of the laws of nature governing the world. Do they, so to speak, uniquely fix what will happen next (given the past and current state of the world); or do they allow e.g. for irreducibly chancy events? And that is surely an empirical question. We can't settle from the armchair whether (i) we live in a "classical" world where the laws make the world...

Oh, good graduate students most certainly make contributions: they get good papers published in good journals. And if that doesn't count as "making a contribution", then very few of us make contributions. (Our students make contributions in another important way too: they teach their teachers, keep us enthused, prompt better work from us too.) Of course, few pieces published by graduate students make stunning advances: but then few pieces published by anyone make stunning advances (in philosophy or in any other discipline).