Let's narrow the question a bit: can arithmetic be reduced to logic? If arithmetic can't be so reduced, then certainly mathematics more generally can't be. What would count as giving a reduction of arithmetic to logic? Well, We would need to give explicit definitions (or perhaps some other kind of bridging principles) relating the concepts of arithmetic to logical concepts. Otherwise we won't get arithmetical concepts into the picture at all. We would need to show how the theorems of arithmetic can in fact be derived from logical axioms plus those definitions or bridge principles. But what kind of definitions in terms of what sorts of concepts are we allowed at step (1)? And what kinds of logical principle can we draw on at step (2)? Suppose you think that the notion of a set is a logical notion. And suppose that you define zero to be the empty set, one to be the set of all singleton sets, two to be the set of all pairs, and so on. We can then define the...

Opinions are only worth as much as the reasons they are based on. If the reasons are no good, the opinions don't deserve respect -- indeed, they deserve to be vigorously criticized. And that applies as much to religious opinions as any other kind of opinion. Some Christians have a thought out position sustaining an admirable ethical way of life; other Christians have frankly batty superstitious reasons for holding a toxic mix of deeply unpleasant views that are a disgrace to humanity. (Similar things, of course, can be said about non-Christians!) Whether your teacher's religious views deserve any respect rather depends on which camp she is in. And, indeed, as you remark, just because a set of opinions are supposed to be "religious" gives those views no special claim on our respect at all ("that's my religious belief" is not an argument -- it just deserves the riposte "so what?"). For a terrific essay related to these matters, freely downloadable, read my colleague Simon Blackburn's ' ...

Writing philosophy so very often involves uncomfortable compromises. If you put in all your own doubts and reservations, signal all the places where -- as you well realize -- objections might be raised, indicate all your silent assumptions, and so on, then your essay or paper or book would be pretty unreadable. But if you just silence the inner voice that keeps saying "But on the other hand ..." and go for the unqualified bold sweep and the big idea, you feel you are cheating your reader (and yourself). So you try for some middle way. And when it comes out in cold print -- even if you don't immediately spot some horrible mistake you now have to live with -- it is difficult not to worry that, after all your efforts, you've got the compromises all wrong, and the piece doesn't really convey quite what you intended. So yes, it is easy to "hate" your own writing ... Or is that just me?!

Gödel's first incompleteness theorem applied to the arithmetic Q tells us that there is a corresponding Gödel sentence G Q such that, if Q is consistent, it can't prove G Q , and if Q satisfies a rather stronger condition (so-called omega-consistency) then Q can't prove not-G Q either. How do we establish the incompleteness theorem? There is a number of different arguments. But Gödel's original one depends on the very ingenious trick of using numbers to code facts about proofs. And he shows us how to construct an the arithmetical sentence G Q which -- read in the light of that coding -- "says" that G Q is not provable in Q. (So of course we don't want Q to prove G Q or it would prove a falsehood!) Now, Gödel's original proof of the incompleteness theorem, and all the textbook variants, are presented as nearly all mathematics is presented -- i.e. in informal mathematicians' German or English or whatever, with as much detail filled in as is needed to convince. And what's wrong...

Well, of course he's going to notice cute girls. Love might make you blind, but not in that way. But it is, to say the very least, tactless to notice too obviously, let alone to point them out. I suppose his being an insensitive jerk might be compatible with his loving you ... in his way. But whether you want to love a jerk is another matter. I'm struggling though to extract any philosophical juice from the question! I suppose we might, as philosophers, remark that the concept of love is surely analytically tied up with notions of care andconcern for another (love isn't just a "feeling"): so genuine love is incompatible with behaving in too uncaring a way,too unconcerned for the other's feelings. But is commenting on cute girls being "too uncaring" ... or is it just being a thoughtless idiot who can't join up the dots ("he's a man, honey")? Depends on the guy and how he comments, I guess.

Indeed, not all thought is done 'in words'. Sitting in front of the chess board, I'm certainly thinking hard (and it's serious rational planning, not wool-gathering!). But I'm imagining sequences of moves on a board, not going in for inner speech. Likewise, when Roger Federer out-thinks his opponent, he probably isn't giving himself a wordy running commentary in English (or Schweizer-Deutsch) -- how distracting would that be!? Gilbert Ryle long ago wrote much good sense about this. (One fairly characteristic piece is available online here .) But yes, some thought is naturally described as being 'in words'. And not just my written thoughts here on the screen but, so to speak, the private ratiocinations as I was sorting out my ideas and thinking what to say here. My thoughts weren't in some other medium that I then had to translate into words: I was rehearsing these very words "in my head", as we say. But what does that mean? A good question. What we want here is a general story about what...

I have to say that I do think it is simply bizarre that we inflict Plato on high-school students as an introduction to philosophy. We wouldn't dream of starting off physicists by getting them to read Newton, or mathematicians by getting them to read Euclid. Philosophy is hard enough without having to try to interpret work from two and a bit millennia ago. But having got that off my chest, can I recommend Julia Annas's An Introduction to Plato's Republic , which has been around some time but is excellent. It works very well with first-year university students: so try reading it, and not just for its treatment of the theory of forms.

If this question has gone unanswered for a while, that isn't because it is an uninteresting one. On the contrary! It raises a whole range of deep and difficult issues that have been the subject of a vast amount of discussion (from cognitive psychologists as well as arm-chair philosophers) for years. So I hesitate to plunge in. But still, since no one else has responded yet, let me get the ball rolling -- though these remarks are no more than a very preliminary sorting out of some of the issues. For we need to clarify what is meant here by (1) "language", (2) "thought", and (3) "tied up together". (1) What is meant by "language"? A shared natural language like English, or Welsh, or Sanskrit? Or might we more generously count as a language any system of representations which has a syntax (i.e. there are structural rules determining which arrays of elements from system are allowed) and a semantics (there are rules determining what these arrays mean )? Some have argued that we have an innate ...

Surely the question whether pentagons are more like hexagons than circles just invites the riposte: "more like in what respect?". If we are interested in whether figures have straight sides and vertices or lack them, then of course pentagons will get put in the same bucket as hexagons, while circles will go in another bucket (with e.g. elipses and parabolas). It's an objective fact that pentagons are like hexagons (and not circles) in having straight sides and vertices. If we interested in whether we can tile a plane with (regular) figures of a certain kind, then pentagons will be classed with circles (no, you can't tile a plane with those), and hexagons will belong in the other bucket along with e.g. squares and triangles. It's an objective fact that pentagons are like circles (and not hexagons) in that you can't tile a plane with them. So we might say that the bald question "are pentagons more like hexagons than circles?" is incomplete. It needs to filled out (either explicitly or by...

I'm sure that you won't be doing something "horribly wrong" by continuing the relationship, if you both can acknowledge and work with your boyfriend's cast of mind. And it's hardly an uncommon situation you are in (as I'm sure many partners of male academics in the mathematical sciences could ruefully tell you!). Though the situation will probably be harder for you than your boyfriend -- for there will be occasions when his lack of ready perception of your more subtle emotional needs will, almost inevitably, be hurtful (for that is as natural reaction for you as his failures will be for him), and be upsetting however much you can explain things away as due to his mild autism. Can I add to your reading list (this is certainly one for both of you!)? Simon Baron-Cohen's The Essential Difference: Men, Women and the Extreme Male Brain is very readable and very insightful by a leading researcher into autism-spectrum conditions.