Aesthetics isn't my area, but since no one else has responded I'll take a stab at it. To someone who thinks that aesthetic judgments can't be objectively true or false -- someone who thinks that aesthetic judgments are in that respect fundamentally subjective -- I'd pose two questions: (1) We often seem to make objective aesthetic judgments, such as the judgments concerning Bach and Rihanna that you mentioned in your question; why not take those judgments at face value? Why think we have to interpret those judgments as non-objective? (2) If there's a worry that aesthetic judgments can't be objectively true or false, does that worry extend to normative judgments in general , including the judgment that some ways of reasoning are better than others or that some ways of treating people are better than others? If it does, then it's a worry about objective normative judgments in general rather than aesthetic judgments in particular. If it doesn't, then what makes aesthetic judgments less likely to...

You'll find many of your questions about color discussed helpfully in the SEP entry available at this link .

If "√9" refers to the positive square root of 9 (I'm not sure what the convention is concerning the square-root symbol), then I'd say that 3 and √9 are the same object, just as Mark Twain and Samuel Clemens are the same object. (Indeed, the plural verb "are" in each case is a bit of loose talk.) Leibniz's Law (the Indiscernibility of Identicals) therefore implies that everything true of 3 is true of √9, and everything true of Twain is true of Clemens, which seems right to me.

Thanks for sending a follow-up question. Prof. Heck, who knows this territory better than I do, provided helpful corrections and amplifications in his answer to Question 5068 . I recommend taking another look there. You wrote, "The fact that every natural number has a successor is only true for the natural numbers so far encountered (and imagined, I suppose)." The claim that every natural number has a unique successor, like the claim that 1 isn't the successor of any natural number, is an axiom -- a starting point -- rather than a conclusion drawn from examining or imagining data. Your fifth sentence suggests that you know this already. You're quite right that math induction proves its results only given its starting points, but of course that's true of all proofs: all proofs (even proofs that have no premises) rely on essential assumptions. You say that some professional mathematicians would rather accept the existence of a largest natural number than accept the paradoxical features of...

In the category "serious but not too technical for a layperson," I'd include Russ Shafer-Landau's short paperback Whatever Happened to Good and Evil? . It concentrates on metaethics (the fundamental nature of ethics) and moral epistemology (how we might know moral truths, if there are any) rather than normative ethics (particular theories of right and wrong). It's clear, accessible, provocative in places, and enjoyable.

We can use mathematical induction to prove that (i) infinitely many natural numbers exist from the premise that (ii) 1 is a natural number and the premise that (iii) every natural number has a successor. Although it's called mathematical "induction," it's actually deductive reasoning. I take it that (ii) is beyond dispute, and (iii) is at any rate very hard to deny! It won't do to demand proof of (ii) or (iii) before accepting this proof of (i), for if the premises in any proof must themselves have been proven, then we have an infinite regress: nothing could be proven in a finite amount of time. We've therefore proven that infinitely many natural numbers exist. The notation "{1,2,3,4,...}" is just one way of referring to the set containing all and only those infinitely many numbers. It's perhaps a fallible way of referring to that set, because it assumes that the audience knows which number comes next in the series. A more reliable way of referring to the set is "the set whose members are the...

Excellent questions. I'm glad to hear you're looking into this issue. I think philosophers and scientists often throw around talk of "surfaces" much too glibly. I recommend starting your search with the SEP entry on the concept of a boundary, available here . It contains a lot of information relevant to your questions and a bibliography with several useful references.

You asked, "Can we know for sure that the external world exists?" That will depend, of course, on what's required for such knowledge. Some philosophers have said that such knowledge requires a successful proof of the existence of the external world, but many other philosophers (especially in the last few decades) have said that no such proof is required. For those who think a proof is required, G.E. Moore famously (or infamously) offered one: see this link . If you investigate Moore's proof at that link and in other places on the web, I think you'll get a sense of how the proof you offered might be received by various philosophers.

Very interesting (if tantalizingly brief) question. There's reason to think that all human beings rely on their memories for any knowledge they possess. One might think I can know at least some facts about my present-tense experiences without relying on my memory, but what facts could those be? For example, if I'm to know that I have a headache (when I do have one), arguably I must know what counts as a headache, and isn't that something I once learned and now remember? Descartes (1596-1650) was very sensitive to the role of memory in human knowledge. He famously argued that (a) only if you're aware of the existence of a benevolent God do you have sufficient reason to trust your memory, and that (b) without sufficient reason to trust your memory you know virtually nothing. Both (a) and (b), and Descartes's arguments for them, can of course be questioned. You'll find more at this link .

I share your view that such a clause is at best very shoddy. Not being a lawyer, I wonder if the clause is even enforceable; you might ask someone who'd know. It might be unconscionable (a term I wish lawyers applied to more things than they do!) and hence unenforceable, or it might be unenforceable because too difficult in practice for a court to enforce (say, by issuing an injunction forbidding you from accepting a new job!). This may be advice of little value to a desperate job-seeker, but I'd steer clear of firms whose offers come burdened by such clauses.