In relation to my earlier answer, the following article from the Economist may be of interest. It's advertised as follows: "Can the laws of physics change? Curious results from the outer reaches of the universe." The link is

www.economist.com/node/16941123?story_id=16941123&fsrc=nlw|hig|09-02-2010|editors_highlights

This is not exactly what I had in mind, but relevant nonetheless.

BTW, this question is probably best classified under "physics" rather than "mathematics."

For a very extended, and musically informed discussion of your question, I would strongly recommend Roger Scruton's book The Aesthetics of Music. He develops a sophisticated account of the way that we use our imaginations to experience the notes as moving in dance-like ways.

Even if the chords are not presented in the context of a music piece, they are heard in the (more backgrounded) context of music one has heard. Our associations with those pieces of music prime us to hear major versus minor chords in particular ways.

There is also a physical reason for finding major chords to be more settled or stable than minor chords: the wavelengths of a major third match the overtones of the root of a chord more closely than do the wavelengths of a minor third . When we hear a C, for example, it is already producing secondary wavelengths that are those of an E (at a higher octave); the addition of a nearby E thus seems to fit in without added strain.

Your question is very interesting: it is, I think, an instance of a question that might generally be asked of any particular instance of any art: what is it that makes it the kind of work that it is?

To fix ideas, consider the following question, which Stephen Dedalus, the protagonist of James Joyce's Portrait of the Artist as a Young Man, says that he's written down in a book at home: "If a man hacking in fury at a block of wood...make there an image of a cow, is that image a work of art? If not, why not?"

Your question, like Stephen's, has to do with the difference between a genuine artwork and an otherwise identical grouping of sounds, not, however, produced in the context of a work of art.

It therefore seems to me that brilliance alone is not enough for something to qualify as music; the birds outside my window sometimes produce a brilliant series of notes. But the sounds produced by the birds aren't music, even if they are musical, whereas the sounds heard on the soundtrack of The Birds are, it would seem an instance of music? The question, is, however, what makes the latter music and the former not? Is it that there was an artist in the one case and not in the other? Is it that the latter, but not the former, is the product of intention?

These and other answers have been given in work on aesthetics, which, long before Stephen Dedalus wrote in his notebook, was centrally concerned with defining the nature of art, and hence engaging, on even a more general level, the kind of questions that you and Stephen raise about particular arts. In the twentieth century, objections were raised to the very attempt to define the concept of art, on the grounds that the concept did not admit of a definition, but was instead, maybe, a 'family resemblance term', meant to capture a cluster of interrelated features, no one of which was essential or necessary to the concept, but which could overlap and criscross in all sorts of ways, and were nevertheless also not simply amenable to a disjunctive definition.

Might it be the case that individual arts are like that? Or are there criteria for determining what constitutes an instance of a particular art? If the latter, then does the concept of art admit, after all, of definition?

Although Hume does not himself use the terms 'a priori' and 'a posteriori', those categories do, roughly, correspond to the distinction that Hume draws between relations of ideas and matter of fact in the Enquiry Concerning Human Understanding. (The Enquiry Concerning Human Understanding is also referred to as the 'first Enquiry', as I will do in what follows, to distinguish it from the Enquiry Concerning the Principles of Morals, the 'second Enquiry'.) Now, by the by, but interestingly enough, Hume doesn't draw the distinction between relations of ideas and matters of fact, at least explicitly, in the earlier Treatise of Human Nature, much of whose first Book was recast in the first Enquiry, although he does draw a related distinction in Book 1, Part 3, Chapter 1 of the Treatise, between relations that depend on 'intuition' and 'demonstration', and have only to do with ideas, in contrast to other relations, which do not so depend on ideas, and thus do not admit of the sort of certainty characteristic of intuition and demonstration.

Near the beginning of Part 1 of Section IV of the first Enquiry, Hume explains that relations of ideas "are discoverable by the mere operation of thought, without dependence on what is any where existent in the universe," whereas, by contrast, matters of fact "are not ascertained in the same manner...the contrary of every matter of fact is still possible, because it can never imply a contradiction, and is conceived by the mind with the same facility and distinctness, as if every so conformable to reality." Insofar as relations of ideas do not depend "on what is any where existent in the universe," they are akin to Kant's analytic truths, which depend, roughly, only on the meaning of their terms; matters of fact, which do depend on what is existent in the universe, depend on experience, and thus are akin to Kant's synthetic truths. As you quite rightly point out, there is no room in Hume's division between relations of ideas and matters of fact--which is sometimes called 'Hume's fork'--for the category of the synthetic a priori, which seems to straddle the realms of relations of ideas and matters of fact and thus to explode that supposedly exhaustive division.

Now you want to bring these distinctions to bear on Hume's famous thought experiment of the 'missing shade of blue', which he himself took to constitute a prima facie counterexample to his claim that all ideas are derived from impressions, that is, that all thoughts may be traced back to experience. (It's worth noting that in both the first Enquiry and in the Treatise, Hume actually considers the missing shade of blue before he introduces the distinction between judgments based on experience and those not based on experience and hence admitting of certainty. He therefore must not have thought that his 'fork' was related to the missing shade of blue.) The missing shade of blue might seem to constitute a counterexample to this view, because in this case, the thought of a color is arrived at without one having had experience of that previous color. Now Hume himself, after propounding the example, says that although "this may serve as a proof, that not every idea is derived from correspondent impressions...this instance is so singular, that it is scarcely worth our observing, and does not merit, that for it alone we should alter our general maxim." Although Hume may have thought the missing shade of blue "scarcely worth our observing," his readers have given it considerable attention, and you propose that one might understand the example, in Kantian terms, as marking a distinction between imagined and experienced colors, with the latter belonging to the category of the 'analytic a posteriori' and the former belonging to the 'synthetic a posteriori'.

It's not clear to me that this suggestion can work. After all, analytic judgments, for Kant, are not supposed to depend on experience, so it does not seem to me that judgments about colors--when one uses the terms 'analytic' and 'synthetic', one must refer to judgments, what might today be called assertions--can be called 'analytic a posteriori'; synthetic a posteriori judgments, for their part, just are what Kant calls straightforwardly synthetic judgments, since they depend on experience.

But while it's not clear to me that your suggestion can be brought to bear on the particular case of the missing shade of blue, the notion of analytic a posteriori judgments is itself a very interesting one, which Kant himself was not able to envisage, but which may nevertheless constitute a category that merits some consideration. For consider an assertion such as 'Water is H2O'. This is a claim based on experience, but it's necessary. It can't, however, belong to the category of the synthetic a priori, at least as Kant conceives it, since synthetic a priori judgments are supposed to explain the possibility of experience, but not themselves to be objects of experience; the claim can't be analytic, since it's based on experience. So it might seem that, in virtue of its necessity and the fact that it is a claim about experience, that such a claim is best construed as 'analytic a priori'. If this is correct, however, what does this show about Kant's own division of judgments into analytic and synthetic? Might that not have to be rethought in much the same way that Kant himself may be taken to have rethought Hume's fork?

I think you would really enjoy a new anthology from Wiley-Blackwell--Hunting. It is written largely by and for hunters, and looks at the sort of ethical questions you raise in a way you will find hospitable.

I think hunting is extremely difficult to justify. Though once necessary to obtain necessary nutrients, clothing, etc., killing animals to obtain these things is no longer necessary. It doesn't really help justify hunting/fishing to eat what you kill, if you could have eaten something else.

Even assuming it was necessary to eat meat, it would still be problematic to engage in killing as a recreational leisure activity--which is what hunting/fishing are for most people. If the main goal of sport hunting/fishing are having fun, and food is just a byproduct, something odd is going on (as I argue here). But now getting to you question...

Hunters who are concerned about fairness at least see animals as "subjects" instead of merely as "objects." That's all to the good. Fair hunters will probably kill far fewer animals. But should they really think in terms of fairness? Hunting an animal is not a sport involving two competitors, since the animal doesn't participate voluntarily and has no idea what's going on. In a competition between two humans, fairness is mutually beneficial, but that's not necessarily so in the case of hunting and fishing. The "unfair" hunter at a hunting ranch will lure a tame animal to a hunting station, and then shoot him at close range with a powerful rifle. The "fair" hunter might chase a terrified deer for miles, and then shoot him from a distance with a bow and arrow, so the animal dies a slower death. The extra "fairness" in the second case doesn't benefit the deer, and in fact harms him!

I agree with you that all hunting is not equal, and if one is going to hunt, one should do it "the right way." But the right way, it seems to me, is just less wanton and more humane, not necessarily the way that involves concepts of fairness imported from human sport.

It's not clear what it would mean (or what it would take) to show that there was an effect without a cause (unless we just define an effect as something that is caused, in which case there'd be nothing to show). We certainly have not shown (proven) that every event has a cause in the sense that we have not, and could not, pick out the causes of every event that has ever happened. Rather, we tend to assume that all events have causes, except perhaps those people who assume that free choices are uncaused or who assume that there is a first event that was not caused by any prior event. It's an assumption that tends to work for us--that is, it helps us explain things in science and our everyday life--and it is an assumption that does not have any clear counterexamples (but again, it's not clear what a counterexample would look like). So, as you suggest, this thesis of Universal Causation (UC) might be the most justifiable.

However, one might think that a possible counterexample involves the events described by quantum physics, since it looks like those events are indeterministic: given the exact same prior conditions (or causes) and the same laws of physics, more than one effect might occur. For instance, the electron shot at a barrier might go one way or another and nothing explains (or causes?) which way it goes. However, a better way to describe indeterministic events is that they are probabilistically caused. Quantum physics does not entail randomness--rather it describes objective probabilities between events. So, one can say that the set of events that leads to the electron's hitting the barrier causes it to end up in position 1 (with 50% probability) and causes it to end up in position 2 (with 50% probability). Wherever it ends up, it was caused to end up there by prior effects.

OK, all this is just to set up the take-home message which is that the thesis of UC (Universal Causation), which says that every event--or at least every event after the first event--has a cause, does not entail the thesis of determinism. Determinism is the thesis that: Necessarily, given the same prior events and laws of nature, the same later events occur. UC is consistent with indeterministic causation and hence with the falsity of determinism. So, even if we answer your first two questions 'no', that does not mean we have to answer your third question 'yes'.

Finally, "Hard Determinism" is the thesis in the free will debate that says (a) determinism is incompatible with free will, (b) determinism is true, and therefore (c) free will does not exist. Lots of philosophers (like me) are compatibilists who reject (a) (if interested, see my prior responses to questions in the Freedom category). And most philosophers reject (b) since they accept that the dominant interpretation of quantum physics is indeterministic. But many still think determinism, if true, would rule out free will and do so for reasons that make Universal Causation just as threatening, so indeterminism doesn't help. (Basically, they think that our decisions, like everything else are caused by prior events, ultimately by events over which we have no control, so we do not have ultimate control over our decisions, and they think such ultimate control is required for free will--again, I see no reason to think free will requires such unattainable powers!). These philosophers sometimes call themselves "Hard Incompatibilists" (or Skeptics about free will), and they are the descendants of the Hard Determinists.

This is complicated stuff and I've tried to keep it brief. But I hope this helps!

See here for some similar questions and their answers.

Philosophers have expressed wide ranging views on the infinite, and even distinguished different kinds of infinites. In terms of the 'infinite' standing for a sequence of events without end, then (just as there is no greatest possible number) it is difficult for someone to claim to have experienced that (experienced all numbers, none of which is lacking in a greater number), though not perhaps difficult for one to claim to understand it (that is, understanding that there is no greatest possible number) or for someone to have an experience of time or space, along with the feeling that this will never end.

There has been some interesting testimony by some philosophers to have experienced soemthing related that may be of interest. Some philosophers have claimed to experience that which is boundless or, in some sense, eternal. Probably the two most famous philosophers to have spoken and analyzed such experiences are Boethius and Augustine. Boethius spoke of God's eternity (and having some experiential acquaintance with God as eternal) in terms of God possessing the 'whole, simultansous, and complete fruition of a life without bounds' (interminabilis vitae tot simul et perfecta possessio'). This would be different from claiming to experience what you might think of as 'forever' or 'endless'; it is more like experiencing an event so overwhelming and perhaps good that you seem to lose track of future and the past. This has been analyzed by some philosophers as experiencing something that is atemporal or beyond metric time or not bound by it. The philosopher A.E. Taylor in an interesting book in the early part of the last century wrote of the experience of eternity in ways that are (to use your term) intense, but more satisfying than jarring or incredible (not worthy of belief). In one example, he describes 'spending an evening of prolonged enjoyment in the company of wholly congenial friends. The past may be represented for us, if we stay to think of it at all, by whatever happened before the party began, the future -but when we are truly enjoying ourselves we do not anticipate it- by what will happen when the gathering is over. The enjoyment of the social evening has, of course, before and after within itself; the party may last two or three hours. But while it lasts and while our enjoyment of it is steady and at the full, the first half-hour in not envisaged as past, nor the third as future, while the second is going on....' Taylor goes on to defend the coherence and importance of experiences that seem to be in response to a value that we wish to last forever or not be bound by time, a state in which one or more people might be completely present to each other that they would never wish it to end. See Taylor's book The Faith of a Moralist --the title is a bit misleading given what we mean by 'moralist' or 'moralistic' today versus when he wrote the book in 1930. It is a good text for thinking about the experience of values and time. (See especially chapters three to six.)

There are a number of classic works that treat the sorts of issues that you raise. (Interestingly, for what it's worth, relatively few contemporary 'analytic' philosophers have engaged these issues.) Chief among them, perhaps--at least in the Western tradition--are Plato's Republic and the New Testament, both of which, I think, are concerned with the kinds of issues that you mention. More relatively recent works that engage the topics that you mention include Henry David Thoreau's Walden, Martin Heidegger's Being and Time, and Jean-Paul Sartre's Being and Nothingness. (Sartre, of course, is the popularizer of the notion of 'radical choice'.) A very recent book by a living philosopher that treats the issues to which you refer is Susan Wolf's Meaning in Life and What Matters.