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1. ## Calculus Class Epistemology

Actually, it's easy enough to extend to the case of 1 linear factor in the numerator and t distinct linear factors in the denominator. First, assume you have one linear factor in the numerator and three in the denominator. Group any two of the factors in the denominator in a fraction with the numerator and express it as a partial fraction using the rule being discussed. The original factor is then written as a sum of two actors of the form m/(x-A)(x-B ). You can then expand each of these as a partial fraction, and after working out each of the coefficients you find they're given by the same rule. No doubt if I'd thought about the symmetry of the problem it would have fallen even more quickly into my lap. The general result with a linear numerator should follow easily from induction. But for s distinct linear factors in the numerator, that same result doesn't follow in the same way.
2. ## Calculus Class Epistemology

I don't know why it's considered not proven; it took me about ten minutes to prove it for a linear numerator and two linear factors in the denominator--and only that long because I made a mistake in assigning coefficients at one point. There's no loss of generality to factor out all the coefficients of x, so take it in the form (x+A)/{(x+B )(x+C)}. For partial fractions, you express this in the form m/(x+B ) + n/(x+C). Multiply through both sides by the denominator and collect like terms to give the pair of equations m+n=1 and A=mC+nB. Eliminate n from the second using the first to get m=(A-B )/(C-B ), then substitute for m to get n=(C-A)/(C-B ). Now if you take the original fraction without (x+C) in the denominator at x=-C, you get (A-C)/(B-C), which is n, and similarly for m. Now, proving it for s linear factors in the numerator and t>s linear factors in the denominator might be trickier; certainly it's more tedious. In any case, it's a damned handy little rule. I'll have to keep it in mind in the future.
3. ## The Only Two Things The Majority of Women Want.

Where in the world did you get that from her post? I'd have thought it was obvious from the context that she meant "enjoy yourself in your date's company." As it is, that's like saying that since concrete values like types of music, food, and activities vary so widely from person to person, the concept of value is therefore so vague that it's really a stupid, nonsensical floating abstraction. Different people might well find they have nothing in common as fun (one woman might find the relaxed company of self-secure men fun, while someone else might well find logic-chopping in the service of alpha-male doggy-macho one-upmanship the height of fun), but that doesn't mean the concept's vague or a floating abstraction, it means it depends crucially upon individual values. Yeesh.

5. ## Objectivism's Theory of History

I've thought about this for a while, and my rough and ready suggestion is: History is the rational reconstruction of (chains of) past events. Basically, we have evidence of various sorts of what people thought and did (past events, where event is narrower than occurrence; I mean roughly considered actions; the distinction is from R.G. Collingwood's The Art of History) that the historian fits into an interpretive chronological order (reconstruction) using what knowledge we've attained of regularities in human behavior and the world people lived in at the time (hence rational). Any historian has to deal with the fundamental fact that people have free will at the very least in the sense of being able to choose freely among possible means to attain whatever ends they have (and the variability of ends and the knowledge they might have had of the means open to them is another wrinkle); the inference from actions in a given situation to motivation is what makes history an essentially interpretive field--though of course not all interpretations are equal and they can improve over time. And it is this central fact of free will that means history is not strictly speaking a science (in the sense that it doesn't itself strive for universally true statements about human nature), though good history often requires scientific knowledge (not to mention philosophical knowledge as well). I don't consider archeology to be history proper; rather, it's a historical science. That is, roughly speaking, experimental sciences aim to go from the particular to the general through the testing of hypotheses about particular events; historical sciences use general knowledge (in the form of universal statements) to reconstruct what happened in the past (particular statements). Much of geology is a historical science, in this view, as is cosmology and a good deal of evolutionary biology. Historical linguistics is another. I consider them historical sciences rather than history because their primary focus is to use universal scientific statements to order past occurrences and do not have to deal with the interpretation of human events that free will entails. (Though you do get some of that latter in archeology; when it's questions of purely material culture, there is such a limited set of means and ends that it's barely a branch of history, but when you try to reconstruct something of the more social aspects of a culture fom its material remains then it's closer to history, though closer still to anthropology.) It's usually called philosophy of history, or perhaps history of philosophy in the narrow sense (in the broad sense you have the bloviated, bloated maunderings of Hegel, Toynbee & Co. that do strive for a theory of everything human; most historians don't like those). I think philosophy of historical methodology is a more exact term, but not very pretty.
6. ## Contemporary Classical/Romantic Music?

Actually, I don't think this recommendation matches what you're looking for, now that I think about it.
7. ## Contemporary Classical/Romantic Music?

Also, you might make a point of checking out music for winds and symphonic band music. It was a refuge for more traditional American composers during the 50s-70s. A number of fine Italian-American composers, for example, wrote solid, interesting, often beautiful music for symphonic band during those days (Norman Dello Joio, Vincent Persichetti, Vittorio Giannini, and Paul Creston). Among contemporaries, David Gillingham, Frank Ticheli, and David Maslanka have all written extensively and well for symphonic bands. (Eric Ewazen, whom I mentioned before, has also.) Ronald Perera has written some very fine choral and vocal works (The Outermost House, Crossing the Line, Visions). Two contemporary (though not young) composers who have written deliciously for voice are Ned Rorem (of course) and Dominick Argento; they've also written fine instrumental music, and both are considered neo-romantics. Gerald Cohen and Osvaldo Golijov have written beautiful vocal and instrumental music based in Jewish traditional music, and somewhat along those lines Thomas Pasatieri's Letter to Warsaw is very good. All this is, of course, in addition to the standard famous American neo-romantics like Barber, Hanson, and (before 1950) Diamond, the neo-classicals like Piston, and the nationalist composers like Harris and Schuman (when Copland was being tonal, he fit into the first and the third categories); I assume you're already familiar with them.
8. ## Contemporary Classical/Romantic Music?

There are quite a few, many of them fairly obscure. First, Americans. A contemporary you must check out is Eric Ewazen. His style is influenced partly by English folksong; he specializes in music for brass and winds. http://ericewazen.com/ From an older generation and somewhat famous is Robert Ward; another one whose reputation suffered during his lifetime was Alec Wilder, because he straddled the boundary between classical, jazz, and popular song. Lowell Liebermann is extremely good. Also, British classical muic of the 20th century was generally tonal and tuneful. Keeping those requirements in mind, I'd suggest William Alwyn (quite romantic), Malcolm Arnold (romantic in places, neoclassical in others, and often cheeky and fun--though his symphonies are generally quite a bit darker; look for his chamber music [besides the string quartets] and his concertos--both of his clarinet concertos were written for Benny Goodman, for example, and are quite interesting), and Alan Rawsthorne (more of a modernist in the neoclassical strain of Hindemith and Stravinsky, but if I remember correctly he was also committed to a Laborite program of writing music accessible to regular modern listeners). What I've heard of Lennox Berkeley's music is very good, but you'd want to avoid his son Michael's music. Similarly, many Swedish composers up to about 1960 wrote quite traditional music--Hugo Alfven, Kurt Atterberg, Ture Rangstrom, and Vilhelm Peterson-Berger are good lesser-known lights from that period, though they don't count as contemporary, fer sure fer sure. Edward Tubin (Estonian) wrote a series of modern romantic symphonies you might want to sample to see if he's to your taste, and if you do like him (he's tough as granite in places and not always pretty but certainly dramatic and tough-minded) then give Vagn Holmboe (Danish) a try; I consider him one of the finest composers of his century, but he's not light and easy but rather Ibsenesque, if you know what I mean--no compromise in his search for musical truth but broadly tonal. Again, they're both from an older generation and not strictly speaking contemporary, but they might interest you and are not so commonly heard in the States. Let's see...Alla Pavlova has written four very fine symphonies, quite romantic and Russian in sound. Samuel Zyman (Mexican) has written some very fine works as well. Two interesting composers who sound surprisingly alike in places are Peter Sculthorpe (Australian) and Minoru Miki (Japanese); they write music with strains of what I believe is sometimes called a "Pacific" school of composing; both draw on elements of non-western music in essentially western forms (traditional Japanese music for Miki and Indonesian and aboriginal music for Sculthorpe), and in my experience Miki is more consistent than Sculthorpe. If this interests you, Miki's Eurasian Trilogy is the work to look for; it's in three parts taken from Japanese poetic thought, jo-ha-kyu (slow introduction, complex middle section, and a fast conclusion), for which the first is a prelude for Japanese instruments and strings, the second a concerto for koto and orchestra, and the last his Symphony for Two Worlds, which I like very much indeed. There's a CD of Sculthorpe's music on Naxos tht's good too; it includes his didgeridoo concerto "Earth Song." He also has some good string quartets. Reza Vali (born Iranian, now I think an American citizen) is also worth looking for; he writes music in western styles that are quite listenable, but many of them are based on Persian modal music rather than tonal music; again, Naxos has a CD of his music, including a flute concerto, that's quite good, and he's written some fine string quartets as well as a large number of quite listenable settings of Persian folksongs. And finally, if you don't know it, you must find the music of Jean Francaix; he lived through most of the 20th century and composed up until the end (1997); he was the epitome of the witty, urbane, deliciously tuneful French musical craftman with the perfect light touch. He'd be classified as neo-classical rather than romantic. His chamber music is very fine and his concertos are delightful; I especially recommend his Concerto for Two Guitars and his Piano Concerto.
9. ## The American Civil War

This posting of mine is quite unsatisfactory, for while I stand behind the points I made, I should have been much clearer about when I was responding to something Drew1776 wrote versus similar arguments I have heard from others that fit in with his claim that the Civil War was not about slavery. In particular, here. A much fairer way of stating it is that to the extent Drew1776 states the Civil War was not about slavery (and it's worth asking what he and I mean by the Civil War "being about" something) he makes the same error of focusing on some of the details at the expense of the fundamental issues. And more generally, I should have made much clearer that you have to distinguish rigorously between the general question of secession under the Constitution and the specifics of Southern secession. The right to secession from a republic is an interesting issue; but it is largely discussed in the context of the American Civil War, in which secession was inescapably tied up with pro-slavery. This is true, but it's not the best example for my purposes, since the immediate cause of World War II was not a search for Lebensraum; that was one of the subsidiary issues, and in that respect it is comparable to, say, the struggle over tariffs before 1860. A much better example (and of course one more amenable to temperate conversation) would be the cause of World War I--the immediate cause was an assassination, but saying that the war was about the assassination of the Archduke Ferdinand is an error on somewhat the same scale as saying that the Civil War was about preserving the Union, not slavery. You need only ask the same sorts of questions to see this: Why did the various European states stand by their treaty obligations, brooking no compromise, that time and not earlier, when the prospect of a world war was staring them in the face?

12. ## My Room + My Car

Ah, wait, never mind. I thought you might be referring to such things as martial law and stationing federal troops in Maryland, which long predated any justification by the need to eliminate slavery, but I guess you were talking about the entire Civil War.

14. ## Privacy and Sex

Or another conceivable reason, maybe the potential audience isn't willing to pay to watch and you won't put on a free show.
15. ## Relativity question

The basic question is: 25% and 75% of the speed of light relative to what? Velocities aren't simply additive, and to get the velocity of the other spaceship as measured in your reference frame you have to change frames. The frame you're saying measures you as having a velocity of .25c in turn has a velocity of -0.25c in your frame; in that frame the other spaceship is measured as having a velocity -0.75c, which would be measured as -(16/19)c (more generally, [v1+v2]/{1+v1*v2}, with v measured in units of c) in your reference frame. By the same token, you are measured as having a velocity (16/19)c in the other ship's frame. After 19/16 years, yes. Only when the elapsed time is measured in the third reference frame. You might want to work through a space-time diagram with all three observers taken explicitly into account and, as always, be very careful about simultaneity, which is frame-dependent. No, from either point of view 19/16 years have passed. You're confusing yourself by bringing in a third observer without taking that into account in the math and then mingling measurements in his frame, your frame, and the other spaceship's.
16. ## Why pursue pure science

Electron microscopes. Neutron diffraction. The Compton effect (due to the particle aspects of photons) also has some uses in radiation therapy (and astronomy).
17. ## Is there a "puritanical streak" among Objectivists?

For ease of searching, I'll just add that the first quote is the second question posted on November 1, 2006, and the second is the first question posted on March 12, 2007.
18. ## The Art of Fiction and The art f Non-Fiction.

I haven't read the one on fiction, but I have read The Art of Non-Fiction, which is, as you would expect, superb. It's good for improving your ability to read critically, too.
19. ## Who Was The Greatest Military Leader Of All Time?

Nonsense. He brought rapacious governments to settled areas in which the conquered peoples were usually divided up as the personal property of great Mongol lords in exactly the same way as they divided up conquered goods and livestock. In China, for example, there was a ruling elite of Mongols, a second class of Mongol administrators from outside of China (and thus loyal only to the Mongol overlord) called the semu ("colored eyes" in Chinese, usually Uyghurs or Iranians), a third class of subjects of North China (the regions of the former Jin dynasty, which having been conquered 40 years before South China were considered therefore to be more loyal), and then subjects of South China, who were excluded from most influence; below them were the slaves. Throughout the Yuan dynasty (the Mongol state that ruled China, 1279-1368, and the Mongol heartland until around 1400) in China there was a constant tension between the high Mongol aristocrats enslaving what remained of the Chinese population subject directly to the Khan, and the central government trying to maintain a stable tax base; in both cases the Chinese population was seen as little more than property to be squabbled over and to be milked for all they could yield. As for his law code (not "code of individual rights"), the Yasa, it does not survive even in part, but from what is known of it it was hardly a charter for individual rights but rather a draconian code governing relations among the Mongols to ensure swift obedience, certainly not something recognizing any rights of the conquered.
20. ## Who Was The Greatest Military Leader Of All Time?

No. He invaded Khwarezm in 1220 after one of the Khwarezm Shah's governors killed Mongol ambassadors on the grounds they were spies (which Mongol ambassadors frequently were), and when Genghis Khan sent emissaries to demand the governor be put to death, he had the emissaries killed. While the state was quickly overthrown and the Shah forced to retreat until his death, the conquest of all of Persia took another thirty years or so. The Il-Khanate was not the state the Mongols invaded but the subordinate khanate set up by Hulegu, grandson of Genghis Khan, in 1255 to run the conquest of the Islamic world. The Macedonians under Philip had already set in motion the invasion of Anatolia to free the cities of Ionia from Persian rule; this Philip had proclaimed a campaign of all the Greeks to avenge the Persian invasion of Greece a century and more before, and it was as a symbol of attaining this goal that Alexander burned Persepolis. His later campaigns of conquest in Persia were undertaken, at least notionally, to avenge the murder of Darius by one of his lieutenants and restore the Persian empire (because the Greeks had been avenged, Alexander used this as a pretext for dismissing the Greek contingents and going on from there with Macedonian troops and various native auxiliaries), which he intended to rule through a fusion of Macedonian and Iranian culture and peoples, hence his adoption of Iranian court customs that the Greeks and Macedonians considered barbaric and repellent (such as the custom of proskynesis, some form of kneeling to the ruler, which was associated in Greek culture only with the worship of the gods).
21. ## Who Was The Greatest Military Leader Of All Time?

Um, no, his father was poisoned when he was out searching for a bride for him. After that, yes, he, his siblings and half-brothers, and his mother (and one of his father's other wives, though she's almost entirely missing from the Secret History) were exiles. No, he most certainly did not. He started the conquest of the non-Chinese state that ruled the North China Plain, the Jin dynasty (ruled by the Jurchens, ancestors of the Manchus who later ruled China under the Qing dynasty, 1644-1911), but this war was still going on at the time of his death. (He died in 1227, the Jin were finally conquered in 1234.) The conquest of most of China, ruled by the Later Song dynasty, did not occur until 1279 under his grandson Khubilai Khan. As for Russia (more precisely, Kievan Rus'), it was not really an empire at the time but a kingdom consisting of a congeries of cities and their surrounding territories, though I grant its territory was pretty large. The armies of Rus' were defeated by a small army under Subetei in 1223 that had been sent to scout out the area for future invasions, but the area was not conquered by the Mongols during Genghis' lifetime; that only came ten years after his death, 1237-40. The Mongols did conquer parts of Europe during the campaign of 1237-40, and they probably would have conquered much more if the Great Khan at the time, Genghis's son Ogedei, had not died, forcing the leader of the campaign, Batu, to return to Mongolia to ensure his candidate for Ogedei's successor was elected. Who is Sun Zhu? I assume you mean Sunzi (Master Sun), whose personal name was Wu, not Zhu.
22. ## Legal foundation for public decency, lewdness, nudity

"A stimuli"!?! Oh, my eyes! My head! That hurt! And I couldn't just turn my eyes away from my own computer screen either. I think you owe me some damages.
23. ## Should we have family names?

And then you get the sad case of Ivana Long marrying Pete Johnson and deciding to hyphenate the two surnames.
24. ## Spartans: Two – Iran (née Persia): Zero

Re-read the first sentence of the passage you quoted and then consider the second: "Second, Persian cultural accomplishments in literature, the decorative arts and architecture – great portions of them cadged or adapted from other cultures – were more or less nullified with the Arab/Islamic conquest of Persia in the mid-7th century A.D. Under the Muslims, much Persian literature simply vanished." Almost all pre-Islamic Persian literature was lost under Islamic rule, so his second sentence is in fact precisely correct.
25. ## Grammatical error?

That would be mine. Now, on to the serious stuff. First off, I agree with you wholeheartedly. However, I think there's more to say. As you pointed out, the grammatical point is whether (or perhaps which) English speakers distinguish the distributive and collective senses of the plural--that is, whether the plural is used to refer to each member of the group individually or to all of them as a group. Many do, including any number of great native-speaker English writers. Dismissing the distributive sense of the plural (which allows you to distinguish singlar and plural possessions of plural possessors) as ungrammatical does injustice to the many writers who have employed the distinction effectively--and if a dictum from on high prevents the reader from appreciating its use, then it's not a good rule for describing English grammar. It's a subtle distinction, however, and often people don't notice it, much less learn about it, when it's encountered while reading, but it can be used to very sharp effect. Here are some good examples from the Wikiquote page for William Hazlitt, a master of conversational prose, to get an idea of the stylistic issues involved. First: "The slaves of power mind the cause they have to serve, because their own interest is concerned; but the friends of liberty always sacrifice their cause, which is only the cause of humanity, to their own spleen, vanity, and self-opinion." A nice parallelism there, with two sentences joined with a semicolon. You could say that each group has one cause shared by all its members (collective), or possibly that just as power or freedom is various, each member has his own idea of what his cause is (distributive), but I'd say the first reading is more natural. Going to the second half of each sentence, it would be a stretch to say he's implying that all slaves of power have the same interest (collective), though it's possible; and it's not at all sensical to say that all friends of liberty have the same self-opinion, so a distributive reading is required. So there is a contrast between the sense of the plural in each half of either sentence, collective in the first half and distributive in the second. Rather malaprop, no? Indeed no, quite the contrary. Consider the second sentence. Why do the friends of liberty sacrifice their (collective) cause? Because of their individual respective spleens, vanities, and self-opinions (distributive sense). The tension between the cause of the group and the motivation of each individual is brought home more forcefully by the contrast between the plural and singular. Consider the effect if a singular were used in the second sentence: "The friend of liberty always sacrifices his cause...to his own spleen, vanity, and self-opinion." What is lost is the implication that this is a collective sacrifice, or better a mutual one, made as each individual friend of liberty plumps for his own spleen, vanity, and self-opinion. And notice that the spleen, vanity, and self-opinion only come into play because the friends of liberty form the same party and step on each other's toes; just as the sacrifice is mutual, so the grievances that trump the cause are with reference to each other as a group. And since this is contrasted with the cohesion of the slaves of power, who glom onto each other because their individual interests are served by the same cause, the use of the plural for parallelism all the better serves to point up the contrast between the two groups. (There's a secondary issue that the use of the singular in the second sentence allows spleen and vanity to be taken either as mass or count nouns.) You can see this especially when compared with this: "Anyone must be mainly ignorant or thoughtless, who is surprised at everything he sees; or wonderfully conceited who expects everything to conform to his standard of propriety." The singular is effective because it focuses attention on any individual who meets this condition; nothing is gained by speaking of them as a group because they don't act as a group; their mutual relations have nothing to do with their ignorance or thoughtlessness. Second: "Those who make their dress a principal part of themselves, will, in general, become of no more value than their dress." In fact, this sentence is ambiguous--either dress is a mass noun throughout (meaning garb) or the second dress can be taken as singular (a particular woman's dress). Each reading has slightly different connotations--the former implies that they will have no value but their outer appearance, the second that their value reduces to their outfit, and by extension to their success in the futile battle of fashions and social striving. I suspect Hazlitt used the plural precisely to get both shades of meaning. Contrast that with a third: "We all wear some disguise -- make some professions -- use some artifice to set ourselves off as being better than we are..." Surely we don't all wear the same disguise or use the same artifice, so a distributive sense is required here. This allows Hazlitt to contrast the disguise and artifice, which are singular for each of us, with the several various professions we might make, but he could have done the same by making it a singular sentence about "each of us." (I take "each of us" as sufficiently emphatic a parallel with "we all"; "everyone" would be too weak a fit parallel, though it would suit a simple "we.") The use of the plural "we" allows all the possessive pronouns to be first person rather than third person to drive his claim home--as it stands it is much more immediate than "Each of us wears some disguise -- makes some professions -- uses some artifice to set himself off as being better than he is..." Now, against that background consider the sentence from Atlas Shrugged: "The only value men can offer me is the work of their mind." In the first part, "the only value men can offer me," it's still unclear whether men is a collective or distributive plural; are men a unitary group who can offer a collective value or are they a group of distinct individuals who each produces values separately? "The work of their minds" calls to mind a collective project whose value derives from its being cooperative, while "the works of their minds" could either be the individual works of each man separately or the various collective works of several men's minds (or all of the above). "The work of their mind" forces the distributive reading to excellent effect--the only value that all and any men can offer me is the work of their individual minds. Far from being ungrammatical, this sentence uses the resources of English masterfully and subtly.
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