for x,y > 7 twins(x+y) <= twins(x) + twins(y)
in [Math]
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From: Transfer Principle on 15 May 2010 21:44 On May 15, 6:21 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > So Hughes did respond that my theory lacks rigor -- > > since what can be less rigorous than a theory that is > > proved _inconsistent_? > "Lacks rigor" suggests that the theory is too informal, vague, > ambiguous. Your theory was perfectly clear -- clear enough that its > inconsistency was provable.[1] Let's go back to the original context of the comment, which was Burns's question: "And yet, none of the criticisms I have seen of your attempts to wipe away someone's crank-hood have mentioned insufficient rigor or excessive ad-hoc-ness. How do you explain this?" So Burns and Hughes argue that the reason that the theories that I give in order to convince people that a poster isn't a "crank" is _not_ because the theories that I give aren't rigorous. And so, let me attempt to correct my answer and figure out another reason that my theories fail to convince people to avoid such words. Suppose one poster writes "1>0.999..." while another poster writes "1=0.999..." instead. We know that the latter poster doesn't merit a five-letter label, since there's an acceptable and well-established theory, ZFC, in which it's provable. Thus if I can find a theory sufficiently similar to ZFC which proves the former statement, then the former poster can avoid the label as well. And since the theories that I have posted so far have failed to convince anyone, then the theories that I have posted aren't sufficiently similar to ZFC. So what really would be an acceptable theory to prove that a poster of "1>0.999..." isn't a "crank"? We know that a theory one of whose axioms is just "1>0.999..." will not convince anyone. I will keep trying to find a theory which convinces others that posters of "1>0.999..." don't merit any five-letter labels. Even if it takes a long time to find a such a theory, I'd much rather look for one than call the poster "wrong" or worse -- especially when there are already four or five posters to call the poster "wrong." > "If .999... = 1 then (.999...)/1 should equal 1 > let's see > (.999...)/1 = .999... > [Therefore] .999... still=/= 1" -- An astonishing proof by "S. Enterprize" And Hughes's "random" .sig strikes again! Right in the middle of a .999... discussion, here comes a ..sig which mentions .999... So let's go back to three of the main reasons that someone might post that .999... isn't equal to unity, and see which one describes S. Enterprize the best. (1) Enterprize is discussing an alternate theory T such that T proves "1 > 0.999..." (2) Enterprize knows that ZFC proves "1 = 0.999..." but doesn't like this result, so he talks about how 0.999... ought to be strictly less than unity without referring to any theory T in which it is provable. (3) Enterprize is simply wrong, and doesn't know that ZFC proves "1 = 0.999..." at all. From this lone .sig quote, it's tough to determine which of the three cases is most likely. We do note that the so-called "proof" in the .sig quote is circular (which is a likely reason that Hughes decided to include it in his .sig file in the first place). Thus, it would seem reasonable to eliminate case (1) above, since one would think that the creator of a new theory would know how to avoid circular proofs. So absent other information, cases (2) and (3) seem to be the most likely. Either Enterprize _knows_ that standard theory proves "1 = 0.999..." but doesn't find this result satisfactory, or he mistakenly believes that the two standard/classical reals 0.999... and unity are distinct. (Note that only in this last case would I consider the use of the word "wrong" to describe Enterprize accurate.)
From: Transfer Principle on 15 May 2010 22:02 On May 15, 6:27 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > My claim is that if someone posted a line of _borderline_ > > rigor, a more reputable poster is more likely to be given > > the benefit of the doubt (i.e., have the post treated as > > if it were of full mathematical rigor) than one who has > > received a five-letter label. > Yes, when we've seen that a poster is evidently knowledgeable about > mathematics, then we are likely to assume further mathematical writing > can be made rigorous. If we've seen a poster is not knowledgeable, > then we would prefer more explicit reasoning in his arguments. Thank you for admitting this. Let's go back to an explicit example of this. Months ago, there was a poster named T.H. Ray, who wanted to give a new definition of "function." According to Ray's definition, a function is an object which "inheres changes." Predictably, other posters, including IIRC Hughes himself, criticized Ray's use of "inheres." Naturally, the first thing that Ray was asked to do is define "inheres." One was skeptical that "inheres" is even a real word. Well, guess what? Today on my Page-a-Day word calendar is the following entry: inhere \in-'hir\ v: to be inherent: to be a fixed element or attribute. So "inhere" really is an English word! One might argue that "inhere" is a back-formation of "inherent," but the etymology reveals that "inherent" is actually the adjectival form of the verb "inhere." (This is example where the derived word is more common than the root word.) Suffice it to say that had an established mathematician been the poster, he's less likely to be asked to define "inheres" than a poster like Ray. So what was Ray really saying? He was simply defining a function as an object in which change is _inherent_. In fact, I can see where Ray is coming from. If there is an object whose attributes change with, say, time, then we are saying that the attribute is a function of time. In the real world, functions occur when one attribute can be described by a variable that changes with respect to some other variable (often time). Mathematicians, of course, prefer to be more abstract, so they prefer definitions which refer to ordered pairs and Cartesian products. In that other thread, Ray was challenged for excluding the empty set (a degenerate function), constant functions, and possibly even the identity function. But once again, this shows yet again a disconnect between mathematicians and the real world, for non-mathematicians seldom consider something that doesn't change as a "function," or consider the empty set to be a "set" of anything, much less a function.
From: Transfer Principle on 15 May 2010 22:18 On May 15, 6:27 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > My claim is that if someone posted a line of _borderline_ > > rigor, a more reputable poster is more likely to be given > > the benefit of the doubt (i.e., have the post treated as > > if it were of full mathematical rigor) than one who has > > received a five-letter label. > That is bending the principle of charity past the breaking point, so > that it becomes the pandering assumption that no one is ever wrong, Several times in this thread, I've been described as being "pandering," or failing to respect one's intelligence, if I try avoid telling a poster that he's wrong. But this would seem to imply that if I really wanted to respect others' intelligence, I should tell other posters that they are wrong! I strongly disagree with this idea. I believe that telling a poster that he is wrong is _never_ respectful to his intelligence at all. To tell a poster that he is wrong is implying that he's not intelligent enough to know what the right answer is! Even in case (3) above, in which the word "wrong" to describe the poster can be accurate, I don't believe that saying so respects intelligence (but I don't feel that one has to respect intelligence in every sentence). And of course, the other five-letter words are even more disrespectful to one's intelligence. So I don't see how it is ever _more_ respectful of one's intelligence and _less_ pandering to tell a poster that he is wrong than to tell him that he's wrong. Recall that I am willing to call a poster _wrong_, as long as there aren't four or five other posters in the thread calling him wrong as well. I'd much rather find a way for the poster to be right than _repeat_ (and I emphasize the word _repeat_) that he is wrong.
From: Jesse F. Hughes on 15 May 2010 22:18 Transfer Principle <lwalke3(a)lausd.net> writes: > On May 15, 6:21 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Transfer Principle <lwal...(a)lausd.net> writes: >> > So Hughes did respond that my theory lacks rigor -- >> > since what can be less rigorous than a theory that is >> > proved _inconsistent_? >> "Lacks rigor" suggests that the theory is too informal, vague, >> ambiguous. Your theory was perfectly clear -- clear enough that its >> inconsistency was provable.[1] > > Let's go back to the original context of the comment, > which was Burns's question: > > "And yet, none of the criticisms I have seen of your > attempts to wipe away someone's crank-hood have mentioned > insufficient rigor or excessive ad-hoc-ness. How do > you explain this?" > > So Burns and Hughes argue that the reason that the > theories that I give in order to convince people that > a poster isn't a "crank" is _not_ because the theories > that I give aren't rigorous. And so, let me attempt to > correct my answer and figure out another reason that my > theories fail to convince people to avoid such words. > > Suppose one poster writes "1>0.999..." while another > poster writes "1=0.999..." instead. We know that the > latter poster doesn't merit a five-letter label, since > there's an acceptable and well-established theory, ZFC, > in which it's provable. Thus if I can find a theory > sufficiently similar to ZFC which proves the former > statement, then the former poster can avoid the label > as well. And since the theories that I have posted so > far have failed to convince anyone, then the theories > that I have posted aren't sufficiently similar to ZFC. > > So what really would be an acceptable theory to prove > that a poster of "1>0.999..." isn't a "crank"? We know > that a theory one of whose axioms is just "1>0.999..." > will not convince anyone. > > I will keep trying to find a theory which convinces > others that posters of "1>0.999..." don't merit any > five-letter labels. Even if it takes a long time to > find a such a theory, I'd much rather look for one > than call the poster "wrong" or worse -- especially > when there are already four or five posters to call > the poster "wrong." Yes, we know that's what you'd prefer. Frankly, I don't know why you prefer to do so. It seems to me that, on the contrary, this is disrespectful to the poster when it is apparent that he *meant* to interpret 1 and 0.999... in the usual way. But all of this is beside the point. The problem was not that your theory was not rigorous. It was, as I recall, perfectly rigorous. It happened to be inconsistent. >> "If .999... = 1 then (.999...)/1 should equal 1 >> let's see >> (.999...)/1 = .999... >> [Therefore] .999... still=/= 1" -- An astonishing proof by "S. Enterprize" > > And Hughes's "random" .sig strikes again! Right in > the middle of a .999... discussion, here comes a > .sig which mentions .999... You can put scare quotes wherever you wish, but it was a randomly chosen quote. > So let's go back to three of the main reasons that > someone might post that .999... isn't equal to > unity, and see which one describes S. Enterprize > the best. > > (1) Enterprize is discussing an alternate theory T such > that T proves "1 > 0.999..." > (2) Enterprize knows that ZFC proves "1 = 0.999..." but > doesn't like this result, so he talks about how 0.999... > ought to be strictly less than unity without referring to > any theory T in which it is provable. > (3) Enterprize is simply wrong, and doesn't know that ZFC > proves "1 = 0.999..." at all. > > From this lone .sig quote, it's tough to determine which > of the three cases is most likely. We do note that the > so-called "proof" in the .sig quote is circular (which > is a likely reason that Hughes decided to include it in > his .sig file in the first place). Thus, it would seem > reasonable to eliminate case (1) above, since one would > think that the creator of a new theory would know how to > avoid circular proofs. > > So absent other information, cases (2) and (3) seem to be > the most likely. Either Enterprize _knows_ that standard > theory proves "1 = 0.999..." but doesn't find this result > satisfactory, or he mistakenly believes that the two > standard/classical reals 0.999... and unity are distinct. > > (Note that only in this last case would I consider the > use of the word "wrong" to describe Enterprize accurate.) It rather depends on what one means by "satisfactory". I don't recall the context here, but the circularity of the proof was indeed why I included it in the collection. It was a simple example of bad reasoning, which struck me as humorous. A google groups search on "S. Enterprize" "should equal 1" brings up nothing but one post (by me) on March 24 of this year. This quote has been in my collection from at least Nov. 22, 2004 if not earlier, so it's remarkably unlikely that it has appeared only twice in my posts. Google Groups is broken. If you search on "author:jesse(a)phiwumbda.org Enterprize", another post appears -- again, of course, my .sig and not the original. No idea the original context. Someone more interested might look around Nov. 22, 2004, since that *might* be close to when I first found it. -- Jesse F. Hughes "Now 'pure math' makes sense as well as clearly it's a peacock game, where some of you see it as a way to show you as being highly intelligent and thus more desirable to women." -- James S. Harris
From: Jesse F. Hughes on 15 May 2010 22:31
Transfer Principle <lwalke3(a)lausd.net> writes: > On May 15, 6:27 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Transfer Principle <lwal...(a)lausd.net> writes: >> > My claim is that if someone posted a line of _borderline_ >> > rigor, a more reputable poster is more likely to be given >> > the benefit of the doubt (i.e., have the post treated as >> > if it were of full mathematical rigor) than one who has >> > received a five-letter label. >> Yes, when we've seen that a poster is evidently knowledgeable about >> mathematics, then we are likely to assume further mathematical writing >> can be made rigorous. If we've seen a poster is not knowledgeable, >> then we would prefer more explicit reasoning in his arguments. > > Thank you for admitting this. > > Let's go back to an explicit example of this. Months ago, > there was a poster named T.H. Ray, who wanted to give a new > definition of "function." According to Ray's definition, a > function is an object which "inheres changes." > > Predictably, other posters, including IIRC Hughes himself, > criticized Ray's use of "inheres." Naturally, the first > thing that Ray was asked to do is define "inheres." One > was skeptical that "inheres" is even a real word. > > Well, guess what? Today on my Page-a-Day word calendar is > the following entry: > > inhere \in-'hir\ v: to be inherent: to be a fixed element > or attribute. > > So "inhere" really is an English word! One might argue > that "inhere" is a back-formation of "inherent," but the > etymology reveals that "inherent" is actually the > adjectival form of the verb "inhere." (This is example > where the derived word is more common than the root word.) > > Suffice it to say that had an established mathematician > been the poster, he's less likely to be asked to define > "inheres" than a poster like Ray. I didn't ask Ray for a definition of "inheres". I asked what he meant by the whole phrase. I surely didn't doubt that "inheres" was an English word. > So what was Ray really saying? He was simply defining a > function as an object in which change is _inherent_. In > fact, I can see where Ray is coming from. If there is an > object whose attributes change with, say, time, then we > are saying that the attribute is a function of time. In > the real world, functions occur when one attribute can > be described by a variable that changes with respect to > some other variable (often time). But he (and we) were talking about mathematical functions and his definition made no sense at all. For what it's worth, it was this exchange which formed my opinion of T.H. Ray (and which seems consistent to this day). He's not a crank, so much as a blowhard. He talks about things as if he understands them, when he does not. He doesn't try to revolutionize mathematics, so much as pretend to grasp it. > Mathematicians, of course, prefer to be more abstract, > so they prefer definitions which refer to ordered pairs > and Cartesian products. In that other thread, Ray was > challenged for excluding the empty set (a degenerate > function), constant functions, and possibly even the > identity function. But once again, this shows yet again > a disconnect between mathematicians and the real world, > for non-mathematicians seldom consider something that > doesn't change as a "function," or consider the empty > set to be a "set" of anything, much less a function. But the topic was the mathematical notion of function, not some other notion of function. Either Ray was mistaken about what everyone else was discussing, or he was ignorant of the nature of mathematical function. In either case, he was simply wrong. I know that's a shocking conclusion to you, but it's the evident truth. -- "You got more out of it than I put into it last night. Who were you thinking of when we were loving last night?" -- Texas Tornadoes |