August 2010 - mathfun@mailman.xmission.com

Re: [math-fun] left vs. right
by Dan Asimov 30 Aug '10

30 Aug '10

No problem here with left and right, but sometimes I stay fixated on viewing another location as if from where I currently am, so that if it's south of me I'll think of its north end as nearer to me, its south end farrer*. Then when switching to an actual map this can be momentarily disconcerting. --Dan _____________________________ * Well, it *oughta* be a word. _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Does Your Language Shape How You Think?
by Dan Asimov 30 Aug '10

30 Aug '10

Here's a 1992 paper, "Magnetite biomineralization in the human brain": < http://www.ncbi.nlm.nih.gov/pmc/articles/PMC49775/ >, from the Proceedings of the National Academy of Sciences. >From the abstract: << Although the mineral magnetite (Fe3O4) is precipitated biochemically by bacteria, protists, and a variety of animals, it has not been documented previously in human tissue. Using an ultrasensitive superconducting magnetometer in a clean-lab environment, we have detected the presence of ferromagnetic material in a variety of tissues from the human brain. >> --Dan ------------- Veit wrote: << I wrote: << Apparently the presence of magnetite has recently been confirmed in human brains. >> References? I found a paper reporting magnetite in sinus bones -- not within the actual brain. It's not clear how magnetic torques would be detected by neurons or glial cells. >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Does Your Language Shape How You Think?
by Dan Asimov 30 Aug '10

30 Aug '10

Apparently the presence of magnetite has recently been confirmed in human brains. I wonder if the groups that seem to always know which way is N, S, E,or W are those that can sense directions via the Earth's magnetic field. --Dan Hans wrote: << Interesting article, I thought, by Guy Deutscher in Thursday's N.Y. Times: . . . "In order to speak a language like Guugu Yimithirr, you need to know where the cardinal directions are at each and every moment of your waking life." >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Does Your Language Shape How You Think?
by Hans Havermann 30 Aug '10

30 Aug '10

Interesting article, I thought, by Guy Deutscher in Thursday's N.Y. Times: http://www.nytimes.com/2010/08/29/magazine/29language-t.html?_r=1&partner=r… "In order to speak a language like Guugu Yimithirr, you need to know where the cardinal directions are at each and every moment of your waking life." "... some languages, like Matses in Peru, oblige their speakers, like the finickiest of lawyers, to specify exactly how they came to know about the facts they are reporting... If a statement is reported with the incorrect 'evidentiality', it is considered a lie."

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Re: [math-fun] Does Your Language Shape How You Think?
by Dan Asimov 30 Aug '10

30 Aug '10

Yes, it was interesting. When I read it yesterday, it seemed at the beginning -- where the author was saying that much of what Benjamin Lee Whorf had written was wrong -- that he was insisting that the language you speak does *not* shape how you think. But later the article gave a number of examples that seemed to confirm that it does. So I'm puzzled as to what the author believes. I have no doubt that aspects of English shape how we think. A striking example is that when you say you "do not believe" or "do not like" X, your not at all saying merely that "It is false that I believe (or like) X." Instead you're saying you disbelieve, or dislike, X. It's as if English is pushing you to have an opinion. You have to go out of your way to express the fact that you're just not yet convinced of something, or don't yet know how much you like it. I don't know how many other languages have the same issue. --Dan << Interesting article, I thought, by Guy Deutscher in Thursday's N.Y. Times: http://www.nytimes.com/2010/08/29/magazine/29language-t.html?_r=1&partner=r… "In order to speak a language like Guugu Yimithirr, you need to know where the cardinal directions are at each and every moment of your waking life." "... some languages, like Matses in Peru, oblige their speakers, like the finickiest of lawyers, to specify exactly how they came to know about the facts they are reporting... If a statement is reported with the incorrect 'evidentiality', it is considered a lie." >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Lexical distance?
by Dan Asimov 28 Aug '10

28 Aug '10

Marc wrote: << Usually when you sum a series, for pi, say, the leftmost digits tend to settle down first. Consequently, the approximations move closer in distance to the limit, and we say it converges in the familiar delta-epsilon way. Now instead of summing a series, imagine a process where the digits are all fizzing around, maybe even chaotically, but as time goes by a greater percentage (whatever that means) match the digits of the limit value. For example imagine each bit gets randomly flipped from its correct value with decreasing frequency. That feels a lot like what I mean intuitively by convergence, but it's all a bit sketchy. So I'm interested in learning of relevant prior art, best practices, or simply better ways to talk about this. >> Given an oo-string s and sequence of oo-strings {s_n}, let A_n \sub Z+ be the set of places for which s and s_n agree. You want A_n to converge in some sense to all of Z+. Let's see, in purely set theoretic terms is this equivalent to saying that for every place k, there is some n = n(k) such that for j >= n(k) we have that k is in the set A_j ??? This is just what's going on when a sequence of points in the Cantor set K = {0,1}^oo converges to the point 1^oo, where the points of the Cantor set can be identified with subsets of Z+ as their indicator functions. So it's the same thing as when points in K converge under that "Hilbert" metric of dist(A_n, Z+) = sqrt( Sum_{k=1..oo} D(s(k), s_n(k))^2 / n^2 ), where t(k) is the kth component of the oo-string t, and D is the Kronecker delta. --Dan or maybe more to the point: Marc, is this what you want to happen? _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Lexical distance?
by Dan Asimov 27 Aug '10

27 Aug '10

Besides comparing identical places in two strings, one might also care how close they are to being rearrangements of each other. E.g., one might want to consider XYXYXY and YXYXYX much closer than XYXYXY and ZZZZZZ, even though both pairs differ in every place. Assuming finite strings from a finite alphabet in which all pairs of letters are equally "distant", one might define lexical distance as the fewest "moves" to go from one string to another, where a move is either 1) the transposition T of two adjacent string elements, or 2) the substitution S of one letter by another in the same place. In general the two types of moves would be weighted differently, depending on the application. D = p*(#T) + (1-p)*(#S) For an alphabet where letters have various distances, this could be changed slightly to reflect that. --Dan Marc wrote: << A very rough kind of ³lexical distance² between strings (possibly infinite) is just the number of symbols by which they differ. >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Lexical distance?
by Dan Asimov 27 Aug '10

27 Aug '10

Regarding convergence, only a topology is required, and assuming a finite alphabet A with the discrete topology, the space of infinite strings J takes on the topology of the direct product J = A x A x A x . . .. This makes J homeomorphic to the Cantor set, and is of course unchanged under any permutation of the factors. This topology can be realized by a metric, but unfortunately there's a loss of symmetry: I don't think there's any metric such that all self-homeomorphisms are isometries. One metric realizing the above topology -- so at least convergence in the metric equals convergence in the topology -- is to let the nth alphabet factor have all pairs of letters at a distance of 1/n, and then to define distance by the usual Euclidean formula -- square root of sum of squares of distances in all factors. Of course the maximum possible distance is then pi/sqrt(6). --Dan Marc wrote: << . . . I'm trying to grok processes where infinite strings can be said to converge element-wise, but the strings themselves are only roughly inter-related--having Hamming distance, but no meaningful lexical order. >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Lexical distance?
by Marc LeBrun 27 Aug '10

27 Aug '10

A very rough kind of ³lexical distance² between strings (possibly infinite) is just the number of symbols by which they differ. Is there an Official Name for this?

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Re: [math-fun] Lexical distance?
by Paul Reiners 26 Aug '10

26 Aug '10

Hamming distance, I think. On Aug 26, 2010 6:47 PM, "Marc LeBrun" <mlb(a)well.com> wrote: A very rough kind of ³lexical distance² between strings (possibly infinite) is just the number of symbols by which they differ. Is there an Official Name for this? _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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