July 2009 - mathfun@mailman.xmission.com

Re: [math-fun] math, existence, and God
by Dan Asimov 08 Jul '09

08 Jul '09

Forgive me if I'm repeating myself, but I feel inclined to reply to the one part of what Jim quoted that I understand -- a quote from an earlier post by Andy: << Back on March 25, Andy Latto wrote: >My view is that philosphers often worry too much about ontology (which >things "exist" and which don't), when it really isn't a very >interesting or important question, and it's just the nature of our >language that misleads us into thinking it is. >> Personally, I find existence (of certain things discussed in philosophy, anyway) to be of the utmost interest to me, much as is the case in mathematics. For example, does determinism exist? Or does perhaps the randomness that seems inherent to quantum mechanics preclude it? Do there exist other universes that are unable to interact with our own in any way? Is the apparent flow of time a real phenomenon -- something about time itself -- or it is only an artifact of our consciousness? Besides mathematical truth, there seems to exist a physical reality and an experiential reality (at least for each part of the universe, like a living person, capable of having experiences). Do these both truly exist? Are they merely two facets of the same thing? --Dan _____________________________________________________________________ "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] math, existence, and God
by Dan Asimov 08 Jul '09

08 Jul '09

A burning question: ---------------------------- Phlogiston or no phlogiston, is it really true that a burning log does not release things? It certainly gets a lot smaller! --Dan Andy wrote: << There is a sense in which phlogiston exists, because the theory that things are flammable because they contain phlogiston, which is released by burning, exists. There is also a sense in which phlogiston does not exist, and the statement "there's no such thing as phlogiston; when things burn, they are combining with oxygen, not releasing something; phlogiston (the thing which is released when something is burned) does not exist" is a true statement. >> _____________________________________________________________________ "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] math, existence, and God
by James Propp 08 Jul '09

08 Jul '09

Dan Asimov writes: >I seem to have missed the original post. Jim, could you please >repost the original one, since I'm having trouble making sense >of this without more context. Here's what I posted back on April 28 (with a minor typo corrected): ************************************************************************* Back on March 25, Andy Latto wrote: >My view is that philosphers often worry too much about ontology (which >things "exist" and which don't), when it really isn't a very >interesting or important question, and it's just the nature of our >language that misleads us into thinking it is. [followed by a whole bunch of stuff, all of which I agree with and think is expressed extremely well, including a very trenchant thought-experiment about mental experience] Have any theologians tried to develop a theology without ontology, along lines suggested by 20th century mathematics? For instance, if one believes that the differing moral verdicts that two people reach are in principle reconcileable by way of a sufficient amount of dialogue and/or shared experience, one might define God as a kind of projective limit in the category of people-who-could-exist, in which the existence of the limit is not as important as what it entails about mappings between finite beings. This actually brings us back to qualia: Presumably you and I could reach agreement about a whole lot of things if I could experience your life from birth to the present day from your own perspective, and vice versa; but what does it mean for one person to have the experiences of another? I suspect that some science fiction writer must have explored this avenue for the future evolution of religion, even if the theologians haven't! I also suspect that some category theorists have thought along similar lines. Jim Propp ************************************************************************* Jim

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Re: [math-fun] math, existence, and God
by Dan Asimov 07 Jul '09

07 Jul '09

I seem to have missed the original post. Jim, could you please repost the original one, since I'm having trouble making sense of this without more context. Thanks, Dan Jim wrote: << I asked: >Have any theologians tried to develop a theology without ontology, >along lines suggested by 20th century mathematics? . . . . . . >> _____________________________________________________________________ "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] math, existence, and God
by James Propp 06 Jul '09

06 Jul '09

I asked: >Have any theologians tried to develop a theology without ontology, >along lines suggested by 20th century mathematics? As another example of "lines suggested by 20th century mathematics" I'll mention the theory of generalized functions, also called the theory of distributions, and the more recent wrinkle of Gaussian random fields. No longer do we specify a function by specifying what its values at specific points are; rather, we say what you get when you integrate the function against a test function. We do not concern ourselves with whether the Dirac delta function or other generalized functions "exist"; if we know how to integrate our function against every test function in some class, then we know everything about our function that we need to, and questions of existence become irrelevant. (A similar story can be told about the umbral calculus, which became respectable when most of it was reduced to linear algebra.) Of course, those who feel more secure when they can imaginatively construct the objects of their mathematical thought can (and do) construct generalized functions in a suitable dual space. But this just formalizes the original, operationalist insight that questions about what something *is* are better replaced by questions about what something *does*, or how something *interacts* with other things. Having a concrete model (like the dual space construction) is nice, especially because it affords a proof of consistency of all the properties you want these objects to have; but once you have established consistency, you can (and often do) ignore out the model and work in an operational (or, some might say, metaphysical) mode. In this paragraph I was going to close by talking about a belief that I suspect many modern mathematicians hold, namely that any interesting and consistent axiom system is going to turn out to be *about* something, and that asking whether those somethings exist in the mathematical universe is a senseless question (the answer is always "yes"), but I'd like to go a step further: Even before you've come up with axioms, when you're just messing about with objects of thought whose properties are a bit vague, and instead of propositions you're working with formal procedures or heuristics, if you start to find that what you're doing has a coherence to it, then I believe (and I suspect many other folks do to) that it is likely that one has gotten hold of fragments of a self-consistent theory that is yet to be created. Come to think of it, this lesson predates the 20th century; the calculus is a great example. So I wouldn't dismiss out-of-hand an interesting kind of theology that exhibited some logical inconsistencies as long as some non-trivial and surprising form of self-corroboration was also going on. Jim Propp

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[math-fun] Addition formula paper
by Mike Stay 06 Jul '09

06 Jul '09

I'm looking for a paper I read once (more than six years ago) on addition formulas. It had the tangent addition formula as a special case: atan(x) + atan(y) = atan((x+y)/(1-xy)) as well as log log(x) + log(y) = log(xy) and various other ones. The idea was to find a function p and an operator @ such that p(x) + p(y) = p(x@y), and they showed how to come up with a large class of these. It also, as I recall, had something to do with intersecting chords of a conic. Does anyone know what paper this was, or could anyone suggest keywords to google for it? -- Mike Stay - metaweta(a)gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com

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[math-fun] Forwarded mail....
by Richard Guy 06 Jul '09

06 Jul '09

For info on the behavior of aliquot sequences, see Richard K.~Guy \& J.~L.~Selfridge, What drives an aliquot sequence? {\it Math.\ Comput.}, {\bf 29}(1975) 101--107; {\it MR} {\bf 52} \#5542. For the latest I have on 46758 or 99225 [assuming I've made no errors] see the following, where %x shoud be read as s^{758+x)(99225) = s^{759+x}(46758} and s(n) = sigma(n) - n. The max so far is x = 10 with 109 decimal digits. It is presently equipped with a down-dribbler, 2^2 without 7. x = 18 will have only 108 digits. What are p and q ? %10 = 2634352957536601215651504310367034664237525994841454140912074056001634101746270450889034819192963235153851324 ? sigma(%10)-%10 %11 = 4375257663618965969838703886908135210362400648826184831403998005722029284842736914906603629552172468620027076 ? sigma(%11)-%11 %12 = 3812107258892467560724974268427776342559095577235276386026267812986314208130577642999593014216712259224932924 ? sigma(%12)-%12 %13 = 3357281129948365843366244433222476932983757953562658831902107291213172363666597435003856377831158249456856516 ? sigma(%13)-%13 %14 = 2518171581657269542751815956477757174750853819031465204680695087344865157789907955347820287620116570491943484 ? sigma(%14)-%14 %15 = 1888629123883173758068189431062891256470513916997092605609774658685559002236728244577771226723234026392998116 ? sigma(%15)-%15 %16 = 1416471844370396313297145002601020142112938329621791431806267389982998189766074091264106125133452654937155868 ? sigma(%16)-%16 %17 = 1062353883280402701838073792504970617432775483305445625911321617689753946299262182125924369941690102569916132 [= 2^2*15178595752127*230389769006487808825024129*p*q] R.

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Re: [math-fun] MIDI fun with Lisp, Macsyma & Trigonometry
by Henry Baker 05 Jul '09

05 Jul '09

As is usual in these kinds of things, a little insight makes up for lots of calculation. My original parameterization is misleading. Instead of a+b*sin(w*t), a>b>0, for instantaneous frequency, when the max frequency is a+b and the min frequency is a-b, the appropriate parameterization should be the _ratio_ of the min & max. More particularly, if min=a-b and max=a+b, then min/max=(a-b)/(a+b)=q, for some parameter 0<q<1. Furthermore, by using cos(w*t) instead of sin(w*t), we get the max frequency at t=0. We now have a relatively trivial mapping of unit circle to the unit circle: 2*atan(sqrt(q)*tan(alpha/2)), which happens to be a Mobius mapping. The derivative of this mapping wrt alpha gives us the expansion and/or contraction ratio at various points around the circle. The points on this unit circle are squeezed by sqrt(q) near alpha=0 and expanded by sqrt(q) near alpha=pi, and the ratio of these is sqrt(q)^2=q. So there is a geometric version of this problem: position the unit circle on the complex plane & consider the projection of the point -1+0i through a point P on the unit circle onto the vertical line at +1+0i. We stretch this vertical line by sqrt(q) & backproject to find the intersection of the circle again. That intersection is the location of P', the image of P under this mapping. >Date: Thu, 02 Jul 2009 05:02:41 -0700 >From: Henry Baker <hbaker1(a)pipeline.com> > >I ran into unexpected difficulties trying to program what I thought would have been a simple, straightforward MIDI file. > >But the difficulties at least rewarded me with some successful Macsyma interactions. > >I was trying to write a simple program to compose a MIDI (music) file that would have a simple ticking metronome clock that varied its tempo in a strictly sinusoidal pattern for one full period of the sinusoid. > >In particular, if we define the "instantaneous frequency" as being proportional to the inverse of the period between ticks, then a plot of the frequency w.r.t. time should provide points that lie on one full period of a sine wave. > >We can thus describe the frequency function as freq(t)=a+b*sin(w*t), where a>b>0. > >All you need to know about MIDI for this problem is that the output for this file basically consists of a sequence of numbers that describe the length of time to delay until the next tick, so the MIDI file is simply a sequence of _periods_. > >The major problem is to compute the locations of the ticks along the real time line, or more elegantly, around the unit circle. > >[It is relatively easy to get confused in solving this problem and inadvertently produce "phase modulation" instead of "frequency modulation". Although they might sound similar, they aren't the same.] > >There may be an elegant geometric way to see the answer to this problem, but I couldn't come up with it offhand. > >I'm not going to give away the answer here, in case someone wants to use this as an math/programming exercise, but I learned some interesting trig & Macsyma lessons. > >The main question that one would want to answer is "how many ticks do you need?" Obviously, the "average" frequency is "a", but knowing the average frequency isn't the same as knowing the average period. The answer is very elegant, but at least to my current level of understanding, not a priori obvious. > >The cool thing about this problem is that almost everything about this problem is solveable in closed form using Macsyma, but it pushes Macsyma's trig capabilities to the limit (pun intended). In particular, it is possible to develop a closed form formula period(i) that gives you the delays for each integer i. Or more elegantly, there is a closed form mapping between equally-spaced points around the unit circle and the distorted version of the "same" points around the unit circle s.t. their periods satisfy the constraint above. > >There are also some programming difficulties encountered with different branches of the various inverse trig functions involved in the solution of this problem. Thus, although the formula still "works", one must deal with certain branch issues that screw up the simple programming of this closed form solution. > >In working with Macsyma on this problem, I became quite fond of the "trigrat" simplification function, which is now my new best friend. > >-- > >[I'll probably now be embarrassed by someone on this list who comes up with a much more elegant "geometric" solution than mine.]

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[math-fun] MIDI fun with Lisp, Macsyma & Trigonometry
by Henry Baker 02 Jul '09

02 Jul '09

I ran into unexpected difficulties trying to program what I thought would have been a simple, straightforward MIDI file. But the difficulties at least rewarded me with some successful Macsyma interactions. I was trying to write a simple program to compose a MIDI (music) file that would have a simple ticking metronome clock that varied its tempo in a strictly sinusoidal pattern for one full period of the sinusoid. In particular, if we define the "instantaneous frequency" as being proportional to the inverse of the period between ticks, then a plot of the frequency w.r.t. time should provide points that lie on one full period of a sine wave. We can thus describe the frequency function as freq(t)=a+b*sin(w*t), where a>b>0. All you need to know about MIDI for this problem is that the output for this file basically consists of a sequence of numbers that describe the length of time to delay until the next tick, so the MIDI file is simply a sequence of _periods_. The major problem is to compute the locations of the ticks along the real time line, or more elegantly, around the unit circle. [It is relatively easy to get confused in solving this problem and inadvertently produce "phase modulation" instead of "frequency modulation". Although they might sound similar, they aren't the same.] There may be an elegant geometric way to see the answer to this problem, but I couldn't come up with it offhand. I'm not going to give away the answer here, in case someone wants to use this as an math/programming exercise, but I learned some interesting trig & Macsyma lessons. The main question that one would want to answer is "how many ticks do you need?" Obviously, the "average" frequency is "a", but knowing the average frequency isn't the same as knowing the average period. The answer is very elegant, but at least to my current level of understanding, not a priori obvious. The cool thing about this problem is that almost everything about this problem is solveable in closed form using Macsyma, but it pushes Macsyma's trig capabilities to the limit (pun intended). In particular, it is possible to develop a closed form formula period(i) that gives you the delays for each integer i. Or more elegantly, there is a closed form mapping between equally-spaced points around the unit circle and the distorted version of the "same" points around the unit circle s.t. their periods satisfy the constraint above. There are also some programming difficulties encountered with different branches of the various inverse trig functions involved in the solution of this problem. Thus, although the formula still "works", one must deal with certain branch issues that screw up the simple programming of this closed form solution. In working with Macsyma on this problem, I became quite fond of the "trigrat" simplification function, which is now my new best friend. -- [I'll probably now be embarrassed by someone on this list who comes up with a much more elegant "geometric" solution than mine.]

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Re: [math-fun] Addition formula paper [Weierstrass]
by Marc LeBrun 02 Jul '09

02 Jul '09

>=Mike Stay > [...] the problem: given a function (+) of two variables x,y, find a > function p() such that > p(x) + p(y) = p(x (+) y) > Somehow they got from x (+) y = xy to p'(x) = 1/x and then integrated > to get p = log. Mike, I would be quite interested in seeing this paper when you locate it. From Melzak[*] I learnt what I style "Weierstrass's Elliptic Prison": "A function f(x) is said to possess an algebraic addition formula if there is a polynomial P...such that P( f(x+y), f(x), f(y) ) = 0 ...The theorem of Weierstrass states that the only [such] functions are elliptic functions and their special degenerate cases such as trigonometric, hyperbolic and exponential functions." So exo-elliptics spawn exotic additions (+) (née plus ultra<;-) [*] Z. A. Melzak, Companion to Concrete Math. 1973 p56--but no citation to Weierstrass (not that I'd be able to follow it<;-)

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