March 2018 - mathfun@mailman.xmission.com

[math-fun] Math-Fun topics
by rcs＠xmission.com 02 Mar '18

02 Mar '18

We've had some new folks join us recently, who might be unaware of some of my arbitrary rules. The topics of politics and global warming are banned. Rich

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Re: [math-fun] next breakthrough?
by Henry Baker 02 Mar '18

02 Mar '18

I should have just gone ahead and indicated why undecidability scales. Basically, a TM with a given amount of time can *diagonalize* over all TM's who use a bit less time. "Diagonalize" means emulate and then disagree with. Similarly, a TM with a given amount of space can *diagonalize* over all TM's who use a bit less space. "A bit less" is a pretty small delta, because TM emulation can be done with only a small slowdown in time and a very small increase in space. With modern so-called "virtual machines" (VM's) and "containers", most web servers run as VM's on larger hardware. Indeed, "containers" are coming to web browsers, and there are some operating systems that are Russian dolls of VM's. At 10:01 PM 3/1/2018, Henry Baker wrote: >True, but TM complexity scales extremely rapidly with time and/or space (amount of tape), so even "approximations" to large amounts of time and space would require googols of universes to compute. > >So undecidability is pretty smooth/continuous in those dimensions. > >This complexity scaling is far above and beyond the P=NP issue, which gets you only one exponential -- at most. > >100GBytes of RAM on big server machines is quite common. That's quite enough "tape"/"space" for serious exponential stacking. > >At 06:10 PM 3/1/2018, Mike Speciner wrote: >>We don't build Turing machines. We build finite state machines, admittedly with very large numbers of states. So, technically, although not practically, undecidability doesn't apply. >> >>--ms >> >>On 01-Mar-18 20:17, Henry Baker wrote: >>>By "consequential", I think that you may mean "consequential" to >>>a large fraction of the population -- e.g., electrical engineers >>>and physicists are required to work with complex numbers. >>>Consequential enough to include in high school curricula. >>> >>>I'm still waiting for undecidable/non-Turing-computable to make >>>its way into the bulk of society; a lot of the incredibly stupid >>>comments by politicians might be avoided if they had had any >>>education in the *limits* of logical systems and mathematics. >>> >>>For example, no amount of "nerding harder" will enable computer >>>scientists to design a computer such that an arbitrary input >>>won't cause the computer do something "bad". >>> >>>It took physicists many centuries to convince a large fraction >>>of the population about the impossibility of perpetual motion >>>machines, so perhaps we may be facing the same thing here. >>> >>>Curiously, we have *more* trust in undecidability than we >>>have in the impossibility of perpetual motion, because the >>>"laws" of physics are still empirical facts, which might be >>>destroyed with the very next experiment, while undecidability >>>is a function of logic itself, and requires no reference to >>>the "real world". >>> >>>Regarding breakthroughs that are not yet known even to the >>>best of mathematicians, I would say that things related to >>>P=NP and cryptography are pretty high on the list. For example, >>>we are close to trusting cryptography hash functions with all >>>the money in the world (literally!), yet we have no *proof* >>>that these functions are hard to invert or hard to find >>>collisions for. With this much crypto currency in circulation, >>>there's obviously no need for a "Millennium Prize", since a >>>constructive counter-example would produce the richest person >>>in the world! >>> >>>With the recent advances in AI, we might also ponder the "end >>>of mathematics", but not for the usually suggested reason. AI >>>could end mathematics -- not by doing mathematics better, but >>>by doing enough other things better that no one needs mathematicians >>>anymore. This is where we seem to be headed with chess and Go; >>>once machines are so much better, everyone loses interest in the >>>games so no one plays anymore. >>> >>>At 03:47 PM 3/1/2018, Bernie Cosell wrote: >>>>i know this is vague, but my wife was reading about the history of >>>>i and wondered what the next great fundamental conceptual breakthrough >>>>might be. it took something like 1500 years for complex numbers to >>>>get understood and now they're used everywhere. it's now been >>>>about 250 years and she was wondering what might be the next >>>>breakthrough be that could be as consequential as complex numbers are.

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Re: [math-fun] next breakthrough?
by Henry Baker 02 Mar '18

02 Mar '18

By "consequential", I think that you may mean "consequential" to a large fraction of the population -- e.g., electrical engineers and physicists are required to work with complex numbers. Consequential enough to include in high school curricula. I'm still waiting for undecidable/non-Turing-computable to make its way into the bulk of society; a lot of the incredibly stupid comments by politicians might be avoided if they had had any education in the *limits* of logical systems and mathematics. For example, no amount of "nerding harder" will enable computer scientists to design a computer such that an arbitrary input won't cause the computer do something "bad". It took physicists many centuries to convince a large fraction of the population about the impossibility of perpetual motion machines, so perhaps we may be facing the same thing here. Curiously, we have *more* trust in undecidability than we have in the impossibility of perpetual motion, because the "laws" of physics are still empirical facts, which might be destroyed with the very next experiment, while undecidability is a function of logic itself, and requires no reference to the "real world". Regarding breakthroughs that are not yet known even to the best of mathematicians, I would say that things related to P=NP and cryptography are pretty high on the list. For example, we are close to trusting cryptography hash functions with all the money in the world (literally!), yet we have no *proof* that these functions are hard to invert or hard to find collisions for. With this much crypto currency in circulation, there's obviously no need for a "Millennium Prize", since a constructive counter-example would produce the richest person in the world! With the recent advances in AI, we might also ponder the "end of mathematics", but not for the usually suggested reason. AI could end mathematics -- not by doing mathematics better, but by doing enough other things better that no one needs mathematicians anymore. This is where we seem to be headed with chess and Go; once machines are so much better, everyone loses interest in the games so no one plays anymore. At 03:47 PM 3/1/2018, Bernie Cosell wrote: >i know this is vague, but my wife was reading about the history of >i and wondered what the next great fundamental conceptual breakthrough >might be. it took something like 1500 years for complex numbers to >get understood and now they're used everywhere. it's now been >about 250 years and she was wondering what might be the next >breakthrough be that could be as consequential as complex numbers are.

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[math-fun] Everybody needs a conversation stopper.
by Bill Gosper 02 Mar '18

02 Mar '18

Sum[ArcTanh[√3Tan[t/(-2)^n]], {n,∞}] == -ArcTanh[Tan[t]/√3] Actually, this clearly must telescope, with a presumably easy inductive verification. It might even be worthwhile for Mathematica to check for summands and prodands of the form f(index+freevariable). --rwg

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[math-fun] packing polyominoes?
by Mike Stay 01 Mar '18

01 Mar '18

My family recently gave me some packing puzzles for my birthday. One is a rectangle packing puzzle, where the rectangles' edges remain parallel to the sides of the tray; there are some pretty obvious algorithms for solving that one. The other is "Stewart's Coffin", which Martin Gardner pronounced "...the finest dissection puzzle of all time. It looks easy but is fiendishly difficult." It involves four laser-cut polyominoes and a tray that is just slightly too small in either direction for easy placement of the polynomials. Clearly the puzzle involves placing the pieces at an angle, but exactly which angle and how to arrange the pieces is the hard problem. Are there algorithms for solving a puzzle like this other than brute force? -- Mike Stay - metaweta(a)gmail.com http://www.math.ucr.edu/~mike http://reperiendi.wordpress.com

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[math-fun] next breakthrough?
by Bernie Cosell 01 Mar '18

01 Mar '18

i know this is vague, but my wife was reading about the history of i and wondered what the next great fundamental conceptual breakthrough might be. it took something like 1500 years for complex numbers to get understood and now they're used everywhere. it's now been about 250 years and she was wondering what might be the next breakthrough be that could be as consequential as complex numbers are. /bernie\ -- Bernie Cosell bernie@fantasyfarm. com -- Too many people, too few sheep --

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Re: [math-fun] next breakthrough?
by Hilarie Orman 01 Mar '18

01 Mar '18

What things do we know about today but don't realize how important they will be in the future? A guessing game. Conway numbers. Quantum automata. Leave everything to computers; ask AlphaZero to explain it to you. DNA customization melded with silicon and the creation of the Superhuman. Fractals. Continued fractions. Probablistic arithmetic. Catastrophe theory. Dystopian theory. The end of time. Hilarie

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