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[math-fun] Stupid Bell 103 modem question
by Henry Baker 04 May '18

04 May '18

Did anyone ever get this modem to digitally communicate via tin cans and a wire? It might be a fun high school science project, now that the 103 is now just open source software on typical Windows/Mac/Linux laptops.

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Re: [math-fun] Normal subgroups
by Dan Asimov 04 May '18

04 May '18

Jim Propp wrote: ----- for large n, if you pick a group G of order n uniformly at random ----- I wonder how to define a probability on groups of order n. And in particular: If there are several ways to do this, do they result in the same probabilities? —Dan

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Re: [math-fun] 60-Year-Old Maths Problem Partly Solved By Amateu
by Dan Asimov 04 May '18

04 May '18

This is beyond a doubt the same result discussed here recently: ----- The chromatic number of the plane is at least 5. ----- In fact there's now a polymath project, led in part by Terry Tao, to try to simplify the proof or improve this bound. —Dan P.S. No offense, but this is a rather poor description of the problem. It doesn't even mention the plane! But then again, math is not The Grauniad's strong point: ----- The problem is as follows. Imagine a collection of dots connected by > lines. The dots can be arranged any way at all, the only rule is that > all the connecting lines must be of equal length. For instance, in a > square the diagonal would not be joined up, but the outer edges would > be. Now, colour in all the dots so that no two connected points have the > same colour. How many colours are required. For a square, the answer > would be two. But the Hadwiger-Nelson problem asks what the minimum > would be for any configuration -- even one that extends across a plane > of infinite size./ -----

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[math-fun] 60-Year-Old Maths Problem Partly Solved By Amateu
by Ray Tayek 04 May '18

04 May '18

https://science.slashdot.org/story/18/05/04/1521239/60-year-old-maths-probl… Posted by msmash on Friday May 04, 2018 @11:22AM from the unravelling-mysteries dept. An amateur mathematician has made the first breakthrough in more than 60 years towards solving a well-known maths problem. From a report:/Aubrey de Grey, who is more widely known as a maverick biologist intent on extending the human lifespan, has taken the academic world by surprise after announcing a new solution to the so-called Hadwiger-Nelson problem <https://www.theguardian.com/science/2018/may/04/60-year-old-maths-problem-p…>. The problem <http://mathworld.wolfram.com/Hadwiger-NelsonProblem.html> sounds deceptively simple, but despite some professionals spending years trying to crack it, progress has stalled since shortly after the puzzle was first posed in 1950. "Literally, this is the first progress in more than 60 years," said Gil Kalai, a mathematician at Hebrew University of Jerusalem. The problem is as follows. Imagine a collection of dots connected by lines. The dots can be arranged any way at all, the only rule is that all the connecting lines must be of equal length. For instance, in a square the diagonal would not be joined up, but the outer edges would be. Now, colour in all the dots so that no two connected points have the same colour. How many colours are required. For a square, the answer would be two. But the Hadwiger-Nelson problem asks what the minimum would be for any configuration -- even one that extends across a plane of infinite size./ -- Honesty is a very expensive gift. So, don't expect it from cheap people - Warren Buffett http://tayek.com/

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[math-fun] paleocybernetic obscurity
by Bill Gosper 04 May '18

04 May '18

(Old timers only.) Identify: qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6qz6... --rwg

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Re: [math-fun] Set of all sets question
by Dan Asimov 03 May '18

03 May '18

The philosophical problem concerning me here is nothing new. Andy has focused things by pointing to the *union* of the A(beta). It just feels intuitively that given a bunch of nested sets (that *exist*), then their union "ought to" exist as well. And the fact that in some systems the set of all sets can exist (as a class) lends credence to that intuition. —Dan ----- On Thursday, May 3, 2018, Andy Latto <andy.latto(a)pobox.com> wrote: > On Thu, May 3, 2018 at 4:19 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: > > QUESTION: > > --------- > > How can it be that each A(beta) exists, and for that matter any subset > of them exists, > > but all of them cannot exist at once (without contradiction)? > > I don't understand what you think is a contradiction. All of them can > and do exist at once. Their union does not exist (if using ZF) or is a > class but not a set (if using Goedel-Bernay), but this is not a > contradiction, because the union axiom only says that given a *set* of > sets, the union set also exists. It does not guarantee the existence > of the union of any class of sets exists. ... -----

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[math-fun] Set of all sets question
by Dan Asimov 03 May '18

03 May '18

We know that the Class of all sets Sets cannot be a set, because this would imply the existence of the set X = {s in Sets | s is not a member of s}, but X belongs to X if and only if it doesn't, hence Contradiction. But we are told that it makes perfect sense to speak of all sets no larger than some fixed cardinal number = beta. Let the set of all sets that are no larger than the cardinal beta be denoted by A(beta): A(beta) = {s in Sets | card(s) <= beta}. QUESTION: --------- How can it be that each A(beta) exists, and for that matter any subset of them exists, but all of them cannot exist at once (without contradiction)? —Dan

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Re: [math-fun] Is the Continuum Hypothesis a) really true or really false, or b) something else ?
by hbaker1 02 May '18

02 May '18

But that's the heart of the incompleteness theorem: *any* theory strong enough to describe number theory or Turing Machines has certain true statements that have no (finite) proofs. Recall that proofs are finite sequences generated by non-deterministic machines from axioms and rules of inference in an attempt to list all theorems. By a strong analogy to diagonalization of an attempt to list all of the numeral sequences of the real numbers, we can construct a new real number not on the list; and hence we can construct a new truth not on the list of provable theorems. -----Original Message----- >From: Cris Moore <moore(a)santafe.edu> >Sent: Apr 30, 2018 10:42 PM >To: hbaker1 <hbaker1(a)pipeline.com>, math-fun <math-fun(a)mailman.xmission.com> >Subject: Re: [math-fun] Is the Continuum Hypothesis a) really true or really false, or b) something else ? > >Right, the Halting Problem is asymmetric: there’s a finite proof of halting if it halts, but not necessarily a finite proof that it runs forever if it runs forever. > >Cris > >> On Apr 30, 2018, at 11:02 PM, hbaker1 <hbaker1(a)pipeline.com> wrote: >> >> Turing Machines which halt: "recursively enumerable" == "RE" >> Turing Machines which don't halt: complement of RE. >> >> Turing proved that they aren't the same sets. >> >> Thus, RE sets are kind of "fractals", in that their boundaries are a little difficult to determine -- at least from one direction. >> >> -----Original Message----- >>> From: Cris Moore <moore(a)santafe.edu> >>> Sent: Apr 30, 2018 2:28 PM >>> To: math-fun <math-fun(a)mailman.xmission.com> >>> Subject: Re: [math-fun] Is the Continuum Hypothesis a) really true or really false, or b) something else ? >>> >>>> On Apr 30, 2018, at 3:21 PM, Cris Moore <moore(a)santafe.edu> wrote: >>>> >>>> However, we should be careful, since even questions about the halting of Turing machines — which are certainly first-order claims about the integers — can be independent of ZFC: https://www.scottaaronson.com/busybeaver.pdf <https://www.scottaaronson.com/busybeaver.pdf> <https://www.scottaaronson.com/busybeaver.pdf <https://www.scottaaronson.com/busybeaver.pdf>> >>> >>> Sorry, continuing: note that if a Turing machine halts in finite time, there is a finite proof of that fact — namely, run it and see what happens. But if it runs forever, proving that it does so might be difficult. >>> >>> But surely there is a truth of the matter here! The Turing machine will either run forever, or it won’t. I find it hard to be a pluralist here. >>> >>> - Cris

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[math-fun] Radix-r numbers; |r|=1, r=log(ik)
by hbaker1 02 May '18

02 May '18

Approximately represent complex numbers in radix r, where r=log(ik), for some integer k>0, i=sqrt(-1). Several choices of numerals: {0,1}, {0,i}, {0,1,i}, {0,+-1,+-i}. So |r|=1, and we have good approximations to *random walks*. This is not a particularly efficient number system, so is there an efficient algorithm to compute an approximation?

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Re: [math-fun] Is the Continuum Hypothesis a) really true or really false, or b) something else ?
by hbaker1 02 May '18

02 May '18

I would imagine that it would be impossible for a CH-denier to get tenure in any major American university math department today. I suspect that YouTube and Twitter will suppress any postings by CH-deniers, as well. CH-deniers disrupt the entire Hilbert Program.

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