April 2018 - mathfun@mailman.xmission.com

Re: [math-fun] Continuum Hypothesis
by Dan Asimov 30 Apr '18

30 Apr '18

I've known others with views similar to Adam's. I'm not sure I understand the reason for treating (say) the set of real numbers differently from the integers, though. The reason I believe that [the Twin Prime Conjecture is either really true or really false] is that I can imagine a way of verifying the truth status of TPC by a census of all numbers of the form p+2 where p is prime — at least in principle. But likewise, I can imagine in principle a verification of the truth status of the Continuum Hypothesis. Suppose it's true. Then for every subset X of the reals: we need to check, for every function f : X —> Z *and* for every function g : X —> R, that *at least one of these* is a bijection. (Which itself can be described as a kind of transfinite induction.) Because I can imagine such a verification (whether of CH or not-CH), I find it impossible to imagine that the truth status of CH is indeterminate. And the concept of a "subset of the reals" seems quite well-defined to me. Flip a coin once independently for each real number, and you get some subset; every subset is obtainable in that way. Anyhow, that's my story and I'm sticking to it. —Dan Adam Goucher wrote: ----- My own philosophical position is Platonist/realist about first-order properties of the natural numbers, and formalist about everything else. So I believe Con(ZFC) has a definite truth-value (namely 'true'), but CH does not have a definite truth-value. This position means that, unlike a formalist, I reject the perfectly consistent theory "ZFC + ¬Con(ZFC)" as nonsensical on the basis that it proves false statements about the naturals (such as ¬Con(ZFC)). But I'll happily work in either "ZFC + CH" and "ZFC + ¬CH", as they are both (as far as we know) arithmetically sound. If you translate this position into an invented/discovered dichotomy, and then into German, then it simplifies to Kronecker's quotation "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk". -----

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[math-fun] Is the Continuum Hypothesis a) really true or really false, or b) something else ?
by Dan Asimov 30 Apr '18

30 Apr '18

The twin prime conjecture (TPC) says there are infinitely many pairs of prime numbers separated only by 2, like 3 and 5, 5 an 7, 11 and 13, etc. No one knows if it's true or false. But whichever is the case, it might not be possible to *prove* this fact. A similar situation is the Continuum Hypothesis (CH), which states that there is no infinity strictly larger than the size of the integers and strictly smaller than the size of the real numbers. Kurt Goedel proved about 1940 that CH was consistent with the axioms of set theory (ZF, for Zermelo-Frankel), and Paul Cohen proved in 1963 that the negation of CH was *also* consistent with ZF. So in this case there is *no doubt*: There exists neither a proof of CH, nor a proof of its negation ~CH, purely based on the axioms ZF+AC of set theory. Question: --------- So, does that mean that CH is neither really true nor really false, but some third option? Or what? I'm curious what people think about this. —Dan

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Re: [math-fun] Continuum Hypothesis
by John Golden 30 Apr '18

30 Apr '18

I’ve been loving this discussion. With students (in a capstone course) we talk about Cantor and Gödel in sequence, and the CH comes off as an example of incompleteness. I.e. it could be true or false but unverifiable. So you have to change the axioms if you want it to be one or the other and provable. Am I propagating a misconception? On the other hand, most mathematicians with whom I’ve discussed this have a personal belief that it’s true or not. It often segues into a discussion on invented or discovered. I remember a great weeklong lecture from Alain Connes on why math is discovered, in which I think he identified as a CH believer. Hardcore invented types seem to stick to the CH being unverifiable, nothing more to say.

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[math-fun] When is base 2 not binary
by Lucas, Stephen K - lucassk 30 Apr '18

30 Apr '18

New to the list, so I hope this will go in the right place. But replying to Joerg Arndt’s listing of the first few natural numbers in base 2+i using the digits {0,+1,-1,+i,-i}: The problem with this digit set is that it is not immediately obvious what the rule is to go from number to number adding one. Here is another idea: use the digits {-1,0,1,2,3}. And base 2+i has the carry rule [-1,+4,-5], which says if a digit is 4 or greater, subtract 5 from it, add 4 to the next most significant, subtract one from the next next most significant. One way of verifying this is to see that (-1)(2+i)^{k+2}+4(2+i)^{k+1}-5(2+i)^k=0. For example, with least significant digits on the right, 1 = 1, 2 = 2 3 = 3 4 = (-1)4(-1) = (-1)3(-1)(-1) 5 = (-1)3(-1)0 6 = (-1)3(-1)1 7 = (-1)3(-1)2 8 = (-1)3(-1)3 9 = (-1)3(-1)4 = (-1)23(-1) 10 = (-1)230 11 = (-1)231 12 = (-1)232 13 = (-1)233 14 = (-1)234 = (-1)17(-1) = (-2)52(-1) = (-1)202(-1) and so on. A couple of notes: these digits are exactly the same representing natural numbers in base 2-i as well. This is current (!) joint work with Jim Propp on digits representing numbers in different bases simultaneously. And in base 2+i, you can extend the classic work of Katai and Szabo and represent Gaussian integers in base 2+i. If you limit yourself to nonnegative digits, they proved only -n+i works, not n+i. But a negative digit does the job! Steve

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Re: [math-fun] Continuum Hypothesis
by Guy Haworth 30 Apr '18

30 Apr '18

I met Marcus du Sautoy at the Royal Society in London once: I hasten to add that I was a guest of Fellow Ian Mills (he of the 'New SI') to hear Marcus' talk. Marcus explained to Ian that he was succeeding Richard Dawkins as the Simonyi Professor for the Public Understanding of Science. I was half way through saying "Well, at least there's only one mathematics" when I realised this was not true. The Continuum Hypothesis stands beyond the reach of 'the axioms', it can either be a new axiom or not. That gives you two flavours of mathematics. Marcus said that real mathematics had been done down both tracks. I haven't seen any maths done down the 'assuming a third infinity' track. [ Can we prove that if Infinity1 =/= Infinity2, then 2^Infinity1 =/= 2^Infinity2 ? ] g

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[math-fun] Another weird leap year rule
by Keith F. Lynch 29 Apr '18

29 Apr '18

Some years ago, I asked about a calendar whose leap year rule was that you add a day for every odd number the year number is divisible by and subtract a day for every even number the year number is divisible by. The average year length was an interesting and unexpected number. I've finally come up with another weird leap year rule that gives an interesting and unexpected average year length. The rule is to add a day for every way in which the year number is the sum of two squares. (Squares include 0 and 1.) For instance 2018 is the sum of two squares in just one way, 13^2 + 43^2, so it would have one leap day. 2019 is not the sum of two squares, so it would have no leap days. 2005 would have two leap days because 2005 = 18^2 + 41^2 = 22^2 + 39^2. 1850 would have been the latest year with exactly 3, and 1885 would have been the most recent year with exactly 4. So, what is the average year length under this rule? As before, I already know the answer.

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Re: [math-fun] radix r numbers, |r|<1
by Keith F. Lynch 28 Apr '18

28 Apr '18

Henry Baker <hbaker1(a)pipeline.com> wrote: > What I was really trying to get at was irrational/transcendental > bases r, |r| close to 1, and what "numerals" to use. Radix r numbers and radix 1/r numbers are identical, except written in reverse order. For instance in radix 1/10, this is the year .8102 and one third is ...33333.0 Of course real numbers (unlike p-adics) that go on forever to the *left* look a little weird. You can't use fewer than two numerals. You might as well call them 0 and 1. There will in general be many ways to represent the same number. Of special interest are those radices (1<r<2) in which all positive integers have terminating representations. These include radix sqrt(2) and radix phi, but do not include any rational radices such as 1.1. I'm pretty sure they don't include any transcendental radices either. In radix phi, if you have the constraint that no two "1" digits can be adjacent, there's a unique representation for every number. That could be useful for encoding data on long-chain hydrocarbon "tapes," using single bonds for 0s and double bonds for 1s, given that it's unstable to have two adjacent double bonds. Another application is where the cost of 0 bits and 1 bits is unequal. For instance the SF author Charles Stross has proposed using C12 and C13 atoms for 0s and 1s to store data in diamond form. (Both isotopes are stable.) It would be extremely compact and extremely durable. Since there are 90 C12 atoms for every C13 atom, if the goal isn't to store data as compactly as possible, but to store as much data as possible in the total available carbon, it would make sense to use a radix in which 0 bits were 90 times more common than 1 bits. The old five-bit ITA2 teletype code, often wrongly called Baudot, treated 1-bits as more expensive than 0-bits (since the former cause more wear on mechanical teletypes), hence gave the most common letters the codes with the fewest 1-bits. This had the interesting side effect of making ITA2 a rich source of bitwise palindromes, like Morse code but unlike ASCII.

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Re: [math-fun] radix r numbers, |r|<1
by Henry Baker 28 Apr '18

28 Apr '18

Oops! Posted too soon. What I was really trying to get at was irrational/transcendental bases r, |r| close to 1, and what "numerals" to use. It appears that when 0.5<|r|<1.5, we have to leave digit-0 "holes". E.g., suppose for the moment that r=1.1. It will take (1.1)^8 = ~2.14 before we will need another "1" digit. So we get representations with subsequences of 0's in them. Ditto for r=0.9, where it will take 0.9^(-7) = ~2.09. At 09:16 AM 4/28/2018, Henry Baker wrote: >Positional number systems that can express *every* number >rely on the fact that the geometric series > > 2 3 4 5 6 7 8 9 10 >(%o7)/T/ 1 + r + r + r + r + r + r + r + r + r + r + . . . > >*diverges* for |r|>=1. > >But what if we don't care about representing *every* number? > >What if we only care about the interval |x|<2 ? > >In this case, r=1/2 should work, right? > >Then 1.5 = 1+1/2 = "1.1"; 1.25 = 1+1/4 = "1.01"; etc. > >A greedy algorithm should produce an acceptable representation.

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[math-fun] radix r numbers, |r|<1
by Henry Baker 28 Apr '18

28 Apr '18

Positional number systems that can express *every* number rely on the fact that the geometric series 2 3 4 5 6 7 8 9 10 (%o7)/T/ 1 + r + r + r + r + r + r + r + r + r + r + . . . *diverges* for |r|>=1. But what if we don't care about representing *every* number? What if we only care about the interval |x|<2 ? In this case, r=1/2 should work, right? Then 1.5 = 1+1/2 = "1.1"; 1.25 = 1+1/4 = "1.01"; etc. A greedy algorithm should produce an acceptable representation.

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[math-fun] Verification of the result and sigma estimation
by françois mendzina essomba2 28 Apr '18

28 Apr '18

>From what I have previously posted: J[4*n]=((4*n-3)/(4*n-2))* J[4*n-4] ; J[4*n+1]=((4*n-2)/(4*n-1))* J[4*n-3] ; J[4n+2]=((4*n-1)/(4*n))* J[4*n-2] ; J[4*n+3]=((4*n)/(4*n+1))* J[4*n-1] ; Verification of the result and sigma estimation J[4*n+3]=((4*n)/(4*n+1))* J[4*n-1] ; J[4*n+4]=((4*n+1)/(4*n+2))* J[4*n] ; By expressing the two preceding expressions according to J [3] and J [4], we will have: J[4*m+3]=J[3]*product(((4*n)/(4*n+1)),n=1..m) ; J[4*m+4]=J[4]*product(((4*n+1)/(4*n+2)),n=1..m) ; We have, on the same principles as for the sinus function: Limit(J[4*m+3],m=infinity) = Limit( J[4*m+4],m=infinity) ; Which corresponds to : J[3]*product(((4*n)/(4*n+1)),n=1..infinity)= J[4]*product(((4*n+1)/(4*n+2)),n=1..infinity) ; we obtain : J[4]/J[3]= product(((4*n)*(4*n+2)/(4*n+1)^2),n=1..infinity) ; we have : J[3]=1 et J[4]=(1/2)*J[0]=(1/4)*sigma Finally i find (1/4)*sigma = product(((4*n)*(4*n+2)/(4*n+1)^2),n=1..infinity) ; FME...

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