[math-fun] Numbers Aplenty
Giovanni Resta's new website Numbers Aplenty<http://www.numbersaplenty.com/>is an impressive implementation of the old idea that every natural number is interesting.
All natural numbers are interesting because, if there were an uninteresting number, there would be a smallest one, and that number would be interesting by virtue of being the smallest uninteresting number. -- Gene
________________________________ From: W. Edwin Clark <wclark@mail.usf.edu> To: math-fun <math-fun@mailman.xmission.com> Sent: Sunday, January 5, 2014 11:20 AM Subject: [math-fun] Numbers Aplenty
Giovanni Resta's new website Numbers Aplenty<http://www.numbersaplenty.com/>is an impressive implementation of the old idea that every natural number is interesting. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
The website looks great!
All natural numbers are interesting because, if there were an uninteresting number, there would be a smallest one, and that number would be interesting by virtue of being the smallest uninteresting number.
(Easy) exercise: Show how Gene's argument fails in first-order logic. (Hint: Russell.) Charles Greathouse Analyst/Programmer Case Western Reserve University On Sun, Jan 5, 2014 at 3:21 PM, Eugene Salamin <gene_salamin@yahoo.com>wrote:
All natural numbers are interesting because, if there were an uninteresting number, there would be a smallest one, and that number would be interesting by virtue of being the smallest uninteresting number.
-- Gene
________________________________ From: W. Edwin Clark <wclark@mail.usf.edu> To: math-fun <math-fun@mailman.xmission.com> Sent: Sunday, January 5, 2014 11:20 AM Subject: [math-fun] Numbers Aplenty
Giovanni Resta's new website Numbers Aplenty<http://www.numbersaplenty.com/>is an impressive implementation of the old idea that every natural number is interesting. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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On the contrary, Gene's argument is perfectly logical. Its absurd conclusion shows that arithmetic is inconsistent. --Dan Charles Greathouse wrote: Gene Salamin wrote:
All natural numbers are interesting because, if there were an uninteresting number, there would be a smallest one, and that number would be interesting by virtue of being the smallest uninteresting number.
(Easy) exercise: Show how Gene's argument fails in first-order logic. (Hint: Russell.)
Show us how Gene's argument fails. -- Gene
________________________________ From: Charles Greathouse <charles.greathouse@case.edu> To: Eugene Salamin <gene_salamin@yahoo.com>; math-fun <math-fun@mailman.xmission.com> Sent: Sunday, January 5, 2014 12:28 PM Subject: Re: [math-fun] Numbers Aplenty
The website looks great!
All natural numbers are interesting because, if there were an uninteresting number, there would be a smallest one, and that number would be interesting by virtue of being the smallest uninteresting number.
(Easy) exercise: Show how Gene's argument fails in first-order logic. (Hint: Russell.)
Charles Greathouse Analyst/Programmer Case Western Reserve University
On Sun, Jan 5, 2014 at 3:21 PM, Eugene Salamin <gene_salamin@yahoo.com> wrote:
All natural numbers are interesting because, if there were an uninteresting number, there would be a smallest one, and that number would be interesting by virtue of being the smallest uninteresting number.
-- Gene
It's a sleight of hand, an antinomy. If you fix any definition of Interesting(x) for which not Interesting(n) for some natural numbers n, you can find the least natural number which is not Interesting. But then you move to a higher-order Interesting'(x) which is based on Interesting(x). In particular this is an instantiation of the Berry paradox ("the smallest positive integer not definable in fewer than twelve words") first published by Russell, and not too dissimilar to Russell's paradox itself. Charles Greathouse Analyst/Programmer Case Western Reserve University On Sun, Jan 5, 2014 at 5:10 PM, Eugene Salamin <gene_salamin@yahoo.com>wrote:
Show us how Gene's argument fails.
-- Gene
________________________________ From: Charles Greathouse <charles.greathouse@case.edu> To: Eugene Salamin <gene_salamin@yahoo.com>; math-fun < math-fun@mailman.xmission.com> Sent: Sunday, January 5, 2014 12:28 PM Subject: Re: [math-fun] Numbers Aplenty
The website looks great!
All natural numbers are interesting because, if there were an uninteresting number, there would be a smallest one, and that number would be interesting by virtue of being the smallest uninteresting number.
(Easy) exercise: Show how Gene's argument fails in first-order logic. (Hint: Russell.)
Charles Greathouse Analyst/Programmer Case Western Reserve University
On Sun, Jan 5, 2014 at 3:21 PM, Eugene Salamin <gene_salamin@yahoo.com> wrote:
All natural numbers are interesting because, if there were an uninteresting number, there would be a smallest one, and that number would be interesting by virtue of being the smallest uninteresting number.
-- Gene
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Charles, I understand your first paragraph, which I think nails the interesting-number (IN) paradox. But what is your unraveling of the Berry paradox, and why is it equivalent to IN? --Dan On 2014-01-06, at 7:22 AM, Charles Greathouse wrote:
It's a sleight of hand, an antinomy. If you fix any definition of Interesting(x) for which not Interesting(n) for some natural numbers n, you can find the least natural number which is not Interesting. But then you move to a higher-order Interesting'(x) which is based on Interesting(x).
In particular this is an instantiation of the Berry paradox ("the smallest positive integer not definable in fewer than twelve words") first published by Russell, and not too dissimilar to Russell's paradox itself.
The Berry paradox is just the same: for some fixed definition of definitions-from-words you can find the smallest natural number not among them, but the problem then makes use of a higher-order predicate making use of the original definitions-from-words. Charles Greathouse Analyst/Programmer Case Western Reserve University On Mon, Jan 6, 2014 at 11:00 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Charles, I understand your first paragraph, which I think nails the interesting-number (IN) paradox.
But what is your unraveling of the Berry paradox, and why is it equivalent to IN?
--Dan
On 2014-01-06, at 7:22 AM, Charles Greathouse wrote:
It's a sleight of hand, an antinomy. If you fix any definition of Interesting(x) for which not Interesting(n) for some natural numbers n, you can find the least natural number which is not Interesting. But then you move to a higher-order Interesting'(x) which is based on Interesting(x).
In particular this is an instantiation of the Berry paradox ("the smallest positive integer not definable in fewer than twelve words") first published by Russell, and not too dissimilar to Russell's paradox itself.
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Can anyone explain why Mathematica is complaining here? It seems all the arguments of Prime[] in In[1] are obviously integers. It's not my fault if it uses floats. (And is there any way to coax it to get the exact answer, whose value I leave as a little puzzle?) ----- In[1]:= 1/2 + Sum[(1/Prime[n])Product[(1-1/Prime[k]),{k,1,n-1}],{n,2,Infinity}] 1 Product[1 - --------, {k, 1, -1 + n}] 1 Prime[k] Out[1]= - + Sum[-------------------------------------, {n, 2, Infinity}] 2 Prime[n] In[2]:= N[%] Prime::intpp: Positive integer argument expected in Prime[17.]. Prime::intpp: Positive integer argument expected in Prime[18.]. Prime::intpp: Positive integer argument expected in Prime[19.]. General::stop: Further output of Prime::intpp will be suppressed during this calculation. Out[2]= 0.897804 ----- --Dan
Certainly, all natural numbers are interesting, but (it seems to me) they are not all equally interesting (or popular or useful or talked-about). So, there's no least-interesting number, but are there less-interesting numbers? Without trying to establish a hierarchy, are there numbers that, by having minimal membership in other groups, are less interesting? Kerry On Sun, Jan 5, 2014 at 1:21 PM, Eugene Salamin <gene_salamin@yahoo.com>wrote:
All natural numbers are interesting because, if there were an uninteresting number, there would be a smallest one, and that number would be interesting by virtue of being the smallest uninteresting number.
-- Gene
________________________________ From: W. Edwin Clark <wclark@mail.usf.edu> To: math-fun <math-fun@mailman.xmission.com> Sent: Sunday, January 5, 2014 11:20 AM Subject: [math-fun] Numbers Aplenty
Giovanni Resta's new website Numbers Aplenty<http://www.numbersaplenty.com/>is an impressive implementation of the old idea that every natural number is interesting. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I had a similar thought to Kerry's: for some given collection of properties (like those in this site), weight them by how often they show up in the first 10^15 terms (the site's limit) and add up the weights. Which number(s) have the lowest score? A possible family of weighting functions: f_k(n) = n^-k, where n is the number of occurrences in those terms. So f_0(k) counts properties without regard to how often the occur, f_1(n) counts a property that happens 4 times as often 1/4th the weight, f_{1/2}(n) counts it with 1/2 the weight, etc. Charles Greathouse Analyst/Programmer Case Western Reserve University On Sun, Jan 5, 2014 at 3:31 PM, Kerry Mitchell <lkmitch@gmail.com> wrote:
Certainly, all natural numbers are interesting, but (it seems to me) they are not all equally interesting (or popular or useful or talked-about). So, there's no least-interesting number, but are there less-interesting numbers? Without trying to establish a hierarchy, are there numbers that, by having minimal membership in other groups, are less interesting?
Kerry
On Sun, Jan 5, 2014 at 1:21 PM, Eugene Salamin <gene_salamin@yahoo.com
wrote:
All natural numbers are interesting because, if there were an uninteresting number, there would be a smallest one, and that number would be interesting by virtue of being the smallest uninteresting number.
-- Gene
________________________________ From: W. Edwin Clark <wclark@mail.usf.edu> To: math-fun <math-fun@mailman.xmission.com> Sent: Sunday, January 5, 2014 11:20 AM Subject: [math-fun] Numbers Aplenty
Giovanni Resta's new website Numbers Aplenty<http://www.numbersaplenty.com/>is an impressive implementation of the old idea that every natural number is interesting. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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What is the smallest value of n such that n+1 appears in more of the increasing sequences in the OEIS than n does? The reason I want to restrict attention to increasing sequences in the OEIS is that these correspond to interesting subsets of the positive integers. I suppose if anyone wants to answer my question with the word "increasing" omitted, I'd be interested in that too. Conjecture: The n that you get is the same for both versions of my question. Refined conjecture: in both cases, n is 11. Jim Propp On Sunday, January 5, 2014, Charles Greathouse wrote:
I had a similar thought to Kerry's: for some given collection of properties (like those in this site), weight them by how often they show up in the first 10^15 terms (the site's limit) and add up the weights. Which number(s) have the lowest score?
A possible family of weighting functions: f_k(n) = n^-k, where n is the number of occurrences in those terms. So f_0(k) counts properties without regard to how often the occur, f_1(n) counts a property that happens 4 times as often 1/4th the weight, f_{1/2}(n) counts it with 1/2 the weight, etc.
Charles Greathouse Analyst/Programmer Case Western Reserve University
On Sun, Jan 5, 2014 at 3:31 PM, Kerry Mitchell <lkmitch@gmail.com<javascript:;>> wrote:
Certainly, all natural numbers are interesting, but (it seems to me) they are not all equally interesting (or popular or useful or talked-about). So, there's no least-interesting number, but are there less-interesting numbers? Without trying to establish a hierarchy, are there numbers that, by having minimal membership in other groups, are less interesting?
Kerry
On Sun, Jan 5, 2014 at 1:21 PM, Eugene Salamin <gene_salamin@yahoo.com<javascript:;>
wrote:
All natural numbers are interesting because, if there were an uninteresting number, there would be a smallest one, and that number would be interesting by virtue of being the smallest uninteresting number.
-- Gene
________________________________ From: W. Edwin Clark <wclark@mail.usf.edu <javascript:;>> To: math-fun <math-fun@mailman.xmission.com <javascript:;>> Sent: Sunday, January 5, 2014 11:20 AM Subject: [math-fun] Numbers Aplenty
Giovanni Resta's new website Numbers Aplenty<http://www.numbersaplenty.com/>is an impressive implementation of the old idea that every natural number is interesting. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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On the topic of small interesting numbers: What's the smallest n>2 such that no auto manufacturer has made a hubcap with n-fold symmetry? Or equivalently, if you were to write an illustrated children's book on the natural numbers with each number rendered as an actual hubcap, how many pages would such a book have? Now that I've forever changed your life, when walking along a line of parked cars, here are some tips for speed-hubcapping: Deciding that n is even is easy, and the eye is also good at noticing perpendicular mirror lines to determine divisibility by 4. Then deciding between 12, 16, 20 or 10, 14, 18 is easy because the gaps are large. The case of odd n is harder. While I can quickly recognize 5, 7, and even 9 (as three 3's), starting will 11 I usually end up just counting (and then having to run to catch up with the people I'm walking with). -Veit On Jan 5, 2014, at 5:46 PM, James Propp <jamespropp@gmail.com> wrote:
What is the smallest value of n such that n+1 appears in more of the increasing sequences in the OEIS than n does?
The reason I want to restrict attention to increasing sequences in the OEIS is that these correspond to interesting subsets of the positive integers. I suppose if anyone wants to answer my question with the word "increasing" omitted, I'd be interested in that too. Conjecture: The n that you get is the same for both versions of my question. Refined conjecture: in both cases, n is 11.
Jim Propp
Lots of hubcaps<https://www.google.com/search?site=imghp&tbm=isch&source=hp&biw=1280&bih=847&q=Hubcaps&oq=Hubcaps&gs_l=img.3..0l10.2591.5550.0.6038.7.6.0.1.1.0.97.481.6.6.0....0...1ac.1.32.img..0.7.493.Xp3soaiR-SE> I see one that has n-fold symmetry for all n. So there is no smallest n not made. I guess you mean has symmetry group C_n or D_n. On Tue, Jan 7, 2014 at 9:21 AM, Veit Elser <ve10@cornell.edu> wrote:
On the topic of small interesting numbers:
What's the smallest n>2 such that no auto manufacturer has made a hubcap with n-fold symmetry?
Or equivalently, if you were to write an illustrated children's book on the natural numbers with each number rendered as an actual hubcap, how many pages would such a book have?
Now that I've forever changed your life, when walking along a line of parked cars, here are some tips for speed-hubcapping:
Deciding that n is even is easy, and the eye is also good at noticing perpendicular mirror lines to determine divisibility by 4. Then deciding between 12, 16, 20 or 10, 14, 18 is easy because the gaps are large.
The case of odd n is harder. While I can quickly recognize 5, 7, and even 9 (as three 3's), starting will 11 I usually end up just counting (and then having to run to catch up with the people I'm walking with).
-Veit
On Jan 5, 2014, at 5:46 PM, James Propp <jamespropp@gmail.com> wrote:
What is the smallest value of n such that n+1 appears in more of the increasing sequences in the OEIS than n does?
The reason I want to restrict attention to increasing sequences in the OEIS is that these correspond to interesting subsets of the positive integers. I suppose if anyone wants to answer my question with the word "increasing" omitted, I'd be interested in that too. Conjecture: The n that you get is the same for both versions of my question. Refined conjecture: in both cases, n is 11.
Jim Propp
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On Jan 7, 2014, at 11:11 AM, "W. Edwin Clark" <wclark@mail.usf.edu> wrote:
I see one that has n-fold symmetry for all n. So there is no smallest n not made. By "n-fold symmetry" I meant that the maximal symmetry group is C_n or D_n.
C_n Hubcaps have to be made in two forms, one for each side of the car.
I guess you mean has symmetry group C_n or D_n.
Lets not overlook drinking glasses. I've seen everything up to 12. --Rich ------ Quoting Veit Elser <ve10@cornell.edu>:
On the topic of small interesting numbers:
What's the smallest n>2 such that no auto manufacturer has made a hubcap with n-fold symmetry?
Or equivalently, if you were to write an illustrated children's book on the natural numbers with each number rendered as an actual hubcap, how many pages would such a book have?
Now that I've forever changed your life, when walking along a line of parked cars, here are some tips for speed-hubcapping:
Deciding that n is even is easy, and the eye is also good at noticing perpendicular mirror lines to determine divisibility by 4. Then deciding between 12, 16, 20 or 10, 14, 18 is easy because the gaps are large.
The case of odd n is harder. While I can quickly recognize 5, 7, and even 9 (as three 3's), starting will 11 I usually end up just counting (and then having to run to catch up with the people I'm walking with).
-Veit
On Jan 5, 2014, at 5:46 PM, James Propp <jamespropp@gmail.com> wrote:
What is the smallest value of n such that n+1 appears in more of the increasing sequences in the OEIS than n does?
The reason I want to restrict attention to increasing sequences in the OEIS is that these correspond to interesting subsets of the positive integers. I suppose if anyone wants to answer my question with the word "increasing" omitted, I'd be interested in that too. Conjecture: The n that you get is the same for both versions of my question. Refined conjecture: in both cases, n is 11.
Jim Propp
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What about coins? Jim Propp On Tuesday, January 7, 2014, wrote:
Lets not overlook drinking glasses. I've seen everything up to 12. --Rich
------ Quoting Veit Elser <ve10@cornell.edu>:
On the topic of small interesting numbers:
What's the smallest n>2 such that no auto manufacturer has made a hubcap with n-fold symmetry?
Or equivalently, if you were to write an illustrated children's book on the natural numbers with each number rendered as an actual hubcap, how many pages would such a book have?
Now that I've forever changed your life, when walking along a line of parked cars, here are some tips for speed-hubcapping:
Deciding that n is even is easy, and the eye is also good at noticing perpendicular mirror lines to determine divisibility by 4. Then deciding between 12, 16, 20 or 10, 14, 18 is easy because the gaps are large.
The case of odd n is harder. While I can quickly recognize 5, 7, and even 9 (as three 3's), starting will 11 I usually end up just counting (and then having to run to catch up with the people I'm walking with).
-Veit
On Jan 5, 2014, at 5:46 PM, James Propp <jamespropp@gmail.com> wrote:
What is the smallest value of n such that n+1 appears in more of
the increasing sequences in the OEIS than n does?
The reason I want to restrict attention to increasing sequences in the OEIS is that these correspond to interesting subsets of the positive integers. I suppose if anyone wants to answer my question with the word "increasing" omitted, I'd be interested in that too. Conjecture: The n that you get is the same for both versions of my question. Refined conjecture: in both cases, n is 11.
Jim Propp
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42 was used as it's the most boring number chosen by John Cleese for a sketch as recalled by Douglas Adams (though Adams had previously said that it was just a random choice). On 5 Jan 2014, at 20:31, Kerry Mitchell wrote:
Certainly, all natural numbers are interesting, but (it seems to me) they are not all equally interesting (or popular or useful or talked-about). So, there's no least-interesting number, but are there less-interesting numbers? Without trying to establish a hierarchy, are there numbers that, by having minimal membership in other groups, are less interesting?
Kerry
On Sun, Jan 5, 2014 at 1:21 PM, Eugene Salamin <gene_salamin@yahoo.com>wrote:
All natural numbers are interesting because, if there were an uninteresting number, there would be a smallest one, and that number would be interesting by virtue of being the smallest uninteresting number.
-- Gene
What one might denote the "Hardy-Ramanujan (1729)" effect. I recently came across a reference in my own notes to "5x2 map-foldings (1980)". A brief and fruitless search for the elusive reference was aborted when it dawned on me that 1980 is the number of different ways in which a 10-leaved cartographical almanac may completely folded up. WFL On 1/5/14, W. Edwin Clark <wclark@mail.usf.edu> wrote:
Giovanni Resta's new website Numbers Aplenty<http://www.numbersaplenty.com/>is an impressive implementation of the old idea that every natural number is interesting. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (10)
-
Charles Greathouse -
Dan Asimov -
David Makin -
Eugene Salamin -
Fred Lunnon -
James Propp -
Kerry Mitchell -
rcs＠xmission.com -
Veit Elser -
W. Edwin Clark