Comments on: The end of the number line http://bit-player.org/2008/the-end-of-the-number-line An amateur's outlook on computation and mathematics. Fri, 25 Jul 2008 16:49:51 +0000 http://wordpress.org/?v=1.5.2 Comment on The end of the number line by: Stephen Meskin http://bit-player.org/2008/the-end-of-the-number-line#comment-1621 Sat, 23 Feb 2008 03:13:37 +0000 http://bit-player.org/2008/the-end-of-the-number-line#comment-1621 The Sharkovski Theorem is rather pretty because it is simply stated and deep. Stating it in terms of an ordering of N is an "artistic" touch. The relationship between mathematics and art is often discussed in the sense of how one can use math to produce artistic visual or musical effects. I have been wondering about the internal aesthetics of math; of mathematical statements or proofs that are beautiful; math that is beautiful based on its mathematical content. The three statements below that I have quoted from your article, indicate that awareness of this kind of beauty is part of your work (and the work of many good mathematicians). "It’s a surprisingly sweet little function, concise and simple,..." "It’s entirely possible to do that, but the resulting program is ugly and artificial." "That’s rather pretty,..." As you can see, I am not just saying that the beauty in mathematics arises because mathematicians seek truth, and truth is beauty. But rather there are are some mathematical truths that are beautiful and others that are not. This is probably not a new idea. Can you refer me to writings on this subject? If not, you may want to write a column on it yourself. The Sharkovski Theorem is rather pretty because it is simply stated and deep. Stating it in terms of an ordering of N is an “artistic” touch.

The relationship between mathematics and art is often discussed in the sense of how one can use math to produce artistic visual or musical effects. I have been wondering about the internal aesthetics of math; of mathematical statements or proofs that are beautiful; math that is beautiful based on its mathematical content.

The three statements below that I have quoted from your article, indicate that awareness of this kind of beauty is part of your work (and the work of many good mathematicians).

“It’s a surprisingly sweet little function, concise and simple,…”
“It’s entirely possible to do that, but the resulting program is ugly and artificial.”
“That’s rather pretty,…”

As you can see, I am not just saying that the beauty in mathematics arises because mathematicians seek truth, and truth is beauty. But rather there are are some mathematical truths that are beautiful and others that are not. This is probably not a new idea. Can you refer me to writings on this subject? If not, you may want to write a column on it yourself.

]]> Comment on The end of the number line by: Jim Ward http://bit-player.org/2008/the-end-of-the-number-line#comment-1615 Thu, 14 Feb 2008 13:39:50 +0000 http://bit-player.org/2008/the-end-of-the-number-line#comment-1615 One difference between "2, 1, 4, 3, 6, 5, 8, 7, …" and "1, 3, 5, …, 2, 4, 6, …" is that "..." means "do forever". So you need 2 "do forever" statements. One difference between “2, 1, 4, 3, 6, 5, 8, 7, …” and “1, 3, 5, …, 2, 4, 6, …” is that “…” means “do forever”. So you need 2 “do forever” statements.

]]> Comment on The end of the number line by: unekdoud http://bit-player.org/2008/the-end-of-the-number-line#comment-1610 Thu, 07 Feb 2008 14:38:21 +0000 http://bit-player.org/2008/the-end-of-the-number-line#comment-1610 I suppose the 4th approach derives from trying to find the element before 3. Tracing the program gives 1. I suppose the 4th approach derives from trying to find the element before 3. Tracing the program gives 1.

]]> Comment on The end of the number line by: Seb http://bit-player.org/2008/the-end-of-the-number-line#comment-1609 Thu, 07 Feb 2008 01:12:01 +0000 http://bit-player.org/2008/the-end-of-the-number-line#comment-1609 Does anybody know if there is a statement similar to Sharkovski's theorem but for cellular automata? Does anybody know if there is a statement similar to Sharkovski’s theorem but for cellular automata?

]]> Comment on The end of the number line by: Barry Cipra http://bit-player.org/2008/the-end-of-the-number-line#comment-1606 Wed, 06 Feb 2008 22:09:21 +0000 http://bit-player.org/2008/the-end-of-the-number-line#comment-1606 It occurs to me it's also of interest to find a good recursive definition for the "smaller than" function for the Sharkovski ordering -- i.e., the function S(a,b) that returns a or b, depending on which one appears first. Here's a not-so-succinct stab at doing so: <pre> function S(a,b) if a==1 then return b elseif b=1 then return a elseif odd(ab) then return min(a,b) elseif odd(a) then return a elseif odd(b) then return b else return 2 * S(a/2,b/2) </pre> Note, the function min(a,b) has its own recursive definition when a and b are positive integers: <pre> function min(a,b) if a==1 then return a elseif b==1 then return b else return 1+min(a-1,b-1) </pre> It occurs to me it’s also of interest to find a good recursive definition for the “smaller than” function for the Sharkovski ordering — i.e., the function S(a,b) that returns a or b, depending on which one appears first. Here’s a not-so-succinct stab at doing so:

function S(a,b)
  if a==1
    then return b
    elseif b=1
      then return a
      elseif odd(ab)
        then return min(a,b)
        elseif odd(a)
          then return a
          elseif odd(b)
            then return b
            else return 2 * S(a/2,b/2)

Note, the function min(a,b) has its own recursive definition when a and b are positive integers:

function min(a,b)
  if a==1
    then return a
    elseif b==1
      then return b
      else return 1+min(a-1,b-1)
]]> Comment on The end of the number line by: Seb http://bit-player.org/2008/the-end-of-the-number-line#comment-1605 Wed, 06 Feb 2008 19:48:54 +0000 http://bit-player.org/2008/the-end-of-the-number-line#comment-1605 Sharkovski's ordering makes the statement of the associated theorem in dynamics very simple. Other orderings may make the statement more complicated but still be acceptable. Sharkovski’s ordering makes the statement of the associated theorem in dynamics very simple. Other orderings may make the statement more complicated but still be acceptable.

]]> Comment on The end of the number line by: David S. Mazel http://bit-player.org/2008/the-end-of-the-number-line#comment-1604 Wed, 06 Feb 2008 04:17:41 +0000 http://bit-player.org/2008/the-end-of-the-number-line#comment-1604 To read this article, and note how nicely Brian interlaces the sequences with code, reminds me of Chaitin's ideas of algorithmic complexity. When we see these various sequences, it is interesting and indeed, a method for classification, to ask: How can we code a sequence and with that coding, say something about the sequence? I think Brian does a wonderful job of connecting these ideas (and I wish I had more time to expand my thoughts here) and I hope that in future posts, he'll explore what he's started here even more. To read this article, and note how nicely Brian interlaces the sequences with code, reminds me of Chaitin’s ideas of algorithmic complexity. When we see these various sequences, it is interesting and indeed, a method for classification, to ask: How can we code a sequence and with that coding, say something about the sequence?

I think Brian does a wonderful job of connecting these ideas (and I wish I had more time to expand my thoughts here) and I hope that in future posts, he’ll explore what he’s started here even more.

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