Wikipedia:Reference desk/Archives/Mathematics/2011 August 7

= August 7 =

How natural and universal is mathematics?
Is there anything in mathematics that has been invented by humans? To illustrate my question: suppose that on some planet there is an extraterrestrial civilisation of aliens whose brains function in a way very similar to how ours do, and the two civilisations - ours and theirs - have been developing parallel with each other, but in ignorance with each other. Then for the aliens 2 + 2 will still be 4, log5(125) will still be 3, negative numbers will likely be identical to ours, and the ratio of a circle's circumference to its diametre will still be a constant approximating to 3.14159. Is there anything in our mathematics, and especially in the basics of mathematics usually covered in school courses, that might be different for them? Of course, exceptions would be nomenclature, terminology, symbol denomination, and a possible use of a notation other than the base-10 positional notation. --Theurgist (talk) 15:59, 7 August 2011 (UTC)
 * Anything that is not computable can be different. Count Iblis (talk) 17:22, 7 August 2011 (UTC)
 * Oh, I totally disagree. Can the collection of Turing machines that halt be different?  But the halting problem is not computably decidable. --Trovatore (talk) 19:56, 7 August 2011 (UTC)
 * Yes, but anything we can do that has rigorous meaning is still formally describable. If we compute an integral over some interval, what we do is perform a finite number of formal manipulations. The standard interpretation that we're integrating over an uncountable number of real numbers, most of which are not formally describable is just a fairy tale. That fairy tale can be changed, what matters is that we can at most play with a countable (and in practice finite) number of formal rules. Count Iblis (talk) 21:55, 7 August 2011 (UTC)
 * Again, I totally disagree with you. The uncountable infinite collections are in fact real.  Their existence is a falsifiable hypothesis, not falsified, with explanatory power, and is the best current explanation of the observed facts.  --Trovatore (talk) 22:27, 7 August 2011 (UTC)
 * Whether math is empirical or a priori is a question that's debated in philosophy of mathematics. If it is empirical then there could be totally different math that we somehow can't conceive of.  But I don't think it's a question you're going to find a definite answer to.  That said, most of our math is based on efforts to model our empirical observations of the universe.  If these extraterrestrials are in the same universe, with the same physics, I would expect they might develop a similar looking system.  That's purely speculation. Rckrone (talk) 17:39, 7 August 2011 (UTC)
 * Empirical and a priori are not necessarily in conflict. Just because something must be the same in all possible worlds (a priori) doesn't mean that your means of finding out about it can't be empirical.  I think we have an article on quasi-empiricism in mathematics; I don't think Quine or Putnam would allege that mathematics could be different than it is. --Trovatore (talk) 19:27, 7 August 2011 (UTC)
 * Alien ways of thought may not be identified as mathematics. Bo Jacoby (talk) 18:49, 7 August 2011 (UTC).


 * Yeah, a naïve question on my part, indeed. I didn't expect, though, that my illustration with the aliens would be taken so literally. The question was prompted by my observation that however advanced and complex humans' mathematic gets, it always analyses the environment's actual properties. Before asking, I tried searching Wikipedia for some articles like natural mathematics and some others, but they didn't exist. If the answer to this query requires speculation, then I should take it elsewhere, because doing speculation is not the RD's job. I'm not too good at mathematics, so you can expect some more naïve questions by me here :) Thanks for the replies. --Theurgist (talk) 19:11, 7 August 2011 (UTC)
 * Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk. (God made the integers, all the rest is the work of man) — Leopold Kronecker
 * Kronecker, of course, should not be quoted approvingly here. He was quite wrong.  --Trovatore (talk) 20:51, 7 August 2011 (UTC)
 * ... so you'd give him a delta grade for that statement?  D b f i r s   23:26, 7 August 2011 (UTC)

Ich stimme  zu. As a  Christian,  I  believe  God  made  more  than  the  Integers,  but  I  can  understand  the  intention  of  Herr Doktor  Kroenecker's  statement. Isaiah 28 makes  it  quite  plain  that  any  clever  idea  man  has,  God  has  in  many  ways  showed  it  to  him,  otherwise  he  would  know  nothing. Daniel Chapter 12 speaks  of  later  times ( perhaps now )  when men  shall  run  to  and  fro  and  knowledge  shall  be  increased -  written  about  500 or  so  years  B.C.  This  part  about  us  running  perhaps  a  reference  to  our  fast  paced life styles. The prophet Nahum gives  a  for seeing  of  a  later event  from  his  time  of  an  attack  on  Nineveh  now  historical  to  us,  which  can  also  be   a  dual  reference  in  which  he  says  in   Nahum 2:4   :   "The CHARIOTS shall rage in the streets, they shall justle one against another in the broad ways: they shall seem like torches, they shall run like the lightnings." I believe  God's  hand  is  in  the  fact  the  Fibonacci  Numbers  are  naturally  occurent  in  Nature,  in  things  such  as  the  orientation  of  pineapple  seeds  and  leaves  on  plants,  and  the  fact  the  earth  is  on  a  23.5  degree  tilt,  any  more  or  less  could  not  be  good -  but  that  is  my  opinion. We know  of  the  debate  as  to  whether  Newton  invented  calculs  or  Leibniz  die  Rechnung  erfunden  hat,  oder  die  beide  sie  entdeckten -  that  is,  whether  they  made  it  or  just  found  it. Certainly we  can  and  should  think  for  ourselves,  and  anyone  who  genuinely  believes  in  God  should  still  have  enough  of  their  own  volition  and  free  thought  to   work  out  a  lot  of  things  for  themselves. In reference  to  the  mention  of  Pi,  I  have  a question. How is  it  Pi  is  an  irrational  number,  when  it  exists  in  and  of  itself  as  a  ratio ? Even if  as  is  said,  it  cannnot  ( as  yet  )  be  expressed  as  the  ratio  of  two  wholes,  how  do  they  know  so,  and  how  is  it  the  proof  of  its  irrationality  works  when  by  definition  Pi  is  only  what  we  say  it  is - the ratio  of  a  circle circumference  to  its  diameter ? Chris the Russian Christopher Lilly  02:57, 8 August 2011 (UTC)
 * If you'd like to see some proofs, there is an article: Proof_that_π_is_irrational. Rckrone (talk) 05:35, 8 August 2011 (UTC)

Thank You,  that  was  very  interesting. Chris the Russian Christopher Lilly  06:46, 9 August 2011 (UTC)