Re: Re: Re: Bug Report - Two numerical values for a same variable

*To*: mathgroup at smc.vnet.net*Subject*: [mg54365] Re: [mg54300] Re: [mg54271] Re: Bug Report - Two numerical values for a same variable*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Sat, 19 Feb 2005 02:32:15 -0500 (EST)*References*: <00ed01c512b0$2f242850$6400a8c0@Main> <curpbn$r28$1@smc.vnet.net> <200502150438.XAA29728@smc.vnet.net> <200502161936.OAA19223@smc.vnet.net> <bc260b75189ab899a75c3a5c65dc0bcf@gmail.com> <4214B60F.2050207@math.umass.edu> <7264774360d595ba92f3e713df810465@mimuw.edu.pl>*Sender*: owner-wri-mathgroup at wolfram.com

To make this absolutely precise: the distinction is that between syntax and semantics. For example 1[2, 3] is a "legal" Mathematica expression since it has the correct syntax but semantically it is meaningless. Rational[a,b] or Rational[Pi,E] or Complex[Pi,E] are all valid, syntactically correct expressions but they are all semantically meaningless. Andrzej Kozlowski On 18 Feb 2005, at 08:54, Andrzej Kozlowski wrote: > Obviously this was meant as an illustration of how things "would have > worked" had Rational[2,3] not been an Atom! Note the "would have"! > Rational[a,b] is just an undefined expression with head Rational and > is not an atom. > > Andrzej > > > On 17 Feb 2005, at 16:19, Murray Eisenberg wrote: > >> Given that Rational[a, b] is, as you say, meaningless, I'm not yet >> convinced by your reasoning on grounds of Mathematica consistency! >> >> Andrzej Kozlowski wrote: >>> The reasons why Rational[2,3] or Complex[2,3] are atoms are nor >>> really mathematical but come from considering the structure of >>> expressions in the Mathematica language and way expressions are >>> transformed by various structured operations. To see what I mean >>> consider the following list >>> ls = {1, 2, 3/4, 5 + 6*I, Rational[a, b]}; >>> Note that 3/4 evaluates to the atom Rational[3,4] but Rational[a,b] >>> is not an atom and is actually meaningless. >>> I think most people would agree that we would like all the numbers >>> 1.2,3/4 and 5+6I to be treated "in the same way" by various >>> structured operations that accept level specifications, such as like >>> Map, Apply etc. This is indeed the case. Consider >>> In[2]:= >>> Map[g, ls, -1] >>> Out[2]= >>> {g[1], g[2], g[3/4], g[5 + 6*I], g[Rational[g[a], g[b]]]} >>> You can see the difference between the treatment of the atomic >>> Rational[3,4] (or Complex[5,6]) and non-atomic Rational[a,b]. This >>> and similar reasons justify treating Rational[3,4] and Complex[5,6] >>> as atoms. This is a quite different issue from the one whether >>> rationals or complex numbers are in some mathematical sense "atoms" >>> or not. A case can be made both for the "yes" and the "no" answer, >>> but it has nothing to do with the reason why in Mathematica >>> Rational[2,3] and Complex[5,6] are atoms. >>> Andrzej Kozlowski >>> On 16 Feb 2005, at 20:36, Murray Eisenberg wrote: >>>> The manipulations below are precisely what's so confusing about >>>> Rational >>>> objects (and Complex objects) being atoms. If >>>> >>>> 1/2 /. Rational[x_, 2] -> Rational[x, 7] >>>> >>>> works, then why not the following? >>>> >>>> Part[Rational[1, 2], 2] >>>> >>>> I can "believe" that integers and reals (and maybe strings) are >>>> atoms; >>>> but believing that rationals and complex numbers are atoms is a hard >>>> thing to swallow! >>>> >>>> This has always bothered me -- and hence given me trouble trying >>>> to, um, >>>> rationalize this to students when I've taught Mathematica. >>>> >>>> >>>> Scott Hemphill wrote: >>>> >>>>> DrBob <drbob at bigfoot.com> writes: >>>>> >>>>> >>>>>> That explains it, but only in the sense that "things fall down" >>>>>> is a theory of gravity. Why should Rationals be atomic, for >>>>>> goodness sake? And how did I use Mathematica all this time >>>>>> without hearing about it? >>>>>> >>>>>> Sigh... >>>>> >>>>> >>>>> >>>>> >>>>> In[1]:= FullForm[1/2] >>>>> >>>>> Out[1]//FullForm= Rational[1, 2] >>>>> >>>>> In[2]:= 1/2 /. Rational[x_,2] -> Rational[x,7] >>>>> >>>>> 1 >>>>> Out[2]= - >>>>> 7 >>>>> >>>>> Scott >>>> >>>> >>>> -- >>>> Murray Eisenberg murray at math.umass.edu >>>> Mathematics & Statistics Dept. >>>> Lederle Graduate Research Tower phone 413 549-1020 (H) >>>> University of Massachusetts 413 545-2859 (W) >>>> 710 North Pleasant Street fax 413 545-1801 >>>> Amherst, MA 01003-9305 >>>> >>>> >> >> -- >> Murray Eisenberg murray at math.umass.edu >> Mathematics & Statistics Dept. >> Lederle Graduate Research Tower phone 413 549-1020 (H) >> University of Massachusetts 413 545-2859 (W) >> 710 North Pleasant Street fax 413 545-1801 >> Amherst, MA 01003-9305 >

**References**:**Re: Bug Report - Two numerical values for a same variable***From:*Scott Hemphill <hemphill@hemphills.net>

**Re: Re: Bug Report - Two numerical values for a same variable***From:*Murray Eisenberg <murray@math.umass.edu>