> On Mar 24, 2014, at 5:42 AM, Joachim Durchholz <[email protected]> wrote: > > Am 24.03.2014 01:58, schrieb Richard Fateman: >> >> Now we must address what is meant by integer. In common lisp, integer >> meansarbitrary precision integer. > > Consequently, every rational number IS an integer/integer. >> [...] > > It seems to me the meanings of words in sympy should correspond to > > their meanings in mathematics not some hack mockery of the word that > > appears in one or even several programming languages. > > Your position seems inconsistent - you derive your idea about the nature of > rational numbers from Common Lisp, yet you denounce exactly that kind of > reasoning as "some hack mockery of the word that appears in [...] programming > languages". > >> It seems to me the meanings of words in sympy should correspond to their >> meanings in mathematics not some hack mockery of the word that appears in >> one or even several programming languages. The data structure for IEEE >> double-float is used to represent a subset of the rational numbers. > > To paraphrase, IEEE numbers are some hack mockery of the word that appear in > hardware implementations. > >> You could also have a data structure for "rational as the ratio of two >> arbitrary precision integers,." > > That's been found fairly useful for symbolic programming. > > Isn't that what SymPy does? > > ---- > > Just to demonstrate how far away "meaning in mathematics" is from any > computerized representation, including that of Common List, here's a short > (and probably slightly wrong) description of what rational numbers "are": > > Going down to the very foundations, rational numbers "are" the minimum model > that satisfies the field axioms (see model theory for the definition of > "minimum" and why "the" minimum exists for these axioms).
You also need to add a characteristic 0 condition. Otherwise, the smallest field is the trivial field. Aaron Meurer > > Most importantly, rationals "are" not a pair of integers, because for integer > pairs, (1,2) != (2,4), but for rationals, 1/2 = 2/4. > Integer pairs are merely a possible *representation* of rationals. > > And now, for fun, the real numbers: > > These "are" the minimum model that satisfies the total and the dense ordering > axioms. > No mention of field properties. The real numbers just happen to have the > rational numbers embedded (where "embedded" is a term with a strict > definition in model theory). > > -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/sympy. > To view this discussion on the web visit > https://groups.google.com/d/msgid/sympy/53300BF9.4010001%40durchholz.org. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/sympy. To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/4104032825062653343%40unknownmsgid. For more options, visit https://groups.google.com/d/optout.
