Tim Peters a écrit : > [Alex Martelli] > ... >>> In mathematics, 1 is not "the same" as 1.0 -- there exists a natural >>> morphism of integers into reals that _maps_ 1 to 1.0, but they're still >>> NOT "the same" thing. And similarly for the real-vs-complex case.
- but does there exists any sense of "mathematics object identity" that's not built out of similar maps ? IOW, if - abstracting from your example - I admit as a rule that is sufficient to deny identity, to point out that two objects are "merely" related by a natural map, isn't it then the case that I can display an infinite quantity of *distinct* clones of any mathematical object or structure that I can think of ? Smallscale python illustration : >>> 2<<100 2535301200456458802993406410752L >>> 2<<100 is 2<<100 False > [Xavier Morel] >> I disagree here, 1 and 1.0 are the same mathematical object e.g. 1 (and >> the same as "1+0i"), the difference due to notation only makes sense in >> computer science where integers, real and complex ensembles are disjoin. >> In mathematics, Z is included in IR which is included in C (note: this >> is not mathspeak, but I have no idea how to say it in english), and this >> notation -- at best -- merely determines the ensemble you're currently >> considering. >> >> There is no "natural morphism" of integers into reals because there is >> no mathematical difference between integers and reals, the real ensemble >> is merely a superset of the integers one. Well, how do you know there should be a difference between (a) "(there exists) a natural morphism from A to B" (b) "A is a subset of B" ? To paraphrase you : "the 'natural morphism' ensemble is merely a superset of the 'subset/superset' one" More to the point, where you view really breaks is when two incompatible extensions of the number tower are possible, like what happens with complex numbers and surreal numbers, which are both rich extensions of real numbers. Or real numbers and transfinite cardinals, viewed both as extending natural numbers. >> >> Or so it was last time i got a math course. > > This all depends on which math course you last took ;-) I often wonder whether young and adventurous minds could be more efficiently seduced to mathematics by enlisting them to interstellar travel. The idea goes thus. First note that by adjusting (natural) language courses one may end up with home-grown pupils equivalent to pupils taken from another continent; for instance you may teach arab to young americans and obtain something more in the likeness of an arab. Then argue that by adjusting the mathematics curriculum, one may obtain similar effects, but with displacements on the scale of interstellar travel rather than intercontinental travel ! [Tim Peters] > You have more a physicist's view here. The simplest case is real versus > complex, > where even a physicist <wink> can accept that a complex number, > formally, is an ordered pair of real numbers. Well, I suppose you've heard of Paul Erdoes's notion of a "Supreme Fascist" dictator of mathematics who owns a Book of All the Best Proofs ? It's easy to extend Him with a -WarpDrive- method of the form "Any teacher teaching A before B shall be sentenced to death". The spacedrive effect would be obtained by an appropriate choices of A and B divergent from our current habits. Of course one could expect many choices to lead nowhere - or at least to rough rides. But, I surmise that a promising and most fascinating pick would be A=real, B=complex. ... > in a computer all numerics suck, Sets suck. Cheers, BB -- 666 ?? -- 666 ~ .666 ~ 2/3 ~ 1 - 1/3 ~ tertium non datur ~ the excluded middle ~ "either with us, or against us" ! -- http://mail.python.org/mailman/listinfo/python-list