More than you ever wanted to know about objects [was: Is everything a refrence or isn't it]

[Boris Borcic]

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" !