| Mark Nicholls 2005-05-16, 8:57 am |
|
> "Jerzy Karczmarczuk" <karczma@info.unicaen.fr> writes:
>
>
> I'm quite impressed that your kindergarden teacher was able to
explain
> semantics to you. I doubt that mine could have to me. ;-)
>
> In any case, thank you for the explanation of the essence of how
> category theory applies to the above discussion. It was most
> enlightening and removes some of the extraneous heat from the
> discussion.
>
> I have a piece of paper certifying (to some extent) me as a
> mathematician.
oh good, I'm not claiming it means I'm right...just that I once sat in
rooms where people droned on about such things, and then regurgitated
some of it out again.
> However, I was not aware of category theory nor its
> implications. Thus, to me, the point of Z as a subset of Q, was a
> trivial and obvious construction, there is a "natural isomoprhism"
> between Z and a particular subset of the rationals.
hmmmmm,
is there?
isn't there a natural isomorphism between Z and the subring of Q?
if it's just a set mappings, that's pretty useless, Z thus becomes a
set of unrelated labels...which happen to be coincident.
if it's a mapping between fields, then that doesn't hold.
I don't like removing the field axiom and still claiming the subset to
be a subset of a field.....I would map out to a ring, and then show
equivalence.
It may seem petty, but a member of the corresponding 'subset' of R has
all sorts of properties that Z doesn't....i.e. I can quite happily talk
about limits, and thus functions and their differentials.....I am not
quite so certain about Q (it obviously doesn't contain all it's limit
points), I can do some sort of logic based on the limits in Q, but it's
dodgy ground....I certainly can't do anything of the sort in Z.
> In that the
> subset of the rationals obeys all the axioms the integers for the
> integer "operations". The fact the the integers can be mapped to
> other subsets of the rationals (or that the rationals can be mapped
to
> the integers by other functions) is mostly irrelevant for that point
> of view. At least in my day, one could graduate from college with a
> mathematics degree, understanding that mapping, at not know that one
> was missing something (i.e. category theory).
>
> If one considers semantics, which I believe is Mike Nichols point,
the
> difference can become relevant. There are operations one can perform
> on rationals that one cannot apply to integers (and if one applies
> such functions on the rationals which are naturally isomorphic to the
> integers, one leaves that subset of the rationals).
>
I think there is at least a danger of assuming Z is a subset of Q, I am
not apposing defining 'properties' on member of Q like
'IsCoincidentWithInteger', but if Z is a subset of Q, is Q a subset of
R, and Z a subset of R....and what happens when we defined functions,
and take derivatives and evaluate over Z.....oooooo......it sends a
shiver down my spine.
> Taking this back to the circle/elispe question. I myself would
expect
> "objects" to have identities (and the types would apply to the
> objects, not to the variables referencing them). Thus, if I have the
> following pseudo-code, I would expect the assertion to pass. Others
> would not, as they would prefer that variables rather than objects
> would have type.
I have no fixed opinion, to me objects have types, but thats just what
I'm used to....my worry is that if you cannot get Z and Q right, you
don't stand a chance with ellipses and circles.
>
> Circle c = circle with origin (x,y) and radius (r).
> Ellipse e = c; // as opposed to e = ellipse(c) -- see below
> e.majoraxis = 2*r;
> assert(e.minoraxis == 2*r); // rather than (e.minoraxis == r)
> // e is still a circle and its major and minor axises (I can't spell
in Latin)
> // are the same "by definition/object type identity" becase e is a
"circle";
> // therefore, changing one changes both (even though that seems odd
for an
> // Ellipse). However, if I wanted Ellipse like behavior (two
independent
> // axises), I would have created that type of object initially or
created
> // an ellipse from the circle (e.g. ellipse(c)), such that the new
object
> // just happened to have two equal (but actually independent) axises.
>
> Note, I specifically choose to program in a subset of my programming
> language of choice to get this behavior, such that the subset of the
> language makes these properties true.
>
> To follow the analogy, that is, I use the subset of Q where I can
> successfully apply the axioms of Z to the elements when using the
> functions of Z. I do not use the additional functions of Q in my
> dialect. I never leave the closed subset of Q which naturally maps
to
> the elements of Z. The functions which would take me out of that
> closed subset are not part of my subset of the language. Others may
> wish to use functions which would expose them to the entire set of Q,
> I would not. I want just the elements of Q which can be mapped to Z
> and have the properties of Z hold. I am willing to forgo applying
any
> functions where that would not be true.
>
> In the C++ OO world, that means I have to treat all my functions as
> "virtual" (and access everything through references), such that the
> objects have "type identity". If one can do that in C++, which is
> naturally a "variable typed language" as opposed to an "object typed"
> one, one should be able to do it in less maligned languages. It is
> simply a matter of discipline. And like any discipline, if practiced
> long enough, it becomes a habit and no longer a burden.
>
> Thus, for me, it is conventinent to think of Z as a subset of Q, even
> though that means I have many functions of Q which don't apply. I
> don't even think about the existence of those functions.
if you don't think about the existence of them, then why embed Z in
domain that supports them.
> When I see
> "2+3", I assume that I'm using an addition that maps onto the
addition
> I know and love from Z. Q wouldn't be nearly as interesting (for
> adding) if corresponding function did not exist.
>
take the function F(x,y) = x + y
i.e. F(2,3) = "2+3"
can you do partial derivatives?
do these partials apply, if x,y are in Z?
have I at any point left the closed set of { -2,-1,0,1,2,3.... }
the answer (to me) depends on what you mean by { -2,-1,0,1,2,3.... }.
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