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Topic: Sixth grade math
Replies: 115   Last Post: Feb 15, 2010 5:36 AM

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ostap_bender_1900@hotmail.com

Posts: 681
Registered: 2/1/08
Re: Sixth grade math
Posted: Feb 8, 2010 7:31 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On Feb 8, 4:15 am, "T.H. Ray" <thray...@aol.com> wrote:
> Ostap Bender wrote
>
>
>

> > On Feb 7, 2:11 am, "T.H. Ray" <thray...@aol.com>
> > wrote:

> > > Ostap Bender wrote
>
> > > > On Feb 6, 5:41 am, "T.H. Ray" <thray...@aol.com>
> > > > wrote:

> > > > > Ostap Bender wrote
>
> > > > > > On Feb 5, 11:11 am, "T.H. Ray"
> > <thray...@aol.com>
> > > > > > wrote:
> > > > > > > MoeBlee wrote
>
> > > > > > > > On Feb 5, 10:50 am, "T.H. Ray"
> > > > <thray...@aol.com>
> > > > > > > > wrote:
>
> > > > > > > > > Speaking only for my own
> > characterization,
> > > > I
> > > > > > find
> > > > > > > > no
> > > > > > > > > conflict between the set theoretic
> > > > definition
> > > > > > and
> > > > > > > > > mine.
>
> > > > > > > > What was your particular characterization
> > > > again?
> > > > > > If
> > > > > > > > it's not
> > > > > > > > extensional then it's in conflict with

> > the
> > > > set
> > > > > > > > theoretic definition.
>
> > > > > > > It's both extensional and
> > intensional--though
> > > > > > > incomplete, as I pointed out--to define
> > > > function as
> > > > > > the
> > > > > > > transformation of one set of numbers to
> > > > another.
> > > > > >  The
> > > > > > > property of relation inheres in every
> > function.
>
> > > > > > > > In ordinary mathematics, one may be
> > called to
> > > > > > prove
> > > > > > > > that something
> > > > > > > > (call it 'C') is or is not a function,

> > and
> > > > the
> > > > > > way to
> > > > > > > > do that is show
> > > > > > > > that C is or is not a relation such that

> > for
> > > > all
> > > > > > x,
> > > > > > > > y, z, if <x y> and
> > > > > > > > <x z> in C, then y=z. On the other hand,

> > with
> > > > > > these
> > > > > > > > various informal
> > > > > > > > definitions, what even IS the

> > mathematical
> > > > (and
> > > > > > > > compatible with
> > > > > > > > classial mathematics, as that is the

> > context
> > > > of
> > > > > > the
> > > > > > > > ordinary
> > > > > > > > mathematics a sixth-grader will go on to

> > > > study in
> > > > > > > > college) means that
> > > > > > > > one would prove that something is or is

> > not a
> > > > > > > > function?
>
> > > > > > > Sure.  See my explanation to Ostap Bender
> > as to
> > > > why
> > > > > > > his description that assigns properties is
> > not
> > > > a
> > > > > > > function, absent a relation between sets.
>
> > > > > > Which explanation? Of which description?
>
> > > > > > Are you saying that my relation f: C -> Q,
> > where
> > > > C =
> > > > > > {New York, San
> > > > > > Francisco, Los Angeles} and f = {{New York,

> > 35},
> > > > (San
> > > > > > Francisco, 55),
> > > > > > {Los Angeles, 70}} is NOT a function? Why?

>
> > > > > Better to ask you, why you think it _is_ a
> > > > function,
> > > > > since I have already explained why not.
>
> > > > I am sorry but I never saw how anything you wrote
> > > > proves that the
> > > > above is not a function.

>
> > > > >  What properties
> > > > > of a function do you think this assignment of

> > > > values
> > > > > has?  I need to know what you do not understand
> > of
> > > > my
> > > > > previous explanation.
>
> > > > I am an old-fashioned man and still operate under
> > the
> > > > following
> > > > definitions:

>
> > > > From Wiki:
>
> > > > In mathematics, a function is a relation between
> > a
> > > > given set of
> > > > elements called the domain and a set of elements
> > > > called the codomain.
> > > > The function associates each element in the

> > domain
> > > > with exactly one
> > > > element in the codomain.

>
> > > > A binary relation f between two sets A and B is a
> > > > subset of A × B.

>
> > > > Thus, as Bill Dubuque puts it, a function f: A ->
> > B
> > > > is a single-valued
> > > > total binary relation between sets A and B.

>
> > > The set-theoretic definition of function  demands a
> > > relation that allows transforming a set of values

> > into one
> > > common value.
>
> > What does this mean?
>
> > BTW, why do you like the term "to transform" so much?
> > "To transform"
> > means "to change", doesn't it? Do all functions
> > actually change their
> > domains? I just don't see that as a good metaphor.

>
> It isn't a metaphor at all.  It's a property that
> inheres in every function.  It's a necessary condition.
>


Necessary condition for what? Could you please give precise
mathematical definitions of what you are talking about?

>
> The range of values within the domain have to obey
> a relation between sets that defines the function.
>


Hoe does all this relate to what I wrote:

""To transform" means "to change", doesn't it? Do all functions
actually change their domains? I just don't see that as a good
metaphor.


Take, for example, the function Y that maps people to their eye
colour. Does Y transform me into an eye colour? I don't think so. I
don't feel transformed. I still feel like a man, not a colour.

>
> > > We can order all kinds of relations that
> > > aren't functions.  Think of it in terms of natural
> > > language, in which which a syntactically correct
> > > statement does not necessarily confer meaning.  In

> > your
> > > example, the temperatures associated with cities
> > are
> > > not arbitrary assignments; you wouldn't match
> > cities
> > > with temperature at random.  It is not the mere
> > fact
> > > that one number corresponds to one city that
> > defines a
> > > function.  If domain = city and codomain =
> > temperature,
> > > the function that changes one set of temperatures
> > to
> > > another is climate, i.e., a fixed point relation
> > among
> > > cities that assigns one temperature to each city.
>
> > Well, if you prefer to see that a function is not
> > just a report of
> > what got assigned to what, but should be viewed as a
> > physical
> > mechanism/process that actually determines this
> > assignment - well, it
> > is a valid view.

>
> It's not physical, though the mathematics models many
> physical processes.  Point is, unless there is a
> property of transformation
>



What is "a property of transformation"?

Maybe you could give me the exact mathematical definitions of your
model?

>
> associated with the relation
> between sets (that's where the "machine" and "black
> box" analogies come in), there is no means of defining
> the function.  Your order relation, assigning
> temperatures to the set of cities, is not a function
> until or unless you assume a set of temperatures and
> define a relation by which the temperatures take a
> value unique to each city.
>


Why does the temperatures have to be unique to each city? Why can't
two different cities have the same temperature today?

>
> > BTW, what does the term "fixed point relation among
> > cities" mean?

>
> By Brouwer's fixed point theorem, temperature and other
> climatic properties, such as barometric pressure, range
> over the earth's surface in such a way that one fixed
> point of the topology
>


What's "fixed point of the topology"?

>
> assures repeated return to the
> point, i.e., a constant relation between the antipodes.
>


Who are the "antipodes" here and what is "a constant relation between
the antipodes"? Constant in what sense?

>
> This physically means that identical weather conditions
> at all times exist at opposite points of the sphere;
>


But this is not true, is it? Do points on opposite sides of the Globe
always have exactly the same weather conditions?

>
> we
> can't actually calculate with accuracy where those points
> are, because they change continuously.  
>


What do you mean by "those points"?

>
> The cities, which
> occupy fixed points in that domain,
>


In what domain?

>
> acquire unique
> values consistent with that continuous function.
>


Which function?

>
>  Thus,
> a fixed point relation.
>


Your exposition seems to me like a Willard Quine lecture: I understand
individual words, but have no idea what meaning you assign to them.

Please understand: I am NOT a philosopher. I need clarity.




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