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Re: Sixth grade math
Posted:
Feb 9, 2010 7:25 PM
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On Feb 9, 4:02 am, "T.H. Ray" <thray...@aol.com> wrote:
> Ostap Bender wrote
>
>
>
> > 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?
>
> If you don't know what the logical terms necessary and
> sufficient mean, I can't help you.
>
Is it necessary for you to act as an insulting jackass? Suffices to
say that I have been familiar with the terms necessary and sufficient
since 8th grade.
What I referred to is your sentence:
>
> > > It's a necessary condition.
>
If you were so much more of a logician than I, you would know that
when you say that something is "a necessary condition", you must
specify FOR WHAT it is a necessary condition.
I recommend that you familiarise yourself with the concept of
"necessary condition".
>
> > > 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.
>
> I'm afraid your concrete thinking won't get you to the
> abstract meaning of function. Your particular eye
> color has nothing to do with the set of the population
> that maps to the set of eye colors distributed among the
> population.
>
What is this "set of the population" that you refer to? Do you belong
to it? How about my barber?
The bottom line is that I consider eye colour to be a legitimate
function that maps each mammal to his/her eye 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"?
>
> Betweeness is one term that is often used.
>
Is this the best you can do? Are you familiar with the concept of
"definition"?
>
> > 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"?
>
> Look--if you want to know the fixed point theorem, go
> learn it. A good popular source is John Casti's
> Five Golden Rules.
>
I am familiar with the fixed point theorem since my freshman year in
college. And I am familiar with the concept of "fixed point of a
function". I am also familiar with the concept of "fixed point
property of a topological space". What I am unfamiliar with is the
concept of "fixed point of THE topology".
In any case, let me remind you that the title of this thread is: Sixth
grade math. Here we are discussing how to explain the concept of a
function to an average 6th grader. Do you think that your explanations
here are indeed a good way to familiarise 11-year-pld children with
what a function is?
Is this what you are going to tell the 6th grade child: "If you want
to know the fixed point theorem, go learn it. A good popular source
is John Casti's Five Golden Rules"?
>
> > > 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?
>
> Yep. They do.
>
No, they don't. I asked my friend meteorologist to make sure, and he
told me that they don't.
>
> > > 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?
>
> The two dimensional surface of a 2-sphere; i.e., the
> Earth.
>
What do you mean by "The cities, which occupy fixed points on the
Earth"? Are there cities that occupy fixed points on the Earth and
cities that don't? Are you talking about mobile cities?
>
> > > 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.
>
> I don't assign meaning. I abstract it.
>
Is this what you do? You abstract meaning? A very interesting
pastime.
>
> > Please understand: I am NOT a philosopher. I need
> > clarity.
>
> I'm not a philosopher, either. I regret that you
> do not understand me, and I apologize for my lack of
> clarity.
>
Somehow I feel that you are avoiding precision on purpose. I fail to
se why a person would make no effort to make himself to be understood
by others.
One of the most influential films that I saw in childhood had the
following episode. The students were to write a composition titled
"What is Happiness". The hero's composition consisted of only one
sentence: "Happiness is when you are understood".
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