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Re: Sixth grade math
Posted:
Feb 10, 2010 9:05 PM
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On Feb 10, 4:36 am, "T.H. Ray" <thray...@aol.com> wrote:
> Ostap Bender wrote
>
> > 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.
>
> Good. And I didn't mean to be insulting. Sorry.
>
> > 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.
>
> It seems I have to keep repeating myself. I thought
> I was clear that function was not defined until the
> (necessary)relation between setsw as defined.
>
"Clear" is the last word I and most other people here would apply to
your exposition of your own thoughts. Your explanations are the most
cryptic I have ever seen.
>
> > I recommend that you familiarise yourself with the
> > concept of
> > "necessary condition".
>
> Will do.
>
> > > > > 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?
>
> I do, and my barber as well--I cut my own hair.
>
But you didn't answer the ACTUAL question: What is this "set of the
population" that you refer to?
>
> > The bottom line is that I consider eye colour to be a
> > legitimate
> > function that maps each mammal to his/her eye colour.
>
> Yes, but you're leaving out the necessary relation
> between sets that allows a mapping. If those terms of
> existence were not important, any relation would be a
> function.
>
But how does your general amorphous paragraph above relate to the "eye
colour" function? Can't you be concrete?
>
>
> > > > > > > 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"?
>
> Oh, apparently not.
>
So, 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"?
>
> > > 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".
>
> I gave you a _very_ well known result of Brouwer's fixed
> point theorem that you claim to know so well. Here,
> try this:
>
> http://books.google.com/books?id=M-qK8anbZmwC&pg=PA395&lpg=PA395&dq=w...
>
I understand that you are referring to the Meteorology Theorem that
shows that "somewhere on the Earth, there is a pair of antipodal
points having simultaneously the same temperature and pressure".
Hardly a surprising result. What i don't understand is how this result
relates to the general question of what is and what isn't a function
in mathematics. Why did you bring this theorem up in our discussion?
>
> > 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?
>
> No. You'd have to go back to my original comments to
> get my opinion on that.
>
> > 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.
>
> He's wrong. To be precise, temperature and barometric
> pressure are identical, as I first made clear.
>
Only for SOME point, not ALL points. Where did you say "some"?
>
> > > > > 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".
>
> It's a rather brash extrapolation, isn't it, that
> because you do not understand me, no one does.
>
Am I the only one here in this thread who says that they find you hard
to understand? Just look around.
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