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Re: CL # 4, Some comments
Posted:
Nov 11, 1996 11:18 AM
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On Mon, 11 Nov 1996, Tad Watanabe wrote:
> > [Toom:] Here Ralph takes for granted that every mathematical question
> > has exactly one correct answer.
> >
> > [Raimi] Yes, if you include "I don't know", and "This question is not a
> > mathematical question."
> >
> > [Watanabe] How about "There is no answer," or "There is insufficient
> > information"? Or, are these in the second category?
> >
> > Raimi: Sure
>
> [Watanabe:] OK, which question are you answering "sure" to???
>
Me: I meant, "Sure; these are also possibly correct (unique) answers."
I should add that strictly speaking the second of those "answers" might
contain more information than is needed, reducing to the first when
the extra information is removed. However, I can imagine a question for
which the first answer is incorrect and the second correct, e.g. "Is there
enough information [here given] to produce a single numerical answer?"
[Raimi:]
> > I do not mean to invoke grammar as an end run around the question
> > of whether there is or is not a correct answer, unique, that is. The
> > question of the zeros of x^2-5x+6 has a unique answer in that when a
> > person has offered the answer one can be certain whether that answer is
> > correct or not. Whether the answer names two or a thousand numbers is
> > just not relevant to this criterion of correctness.
[Watanabe:]
> I thought this discussion was not on whether or not there is a correct
> answer in math problem but whether or not there is always ONE correct answer. >
[Me:] That's right. Naming a thousand numbers can constitute the unique
answer to a mathematical question, e.g. "Find the roots of
(x-1)(x-2)...(x-1000)=0." But then, the question, "Name *a* root of that
polynomial" does have several answers, and I will have to allow that to be
a mathematical question, although mixed in with a question of the tastes
of the respondant. What I had in mind were questions that depend in no
way on the tastes or desires of the respondant. I will confess I was too
hasty, in this connection, in saying mathematical questions have unique
answers. I really meant something else, which is that the answers are
uniquely either true or false (whether or not they depend in part on the
tastes of the respondant), and not partly one and partly the other.
[Raimi:]
> > I will allow that the standards of mathematical truth have been
> > established within a society that judges whether or not they have been
> > met in a certain case, but that is certainly not what one means by saying
> > that certain questions are negotiable, as in the example I gave of
> > whether one or another scheme of measurement is more or less convenient.
> > This is a question in which mathematics figures strongly, of course, but
> > it is not in itself purely mathematical. So too, the question of whether
> > my present philosophical stance is or is not correct is not a
> > mathematical question, even though I would not want to discuss it with
> > people ignorant of mathematics.
[Watanabe:]
> It seems like a lot depend on our PHILOSOPHICAL view of what MATHEMATICS
> is. I'm not sure trying to separate them out is useful.
[Me:] I believe it is useful. Children should always be able to
distinguish the behavior of numbers from the behavior of apples, and to
realize that one is a model for the other, and can therefore sometimes be
misleading. I don't believe you are in any doubt about what the
properties of numbers are, with addition, multiplication, etc., and how
addition sometimes models the mixing of two fluids and sometimes not. The
failure of the model in the case of "adding" alcohol to water does not
imply any ambiguity in the problem "Find 3+5", and children should learn
this early.
Ralph A. Raimi Tel. 716 275 4429, or (home) 716 244 9368
University of Rochester Fax 716 244 6631
Rochester, NY 14627 Homepage: http://www.math.rochester.edu/u/rarm
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