From Aaron Sloman Sun Jul 21 12:25:12 BST 1996
To: PSYCHE-D@IRIS.RFMH.ORG
Subject: Avoiding "seven" (Was Avoiding the noun "consciousness")
In response to my earlier remarks about the confusions in thinking that
there's one thing that we refer to when we use the noun "consciousness"
Selmer Bringsjord makes a useful and interesting
comment:
> Date: Mon, 24 Jun 1996 21:42:45 -0400
>
> BANISH 'CONSCIOUSNESS'? YES!!
>
> I have been converted! --- not only by the specific recommendation
> that we avoid *this* dirty little noun, but by the valor that underlies
> this recommendation. Let's not be frightened away from cleaning our
> *entire* house... Accordingly, I have come to the conclusion that
> *all* those ferociously vague nouns we have been forced to hear since
> Kindergarten ought henceforth to be firmly and irretrievably banished.
> One such culprit that I find particularly irksome is 'seven.' How
> is it that we go merrily along referring to seven as it it were an
> object?
Actually you have chosen an excellent example.
I believe that one of the reasons why so many kids never develop
their mathematical potential in school is that their teachers fail
to appreciate the fact that in different contexts they are referring
to different things despite the use of the very same noun.
Thus "seven" can be
1 One of a sequence of arbitrary symbols (one, two, three,...)
which we get kids to memorise, because having done so they can
use them for a variety of useful activities
2 The answer to a question about what happens when you correlate
production of elements of that sequence with successive
identification of objects in a named set, till the set is
exhausted (e.g. How many buttons in that box?)
3 An abstract property of a discrete set, i.e. its cardinality
4 An ordinal, i.e. specifying location in an ordered series
(which seat did he sit on? The seventh)
5 Something that can be combined with a type of unit (inches,
grams) to produce a continuous measure of properties of objects
(how wide is the paper? seven inches).
6 Something that can characterise change, i.e. express a relation
between things that exist at different times (e.g. How much
has he grown? How far did he move? seven inches)
7 Something that can be used to describe an increment (in
discrete or continuous) abstract domains (What do you have
to add to this to get that? seven). Here seven has the
characteristics of a one-dimensional vector.
8 Something to which operators can be applied, such as plus and
times, and which can also be the result of such an operator.
Here seven is an element in a mathematical group, or a field.
9 A (continuous) measure of area, even when you cannot identify
discrete units in that area and count them (how many square
inches does that circle occupy: seven)
7 Something which can be negated. But when you negate it which
of the above are you negating, and why can't they all be negated?
E.g. howcome your movement can be minus seven inches west, and your
bank balance can be minus seven rubles but the area of a
polygon can't be minus seven square inches? (Or can it? Let it
depend on whether you are going clockwise or counter clockwise
round the periphery?)
8 A denominator in a fraction
.... and lots more....
Now here's my conjecture: the vast majority of primary school
teachers have no conception of the diversity of concepts with the
same name ("seven") that they are expecting the kids to learn,
because they were once indoctrinated with the idea that the number
seven is actually just one thing and if you can't grasp that then
you must just be stupid.
(A few of them will have been lucky enough to meet some of the
relevant distinctions, e.g. between the natural number seven, the
integer seven, the rational number seven, the real number seven, the
complex number seven (with zero imaginary component), or a subset of
these, but will not really know how to relate this to the problems
of teaching.)
The kids meanwhile have very powerful conceptual apparatus for
(unconsciously) picking up all sorts of abstract concepts often without
even being taught them (e.g. passive grammatical forms, subjunctive
conditionals, grasping the difference between a sheep and a dog etc.)
but also noticing differences and inconsistencies and enjoying
utterances like "I can see the sea".
^^^ ^^^
They thus get very confused when the teacher fails to comment on
some of the differences and inconsistencies that they are
(unconsciously) detecting, or when teachers give unclear or even
wrong answers to questions that they (the teachers) do not
understand properly.
As a result the children give answers that are "wrong" according to
the teacher. They are then branded stupid. This causes negative
reinforcement of the whole process and mathematics becomes something
horrible and difficult, with huge amounts of wasted human potential
for development and enjoyment of life.
I wonder how many readers of this list can give a coherent answer to
the question why multiplying two negative numbers should give a
positive number, for example. Make sure you specify to which type of
number seven your answer is relevant.
Anyhow, I must thank Selmer for drawing our attention to the deep
similarities between the problems of consciousness and problems of the
number seven.
Fortunately primary school courses on consciousness are not yet
commonplace, so it is only people who go to university who get damaged
by the noun "consciousness" ....
> Bertrand Russell warned that doing good philosophy takes toil. Making
> pariahs out of terms is pretty darn easy, no?
Russell was one of those who helped us appreciate some of the subtleties
about numbers mentioned above. He was not equally insightful about
everything.
Cheers
Aaron
===
Aaron Sloman, ( http://www.cs.bham.ac.uk/~axs )
School of Computer Science, The University of Birmingham, B15 2TT, England
EMAIL A.Sloman@cs.bham.ac.uk
Phone: +44-121-414-4775 (Sec 3711) Fax: +44-121-414-4281