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