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Re: [xmca] a minus times a plus

See, whatever people say about my suggestion:

One of the two numbers in the minus minus multiplication can be viewed as an arrow pointing to the number on the number line. The other one is an operator----this is the same idea as multiplication as multiple addition!!!!! 3 x 5 = 5 + 5 + 5, where 5 is the arrow and 3 is the operator that tells you you got to put three arrows in a line toe to head.

So -2 x -5, when -5 is an arrow pointing to the left from 0 length 5, by the -2, is stretched times 2 (makes - 10) and ,because of the -, is inverted to point to the opposite direction, that is, 10.

Because of coommensurability, the reasoning can be made in the other direction, too, so -2 is the arrow pointing to the left, 5 is the multiplier, and the - is the reflection operation.



On 30-Apr-09, at 8:39 AM, Mike Cole wrote:

Yes, right David. Very interesting.
I am left, however, without a practical procedure for help the teen who is confusing addition/subtraction and multiplication (never mind division!).

The web has some nice number line demos that can really help with positive
and negative numbers along a single number line but the apps are
all addition/subtraction.  Where is the app for multiplication??

On Wed, Apr 29, 2009 at 6:49 PM, David Kellogg <vaughndogblack@yahoo.com>wrote:

Mike, Eugene:

In some languages, a double negative is an affirmative (e.g. the Chinese
hit song "Bushi Wo Bumingbai", which means "It's not that I don't
understand"). In other languages, a double negative is a negative (e.g.
French, which uses the "ne pas" construction and shows a fondness for
intensifying rather than negating double negatives in lots of other ways).

As the bastard tongue of bastards, English is somewhere in between. In my
examples, I deliberately cut out the following sequence:

a) It's worth nothing.
b) It's NOT worth nothing.
c) It ain't wort' nuttin'.

You can see that a) is a simple negative and b) is a CHINESE style double
negative, but c) is a FRENCH double negative.

Now, if we go any further (e.g. the kinds of triple and quadruple negatives
you get in something like "Nothin' ain't worth nothin' hon if it ain't
free") then we see that natural language (in numbers of negators over two
and even just with two negators) tends to use negation as an adverbial
intensifier and not really as a mathematical or logical operator.

Language is what it is because it does what it does. There is an expansion of the Arab proverb which I well remember from my days on the street in Algeria: "Me against my brother, me and my brother against my cousin, and me, by brother and my cousin against you, you kafir (Kabyle, Jew, communist,
Tunisian, etc.)!"

You can see that here the negation of the negation actually creates HIGHER forms of solidarity rather than simply reversing the lower forms. You can also see that none of them are particularly high. One can actually begin to sympathize with Wolff-Michael's assertion, that Derek Melser claims not to be able to see, to the effect that labor movements create solidarity by
fencing out rather than fencing in.

(I think what Wolff-Michael denies by this assertion is precisely that the
working class has historic tasks that are capable of uniting all the
oppressed and fencing out precisely those who might open the gates to the
oppressors. This is a fairly common form of denial, particularly among
academics, who are not always that careful about closing the political fence
gate after themselves.)

In order to get to the idea of negation as a reversible operator rather than negation as an adverbial intensifier, we need a refined, more abstract, more scientific model. This is why linguistic models really will muddle up our mathematical understandings at some point, Mike, though I agree that they are "bonnes a penser" at lower levels (and of course I am a hopeless
slave of language in the way I think about mathematics myself).

You know the hoary old linguist's joke about negation (and if you don't I retell it mercilessly in my "Commentary" in the current MCA). A linguistics professor explaining negation to a sleepy room of undergraduates: "A double negation is a negation in French, but it's an affirmation in English. This
makes us rather doubtful of Chomsky's claim that language is based on
cognitive universals. However," he continued brightly, "there is no known
language in which a double affirmation is a negation!"

"Yeah," said someone in the back of the room. "Right."

David Kellogg
Seoul National University of Education

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