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November 4, 2012

I am on a mission to see how bland I can get before it gets boring (funded in part by my desire to examine the co-incidence of boring with bland.  Just said no to each and every one of the 7 rasa.

I got one number.  You, then, gave me one number.  Now I multiply my number with yours to get one number.  What we have here are three numbers: two factors to the left of ‘=’ and to its right is one product.

Imagine 7 without the serifs routinely deployed to discipline the wayward attention.  Product is positioned at the intersection of the vertical and horizontal line segments; factors are at the other end points: one to the bottom and one to the left of the visually-centered product.

Division is different from multiplication.  If I have one number and you have one number, then I can divide your number with mine to get one number–but not always: there is more than one number which I can multiply with 0 to produce 0.

Imagine an abstract set 1 = {.} at the bottom end of the vertical line segment of 7 and an abstract set 2 = {., .} at the left end of the horizontal line segment.

Product of two objects in a category is one object of the category.  And there is more–to multiplication.

1 x 2 = 2.  Imagine an abstract set 2 = {., .} at the visual center of 7.  Two arrows–one from each one of the two dots in 2–go to the one dot in 1 = {.} at the bottom end.  Two arrows–one from each one of the two dots in 2–go to one of the two dots in 2 = {., .} at the left end.

There is something about this picture: it’s not the picture of product.

1 x 2 = 2.  I can divide the product on the right with one of the factors on the left to get one number.  This one–finding the one–is what products are good at.

Looking again at 7, we see a pair of maps

f2 -> 1 (downward-pointing vertical arrow)

g2 -> 2 (leftward-pointing horizontal arrow)

Standing at the common domain 2, I look up to the sky, and imagine an abstract set at the center of my gaze.  The abstract set in the sky returns the favor imagining a map to where I am at; all of which looks like a [attentional] filter resting on its side indicative of not-in-use.  I can compose the map h (with sky-high domain) with f to get one map f’ and with g to get one map g’.

What if it rains arrows: instead of one arrow h from some X in the sky to the 2 at the center of 7, we have more than one

h: X -> 2

h’: X -> 2

Can you do something?  Can you find the one map

h: X -> 2

which when composed with the pair

f2 -> 1

g2 -> 2

gave the pair

f’: X -> 1

g’: X -> 2

YES: 1 x 2 = 2; 2/2 = 1

NO: 1 x 0 = 0, 2 x 0 = 0,…

Let X = {., .}.  There are two maps

v: X -> 2

sending each one of the two dots in the domain X to one [same one] dot in the codomain 2


w: X -> 2

sending the two dots in the domain X to two different dots in the codomain 2

In the country that we are calling X there are two places called ‘space’, ‘face’.  Let’s also label the dots in the object of our attention (2): ‘thing’, ‘number’.  The map

v: X -> 2

says both ‘space’ and ‘face’ are things, while the map

w: X -> 2

begs to differ, assigning ‘number’ to ‘space’ and ‘thing’ to ‘face’.

Refreshing the 7, now with 2 = {thing, number} at our fixation spot, into focus, we situate ‘symbol-string’ in the 1 at the bottom and ‘word’, ’0′ in the 2 to the left; with

f2 -> 1

telling us that both ‘thing’ and ‘number’ are symbol strings, and

g2 -> 2

adding that both ‘thing’ and ‘number’ are words.

Forget 7: imagine Y, flip it downside-up.

We have two items

v: X -> 2

w: X -> 2

on the menu to choose from.  The stem-the vertical line segment that branches into two slanted line segments can be v or w.  The sides of the inverted-v are where it all started:

f2 -> 1

g2 -> 2

What difference does it make:

1. if the stem is v

2. if the stem is w

Stem 1 (v: X -> 2)

Both ‘space’ and ‘face’ in X go to ‘thing’ in 2, and from there they head downwards to ‘symbol-string’ in 1 and leftwards to ‘word’ in 2.

Stem 2 (w: X -> 2)

The element ‘space’ in X goes to the element ‘number’ while the element ‘face’ goes to the element ‘thing’ in the codomain 2.  The differences preserved by w are lost in the subsequent travels (down, left).

Two maps

v: X -> 2

w: X -> 2

when composed with a pair of maps

f: 2 -> 1

g: 2 -> 2

result in one pair [of composite maps]

fv = fw: X -> 1

gwgv: X -> 2

If two numbers (1, 2) when multiplied with a number (0) result in one number (0), then I don’t know how to calculate each one of the two numbers (1, 2) by dividing the product with a factor (0/0).

If multiplying a number (1) with a number (2) gives one number (2) and multiplying the same number (1) with another number (3) gives another number (3), then I can divide the product (2) of two numbers (1, 2) with one factor (1) to get the other number (2).

There are many more words that went into the genesis of the ONE in the definition of PRODUCT on page 217 of the Conceptual Mathematics textbook.  Here are some of the diagrams that go with the brief introduction to the definition: ContentConverse

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