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Parenting and Geometry-given laws

December 7, 2012

My dad was a farmer, so was his dad–my grandfather, who long before he turned grand was a teen.  By virtue of being people who worked with land, we had axes and many such heavy farm-paraphernalia at our home.  One afternoon–more like mid-morning, upon returning from the early morning rounds to the farm, my granpa’s dad saw a sizable pile of wooden pieces scattered all over the earthen floor.  With some trepidation my great-grandma acknowledged that the woodsmith did indeed deliver a majestic teakwood cot, and it so happened that the ironsmith also delivered a heavy metal ax, and that their teenage kid chopped the new piece of furniture into quite a number of irregular-sized pieces of wood, which delighted my great-grandpa:  the sight of material evidence of his son’s searching examination of the reach of tools.

Breaking things–without getting all melodramatic about how expensive, etc. it was–be it with a sharp ax in actuality or with clear thought while lying in bed in foetal configuration–is like opening a book–reducing reality into comprehensible sciences.  (I hope science gets to put it all back together one 3 AM.)

I axed a line

into a line of length l, a point of length 0 on its left-end, and another point also of length 0 on its right-end.  One of the good things about looking is that it brings about changes in me but not in the line-segment I’m looking at.  (It would be, I’m afraid, a scary world–things turning beautiful the moment we look at them–to live in.)  Just because I think of the end-points of a line-segment as lines (of length zero), the original line-segment, though it is now clearly made-up of three line-segments of lengths 0, l, and 0 in my thought, is not going to change its length–not even an iota from its original length l i.e.

0 + l + 0 = l

When I think of adding, I can think of adding the left-end point to the line-segment first, and the resultant line-segment to the right-end point

(0 + l) + 0

or

0 + (l + 0)

It’s the same ‘l’ either way

(0 + l) + 0 = 0 + (l + 0)

Kindled by the associative law

(a + b) + c = a + (b + c)

we saw in a simple line-segment, fueled by boredom, our curiosity ventures out–to look–what shall we look–looking at some laid-back rectangle.

Taking a knife, I imagine slicing the said rectangle into two pieces.  The original rectangle had area; so do the two pieces, the sum of whose areas

(a x b) + (a x c)

equals that of the rectangle I’m looking at

a x (b + c)

Summing up, if only our parents didn’t ground us for pulling apart those pricey Christmas gifts, I surmise we would recognize the distributive law (Conceptual Mathematics, page 275)

a x (b + c) = (a x b) + (a x c)

in a rectangle of length ‘a’ and height ‘b + c’.

I hope people don’t press those buttons that summon the nurse-on-duty to their beds if I mess around with addition and multiplication, switching around x, +

a + (b x c) = (a + b) x (a + c)

Taking a = 1, b = 1, and c = 1, I end up with 2 = 4, which I have to admit, is utter nonsense if not patent nuisance.

If you, treading lightly, look at the distributive law

A x (B + C) = (A x B) + (A x C)

and see a sheet of paper reluctantly coming out of the printer, what, then, is wrong with looking at

A + (B x C) = (A + B) x (A + C)

and asking for a corresponding image–may not be a plane surface flashing area, may be one of those impossible visions (of climbing Penrose staircase).  Before I get completely lost in illusions, let me go study the standard map (Conceptual Mathematics, pp. 276-9).

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