Parenting and Geometry-given laws
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
0 + (l + 0)
It’s the same ‘l’ either way
(0 + l) + 0 = 0 + (l + 0)
Kindled by the associative law
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).