So…the number line…you studied it in grade school. Do you remember being told that addition was done by starting at a number and then hopping a certain number of times? And subtraction was the same thing except you hopped the other way?
A bit later on…maybe in high school geometry, your teacher may have said something like, “A line is composed of an infinite number of points”. Remember that?
What about Calculus? Do you know that calculus means stone in Latin? Yeah…a stone…you should be thinking “pebble” really, because people have always found it useful to use pebbles to count things.
COUNT THINGS.
They sorta snuck that one in there. HOPS in grade school, POINTS in high school, and the pinnacle of college math…Calculus…is named after PEBBLES. One gets the impression that numbers are things and that the number line is a really big collection of those things.
Furthermore…these things are absolutely discrete. 17 is a number and so is 8. That’s two different pebbles.
Math is a very authoritative subject. I mean, people question history and poetry. People even question science…but nobody, NOBODY ever doubts math. They may hate it, but they do not doubt it.
The easiest lessons to learn (and the hardest to forget) are the ones that we learn unconsciously. (Most dogs know about 5 commands that their humans formally taught them. But they also know about a gazillion other things that they learned by just living with humans.) So…without even thinking…we have learned that MATH SAYS that reality consists of THINGS which are SEPARATE and DISTINCT.
We acquire a lot of “facts” from this logical and common sense approach:
My car is not a tree.
My car is not your car.
Fairies don’t exist.
My child is not starving.
Those children are starving.
17 is not 8.
(This is an excellent way to remain sane and untroubled.)
But is all this really true?
When we think of reality as a collection of things that can be counted…Is that real?
Georg Cantor told us that infinity can come in sizes…that one infinity might actually be bigger than another infinity. The regular numbers are infinite.
1, 2, 3, 4, 5…infinity.
But if we also include all the crazy numbers that are between the regular numbers (like 2.7 and .000034008 and ¾ and π )then the infinity becomes so big that we can’t even count it! For example, what is the name of the number that comes right after 7?
7.1?
7.0001?
7.0000000001?
You can’t tell me!
Cantor is telling us that the primary characteristic of the real number line is that you cannot count it! He calls this the continuum. No pebbles. No atoms. No THINGS. Just an infinitely smooth continuum.
And is 17 really such an indisputable fact? Consider this number line.
______________________________________________________________
Where is 17 on that line? You can’t tell me. Okay, let me help. How about now? Where is 17?
________________________________0_____________________________
Still don’t know? Okay, how about now?
________________0__1__________________________________________
Now you can work it out. As long as I give you a reference point (0) and a consistent unit (1), then everything else becomes findable. But why did the zero go precisely there? And why was the unit just exactly that long? Because I said so. It was arbitrary. The world of countable things exists because, at some deep and pre-existing level, we make arbitrary decisions.
No doubt that this world of countable things is a useful dream (or a nightmare). But the point is that it is a dream…a projection. The foundations of the world are arbitrary assumptions made in a pre-existing continuum.
The world of countable things is infinite in a smallish sort of way. It is merely plural. Whereas the continuum is singular and infinite in a big sort of way. This is like the difference between time and eternity I suppose.
One of the Three Stooges hits his own hand while trying to use a hammer. Funny…because…it did not happen to “me”.
I hit my own left thumb with a hammer held in my right hand and it is not…funny. Nor is there any rationalization from my right hand along the lines of:
“Oh well, too bad for old Lefty. That must have hurt. I’m certainly glad it didn’t happen to me.”
No. Instead, I drop the hammer. I cover my left hand with my right and pull it into the center of my stomach. I hold it there in that nurturing pose while I go in search of ice or medical attention.
This behavior is simply because I make no distinction between my left thumb and me. I do not regard my left thumb as a thing to be counted. I regard it as “me”.
It is wisdom to know that reality is a continuum. It is compassion to feel it.
And it is helpful to remember that the continuum exists without the numbers, but the numbers don’t exist without the continuum.
John Donne—in the year 1624—Devotions upon Emergent Occasions—Meditation 17
No man is an island entire of itself; every man
is a piece of the continent, a part of the main;
if a clod be washed away by the sea, Europe
is the less, as well as if a promontory were, as
well as any manner of thy friends or of thine
own were; any man's death diminishes me,
because I am involved in mankind.
And therefore never send to know for whom
the bell tolls; it tolls for thee.
