I often bore people with the personally startling story of
when I learned that these life assumptions were extremely important and that the
knowledge subsequently derived from such baseless but expedient assumptions
was knowledge not to be fully trusted as truth but more on an intuitional basis.
When just a young man, the significant moment occurred as I
was confronted with one of Zeno’s paradoxes. Zeno was a Greek philosopher from
the 5th century B.C. There was a grade school level cartoonish
article on the bulletin board in my elementary math class which I am fairly
certain did not mention Zeno by name. I did not know that there was a great
thinker behind the thing. I would love to have that little piece of paper now
because it began a lifelong healthy doubt in all established thought. Zeno
needs to be added to my list of heroes.
An adult version of his paradox is as follows:
The Dichotomy Paradox
“That which is in locomotion must
arrive at the half-way stage before it arrives at the goal.”Aristotle, Phisics
Suppose Homer wants to catch a stationary bus. Before he
can get there, he must get halfway there. Before he can get halfway there, he
must get a quarter of the way there. Before traveling a quarter, he must travel
one-eighth; before an eighth, one-sixteenth; and so on.
The resulting sequence can be represented as:
This description requires one to complete an infinite number of tasks, which Zeno maintains is impossibility. -Wikipedia
When I tell this story I often tell of how much it disturbed
me at the time. I was ever the worrier. Even without an impressive authority to
back it, this thing on the bulletin board was not striking me as particularly humorous
as intended, but was an attack on the “number line” itself which was posted
above the black board of every elementary math class I can remember. Its
importance was preeminent to the class. Do not attack my number line!
Further along in my education, with Zeno nagging me
constantly, I found that our numbering system, or rather numbers in and of
themselves, were derived from one important assumption or “postulate.” This was
the concept of the “spatial point.” Between the point marked “Zero” and the
point marked “One” is an infinite number of divisions. This is because, as I
learned later in geometry, each point on the number line is infinitely small.
Add these infinitely small points together and get an infinitely thin line. I
promise I am not making this up from whole cloth but you should be suspecting
me, because you should suspect everything you believe that you know. Zeno
should be sitting on your shoulder when making any decision.
My mind raced as a young man. If the points are infinitely
small, they cannot really connect together. I have fairly good “spatial
reasoning,” I think. So, from this hazy notion I concluded that the “number”
part of the number line was pretty much based on something called “the
point.” They actually tell you these
things in the beginning but you pass over them and go on to build huge
structures of elegant thought based upon them. It would be different if the
teacher came in every day and stated what needs to be said: “Accepting the
postulate that there is such a thing as a number, let us proceed with the
class.”
After further research to make sure once again that I was
not dreaming in elementary school when they told me that mathematics itself was
fairly a guess, it seems that this postulate has further degraded in its
ability to convince me.
In Wikipedia, the entry on “point”:
In geometry, topology and branches of mathematics, a spatial point is a primitive notion upon which other
concepts may be defined. In geometry, points are zero dimensional, they do not
have volume, area, length or any other higher-dimensional analogue. [emphasis
mine]
And the entry on
“primitive notion”:
In mathematics, logic and formal systems, a primitive notion is an undefined concept. In
particular, a primitive notion is not defined in terms of previously defined
concepts, but is only motivated informally, usually by an appeal to intuition
and everyday experience.
If I were to start with a notion like “Let’s assume there is a
God…” or “Let’s assume no God…” well, you get the point, which is infinitely
large as well as infinitely small, being zero dimensional as it is, or was, or
whatever.
