[email protected] wrote:
-snip-
Thanks Tom! I'll have to check it out when I get home.
I'm still plugging away at the college text books i have, but it never
hurts to double one's efforts.
I find it intriguing that math can become its own 'alternate reality',
but i seriously doubt that I'll take it that far. You never know,
though, heh.
It's a different universe than what we experience, directly. You
experience it entirely within your mind, though there are real effects
at a macro scale in our universe that are 'mathematical' and directly
experienced (kind of) -- I'm referring to our sensations of light,
which is highly mathematical in behavior and where some of that math
behavior can be experienced rather directly, without having to delve
into micro-scale atomic effects indirectly observed primarily through
theories in science.
Think of it kind of as a landscape and world of its own. Highly
unified and coherent, with rules that make sense and are consistently
applied. It's knowable. But there are vast reaches of unexplored
areas, too. As well as intimate details within more known areas
still as yet undiscovered.
You can get lost in this universe. Many do.
A problem posed, at least as early as 450 BCE or so, was expressed by
the Greek, Protagoras. He felt that there were many hindrances to
knowledge of gods, for example -- the obscurity of the entire subject
area born of the fact that we simply aren't gods ourselves as well as
our brief lifetimes. He concluded that there could never be any
absolute truths or eternal standards of right and wrong and that in
the end, the only thing we could say would be particular truths that
would be valid only for the individual attempting to know them. But
nothing universal. He was one of the Greek Sophists of the time.
A few centuries later, the Skeptics had added their own twist to this
-- that all knowledge is derived from sense perceptions and therefore
_must_ be limited and entirely relative. They then deduced from this
realization that people cannot ever prove anything at all. Since it
is given that our sense perceptions are organic, inherently flawed,
and may deceive us, no truths are certain. They then argued that the
rational course of life is to always suspend judgement. They felt
that we could attain peace of mind only by abandoning a fruitless
search for what cannot ever possibly be had.
With this in mind, and thinking closely about our sense perceptions as
organic, imperfect, and quite likely giving us false information...
then what is the solution?? How can anything at all be known?
But let me even pose this as a still more difficult puzzle. When we
try and generalize about things, say to elicit the "true nature" of
related things let's say, how do we do that? We use our own
experiences to guide us. In other words, we might describe one thing
in terms of our sensory impressions of something else. "The Earth is
like a ball," for example, taking something very difficult to fully
apprehend directly (the Earth, which is vast and beyond experiencing
when trapped on the ground) and describing it as something we can hold
and feel and see, easily. But if we are using macro-models of our own
experiences with which to describe other things also in our
experience, then....
Well, what if our senses of nature are lying to us?? What if what our
senses tell us and the general models that we imagine are present as a
result of these senses are wrong? Consistent, perhaps. But
consistently wrong? How do we escape this problem?
We know for a fact that our senses do "lie" to us. For example, it is
obvious from some of our sensations that the Earth does not move. Our
sense of motion works pretty good when we are walking, floating in
water, running, etc. Yet when we just sit in one spot and try to feel
motion, we feel none. But the Earth does move.
What causes things to sink or float? Our senses are completely
lacking a density detector. We can feel weight. We can feel bulk.
But we have nothing to inform us well about the ratio of mass to
volume. This is a scientific concept, developed and derived after
much investigation and effort, to help us understand why things "sink
or float." But we have no direct sensation of it and, more truly,
cannot even invent or recognize the idea of density without a theory
about it (as history shows us well.)
So what do we do? We cannot trust our direct sensations. We cannot
even trust that the models of thinking that we develop as a result of
these questionable sensations are useful for anything more than a
shallow understanding. How do we delve deeply?
How I personally like to "see" or imagine mathematics applying into
physics (describing and predicting our natural universe that relates
to our perceptions of it) is this way:
Imagine a man who has long thought on the nature of things and
realizes the above quandry. Wanting to find some way out, some way to
escape what his senses try to convince him of, to think clearly in
some fashion... he decided to try something new. He goes into a deep
hole, seals up his eyes, covers his other senses, and covers over
himself to try and block out all that the world screams into his ears
and shines into his eyes. He tries to exclude the world. To get rid
of it so that he can think without the distractions and try to gain
some clearer view.
For a thousand years he stays. A thousand years of blocking out his
senses, leaving only his mind as his resource. In the process, he
develops the idea of a point, a line, a plane. None of which have
anything really to do with nature or his sensations of it. Abstract
ideas, instead. He pursues the implications of just a few basic
assumptions whereever that takes his mind. Again, none of this
connects with nature, nor is confused by experiencing it. He
rigorously develops whole tapestries of ideas, investigates various
avenues and their implications, and discovers a new world within his
mind. A precise world and highly unified world. But one that is
unique and apart from ordinary human experience and sensation.
A thousand years in the hole. A thousand years thinking about things
unrelated to nature. A thousand years of developing rigous ideas that
can be communicated precisely to anyone, whether they be in that day
and age or another, whether they be black or white, and no matter what
the fad is or the ambient paradigms of the culture. And where if one
person sees a deduction, another will see the same deduction.
Finally, after all this time, he opens his eyes; exposes his skin;
breathes in the air once again; and looks upon the world once more.
But with a new inner "eye". A new set of tools upon which to judge
this world, to measure it -- with which to observe it. And no longer
will his examinations be tricked by his own sensations and models
developed solely from them.
It's like that, in a sense. There is much more to be filled in here
and I've left gaping holes. But the basic image is there.
Jon
