>if the time for each
>infinitesimal is epsilon(>0) seconds,
Remember your supposition. There are (by definition), an /infinite/
number of infiniitesimals (arbitrarily identifiable points) along
the path. If "epsilon" (I believe you meant small-"tau", but let's
drop the jargon) is some finite amount of time, however minute, then
you have an infinite number of "non-zero units of time". That adds
up to an "infinite" length of time to move the cup some arbitrary
finite distance.
>and distance traversed
>delta inches, then see:
>delta = 0.765(in/sec)*epsilon(sec).
>(at constant speed)
>Now see that this relation holds
>for an infinite # of transits.
This is what I mean about assuming we can use a mathematical
tool as a /literal representation/ of objective reality.
Mathematics can only be what we created it to be: an arbitrarily-
ordered symbolic system to /describe/ what we have /perceived/,
whether what we have perceived are phenomena around us (e.g. motion),
or abstract constructs in our minds (e.g. geometry).
Take the equation above. Someone (say, Newton) observes a
phenomenon (constant motion, for example), and perceives a pattern
in it (distance traveled bears a direct relationship with the
object's speed and its travel time). So far, so good. But the
conclusion, presented this way, isn't very /useful/, is it?
So why don't we formalize it; make it into a tool that we can use.
But how? Well, that's easy. We'll just translate this relationship
using the symbology of mathematical operators already invented
(equality, division, multiplication, etc.).
So, the conclusion goes from the verbal representation of the
logical relationship described above, to: d(m)=s(m/s)*t(s).
Great! So, now we have a formal representation of objective
reality......or do we?
Mathematics has come a long way. It used to be nothing more
than hunters counting possible prey in a herd on their fingers.
This manifestation of mathematics (arithmetic) accurately
represented objective reality /in that context/, because it
didn't attempt to do anything more than ascribe a "whole number"
to an arbitrary grouping of uniquely identifiable objects in their
environment.
Then the concept of real and rational numbers was invented, along
with tools like number lines, and what-not. These things are useful,
because they help us to /model/, and therefore predict, what
happens around us. But they also do something else. They introduce
the concept of factionalism, and infinity. Again, these concepts
make the corresponding mathematical tools useful, in certain contexts,
but (and this is the important part), they therefore imbue, as a
result, the characteristics of zero and infinity into everything
they are used on.
No one would say that it makes sense to hunt a herd of 35.5538
water buffalo, but when we are quantifying and qualifying things
below the level of our immediate perception, it is easy to lose
sight of the built-in assumptions that mathematical treatments
bring with them. Is space quantized? We can't perceive any
quantization directly. Maybe it is, and maybe it isn't. But as
soon as you say "Space must be continuous, rather than discrete,
because I can model it using real numbers on a number line.", then
you've inadvertently applied to the "real world" an assumption,
intrinsic to that mathematical tool, of zero and infinity. It's
easy to do without even realizing it, because the inference is so
automatic, so subtle.
>(no paradox here)
>"Infinity " may be a reality,
>but it isn't a "real number".
Is space and/or time continuous or discrete? Well, before you can
try to answer that logically, you have to realize that the question
contains another baseless assumption. Are there any other alternatives?
Maybe space is partly discrete and partly continuous, and the
distribution of each is random, or maybe patterned, and we haven't
measured or deduced it yet. Who's to say? It /could/ be, because it's
easy to work out a logical (not mathematical) formalization that is
consistent with all the known laws (and all the cutting-edge theories
for that matter).
But the /context/ of the current discussion is not whether those
two possibilities are the only ones, just whether or not space/time
is quantized, and whether or not space/time is continuous. Answering
these questions does not assume there isn't any other alternative.
So, then, what are the /logical/ alternatives? Well, we have space,
time, quantum and continuum, that's two variables with binary values:
- space is continuous and time is quantized
- space is quantized and time is continuous
- space is continuous and time is continuous
- space is quantized and time is quantized
If space is continuous and time is quantized, you have the problem
where you have an infinity of uniquely identifiable positions to
travel through, and a finite (non-zero) time for each transition,
making any travel from A to B take forever.
If space is quantized and time is continuous, you have the problem
where you have an infinity of uniquely identifiable points of time
between any two discrete spatial points, making any travel from A to B
(indeed, any trip from one discrete spatial point to another) take
forever.
If space is continuous and time is continuous, you have the problem
where one trip from A to B might take 7.34129... seconds, and another
trip from A to B might take 2.2227714.... seconds, so you have one
instance where infinity and infinitely-small work out to one value
in one instance, and infinity and infinitelty-small work out to a
completely different value in another instance, etc, etc. Unless the
values of one or both of the infinities changes from one trip to the
next, how can the trip be shorter or longer each time?
However, if space is quantized and time is quantized, the trip
will never take "forever", and you avoid the paradox where "infinity"
has to "change its value" every trip. The difference in trip time in
each case (difference in average speed) is explained as the difference
in the "whole number" of space quanta transitioned per unit time, or
vice-versa.
>George Cantor gave his
>sanity so that we could be free
>from the confusion.
And Jesus gave his life so we could "live" forever.
Lament the tortured mind of Cantor (whoever he was),
but don't let that stop you from thinking for yourself.
I don't. Hell, I'm still reading over my reply, and
thinking "Did I miss something? Are the conclusions
logically consistent?". Oh, well. I'm sure someone will
tell me if I goofed. That's how the dynamics of the memetic
construct called "peer review" weeds out ideas that are
inconsistent with objective reality from human discourse.
>"alas poor George, we hardly knew ye."
>
>...home for non-verbal memes, ty
Dan