Lee Bloomquist's mail on Achilles and the tortoise took me back to my
school days. I remember once that our teacher had presented a problem
which was quite simple-- "At what precise time does the hour hand and the
minute hand of a clock coincide, after 4'o clock".
I had proceeded enthusiastically "Let's see now.. at 4:20 the minute hand
would be at the place where the hour hand was at 4, now in 20 minutes the
hour hand would have moved (5*20)/60 = 1(2/3) minutes, now in 1(2/3)
minutes, the minute hand would be there, but then the hour hand..., it
goes on".
But my teacher had provided a simple answer to the question-- "Let t be
the number of minutes after 4'o clock, that the two hands coincide. Now
since we know the speed of the two hands and their relative starting
positions, it would be straight forward to solve for t".
Well, we *feel* that the minute hand overtakes the hour hand, and here is
precisely that feeling that is articulated mathematically.
Let me call the above as inference (1), as I will come back to it later.
Now regarding the problem of finding the length of a coastline, the
solution varies depending on the resolution of the measuring scale. This
is quite similar to the problem of infinities we frequently encounter. For
example there is this "classical" infinity defined by the "number of
integers in the set of integers". The other infinity (usually called "c")
is the number of real numbers between any two real numbers (say 1 and 2).
Hence the problem of measuring the length of a coastline is not a question
of measurement, but is of "dimensions". If we consider the set of integers
to represent distances in a space of dimension "1", then we can as well
map the entire set of integers between 1 and 2 of the original set. This
would represent the number of caliberations between 1 and 2. We can yet
again map another set of integers between 1 and 2 of this second set, and
so on...
It seems as though that the unit dimension contains so many unit
dimensions within itself. It is easy to see that *any* problem can be
represented as an infinite ("c") dimensional entity which can never be
measured.
However from (1) above, we see that classical mathematical notations does
seem to provide a mathematical solution to what we know intuitively. The
solution strategy here stems from assuming a "solution state" and
calculating when the measured entity reaches this state.
Regarding competition, suppose we had a solution state say "five fold
improvement in quality", we can determine whether the competitive system
is tending towards it or not and then regard whether it is good or bad.
However, I suspect that it is not as straight forward as above. "Quality"
is a multi-dimensional (often non measurable) entity. While we can
probably determine whether a given quality ideal corresponds to the
behavior of the system, it may not be possible to determine whether a
system which does not exhibit the ideal, is tending towards the ideal or
not. To put it theoritically, consider that you have an "ideal" point in
three-dimensional space (x,y,z). Let the competitive system now be in
(x,y',z'). So is it good that one facet of the ideal (x) has been reached,
or is it bad that two facets of the ideal (y,z) have not been reached? The
problem is complicated by the fact that each dimension may not exhibit
orderings of their entities. This makes it impossible to determine whether
the system is "approaching" or "departing" from the ideal, or to calculate
an "error function" (like ranking) which can be fed back into the system.
So after all this, if I were asked "Is competition good or bad?", I'd only
say "By itself, competition is incomplete".
Warm Regards
Srinath
srinaths@usa.net
--Srinath Srinivasa <srinaths@lotus.iitm.ernet.in>
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