16.02.2016
Indhold
Not
mathematics, but philosophy of time.
Zeno’s paradoxes above movement, i.e. that which is
called “Achilles and the tortoise” and “The Dichotomy paradox is not about
mathematics but philosophy of time. Certainly, mathematics enters in this
article, but only to show that it is misplaced.
The point of the heading is that mathematical
analysis of these so-called paradoxes is not necessary to interpret them.
Rendition
of Zeno’s paradoxes of motion.
The
paradox about Achilles and the tortoise is
the paradox Aristotle calls “the so-called ‘Achilles’:
”In
a race the quickest runner can never overtake the slowest, since the pursuer
must first reach the point whence the pursued started, so that the slower must
always hold a lead.” (239b14) ([Kirk 83], p. 272)
The
dichotomy paradox.
Zeno’s dichotomy account” enters as a part of the dichotomy paradox:
“...
The first [paradox] asserts the non-existence of motion on the ground that that
which is in locomotion must arrive at the half-way stage before it arrives at
the goal...” (239b11)
([Kirk 83], p. 270, min kantede parentes.)
Zeno’s
dichotomy account is simply the latter statement:
”that
which is in locomotion must arrive at the half-way stage before it arrives at
the goal” (239b11) ([Kirk 83], p. 270)
Misunderstandings
of the paradoxes.
A frequently advanced argument against these paradoxes
consists in proving that what is called “the sum of the stages passed” equals
the length of the whole stage. For according to the line of thought, this means
that the run finishes.
The argument presupposes that this sum can be understood
as the limit value of the sum of the first n stages for n going towards infinite.
However, this is not
the case, for per definition, a limit value of a sum of n numbers for n going towards
infinite is not a sum but just a value, to which this summing up can come arbitrarily
close without coming farther away for any higher n. Thus, this
proof does not contradict Zeno’s paradox, but rather copies it.
We are only dealing with the fact that a function
converges towards a certain value,
not that this value is reached. Maintaining that this value is a computed result is this to ignore the problem that is presented in Zeno’s
account.
Either we must understand Zeno’s paradox in a
mathematical context, or we must presuppose a common sense view of physics or
physical conditions. However, if we presuppose common sense, there is reason neither
to disprove either Zeno’s paradox about Achilles and the tortoise nor the dichotomy
paradox.
For in advance it accords with common sense that
Achilles catches up the tortoise. We can quite simply use the formula s = v * t,
where s denotes the length of the stretch traversed by the object, v the
velocity of the object, and t denotes the elapsed time. If the length of the
stretch is S, the object reaches the goal at the time T = S / v.
The
error of the easily refutable interpretation.
To presuppose common sense is a case of The easily refutable interpretation This
consists in not choosing the most difficultly refutable interpretation of a
statement one wants to refute:
When criticizing an argument, in the
interest of the truth, we ought not to choose the most easily refutable
interpretation thereof it, but the most difficultly refutable. For when the
most easily refutable interpretation has been refuted, the most difficultly
refutable interpretation has still not been refuted.
A more difficultly refutable interpretation can
be gained by removing criticisable subordinate details of the text that may
lead to a refutation.
It can be advanced again the paradox that it is
about a finite number of steps, or that the runner will reach his goal when his
distance is less than the diameter of physical point, whatever that is.
These objections becomes invalid if the paradox
is interpreted in a more abstract way, as an account that takes place in a
mathematic model added time.
Zeno’s
message.
The decisive point.
It is seen that after each of the described stages
the moved objects are in principle unchanged. When it is added to this that
there are no absolute lengths on the line in a Euclidian space, the situation
is completely unchanged after each of these stages.
Since moment and position follow each other
according to the above state formula, s = v * t, the quoted paradoxes apply
just as well to the elapsed time as for the stretch passed.
Conclusion.
But why is the above stated description s 0 v * t
not just as good as Zeno’s description of stages?
The reason is that it does not treat the
transition from the past to the now, but ignores it: At the moment t = T we are
in the now; before that moment we was on our way to the now, but the transition
to the now is not described.
Contrary to this, Zeno’s paradox describes how we
can come arbitrarily close at the now without being there. It does not ignore
the question about the transition from past to present, but shows that there is
a dualism between past and present.
In the now, we experience the no, but the time
before the now we just remember, or rather it is just memories.
Postscript.
It must be observed that Zeno’s paradoxes also
apply to the single stages, especially to the first and even more especially to
the first of the thus appeared stages. This resembles the alternative
interpretation of the dichotomy paradox.
In order to reach the goal, the object must reach
the midpoint of the stretch. In order to reach the half-way to that point, etc.
Just as above, we can talk about moments and duration like about points and
lengths. We can imagine a future moment arbitrarily close to the now but we
cannot describe the transition out of the now, the transition from the now to
the future. There remains a dualism between the now and the future.
This paradox has neither been clarified nor
understood by common sense.
Litteratur.
[Kirk
83] G.S. Kirk, J.E. Raven, M. Schofield: The Presocratic Philosophers, 2.
ed.,
(Cambridge 1983)
(Cambridge 1983)
[Treatise No 8]
http://philosophical-debate.blogspot.dk/2013/01/treatise-no-8.html
[Zeno]
http://ovemk2.blogspot.dk/2015/05/zenos-runner-paradoxes.html
