Bondfångeri.....

av ole, söndag, maj 16, 2010, 23:21 (5115 dagar sedan) @ Håkan

Det något märkliga är att det i den bok jag har i handen
står:

"The thesis of the treatise (som visar tillbaks åt "On Indivisible Lines")
is that the doctrine of indivisibles espoused by Xenocrates, a
successor of Plato as head of the Academy, is untenable.
The indivisible, or fixed infinitesimal of length or area or
volume, has fascinated men of many ages; Xenocrates thought that
this notion would resolve the paradoxes, such as those of Zeno,
that plaugued mathematical and philosophical thought. ....."

Boyer and Merzbach, A History of Mathematics, Wiley and Sons 1989

Här kan det va språkligt bondfångeri, men jag tycker det
verkar som att dom påståtta paradoxerna är Zenons.................

Det lär nog vara någon tvivel om dom påståtta paradoxernas härkomst.
I Stanford Encyclopedia of Philosophy skriver dom:

"Almost everything that we know about Zeno of Elea is to be found
in the opening pages of Plato's Parmenides. There we learn that Zeno
was nearly 40 years old when Socrates was a young man, say 20. Since
Socrates was born in 469 BC we can estimate a birth date for Zeno
around 490 BC. Beyond this, really all we know is that he was close
to Parmenides (Plato reports the gossip that they were lovers when
Zeno was young), and that he wrote a book of paradoxes defending
Parmenides' philosophy. Sadly this book has not survived, and what
we know of his arguments is second-hand, principally through Aristotle
and his commentators (here I have drawn particularly on Simplicius,
who, though writing a thousand years after Zeno, apparently possessed
at least some of his book). There were apparently
40 ‘paradoxes of plurality’, attempting to show that ontological
pluralism — a belief in the existence of many things rather than
only one — leads to absurd conclusions; of these paradoxes only
two definitely survive, though a third argument can probably be
attributed to Zeno. Aristotle speaks of a further four arguments
against motion (and by extension change generally), all of which
he gives and attempts to refute. In addition Aristotle attributes
two other paradoxes to Zeno. Sadly again, almost none of these
paradoxes are quoted in Zeno's original words by their various
commentators, but in paraphrase."

http://plato.stanford.edu/entries/paradox-zeno/#ParMot

Så till frågan om dom påståtta paradoxen.
Dom är löst upp ser det ut för.

"Finally, we have seen how to tackle the paradoxes using the resources
of mathematics as developed in the Nineteenth century. For a long time
it was considered one the great virtues of this system that it finally
showed how to do without infinitesimal quantities, smaller than any
finite number but larger than zero. (Newton's calculus for instance
effectively made use of such numbers, treating them sometimes as zero
and sometimes as finite; the problem with such an approach is that how
to treat the numbers is a matter of intuition not rigor.) However,
in the Twentieth century Robinson showed how to introduce infinitesimal
numbers into mathematics: this is the system of ‘non-standard analysis’
(the familiar system of real numbers, given a rigorous foundation by
Dedekind, is by contrast just ‘analysis’). And it has been shown by
McLaughlin (1992, 1994) that Zeno's paradoxes can also be resolved
in non-standard analysis; they are no more argument against non-standard
analysis than the standard mathematics we have assumed here. It should be
emphasized however that — contrary to McLaughlin's suggestions — there is
no need for non-standard analysis to solve the paradoxes:
either system is equally successful. (The construction of non-standard
analysis does however raise a further question about the applicability
of analysis to physical space and time: it seems plausible that all
physical theories can be formulated in either terms, and so as far as
our experience extends both seem equally confirmed. But they cannot both
be true of space and time: either space has infinitesimal parts or it doesn't.)

http://plato.stanford.edu/entries/paradox-zeno/#ParMot

Ja, ja, Vad vet jag?? Språkligt bondfångeri??