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Re: [HM] Magnitudes
Posted:
Nov 17, 2003 12:04 AM
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Hans Samelson wrote:
<< All I want is put the greek idea of magnitudes into terms that I
understand or at least am familiar with. (Thus I stayed away from
Descartes.)>>
I don't understand this statement. You are a modern mathematician,
with modern ideas. Descartes is much nearer modern concepts (not just
his notation) than Euclid or Archimedes. Your difficulty seems to be
with the ancient concepts, which Descartes may have paid lip service
to but which he, rather adroitly, handles in such a way as to make it
seem he is dealing with numbers.
<< I find Mueller's approach for plane figured quite unsatisfactory;
he cuts the figures to make rectangles and then combines several
rectangles into one. ... I don't think that is what the Greeks had
in mind. ...
He certainly didn't have in mind converting all these triangles into
rectangles and then attaching ALL these rectangles into one very long
one.>>
I don't follow you. This is exactly what they did. Euclid in books I
and II shows us how to reduce any rectilineal figure to a rectangle.
This has always been the interpretation of Euclid. Kepler stated his
laws in terms of proportion between geometric areas. Newton did the
same thing.
<<As for the physical magnitudes, Length, area,. . . , weight, I think
that is simply a different use of the word and is not related to the
greek idea from Eudoxos-Euclid.>>
How so? Length, area, time, etc. in modern physics are lineal
descendants of the same concepts in Newtonian physics. And Newton's
concepts descendants of Decartes and Galileo. So how can they not be
related? Euclid did not treat of time nor weight, but Archimedes and
others did and they treated them as magnitudes. What else would they
be?
We are discussing concepts here, not names. Newton actually used the
term "quantity", but the concept is the same. Newton defines number in
_terms_ of quantity, explicitly abandoning the ancient definition. So
obviously a quantity is not a number but a magnitude.
This definition held until Dedekind complained that magnitude,
"Grosse", had no clear definition. This may be the source of some of
your confusion. You are looking for a clearly defined (in the modern
manner) concept, when there is none, never has been and perhaps never
will be.
Magnitudes:
Are continuous.
They are of various sorts, or kinds: length, time, etc.
"Geometric" magnitudes have dimension: length, area, volume. Other
magnitudes do not. But this did not come into play in Euclid because
he does not treat of the product of two magnitudes. But if the
product of two magnitudes _is_ considered, it is _never_ of the same
sort. It is higher on Vieta's ladder scale.
Magnitudes are connected to things in the world and the connections
between them are defined in terms of those things. That is the
magnitude called "angle" is associated with the inclination of lines
and one combines and finds the difference of angles, by geometric
means. For this reason, the product of areas, or of volumes and
lengths was considered somewhat illegitimate. But illegitimate ideas
were as numerous as illegitimate children.
The ratio of like magnitudes was a well established notion, which
occurred principally as part of a proportion. Magnitudes of different
kinds are never related in Euclid except in proportion.
The ratio of unlike magnitudes was never recognized as legitimate
until the use of proportion was abandoned, i.e. after Newton.
But ratio of unlike magnitudes were recognized in a left-handed way as
magnitudes of a different sort, e.g. speed.
Newton recognized number as the ratio of like magnitudes and
distinguished whole number, fractional number, and surds.
In essence he allowed ratios to be added and subtracted, despite the
lack of any clear definition of what that meant for any kind of
magnitudes other than length, rectilineal area, certain lunes, and
other things which had by one means or another been reduced to
rectilineal area.
Regards,
Bob
Robert Eldon Taylor
philologos at mindspring dot com
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