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First post! :]

This has been bothering me for a while now:

[Taken from John J. Roche's "The Mathematics of Measurement: A Critical History"]

When physico-mathematicians in the seventeenth century needed to represent their physical quantities mathematically they almost always turned to geometry rather than to numbers.

The author goes on to explain this further, saying how the Ancient Greeks had used geometrical figures to represent physical quantities rather than numbers (influencing the physicists later on) and how they were most appropriate for representation since they denoted "continuous physical quantities".

I don't understand this; what makes physical quantities continuous? Wouldn't you figure them to be discrete considering that they are quantified by units?

More importantly, is the author saying that physicists and mathematicians realized physical relationships geometrically? Or is he just saying that they merely chose to represent physical quantities and their relationships to others with geometrical figures? It seems extremely non-intuitive to view physical quantities as being segments/planes/solids instead of as quantifiable values with defined dimensions. Of course it's easier to view/realize proportionalities between physical quantities through geometrical/graphical means, but there must have been a numerically-based intuition even before this. In other words in finding a relationship between 2 quantities, there must have been a numerically-based line of thought that preceded a geometrical demonstration of it, simply through my reasoning that it is more intuitive.

I wanted to discuss this though, to see if anyone had any input/corrections to make about my thoughts here; I'd love to hear what you guys have to say :]

Here's a direct link to the book I'm referring to (Google Books). The quote stated above is on page 87, and it continues on from there.

Thank you!

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Of course I have never read the classical sources in the original, but in translation it sometime sounds like the classical thinkers did not conceive of "squaring" as an operation on number in its own right, but rather an expression for "finding the area of a square with side of a given length". No idea if that is just me or something real. –  dmckee Jun 16 '12 at 17:38
    
I was thinking more along the lines of proportional thinking between quantities but this is interesting as well. If you read a little further in the link, the author explains that with geometrical figures as representational tools, it was then possible to compare completely "heterogeneous" quantities. This suggests that the Ancient Greeks perceived number as being a scheme for representing just quantity, completely separate from the concept of quantity of something. In other words, since physical quantities had "kinds"/"types" they couldn't be regarded as just an amount. –  ProSteve037 Jun 19 '12 at 22:16

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up vote 4 down vote accepted

Historically, the continuum was philosophically very confusing to people, since the idea of real numbers was obviously ok from the fact that we see geometry, but these real numbers could not be specified by a finite procedure.

If you think about the collection of real numbers, it is very vast. It is uncountable. The collection of names is countable, so there are unnamable real numbers. Any scheme you give for constructing and specifying points is therefore incomplete in some way. In order to formulate the notion of real numbers, you need a notion of set theory, and the required set theory was only developed in the 19th century. Because of this, the concepts of geometry, which is intuitive, were used as a shoddy replacement for real number constructions until something better came along.

This type of thinking is obsolete today. We can formulate real numbers without using geometry, and it is better to think of geometry as a particular case of real number constructions, rather than the other way around.

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Ahh I see. So you're saying that in generalizing their numerically-based findings for all numbers, they had to move away from thinking with numbers because of this fundamental conceptual flaw? In other words, in generalizing a relationship involving quantities, all possible quantities are taken into account and represented geometrically. Am I following correctly? –  ProSteve037 Jun 21 '12 at 23:54
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@ProSteve037: Yes--- their flaw was that they didn't know what a real number was exactly precisely, because an arbitrary real number only admits an infinite description in terms of an actually infinite decimal sequence, or a dedekind cut. So they represented this idea geometrically, because our geometrical intuition allows us to see that an arbitrary quantity of a real number type makes sense. –  Ron Maimon Jun 22 '12 at 0:43
    
I wish I could +1 you but I've only 11 rep haha. Thank you very much for clearing this up for me! :] Also, your profile information says you're self-taught... Could you recommend any books/sources for me regarding this? –  ProSteve037 Jun 23 '12 at 4:35
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@ProSteve037: Everybody is self-taught, because physics formal education is shoddy. I learned this from browsing in libraries, too long ago to remember. But the first chapter of Munkres' Topology has a discussion of how to construct the reals, I think. But you can also do it yourself if you know set theory. "Set theory and the continuum hypothesis" has an excellent discussion of the axioms of set theory, but it is advanced as far as logic. You can read any book called "set theory" from before 1960 and it will contain a discussion of how to form the real numbers. –  Ron Maimon Jun 23 '12 at 5:35
    
Thank you very much! –  ProSteve037 Jun 26 '12 at 17:25

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