First post! :] This has been bothering me for a while now: [Taken from John J. R
ID: 1377126 • Letter: F
Question
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!
Explanation / Answer
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.
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.