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Topic: Alternative area postulate for geometry
Replies: 141   Last Post: Jul 18, 2012 9:39 PM

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 Joe Niederberger Posts: 1,659 Registered: 10/12/08
Re: Alternative area postulate for geometry
Posted: Jun 27, 2012 1:32 PM

People have reasoned about area and volume long before rigorous axiomatics were developed for those intuitive concepts. That's how I understood this inquiry - its obviously not at the level of formal math.

Even in informal math, there are propositions one might accept without further "explanation" (take as at least analogous to "postulate"). I like Cavalieri's principle in this kind of context because its arguably acceptable without proof to thinking persons and its powerful enough to derive some striking results about areas. Cavalieri's principle though, like the one proposed, is agnostic as to what the "unit" ends up being. You won't being able to prove formally or informally that a unit square has area = 1 with those alone.

On a different note though, I think Kirby is 100% correct about the arbitrariness of what ends up being the "unit" of measure for area or volume. "Rectilinear" units are completely conventional, as are rectilinear coordinates for a plane.

I'll note in passing that the wikipedia entry on area says "Every unit of length has a corresponding unit of area, namely the area of a square with the given side length." One could equally truthfully say "and also a different corresponding unit of area that is the area of a tetrahedron with all edges of length one, and also..."
The word "namely" might cause people to think the one unit mentioned is the only, which it isn't.

Later, the same wikipedia article states "The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a definition or axiom." I suppose that could be done, but I've never sat through such a development. Also I think saying it " follows directly from the basic properties of area" is obtuse unless the author has pointed out what those "basic properties" are, which the Wpedia author has NOT done.

I think such a list would include things either like your #1, or Cavalieri. You are correct that #4 is a choice that is made. And the rectangle formula LxW is then "dervivable".

But I disagree that #1 is a "definition" - it is proposition one accepts without proof. Part of that accepting is imlplicit in the fact that area hasn't been defined at all - its left to ituition.

To get more formal and define "area" you have to start thinking about functions from sets of points to real numbers that have the properties you think an "area function" should have.

Joe N

Date Subject Author
6/25/12 Peter Duveen
6/25/12 Robert Hansen
6/25/12 Gary Tupper
6/25/12 Robert Hansen
6/25/12 Gary Tupper
6/25/12 Robert Hansen
6/25/12 kirby urner
6/25/12 Peter Duveen
6/25/12 kirby urner
6/26/12 Robert Hansen
6/26/12 Gary Tupper
6/28/12 Gary Tupper
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6/29/12 Wayne Bishop
6/29/12 Gary Tupper
6/29/12 Wayne Bishop
6/26/12 Joe Niederberger
6/26/12 Joe Niederberger
6/26/12 Peter Duveen
6/26/12 Robert Hansen
6/27/12 Wayne Bishop
6/28/12 kirby urner
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6/26/12 Joe Niederberger
6/26/12 Peter Duveen
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6/27/12 Robert Hansen
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