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Oct. 26th, 2006
Ramble, Part 1: The Pythagorean Project
Oct. 26th, 2006 11:46 amI'd like to begin by taking a step further back. Two of the most fundamental mathematical activities are counting and measuring; they give rise, eventually, to arithmetic and geometry respectively. (The very name of the latter points to this: "geo-metry", "Earth-measure".) I want to distinguish between the outputs of these activities, at least temporarily; I'll reserve the word "number" to refer to the results of counting - in other words, what we would now call positive whole numbers. I'll use the word "magnitude" slightly equivocally, to refer, on the one hand, to geometric objects such as line segments, regions in the plane, or regions in space, and on the other to the measures of these objects - lengths, areas, or volumes. (This equivocation is regrettable, but very nearly standard; consider the uses of such words as "radius" and "hypotenuse".)
Now, numbers and magnitudes resemble each other quite a bit. Numbers can be added and multiplied, and - under the right circumstances - subtracted and divided. (Remember that we're talking only about positive whole numbers!) Similarly, magnitudes of the same type can be added and, sometimes, subtracted, and under the right circumstances multiplication and division are also possible. (The product of two lengths, for example, is an area, and so on.) On the other hand, there are important differences as well. Most notably, numbers are discrete - there are no numbers lying between, say, 4 and 5 - while magnitudes are infinitely divisible - given any two unequal magnitudes of the same type, there are infinitely many magnitudes in between. Despite the differences, it's natural to group numbers and magnitudes together and call them all "numbers"; every civilization I know of has done so, often without comment.
The first attempt to consciously and rigorously unify numbers and magnitudes was that of the Pythagoreans, in ancient Greece. I discuss that attempt under the cut.
( Measuring via counting )
Ramble Contents
Now, numbers and magnitudes resemble each other quite a bit. Numbers can be added and multiplied, and - under the right circumstances - subtracted and divided. (Remember that we're talking only about positive whole numbers!) Similarly, magnitudes of the same type can be added and, sometimes, subtracted, and under the right circumstances multiplication and division are also possible. (The product of two lengths, for example, is an area, and so on.) On the other hand, there are important differences as well. Most notably, numbers are discrete - there are no numbers lying between, say, 4 and 5 - while magnitudes are infinitely divisible - given any two unequal magnitudes of the same type, there are infinitely many magnitudes in between. Despite the differences, it's natural to group numbers and magnitudes together and call them all "numbers"; every civilization I know of has done so, often without comment.
The first attempt to consciously and rigorously unify numbers and magnitudes was that of the Pythagoreans, in ancient Greece. I discuss that attempt under the cut.
( Measuring via counting )
Ramble Contents
