Ramble, Part 1: The Pythagorean Project
Oct. 26th, 2006 11:46 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowI'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.
According to legend, it was Pythagoras who first noticed the mathematical underpinnings of harmony in music. Fired by the importance of numbers (in our sense) in this context, he - or his school - leaped to the conclusion that numbers underlay everything, and their study of geometry, therefore, attempted to interpret measurement purely in terms of counting.
There were three fundamental tools in the Pythagorean approach to geometry: common measure, isometry, and dissection. For the first: let us say that one magnitude measures another if it divides it evenly. Thus, an inch measures a foot, but not a meter. A common measure of two magnitudes is another magnitude which measures both. Since a foot is by definition exactly .3048 meters, one-tenth of a millimeter is a common measure for a foot and a meter. The principle of isometry holds that, if one geometric shape could be moved and placed so as to exactly cover another, then the two shapes have the same magnitude. Finally, dissection is used to determine the magnitude of one object by dividing it into two or more pieces whose magnitudes are already known.
Let me illustrate this by demonstrating a simple proposition in Pythagorean style: two rectangles with equal bases have areas in proportion to their heights. First, find a common measure for the heights of the two rectangles. Then divide each rectangle into sub-rectangles with the same base, but with height equal to the common measure. If the height of the first is m times the common measure and that of the second is n times the common measure, then the first rectangle is divided into n subrectangles and the second into m, and all of the subrectangles have the same area; hence the area of the first is n times the area of the subrectangles, and the area of the second is m times that area. The proposition follows.
It's a short step from that proposition to computing the area of a rectangle as the product of its base and height. Right triangles may then be measured as well; a right triangle, together with a second, 180-degree-rotated copy, forms a rectangle, and hence has half the area of that rectangle. To measure a parallelogram ABCD (with, say, acute angles at A and C), drop the perpendicular from A to CD, with foot at E, and the perpendicular from C to AB, with foot at F. AFCE is then a rectangle, composed of ABCD and two right triangles; thus, we can compute the area of ABCD. Given a triangle ABC, we can, as with right triangles, build a parallelogram from two copies of ABC, and thus measure its area. Finally, any polygon can be partitioned into triangular pieces, and we can therefore compute the area of any polygon.
Note, then, that all of the proportions we speak of are rational - ratios of whole numbers. It is in this sense that the Pythagoreans attempted to interpret all magnitudes in terms of numbers - to subsume measurement into counting, and to view geometry through the lens of arithmetic.
Unfortunately, the attempt fails, because there are magnitudes of the same type which do not have a common measure: they are incommensurable. The first and most famous example is that of the lengths of the side and diagonal of a square: no unit that evenly divides one can divide the other. To phrase this in modern terms: the square root of 2 cannot be written as a ratio of two whole numbers. There are a number of proofs of this fact; here's the best-known. Suppose you write Sqrt[2]=a/b, where a and b are whole numbers. Assume that this fraction is in least terms - that a and b have no common factor. In particular, they can't both be even. Now, a simple manipulation shows that a2=2b2; that tells us that a2 is even. The square of an odd number is odd; therefore a must be even. We can write a=2c, for some number c. But then 2b2=a2=(2c)2=4c2, so b2=2c2. But now b2 is even, so b must be even as well. Now a and b are both even - but that's a contradiction. Therefore, Sqrt[2] cannot be written as a ratio of two whole numbers.
This discovery was catastrophic, and put an end to the Pythagorean project: measurement could not - at least, not in this way - be swallowed up into counting. Nonetheless, the urge to unify the two persisted, and another, more successful attempt followed. That will be the subject of the next post in this series.
Previous Next
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.
According to legend, it was Pythagoras who first noticed the mathematical underpinnings of harmony in music. Fired by the importance of numbers (in our sense) in this context, he - or his school - leaped to the conclusion that numbers underlay everything, and their study of geometry, therefore, attempted to interpret measurement purely in terms of counting.
There were three fundamental tools in the Pythagorean approach to geometry: common measure, isometry, and dissection. For the first: let us say that one magnitude measures another if it divides it evenly. Thus, an inch measures a foot, but not a meter. A common measure of two magnitudes is another magnitude which measures both. Since a foot is by definition exactly .3048 meters, one-tenth of a millimeter is a common measure for a foot and a meter. The principle of isometry holds that, if one geometric shape could be moved and placed so as to exactly cover another, then the two shapes have the same magnitude. Finally, dissection is used to determine the magnitude of one object by dividing it into two or more pieces whose magnitudes are already known.
Let me illustrate this by demonstrating a simple proposition in Pythagorean style: two rectangles with equal bases have areas in proportion to their heights. First, find a common measure for the heights of the two rectangles. Then divide each rectangle into sub-rectangles with the same base, but with height equal to the common measure. If the height of the first is m times the common measure and that of the second is n times the common measure, then the first rectangle is divided into n subrectangles and the second into m, and all of the subrectangles have the same area; hence the area of the first is n times the area of the subrectangles, and the area of the second is m times that area. The proposition follows.
It's a short step from that proposition to computing the area of a rectangle as the product of its base and height. Right triangles may then be measured as well; a right triangle, together with a second, 180-degree-rotated copy, forms a rectangle, and hence has half the area of that rectangle. To measure a parallelogram ABCD (with, say, acute angles at A and C), drop the perpendicular from A to CD, with foot at E, and the perpendicular from C to AB, with foot at F. AFCE is then a rectangle, composed of ABCD and two right triangles; thus, we can compute the area of ABCD. Given a triangle ABC, we can, as with right triangles, build a parallelogram from two copies of ABC, and thus measure its area. Finally, any polygon can be partitioned into triangular pieces, and we can therefore compute the area of any polygon.
Note, then, that all of the proportions we speak of are rational - ratios of whole numbers. It is in this sense that the Pythagoreans attempted to interpret all magnitudes in terms of numbers - to subsume measurement into counting, and to view geometry through the lens of arithmetic.
Unfortunately, the attempt fails, because there are magnitudes of the same type which do not have a common measure: they are incommensurable. The first and most famous example is that of the lengths of the side and diagonal of a square: no unit that evenly divides one can divide the other. To phrase this in modern terms: the square root of 2 cannot be written as a ratio of two whole numbers. There are a number of proofs of this fact; here's the best-known. Suppose you write Sqrt[2]=a/b, where a and b are whole numbers. Assume that this fraction is in least terms - that a and b have no common factor. In particular, they can't both be even. Now, a simple manipulation shows that a2=2b2; that tells us that a2 is even. The square of an odd number is odd; therefore a must be even. We can write a=2c, for some number c. But then 2b2=a2=(2c)2=4c2, so b2=2c2. But now b2 is even, so b must be even as well. Now a and b are both even - but that's a contradiction. Therefore, Sqrt[2] cannot be written as a ratio of two whole numbers.
This discovery was catastrophic, and put an end to the Pythagorean project: measurement could not - at least, not in this way - be swallowed up into counting. Nonetheless, the urge to unify the two persisted, and another, more successful attempt followed. That will be the subject of the next post in this series.
Previous Next
Ramble Contents
no subject
Date: 2006-11-01 04:12 pm (UTC)I've always been particularly partial to the Pythagoreans, since first hearing they let grrls play too. Can you confirm this rumor?
Dolly says if you found some feline friends, you could be a real purrfessor. ;)
no subject
Date: 2006-11-03 06:18 pm (UTC)I'm afraid I can't. I can't disconfirm it either. But there was a curious strain of gender-egalitarianism that popped up in Greece from time to time, e.g. in Plato's Republic. (I have a vague memory that there may have been some Pythagorean influence on Plato, but I don't remember where I saw it, much less whether the source was reliable.)