Ramble, Part 21: Autonomy
Apr. 15th, 2007 02:52 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowAny problem in geometry can easily be reduced to such terms that a knowledge of the lengths of certain straight lines is sufficient for its construction. Thus René Descartes, at the beginning of La Géométrie; and with those words, he inaugurated a new era in mathematics.
The Pythagoreans had attempted to subordinate geometry to arithmetic/algebra; Euclid, more successfully, had made algebra a dependency of geometry. Descartes here proposed a different relationship, one whereby problems in one area could be translated into the terms of the other and studied in that fashion. This implicitly accepts that the two areas are autonomous, each with its own methods and objects of study, and yet also interdependent, capable of aiding one another; and this is a new stance, which has become part of the standard toolkit of the mathematician.
There are a number of interesting aspects to Descartes' ideas here. First, in discussing the relationship between the operations of arithmetic and those of geometry, he writes of
Second, and more important, is the first appearance of something resembling coordinates. Descartes chose to display the power of his methods by using them on a problem first posed by the Greek mathematician Pappus. I won't go into that problem here, except to note that it involved finding points bearing a certain relationship to a given configuration of lines. Descartes therefore begins by supposing that a number of lines l1, l2, and so on are given; l1 and l2 intersect at a point A, l1 and l3 at E, and so forth. Next, he considers a point C, with lines connecting C to B on l1, to D on l2, and so forth, making specified (and not necessarily right!) angles with the given lines. Descartes then writes
A third point, of great importance in the history of mathematics, is, as mentioned, the new autonomy of the two fields of mathematics. It is well-known that "the map is not the territory"; the mathematician's response is to keep a number of different maps handy, and to switch from one to another when it seems appropriate. The different branches of mathematics each have their strengths, and the ability to transfer problems from one domain to another allows a kind of synergy to operate. If the geometric study of a problem stalls out, shifting it to algebra (or to analysis, or combinatorics, or whatever) may open the way to further insight; when that line of attack peters out, a return to geometry or a shift to yet another branch may be fruitful. This is second nature to most mathematicians by now, and a major reason for its phenomenal growth since Descartes' time. (I will mention in passing that, for example, the proof of "Fermat's Last Theorem" (so called, and so misnamed) by Andrew Wiles is geometric in nature. The geometry used is one neither Euclid nor Descartes would have recognized as such, but geometry it is.) Descartes (and Fermat as well, independently) created a hybrid of algebra and geometry; it was only the first. Today we have differential geometry, analytic number theory, combinatorial topology, quantum logic, and dozens of other cross-breed fields of study, each vigorously growing; and it is Descartes and Fermat who set us on this path.
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The Pythagoreans had attempted to subordinate geometry to arithmetic/algebra; Euclid, more successfully, had made algebra a dependency of geometry. Descartes here proposed a different relationship, one whereby problems in one area could be translated into the terms of the other and studied in that fashion. This implicitly accepts that the two areas are autonomous, each with its own methods and objects of study, and yet also interdependent, capable of aiding one another; and this is a new stance, which has become part of the standard toolkit of the mathematician.
There are a number of interesting aspects to Descartes' ideas here. First, in discussing the relationship between the operations of arithmetic and those of geometry, he writes of
taking one line which I shall call unity in order to relate it as closely as possible to numbers, and which can in general be chosen arbitrarily, and having given two other lines, to find a fourth line which shall be to one of the given lines as the other is to unity (which is the same as multiplication)Here, he decisively cuts away the problem of homogeneity not only from algebra, where it had been imported from geometry, and indeed - in order to imitate algebra as well as possible - from geometry itself. Arbitrary multiplications and divisions now become possible, by the simple recourse of choosing a unit of length.
Second, and more important, is the first appearance of something resembling coordinates. Descartes chose to display the power of his methods by using them on a problem first posed by the Greek mathematician Pappus. I won't go into that problem here, except to note that it involved finding points bearing a certain relationship to a given configuration of lines. Descartes therefore begins by supposing that a number of lines l1, l2, and so on are given; l1 and l2 intersect at a point A, l1 and l3 at E, and so forth. Next, he considers a point C, with lines connecting C to B on l1, to D on l2, and so forth, making specified (and not necessarily right!) angles with the given lines. Descartes then writes
I may simplify matters by considering one of the given lines and one of those to be drawn (as, for example, AB and BC) as the principal lines, to which I shall try to refer all the others. Call the segment of the line AB between A and B x, and call BC, y.Here we have what we know today as coordinates, but it should be noted that these are not the familiar rectangular or "Cartesian" coordinates; the angle between AB and BC need not be a right angle. It is important to recognize that Descartes was working as a geometer, trying to adapt algebra to satisfy geometric needs, and the coordinate system he chose was one dictated by the nature of the geometric problem. We are accustomed today to start with a fixed coordinate system, and interpret the geometry in terms of those coordinates, but (and this is a lesson every budding mathematician must learn) it is often more effective to start with the problem and choose the coordinate system to suit.
A third point, of great importance in the history of mathematics, is, as mentioned, the new autonomy of the two fields of mathematics. It is well-known that "the map is not the territory"; the mathematician's response is to keep a number of different maps handy, and to switch from one to another when it seems appropriate. The different branches of mathematics each have their strengths, and the ability to transfer problems from one domain to another allows a kind of synergy to operate. If the geometric study of a problem stalls out, shifting it to algebra (or to analysis, or combinatorics, or whatever) may open the way to further insight; when that line of attack peters out, a return to geometry or a shift to yet another branch may be fruitful. This is second nature to most mathematicians by now, and a major reason for its phenomenal growth since Descartes' time. (I will mention in passing that, for example, the proof of "Fermat's Last Theorem" (so called, and so misnamed) by Andrew Wiles is geometric in nature. The geometry used is one neither Euclid nor Descartes would have recognized as such, but geometry it is.) Descartes (and Fermat as well, independently) created a hybrid of algebra and geometry; it was only the first. Today we have differential geometry, analytic number theory, combinatorial topology, quantum logic, and dozens of other cross-breed fields of study, each vigorously growing; and it is Descartes and Fermat who set us on this path.
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