In the process of mathematical research, dialectical conflicts are of fundamental significance: analytic/synthetic, axiomatic/constructive, … algebraic-geometric….
The development of analytic geometry is usually attributed to Descartes and Fermat. Ancient Greek geometers used relations between curve segments which, from a present-day standpoint, are like equations for the curves in cartesian coordinates. However, the analytic perspective was not fully developed.
Th basic idea of each analytic method is the reduction of a system to a few basic elements. The advantage of simplification gained in this way may possibly be opposed by the disadvantage of complexity in the reconstruction of the system from the basic elements.
In the case of analytic geometry the reduction consists in choosing two perpendicular lines in the plane [or any two intersecting lines!] and determining the position of points on a curve by their distances x,y from these lines. The simplicity of the reduction of the description to a position relative to only two lines is opposed by the complexity of the equation f(x,y)=0. The Greeks, whose thinking was perhaps more synthetic than analytic, preferred to muse many auxiliary lines, in order to attain simple relationships.
The advantage of the analytic method, the reduction of geometric relations to complex quantitative relations — only came to light after algebra had been developed in the East and imported into the West.
Fermat had the idea of analytic geometry in 1629.
Descartes published his geometry in 1637, but had worked on it earlier, perhaps since 1619. Descartes’ _Geometry_ brought classical geometry within the scope of algebra. The book was originally published as an appendix to _Discours de la Method_. Descartes searched for a general method of thinking. Since the only [decent system] was mechanics, mathematics became his means for understanding the universe.
His _Geometry_ actually contains little analytic geometry in the modern sense, no "cartesian" axes and no derivation of the equations of conic sections as in Fermat.
Ancient Greek geometers had considered not only lengths x, y of segments but also their products, such as x², x³, xy, etc. But *they were not regarded as numbers of the same type*.
Descartes abolished this distinction: An algebraic equation became a relation between numbers, an advancing abstraction, which one can regard as the final adoption of the algorithmic tradition of the East by the West.
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Thus it was that the geometry of plane curves became analytic geometry, the investigation of the equations defining curves by algebraic and analytic methods. This was the starting point for two hundred years of development, after which the tendency to synthesis again came to the fore.
I’ve pared down Brieskorn’s words and removed any compliments in the description of Descartes. Why has mathematics been good? Where’s the evidence that it led to “invention”? Where is the evidence that abstraction has been good? Having, over some time, read about the exterior product (wedge operator ^), it seems that the distinction between 1-dimensional and ≥2-dimensional quantities has been reinvented, e.g. electromagnetism exerts 2-dimensional forces, as does mechanical rotation.
