# Has there been a Cartesian revolution in mathematics?

In his book "Méthodes modernes en géométrie", Jean Fresnel wrote:

il ne faut pas se faire d'illusions, Descartes résout des problème de géométrie, non parce qu'il a de la méthode, mais parce qu'il a des idées

(Rough translation: "don't expect too much, Descartes solves geometric problems because he has ideas, not because he has a method".)

The context of this remark was that there are accessible reprints of La Géométrie that students could read for their benefit, but that they shouldn't have too high hopes. However, this casual remark of Jean Fresnel might still capture a general feeling in mathematics, namely that "content" (= ideas and technical details) is at least as important as "method" (= meta mathematics). However, there has indeed been unbelievable progress in mathematics in the time after Descartes. I wonder whether Descartes' "method" had something to do with it or not.

So here is my question: Has there been a Cartesian revolution in mathematics? And if yes, what exactly was so important about the work of Descartes?

Here is my own attempt at an answer: Descartes important contribution was his belief that there exists a "method" that simplifies geometry. This belief is in sharp contrast to the earlier belief passed down by Proclus Lycaeus (8 February 412 – 17 April 485 AD): "It is also reported that Ptolemy once asked Euclid if there was no shorter road to geometry than through the Elements, and Euclid replied that there was no royal road to geometry."

My understanding of Descartes' contribution to geometry is the observation that curves can be expressed as algebraic formulae. If you have taken a bit of schooling, this observation has almost certainly impacted you (e.g. you will learn that plotting y = x^2 with the appropriate coordinate system gives a hyperbola, x^2 + y^2 = 1 gives a circle, ...).

Whether or not you classify this as a "revolution" will of course depend on your definitions, but an argument could be made that the "Cartesian revolution" has already passed. When I think of a problem like, say, angle trisection, it's usually proven in a purely algebraic fashion, much more so than I remember Descartes doing. (NB: This could be due to my own experiences more than an objective view, as Jim points out below.)

In any case, Descartes was certainly a very influential figure, and he helped to establish algebra as the "lingua franca" of mathematics. Whether or not this classifies as a "revolution" I will leave for you to decide.

[Another legacy of Descartes': using x, y, and z as variable names. The story goes: Descartes originally used a, b, and c, but the typesetter ran out of those characters, causing Descartes to switch to the less-frequently used letters at the end of the alphabet.]

• It's not true at all that people rarely use Cartesian-style algebra -- this is one of the basic tools of mathematics, and all mathematicians are adept at using it. Abstract algebra is not a replacement for Cartesian-style algebra: it's just formal reasoning about algebraic structures and computations. – Jim Belk Dec 26 '11 at 19:52
• @Jim: No doubt my thoughts are a reflection of my own research interests. I just felt uncomfortable saying that Descartes' methods were in common use, given how rarely I see them. – Xodarap Dec 27 '11 at 19:04
• @jim: I'm afraid that this isn't quite correct, look at algebraic geometry & algebraic topology... – Mozibur Ullah Apr 18 '12 at 4:30

There was a revolution in mathematics because of Descartes, back in the 1600's. He invented the method of using the mapping from geometric objects to coordinates and algebraic equations, where a quicker rote symbolic proof was possible in comparison to many synthetic proofs (like those in Euclid).

What's great about mathematics is that we learn it all in grade school now. It is almost unconscious and seemingly lame now to even discuss it as a revolution; it's the water we swim through now. I think that it is expressed Fresnel's attitude that in La Géométrie, what we get out of Descartes is not the method at all, but simply some handful of clever results (those applications of the method).

The distinction that Fresnel is trying to draw is not clear. I take it that he means that a method is a clever general way of doing a great many things with little thought versus an idea which is a clever specific idea for a very specific difficult problem (that probably doesn't generalize). If that is the appropriate interpretation of Fresnel's statement, then I disagree with Fresnel; the examples in La Geometrie, may or may not be specific instances of the application of the genera method (I vaguely remember those instances only tangentially being an application of analytic geometry, that is they each had their own idiosyncratic problems and solutions). But even where the distinction works, I think the 'method' is the revolutionary thing rather than the specific solutions to the specfic problems in La Géométrie.

• You wrote "... in La Géométrie, what we get out of Descartes is not the method at all, but simply some handful of clever results". I actually read a bit in "La Géométrie", before asking this question. Descartes really tries to explain the method and even tries to establish some sort of equivalence between algebra and certain geometric constructions (starting with compass and straightedge, and probably continuing with conic sections). He solves "interesting" problems on the way and discusses issues of the algebraic solution procedure (like superfluous solutions). – Thomas Klimpel Dec 23 '11 at 21:03
• @Thomas: I meant to emphasize 'we' in that sentence. That is, 'we' now think of analytic geomtetry (what we now call Descartes' contribution) is obvious, so what -we- get out of his book is just the particular results, the ideas. – Mitch Dec 23 '11 at 21:09

Descartes did start a revolution by turning geometry into algebra, Which allowed the development of calculus. However in contemporary times, there are two very different trends in geometry, one is a reaction & the other a kind of development of Descartes methods:

1. Developing intrinsic methods
These are geometric ideas that do not rely on coordinates explicitly. After all the world as presented to us does not come equipped with coordinates. As an example, the tangent bundle to a manifold can be defined intrinsicly, and the exterior calculus generalises quite nicely the cumbersome vector calculus methods. (There are four methods to define these objects, two of them actually goes via coordinate methods, one 'dynamically', the other 'statically', and the third by algebraic means, the fourth by categorical methods which is probably the most general & most modern perspective).

Hermann Weyl, a highly respected mathematician said: "The introduction of a coordinate system to geometry is an act of violence".

In essence, this trend is in direct opposition to Descartes vision.

2. Algebra as dual to geometry
Coordinates are very special functions on a space. Instead one looks at all functions on a space, and this generates an algebra, and the geometric properties of the space are reflected in the algebra. Interestingly, one can go in the opposite direction, by taking an algebra and turning it in a space. Now its hard to imagine quite what one means by a non-commutative geometry, but its much easier for an algebra.

This can obviously seen as an extension of Descartes way, but also very different from it.

Yes.

I can't imagine how we could have done calculus without the Cartesian system. Also, I can't imagine how we could have done any progress in science without calculus.

Things change with other things. Calculus deals with that.

• Your answer assumes that it's the "Cartesian system" that was invented by Descartes. However, this assumption is not strictly true, because you won't find the definition of the "Cartesian system" in "La Géométrie", nor in any other work by Descartes. And Pierre de Fermat also had similar ideas that could lead to the "Cartesian system", but also failed to really define it (independed of the fact that he didn't publish his ideas). But Descartes at least openly critizises his forerunners for not even trying to invent something along these lines. – Thomas Klimpel Dec 27 '11 at 12:12
• @ThomasKlimpel Can you pinpoint the location where my answer assumes that Cartesian system was invented by Mr. Descartes? – Pratik Deoghare Dec 27 '11 at 12:42
• Sorry, I had the impression you assumed that Cartesian system was invented by Mr. Descartes. I see now how you interpreted the "And if yes, what exactly was so important about the work of Descartes?" part of my question. My own interpretation of that part was that if you consider the "Cartesian system" as a revolution, then what exactly did Descartes contributed to its discovery that couldn't have been contributed by other mathematicians like Fermat. (This site doesn't allow me to change my vote on your answer unless you edit it.) – Thomas Klimpel Dec 29 '11 at 22:31

It's obviously a revolution. Cartesian coordinates allows to solve problems compass and straightedge method was unable to solve like cube doubling, cartesian coordinates are a breakthrough after two thousand years stagnations. I agree on the fact it was not a question of method but a question of idea.

• I'm quite unhappy with this sort of answer. First, you won't find the definition of "Cartesian coordinates" in "La Géométrie". Second, the problem of "cube doubling" not restricted to compass and straightedge was already solved by the Greeks themselves long before Descartes. Already the Greeks wanted to prove that it is impossible, but that was only proved in 1837 by Pierre Wantzel. Third, also Pierre de Fermat discovered similar ideas that could lead to "Cartesian Coordinates", but also failed to really define "Cartesian Coordinates" (independed of the fact that he didn't publish his ideas). – Thomas Klimpel Dec 27 '11 at 12:02
• Frankly, if you refuse to accept Descartes introduced a system called after hime there is no possible discussions, it's a fact accepted by many historians and mathematicians. If you do not find a difference between the very complicated solutions based on conics and algebraic equations there is also a problem in what males a discussion open and fair. I never said Descartes proved the impossibility, I said he offered the means to compute the solution and consequently, the means used by Wantzel later on. Your argument concerning Fermat does not bring anything to the question. You p – Mauceric Dec 27 '11 at 18:14
• I see you make a similar critic to Pratik Deoghare, I believe your argument on the fact you do not find a definition of cartesian coordinates in his work is flawed because Descartes avoided to publish all his ideas by fear of religious retaliations, he has been very chocked by Gallileo trial. What is agreed by almost everybody, except you perhaps, is that is work and ideas gave birth to this system' – Mauceric Dec 27 '11 at 18:32