In his Philosophie de l’algèbre (readable at amazon preview), Jules Vuillemin distinguishes 6 dimensions of interpretation in the Cartesian Architectonics, which depend upon a strict distinction between the synthetical and the analytical. Descartes was of the clear opinion that analysis can only treat particular problems, while synthesis can only proceed within the general. In the view of Descartes, the generality of the synthetical method (composition, demonstration, construction) limits the scope of the analytical method, because the synthetical depends upon conception (general terms, order of elements) and hence remains subject of interpretation.
Vuillemin marks out as the Cartesian starting point, a problem which has been put forward in the 3rd century AC by Pappus: the mathematical method of demonstration depends upon imagining beforehand what is going to be demonstrated as a proof. There are two troubling implication to this: can we use mathematics to proof “imaginary”, “unreal”, “fictitious” “truth”, in short: can we use mathematics as a method of invention?
At the time of Descartes, this was a central interest to several more or less mystical projects of an ars combinatoria, e.g. the Ars Magna of Ramon Llull, or more generally the revived late Renaissance interest in the Aristotelian legacy of not separating clearly topics and analytics in his organon. Topics, therein, is a method of invention, capable of abstracting from the dichotomous opposition of Rhetorics and Logics. For a careful discussion of this late Renaissance interest (in German) cf. Wilhelm Schmidt-Biggemann: Topica Universalis. Eine Modellgeschichte humanistischer und barocker Wissenschaft (Meiner, Hamburg 1983). For an introduction to the Aristotelian heritage cf. the article on Stanford Encyclopedia.
Descartes intention was to clearly separate the analytical, as a method, form any uncritical involvement with topics or rhetorics. Descartes aimed at achieving this to the cost of reducing the complexities of proportionality (involving lines, areas, volumes) to linear segments that can be calculated arithmetically. Cf. Emily R. Grosholz: Cartesian Method and the Problem of Reduction (Oxford Scolarship Online, 1991). Descartes “purification” of the analytical method depends upon formulating the conditions for a problem in terms of a linear equation (involving terms raised to the power of 2, not higher).
(> my line of thought in another post about different scopes of analysis depending on different number “spaces” is closely linked to this problematics)
Here the Cartesian framework for architectonic interpretation:
6 dimensions or architectonic interpretation:
a) Is the problem to be analyzed framed deductively through mediating words/concepts?
b) Is the problem to be analyzed framed inductively through empirical experiments?
- do we seek to understand the causes through relating observable effects? applies to b) speculative.
- do we seek to anticipate effects from assumed causality relations? applies to b) practical.
- do we seek to understand the whole through the parts? applies to b) mathematical. regards the direct operations of addition and multiplication. These operations are synthetical for Descartes.
- do we seek to sort out the parts according to a model of the whole? applies to b) mathematical. regards the indirect operations of subtraction and division. These operations are analytical for Descartes.
As described, Jules Vuillemin develops a framework for algebraic analysis throughout is book which introduces the Cartesian strategy as its starting point. Yet his interest is to emancipate from what he calls “la mathematique matérielle”, and a corresponding understanding of the analytical method, and achieve an understanding of what he calls “la mathématique formelle” (Abstract Algebra) and a corresponding account of the rules underlying it. This emancipation concerns the role of intuition, and a certain dogmatic attitude that remains proper to the mathematical method as long as the role of intuition is playing a constitutive part (according to Vuillemin’s point of view which I find entirely convincing).
Instead, the involved assumptions (in any “theory” of intuition) need to be formulated conceptually and a priori. The following five precepts which Vuillemin formulates are crucial for preventing that such “conceptual synthesis a priori” (if not grounded the Kantian forms of intuition) needs not be “blind”, as Kant held. Vuillemin dedicates an extensive part of the book to discussion the relations and implications to Kant’s a priori synthesis, where Kant gives a transcendental rôle to what he calls “the forms of intuition” (essentially: Euclidean Space and Newtonian (linear) Time) not only to the mathematical method, but to human reasoning at large. Cf. for a discussion of this background (generally, not in relation to Vuillemin): Pirmin Stekeler-Weithofer, Formen der Anschauung, Eine Philosophie der Mathematik (de Gruyter 2008); as well as: Wladimir Velminski, Form, Zahl, Symbol, Leonhard Euler’s Strategien der Anschaulichkeit (Akademieverlag Berlin 2009).
Vuillemin’s stance, with this emphasis on “la mathématique formelle” against that of the “mathématique materielle” is that the mathematical method cannot escape to be dogmatic as long as it does not strip off assumptions that cannot themselves be addressed (challenged) – like the role of intuition – which cannot be criticized and refined). He wants to separate Philosophy from the mathematical methods by affirming that philosophy is responsible for the advancing of dogma, form L. dogma “philosophical tenet,” from Gk. dogma “opinion, tenet,” lit. “that which one thinks is true,” from dokein “to seem good, think” – and can escape to be dogmatist, as the pertaining to doctrines that ought not be explicated and considered, only by separating a mathematical method which is purely formal from the ideas that are settled by the analytical method into objectivity.
(Cf. for a recent and fierce plea in favor of materialist mathematics Alain Badiou, The Concept of Model: An Introduction to the Materialist Epistemology of Mathematics, 2007 ).
Vuillemin introduces 5 precepts which can, he suggests, achieve to maintain this separation if they can be translated to philosophy:
(to be translated from formal mathematics (abstract algebra) to philosophy)
- Lagrange, relating Algebra and Mechanics through generalizing the Cartesian Coordinate space. Lagrange allows for an experimental formulation of variable conditions of a problem. Hence he can work in much greater complexity than Descartes, there is no need to constrain the solution of problems to the Cartesian reduction of polynomials into linearity. It is possible to consider proportional aspects as well (involving lines (x2), areas (x3) volumes (x4), Galois groups (x5), or even higher algebraic structures involving complex functions)).
- Gauss, generalizing demonstrations via constructions by ruler and compass. While for the Cartesian arithmetization of the geometrical method makes sure that there can be a solution to a problem, Gausses’ generalization of the geometrical method allows to be experimental on the level of why, under which assumptions there can be a solution to a problem.
- Abel, inverting the direction of the method. The aim is to find out which axioms a certain theorem actually depends on, and not which axioms allow to demonstrate the validity of a certain theorem. Vuillemin describes this procedure as a hygienic procedure which is necessary for keeping in control the involved assumptions one uses in a proof, not for actually yielding a constructing as demonstration. This inversion of methods allowed to find solutions for the quintic (polynomial terms of the order (x5)) – a problem was was held to be unsolvable before. His inversion of the analytical method consists in explicating the assumed conditions involved through building a synthetical construction of all these assumptions (axioms), in order to analyze whether they are all necessary for supporting a certain theorem.
- Galois, proceeding by adjunction. It assumes that the sense of a solution cannot be determined within the order in which the problem had been formulated and resolved. It alters the notion of “truth” and “necessity” of the mathematical method: truth cannot be achieved in terms of adequation of res ad intellectum. Instead of being representational or semantical, “truth” consists in the totality of the possibilities that are deducible within a certain algebraic structure. Altering the axioms of the structure also alters that which may be “true”. The criterium of a solutions’ adequate representation of a problem dissolves into the criterium of formal compatibility among solution spaces.