Extending a Kantian Dichotomy to a Poincaréan Trichotomy
ABSTRACT: I argue for the possibility of knowledge by invention which is neither á priori nor á posteriori. My conception of knowledge by invention evolves from Poincaré’s conventionalism, but unlike Poincaré’s conventions, propositions known by invention have a truth value. An individuating criteria for this type of knowledge is conjectured. The proposition known through invention is: gounded historically in the discipline to which it belongs; a result of the careful, sincere and objective quest and effort of the knower; chosen freely by the inventer or knower; and, private in its invention but public once invented. I extend knowledge by invention to include the knowledge of the invented proposition by those who do not invent it but accept it as a convention for good reasons. Finally, knowledge by invention combined with a revisionist, Platonist definition of knowledge as actively justified true belief provides a pedagogical model reviving the proactive spirit of the Socratic method with an emphasis on invention and activity and a de-emphasis on information gathering and passivity.
Kant's à priori - à posteriori and analytic - synthetic distinctions inaugurated Modern epistemology and provided the architecture for knowledge in mathematics, science and metaphysics. (1) The product of the two distinctions yields three kinds of knowledge: synthetic à priori, analytic à priori and synthetic à posteriori; analytic à posteriori being impossible. For Kant propositions like; "7+5=12," "all bodies have mass" and "every event has a cause." were synthetic and known à priorily. (2) Post-Kantian philosophy witnessed an attack on the possibility of synthetic à priori knowledge such as the rejections of analysis, geometry and arithmetic as synthetic à priori by Bolzano, Helmholtz and Frege respectively. (3) These were motivated by a fear that Kant's conceptualism, of the mind imposing space and time on the world, may lead to anti-realism, such as that of Husserl's bracketing the existence of the world based on his extensions of Descartes and Kant. (4) Nominalism and idealism are anti-realist but conceptualism and conventionalism need not be. I extend the typology of knowledge by adding knowledge by invention. Many fundamental propositions of mathematics, science and metaphysics hence shift from the realm of synthetic à priori to the realm of knowledge by invention. For Poincaré fundamental definitions of mathematics are neither à priori nor à posteriori, but conventional. I suggest that "conventional" means "known by invention." I will argue in this paper for this unconventional interpretation of Poincaré's conventionalism.