Real Antirealism: The Practical Stance

International audienceTraditional antirealism as well as intuitionism have been construed by reference to the frontier between what is in principle possible and what is not so. While this frontier has all the robustness properties one could dream, it doesn't run through the right place, for a lot of sentences that are in principle decidable are however epistemically transcendent, even w.r.t. a very generous conception of what we human beings are actually able to decide. Then, antirealism, to eliminate these realist vestiges, has to change the map and to focus on what we practically do and can do. Work has been already done on the possible means to achieve this objective by defining epistemically immanent assertability conditions in a logical proper manner (e.g. [1] and [4]), but a great number of questions are still to be investigated in the perspective of the strict antirealism and I will first briefly present some milestones on the feasibilist agenda, as the necessity of overcoming the idealizations of the Kolmogorovian definition of complexity (cf [2]). The remainder of the paper deals with philosophy of mathematics. When the length of proofs is taken into account, as clearly the feasibilist point of view suggests to do, traditional problems in this domain appear in a new light, specially the question of the purity of proofs. Speed-up results show that the Hilbertian ideal of Methodenreinheit is generally out of our reach, even when it is possible in principle. Then, lemmas, which are the open door for impurity (and for the cooperative use of the work already done by other people), are no longer dispensible or eliminable. As purity is, practically speaking, beyond reach, there is no room for implementing Bolzanian thesis that correct proofs should prove propositions by only invoking the propositions to which they owe their own truth (cf [3]). Also, the instrumentalist lessons that have been drawn from conservativity results w.r.t. abstract mathematical notions are very weak in the light of feasibility. As the practical stance rejects the familar notion of something "practically useful but theoretically dispensible," the deflationary significance currently attributed to some conservativity results vanishes. In particular, Field's nominalistic programme, that crucially rests on a recognition-transcendent notion of semantical consequence, is to be considered in a very cautious way. Mathematics cannot be viewed as a mere extractor of empirical juice, for empirical juice never flows quick enough to us human beings when mathematics are absent. To sum up, feasibilism, far from being inconsistent as often claimed, is able, at least I will argue, to provide the less extravagant image of mathematics and science