Abstract
We should now like to draw certain conclusions about general epistemological problems from these reflections on the psychology of mathematics, taking epistemology in the sense of Chapter VII, Section 42, including ontological problems which imply the comparison of logical analyses with genetic data.
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References
F. Enriques, Les concepts fondamentaux de la Science, transi, by Rougier, Paris, 1914.
F. Gonseth, Les mathématiques et la réalité, Paris, 1932, p. 127.
L. Brunschvicg, Les étapes de la philosophie mathématique, Paris, 1912.
In L’enseignement mathématique vol. 30 (1931), translated by Müller.
L. Rougier, Traité de la connaissance, Paris, 1955.
Cf. VII and IX of Etudes épist. génét.
It may be taken for granted that we shall take the term “construction” here in its genetic or general sense, which is to produce or to reconstitute an object of thought, and not in its special mathematical sense, which is to specify an object whose existence is guaranteed by axioms. See note 9 on next page.
Thus, for Couturat, 2 + 3 =5 is only a relation between 2, 3 and 5.
E. W. Beth, in the chapter “Nominalism” of his fine book The Foundations of Mathematics,discusses the problem of construction, in the mathematical sense of the term (the specification of objects whose existence is guaranteed by axioms), and not in the genetic sense (the production of the object) as we do here. He reminds us that in wishing to submit to the limitations of construction in the narrow sense (with a concrete interpretation of the axioms) we do not succeed in reconstituting classical mathematics. As Gödel has shown, we then need supplementary expedients to do this, and, without them, we obtain only a sub-system of classical mathematics, similar to Lorenzen’s mathematics. But it is not this problem which we are discussing here, since we take the term “construction” in the sense of the production of the object. Without needing to discuss the mathematical concept of construction, which we are not competent to do, we confine ourselves to maintaining that even a classical deduction corresponding to a literal interpretation of the axioms (and thus foreign to construction in the limiting mathematical sense) can be interpreted epistemologically, either as an apprehension of a given mathematical reality, or as a construction in the genetic sense. The difference is then the following: in the Platonist interpretation, the axioms consist of a direct apprehension of realities external to ourselves (this is the “strong” branch of our disjunction). In the constructivist interpretation (in the genetic sense), the axioms themselves are constructed, in so far as they are the result of a reflective abstraction starting from operational co-ordinations, and the whole deduction is thus constructive, whether it preserves its classical aspect or is subordinated to the idea of construction in the narrow sense of the term. This is why we maintain that, in fact, even a Platonist needs construction (in the genetic sense), since we do not know (or do not yet know) of any particular faculty which allows us to attain ideal entities which are independent of us, and since, when we think we grasp them, it is after the mind has been at work, and the mind (until we know more about it) always seems to be constructive (in the genetic and not the mathematical sense of the term).
See ‘Logique et équilibre’, Vol. II of Etudes épist. génét.,Etude II.
Inhelder and Piaget, De la logique de l’enfant à la logique de l’adolescent,Paris, 1955.
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Beth, E.W., Piaget, J. (1974). Epistemological Problems with Logical and Psychological Relevance. In: Mathematical Epistemology and Psychology. Synthese Library, vol 12. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2193-6_12
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