The Foundations of Science

Description

EPUB is Maitland's translation. DjVu is Halsted's.

translation of Poincaré's La science et l'hypothèse

Poincaré's views of "mathematical induction" (MI) are in La Science et l'Hypothèse (pt 1, ch. 1 "Sur la nature du raisonnement mathématique"). Duhem's paper explains and refutes Poincaré's views:

Duhem, Pierre Maurice Marie. “La nature du raisonnement mathématique.” Revue de Philosophie 21 (1912): 531–43.

Duhem concludes in favor of syllogistic reasoning (Aristotelian logic?):

Nous pensons avoir suffisamment établi, dans ce qui précède, que la démonstration mathématique se poursuit par voie syllogistique exactement de la même manière que n'importe quelle autre science déductive. Ce qui la distingue des autres sciences déductives, ce n'est pas la forme du raisonnement qu'elle emploie; c'est la nature des notions et propositions auxquelles elle applique ce raisonnement.

We think we've sufficiently established, in the above, that mathematical demonstration proceeds by syllogistic means in exactly the same way as any other deductive science. What distinguishes it from the other deductive sciences is not the form of the reasoning it uses; it's the nature of the notions and propositions to which it applies this reasoning.

You’re correct about Duhem doing a better job on Poincaré. He surely was in a better position to do so and did it well in La nature du raisonnement mathématique. Il a raison (p. 542) :

« Mais les Mathématiques ne sont pas virtuellement contenues dans les seuls axiomes [something many reductionist logicians and computer scientists to this day don’t want to realize] ; elles sont le résultat produit par l’application des axiomes aux définitions. »

The Search for Mathematical Roots is written by a Russellian Bristish mathematical logician, who was also a fairly narrow mathematical historian (however prolific his writings). Symbolic logicians typically rank the piece as a reference, for their purposes.

“Aristotelian logic,” in part at least (I remember that we conversed about this matter before). Even within a deductive approach, modern logic has usefully extended reasonings to non-syllogistic forms (used, for example, to prove things about propositional proof systems themselves, by way of both deductive and inductive/recurrence reasonings, à la Gödel). Also, the complexification and replacement of the ‘subject’ and ‘predicate’ terms of categorical syllogisms by way of ‘functions’ and ‘arguments,’ which Frege brought about, allowed him to analyse the logical connection of statements involving manifold generalities (the analysis of which uses rules of inference and cannot simply assume a merely validating form).

Suffice it to recall that already (before modern extensions) the Stoics had begun pondering and constructing non -syllogistic deductive reasonings, i.e. non-syllogistic conclusive reasonings.