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Reasoning and Science:
Mill claims that deductive reasoning is “empty”: it says nothing new about the world. Everything established in the conclusion of a deductive argument must already be present in the premises (SEP, Mill, 3.1). Therefore, deductive reasoning does not lead to any new knowledge about the world. Furthermore, Mill is radical in his view that mathematics and geometry---areas that lead to acquisition of genuine knowledge---do not employ deductive reasoning. According to Mill, it only appears that mathematics and geometry use deductive reasoning, but on a deeper level, they are using nothing more than inductive reasoning (SEP, Mill, 3.1). The idea that mathematics and geometry ''de facto '' employ inductive reasoning allows him to deny the existence of even this form of knowledge, which Kant and Whewell considered a priori.
Mill holds that “there is in every step of arithmetical and algebraically calculation a real induction, a real inference of facts from facts” (System, VII: 254). He contends that mathematical propositions are not true by definition (System VII: 253); these propositions are not analytic. For example, he thinks that the number two is one plus one not because two is defined as one plus one. On the contrary, two is one plus one, because of induction. We observe, for instance, that one rock and another rock lead to two rocks; similarly, we observe that one swan and another swan lead to two swans, and so on in a multitude of cases. According to Mill, the pattern that one and one lead to two in specific singular instances allow us to generalize that one and one equal two in all cases. Hence, all mathematical and geometric propositions are arrived at and justified through induction (SEP, Mill, 3.4).
