Contrast Between a System of Logic and the Foundation of Arithmetic

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Explain, as clearly as possible, John Stuart Mill's view of arithmetic put forward in his book "A System of Logic" (Book II, Chapter 6) and then present how Gottlob Frege argues against Mill's view in his book entitled "The foundations of Arithmetic". Mill's Argument: Book 2; Chap. 6 We need axioms and definitions in order to structure an argument that points to certainty; each theorem must be solidly grounded on the other. Axioms are solidly grounded on experience. They are the most universal class and the most reliable. The so-called Deductive Sciences are therefore also inductive science in that whilst they rest on pure and empirical experience, they are suppositions from these bases and only hypothetical. They are approximate to the truth but lack the same certainty. This theory however must be tested in order to see whether it is categorically applicable to all sickness including those based on numbers such s Calculus; Arithmetic and Algebra. Every step of arithmetic seems to demonstrate a real inductive progression, such as 2+1 = 3. It is based on solid fact. They are based on evidence. But even here we have inductive with calculations flowing form the fact. Inductions of arithmetic are of two sorts: One purely inductive flowing form the other Some hypothetical where we merely assume that there is equality between all inductions/ theorems: What is commonly called mathematical certainty, therefore, which comprises the twofold conception of unconditional truth

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