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Single Idea 8742

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers ]

Full Idea

Mill says arithmetic has two axioms, that 'things which are equal to the same thing are equal to each other', and 'equals added to equals make equal sums', plus a definition for each numeral as 'formed by the addition of a unit to the previous number'.

Gist of Idea

The only axioms needed are for equality, addition, and successive numbers

Source

report of John Stuart Mill (System of Logic [1843], p.610?) by Stewart Shapiro - Thinking About Mathematics 4.3

Book Ref

Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.95

A Reaction

The difficulty here seems to be the definition of 1, and (even worse for an empiricist), of 0. Then he may have a little trouble when he reaches infinity.

The 20 ideas with the same theme [general ideas about giving arithmetic a formal basis]:

23026 We know mathematical axioms, such as subtracting equals from equals leaves equals, by a natural light [Leibniz]
8737 Kant suggested that arithmetic has no axioms [Kant, by Shapiro]
5557 Axioms ought to be synthetic a priori propositions [Kant]
8742 The only axioms needed are for equality, addition, and successive numbers [Mill, by Shapiro]
13508 Dedekind gives a base number which isn't a successor, then adds successors and induction [Dedekind, by Hart,WD]
16883 Arithmetical statements can't be axioms, because they are provable [Frege, by Burge]
16864 If principles are provable, they are theorems; if not, they are axioms [Frege]
3338 Numbers have been defined in terms of 'successors' to the concept of 'zero' [Peano, by Blackburn]
17964 Number theory just needs calculation laws and rules for integers [Hilbert]
14431 The definition of order needs a transitive relation, to leap over infinite intermediate terms [Russell]
14124 Axiom of Archimedes: a finite multiple of a lesser magnitude can always exceed a greater [Russell]
9939 It is conceivable that the axioms of arithmetic or propositional logic might be changed [Putnam]
9900 For Zermelo 3 belongs to 17, but for Von Neumann it does not [Benacerraf]
9899 The successor of x is either x and all its members, or just the unit set of x [Benacerraf]
10808 Mathematics is generalisations about singleton functions [Lewis]
10608 The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P]
10618 All numbers are related to zero by the ancestral of the successor relation [Smith,P]
10174 Mereological arithmetic needs infinite objects, and function definitions [Reck/Price]
17715 The truth of the axioms doesn't matter for pure mathematics, but it does for applied [Mares]
17312 It is more explanatory if you show how a number is constructed from basic entities and relations [Koslicki]