Combining Philosophers

Ideas for Hermarchus, Robert Audi and Keith Hossack

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4 ideas

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
10674 A plural language gives a single comprehensive induction axiom for arithmetic [Hossack]
     Full Idea: A language with plurals is better for arithmetic. Instead of a first-order fragment expressible by an induction schema, we have the complete truth with a plural induction axiom, beginning 'If there are some numbers...'.
     From: Keith Hossack (Plurals and Complexes [2000], 4)
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
10681 In arithmetic singularists need sets as the instantiator of numeric properties [Hossack]
     Full Idea: In arithmetic singularists need sets as the instantiator of numeric properties.
     From: Keith Hossack (Plurals and Complexes [2000], 8)
10685 Set theory is the science of infinity [Hossack]
     Full Idea: Set theory is the science of infinity.
     From: Keith Hossack (Plurals and Complexes [2000], 10)
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
23621 Numbers are properties, not sets (because numbers are magnitudes) [Hossack]
     Full Idea: I propose that numbers are properties, not sets. Magnitudes are a kind of property, and numbers are magnitudes. …Natural numbers are properties of pluralities, positive reals of continua, and ordinals of series.
     From: Keith Hossack (Knowledge and the Philosophy of Number [2020], Intro)
     A reaction: Interesting! Since time can have a magnitude (three weeks) just as liquids can (three litres), it is not clear that there is a single natural property we can label 'magnitude'. Anything we can manage to measure has a magnitude.