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Ideas for 'On the Question of Absolute Undecidability', 'Parmenides' and 'Metaphysics'

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6 ideas

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
17887 PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner]
     Full Idea: To the extent that we are justified in accepting Peano Arithmetic we are justified in accepting its consistency, and so we know how to expand the axiom system so as to overcome the limitation [of Gödel's Second Theorem].
     From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.1)
     A reaction: Each expansion brings a limitation, but then you can expand again.
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
17891 Arithmetical undecidability is always settled at the next stage up [Koellner]
     Full Idea: The arithmetical instances of undecidability that arise at one stage of the hierarchy are settled at the next.
     From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4)
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
17843 The idea of 'one' is the foundation of number [Aristotle]
     Full Idea: One is the principle of number qua number.
     From: Aristotle (Metaphysics [c.324 BCE], 1052b21)
17850 Each many is just ones, and is measured by the one [Aristotle]
     Full Idea: The reason for saying of each number that it is many is just that it is ones and that each number is measured by the one.
     From: Aristotle (Metaphysics [c.324 BCE], 1056b16)
17851 Number is plurality measured by unity [Aristotle]
     Full Idea: Number is plurality as measured by unity.
     From: Aristotle (Metaphysics [c.324 BCE], 1057a04)
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
9793 Mathematics studies abstracted relations, commensurability and proportion [Aristotle]
     Full Idea: Mathematicians abstract perceptible features to study quantity and continuity ...and examine the mutual relations of some and the features of those relations, and commensurabilities of others, and of yet others the proportions.
     From: Aristotle (Metaphysics [c.324 BCE], 1061a32)
     A reaction: This sounds very much like the intuition of structuralism to me - that the subject is entirely about relations between things, with very little interest in the things themselves. See Aristotle on abstraction (under 'Thought').