Combining Philosophers

Ideas for B Hale / C Wright, Hermann von Helmholtz and Philip Kitcher

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33 ideas

6. Mathematics / A. Nature of Mathematics / 1. Mathematics
6298 Kitcher says maths is an idealisation of the world, and our operations in dealing with it [Kitcher, by Resnik]
12392 Mathematical a priorism is conceptualist, constructivist or realist [Kitcher]
18078 The interest or beauty of mathematics is when it uses current knowledge to advance undestanding [Kitcher]
12426 The 'beauty' or 'interest' of mathematics is just explanatory power [Kitcher]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
12395 Real numbers stand to measurement as natural numbers stand to counting [Kitcher]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / j. Complex numbers
12425 Complex numbers were only accepted when a geometrical model for them was found [Kitcher]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
18071 A one-operation is the segregation of a single object [Kitcher]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
18066 The old view is that mathematics is useful in the world because it describes the world [Kitcher]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
18083 With infinitesimals, you divide by the time, then set the time to zero [Kitcher]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
10624 The incompletability of formal arithmetic reveals that logic also cannot be completely characterized [Hale/Wright]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
8784 Neo-logicism founds arithmetic on Hume's Principle along with second-order logic [Hale/Wright]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
8787 The Julius Caesar problem asks for a criterion for the concept of a 'number' [Hale/Wright]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
10629 If structures are relative, this undermines truth-value and objectivity [Hale/Wright]
10628 The structural view of numbers doesn't fit their usage outside arithmetical contexts [Hale/Wright]
6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
18061 Mathematical intuition is not the type platonism needs [Kitcher]
12420 If mathematics comes through intuition, that is either inexplicable, or too subjective [Kitcher]
12393 Intuition is no basis for securing a priori knowledge, because it is fallible [Kitcher]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
12387 Mathematical knowledge arises from basic perception [Kitcher]
12412 My constructivism is mathematics as an idealization of collecting and ordering objects [Kitcher]
18065 We derive limited mathematics from ordinary things, and erect powerful theories on their basis [Kitcher]
18077 The defenders of complex numbers had to show that they could be expressed in physical terms [Kitcher]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
8788 Logicism is only noteworthy if logic has a privileged position in our ontology and epistemology [Hale/Wright]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
10622 The neo-Fregean is more optimistic than Frege about contextual definitions of numbers [Hale/Wright]
8783 Logicism might also be revived with a quantificational approach, or an abstraction-free approach [Hale/Wright]
12225 Neo-Fregeanism might be better with truth-makers, rather than quantifier commitment [Hale/Wright]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
12423 Analyticity avoids abstract entities, but can there be truth without reference? [Kitcher]
12224 Are neo-Fregeans 'maximalists' - that everything which can exist does exist? [Hale/Wright]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
18069 Arithmetic is an idealizing theory [Kitcher]
18068 Arithmetic is made true by the world, but is also made true by our constructions [Kitcher]
18070 We develop a language for correlations, and use it to perform higher level operations [Kitcher]
18072 Constructivism is ontological (that it is the work of an agent) and epistemological (knowable a priori) [Kitcher]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
18063 Conceptualists say we know mathematics a priori by possessing mathematical concepts [Kitcher]
18064 If meaning makes mathematics true, you still need to say what the meanings refer to [Kitcher]