Combining Philosophers

Ideas for B Hale / C Wright, Immanuel Kant and Anaxagoras

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30 ideas

6. Mathematics / A. Nature of Mathematics / 1. Mathematics
16918 Mathematics cannot proceed just by the analysis of concepts [Kant]
6. Mathematics / A. Nature of Mathematics / 2. Geometry
16930 Geometry is not analytic, because a line's being 'straight' is a quality [Kant]
8739 Geometry studies the Euclidean space that dictates how we perceive things [Kant, by Shapiro]
16919 Geometry rests on our intuition of space [Kant]
8740 Geometry would just be an idle game without its connection to our intuition [Kant]
16899 Geometrical truth comes from a general schema abstracted from a particular object [Kant, by Burge]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
16920 Numbers are formed by addition of units in time [Kant]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic
16929 7+5 = 12 is not analytic, because no analysis of 7+5 will reveal the concept of 12 [Kant]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / c. Potential infinite
9632 Kant only accepts potential infinity, not actual infinity [Kant, by Brown,JR]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
21382 Things get smaller without end [Anaxagoras]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
3343 Euclid's could be the only viable geometry, if rejection of the parallel line postulate doesn't lead to a contradiction [Benardete,JA on Kant]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
8737 Kant suggested that arithmetic has no axioms [Kant, by Shapiro]
5557 Axioms ought to be synthetic a priori propositions [Kant]
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
12421 Kant's intuitions struggle to judge relevance, impossibility and exactness [Kitcher on Kant]
16910 Mathematics can only start from an a priori intuition which is not empirical but pure [Kant]
16917 All necessary mathematical judgements are based on intuitions of space and time [Kant]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
17617 Maths is a priori, but without its relation to empirical objects it is meaningless [Kant]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
16928 Mathematics cannot be empirical because it is necessary, and that has to be a priori [Kant]
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
12458 Kant taught that mathematics is independent of logic, and cannot be grounded in it [Kant, by Hilbert]
2795 If 7+5=12 is analytic, then an infinity of other ways to reach 12 have to be analytic [Kant, by Dancy,J]
12224 Are neo-Fregeans 'maximalists' - that everything which can exist does exist? [Hale/Wright]