Combining Philosophers

Ideas for Luitzen E.J. Brouwer, R.G. Collingwood and Michael Dummett

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21 ideas

6. Mathematics / A. Nature of Mathematics / 1. Mathematics
18119 Mathematics is a mental activity which does not use language [Brouwer, by Bostock]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
9896 A prime number is one which is measured by a unit alone [Dummett]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
18255 Addition of quantities is prior to ordering, as shown in cyclic domains like angles [Dummett]
9191 Ordinals seem more basic than cardinals, since we count objects in sequence [Dummett]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
18247 Brouwer saw reals as potential, not actual, and produced by a rule, or a choice [Brouwer, by Shapiro]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
9895 A number is a multitude composed of units [Dummett]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation
9852 We understand 'there are as many nuts as apples' as easily by pairing them as by counting them [Dummett]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
12451 Scientific laws largely rest on the results of counting and measuring [Brouwer]
18118 Brouwer regards the application of mathematics to the world as somehow 'wicked' [Brouwer, by Bostock]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / c. Potential infinite
15938 Platonists ruin infinity, which is precisely a growing structure which is never completed [Dummett]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
10554 Intuitionists find the Incompleteness Theorem unsurprising, since proof is intuitive, not formal [Dummett]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
9829 The identity of a number may be fixed by something outside structure - by counting [Dummett]
9828 Numbers aren't fixed by position in a structure; it won't tell you whether to start with 0 or 1 [Dummett]
9192 The number 4 has different positions in the naturals and the wholes, with the same structure [Dummett]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
9876 Set theory isn't part of logic, and why reduce to something more complex? [Dummett]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
15939 For intuitionists it is constructed proofs (which take time) which make statements true [Dummett]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
12454 Intuitionists only accept denumerable sets [Brouwer]
12453 Neo-intuitionism abstracts from the reuniting of moments, to intuit bare two-oneness [Brouwer]
8728 Intuitionist mathematics deduces by introspective construction, and rejects unknown truths [Brouwer]
10552 Intuitionism says that totality of numbers is only potential, but is still determinate [Dummett]
8190 Intuitionists rely on the proof of mathematical statements, not their truth [Dummett]