Combining Philosophers
Ideas for Shaughan Lavine, Anaxarchus and Michael D. Resnik
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23 ideas
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
6304
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Mathematical realism says that maths exists, is largely true, and is independent of proofs [Resnik]
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15907
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Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
15942
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Every rational number, unlike every natural number, is divisible by some other number [Lavine]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
15922
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For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
18250
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Cauchy gave a necessary condition for the convergence of a sequence [Lavine]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
15904
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The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
15912
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Counting results in well-ordering, and well-ordering makes counting possible [Lavine]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
15949
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The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine]
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15947
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The infinite is extrapolation from the experience of indefinitely large size [Lavine]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / c. Potential infinite
15940
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The intuitionist endorses only the potential infinite [Lavine]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities
15909
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'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
15915
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Ordinals are basic to Cantor's transfinite, to count the sets [Lavine]
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15917
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Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
15918
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Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
15929
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Set theory will found all of mathematics - except for the notion of proof [Lavine]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
6300
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Mathematical constants and quantifiers only exist as locations within structures or patterns [Resnik]
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6303
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Sets are positions in patterns [Resnik]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
6302
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Structuralism must explain why a triangle is a whole, and not a random set of points [Resnik]
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6295
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There are too many mathematical objects for them all to be mental or physical [Resnik]
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6296
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Maths is pattern recognition and representation, and its truth and proofs are based on these [Resnik]
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6301
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Congruence is the strongest relationship of patterns, equivalence comes next, and mutual occurrence is the weakest [Resnik]
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6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
15935
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Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
15928
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Intuitionism rejects set-theory to found mathematics [Lavine]
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