Combining Philosophers
Ideas for H.Putnam/P.Oppenheim, Richard G. Heck and Tyler Burge
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14 ideas
6. Mathematics / A. Nature of Mathematics / 2. Geometry
16901
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The equivalent algebra model of geometry loses some essential spatial meaning [Burge]
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9159
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You can't simply convert geometry into algebra, as some spatial content is lost [Burge]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
17453
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The meaning of a number isn't just the numerals leading up to it [Heck]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
17457
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A basic grasp of cardinal numbers needs an understanding of equinumerosity [Heck]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
17448
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In counting, numerals are used, not mentioned (as objects that have to correlated) [Heck]
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17455
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Is counting basically mindless, and independent of the cardinality involved? [Heck]
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17456
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Counting is the assignment of successively larger cardinal numbers to collections [Heck]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation
17450
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Understanding 'just as many' needn't involve grasping one-one correspondence [Heck]
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17451
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We can know 'just as many' without the concepts of equinumerosity or numbers [Heck]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
16902
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Peano arithmetic requires grasping 0 as a primitive number [Burge]
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17459
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Frege's Theorem explains why the numbers satisfy the Peano axioms [Heck]
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6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
17454
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Children can use numbers, without a concept of them as countable objects [Heck]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
17458
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Equinumerosity is not the same concept as one-one correspondence [Heck]
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17449
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We can understand cardinality without the idea of one-one correspondence [Heck]
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