Combining Philosophers

Ideas for Leucippus, Agathon and Gottlob Frege

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25 ideas

6. Mathematics / A. Nature of Mathematics / 1. Mathematics
16869 To create order in mathematics we need a full system, guided by patterns of inference [Frege]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
9886 Cardinals say how many, and reals give measurements compared to a unit quantity [Frege]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
18256 Quantity is inconceivable without the idea of addition [Frege]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
8640 We cannot define numbers from the idea of a series, because numbers must precede that [Frege]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
18252 Real numbers are ratios of quantities, such as lengths or masses [Frege]
18253 I wish to go straight from cardinals to reals (as ratios), leaving out the rationals [Frege]
9889 Real numbers are ratios of quantities [Frege, by Dummett]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / l. Zero
9838 Treating 0 as a number avoids antinomies involving treating 'nobody' as a person [Frege, by Dummett]
9564 For Frege 'concept' and 'extension' are primitive, but 'zero' and 'successor' are defined [Frege, by Chihara]
10551 If objects exist because they fall under a concept, 0 is the object under which no objects fall [Frege, by Dummett]
8653 Nought is the number belonging to the concept 'not identical with itself' [Frege]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / m. One
8654 One is the Number which belongs to the concept "identical with 0" [Frege]
8636 We can say 'a and b are F' if F is 'wise', but not if it is 'one' [Frege]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
8641 You can abstract concepts from the moon, but the number one is not among them [Frege]
9989 Units can be equal without being identical [Tait on Frege]
17429 Frege says only concepts which isolate and avoid arbitrary division can give units [Frege, by Koslicki]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / d. Counting via concepts
17427 Frege's 'isolation' could be absence of overlap, or drawing conceptual boundaries [Frege, by Koslicki]
17437 Non-arbitrary division means that what falls under the concept cannot be divided into more of the same [Frege, by Koslicki]
17438 Our concepts decide what is countable, as in seeing the leaves of the tree, or the foliage [Frege, by Koslicki]
17426 A concept creating a unit must isolate and unify what falls under it [Frege]
17428 Frege says counting is determining what number belongs to a given concept [Frege, by Koslicki]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation
15916 Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted [Frege, by Lavine]
17446 Counting rests on one-one correspondence, of numerals to objects [Frege]
9582 Husserl rests sameness of number on one-one correlation, forgetting the correlation with numbers themselves [Frege]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
10034 The number of natural numbers is not a natural number [Frege, by George/Velleman]