Combining Philosophers
Ideas for Lynch,MP/Glasgow,JM, James Robert Brown and Gottlob Frege
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138 ideas
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
9604
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Mathematics is the only place where we are sure we are right [Brown,JR]
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16869
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To create order in mathematics we need a full system, guided by patterns of inference [Frege]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
9622
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'There are two apples' can be expressed logically, with no mention of numbers [Brown,JR]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
9886
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Cardinals say how many, and reals give measurements compared to a unit quantity [Frege]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
18256
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Quantity is inconceivable without the idea of addition [Frege]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
8640
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We cannot define numbers from the idea of a series, because numbers must precede that [Frege]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
18252
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Real numbers are ratios of quantities, such as lengths or masses [Frege]
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18253
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I wish to go straight from cardinals to reals (as ratios), leaving out the rationals [Frege]
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9889
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Real numbers are ratios of quantities [Frege, by Dummett]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / l. Zero
8653
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Nought is the number belonging to the concept 'not identical with itself' [Frege]
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9838
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Treating 0 as a number avoids antinomies involving treating 'nobody' as a person [Frege, by Dummett]
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9564
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For Frege 'concept' and 'extension' are primitive, but 'zero' and 'successor' are defined [Frege, by Chihara]
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10551
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If objects exist because they fall under a concept, 0 is the object under which no objects fall [Frege, by Dummett]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / m. One
8636
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We can say 'a and b are F' if F is 'wise', but not if it is 'one' [Frege]
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8654
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One is the Number which belongs to the concept "identical with 0" [Frege]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / n. Pi
9648
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π is a 'transcendental' number, because it is not the solution of an equation [Brown,JR]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
8641
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You can abstract concepts from the moon, but the number one is not among them [Frege]
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9989
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Units can be equal without being identical [Tait on Frege]
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17429
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Frege says only concepts which isolate and avoid arbitrary division can give units [Frege, by Koslicki]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / d. Counting via concepts
17427
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Frege's 'isolation' could be absence of overlap, or drawing conceptual boundaries [Frege, by Koslicki]
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17437
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Non-arbitrary division means that what falls under the concept cannot be divided into more of the same [Frege, by Koslicki]
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17438
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Our concepts decide what is countable, as in seeing the leaves of the tree, or the foliage [Frege, by Koslicki]
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17426
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A concept creating a unit must isolate and unify what falls under it [Frege]
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17428
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Frege says counting is determining what number belongs to a given concept [Frege, by Koslicki]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation
15916
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Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted [Frege, by Lavine]
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17446
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Counting rests on one-one correspondence, of numerals to objects [Frege]
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9582
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Husserl rests sameness of number on one-one correlation, forgetting the correlation with numbers themselves [Frege]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
9621
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Mathematics represents the world through structurally similar models. [Brown,JR]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
10034
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The number of natural numbers is not a natural number [Frege, by George/Velleman]
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6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
18271
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We can't prove everything, but we can spell out the unproved, so that foundations are clear [Frege]
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6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
9646
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There is no limit to how many ways something can be proved in mathematics [Brown,JR]
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9647
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Computers played an essential role in proving the four-colour theorem of maps [Brown,JR]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
16883
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Arithmetical statements can't be axioms, because they are provable [Frege, by Burge]
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16864
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If principles are provable, they are theorems; if not, they are axioms [Frege]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
17855
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It may be possible to define induction in terms of the ancestral relation [Frege, by Wright,C]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
10625
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Frege had a motive to treat numbers as objects, but not a justification [Hale/Wright on Frege]
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13871
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Frege claims that numbers are objects, as opposed to them being Fregean concepts [Frege, by Wright,C]
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13872
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Numbers are second-level, ascribing properties to concepts rather than to objects [Frege, by Wright,C]
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9816
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For Frege, successor was a relation, not a function [Frege, by Dummett]
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17636
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A cardinal number may be defined as a class of similar classes [Frege, by Russell]
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9953
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Numbers are more than just 'second-level concepts', since existence is also one [Frege, by George/Velleman]
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9954
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"Number of x's such that ..x.." is a functional expression, yielding a name when completed [Frege, by George/Velleman]
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10139
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Frege gives an incoherent account of extensions resulting from abstraction [Fine,K on Frege]
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10028
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For Frege the number of F's is a collection of first-level concepts [Frege, by George/Velleman]
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13887
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Frege started with contextual definition, but then switched to explicit extensional definition [Frege, by Wright,C]
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13897
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Each number, except 0, is the number of the concept of all of its predecessors [Frege, by Wright,C]
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9856
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Frege's account of cardinals fails in modern set theory, so they are now defined differently [Dummett on Frege]
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9902
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Frege's incorrect view is that a number is an equivalence class [Benacerraf on Frege]
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17814
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The natural number n is the set of n-membered sets [Frege, by Yourgrau]
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17819
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A set doesn't have a fixed number, because the elements can be seen in different ways [Yourgrau on Frege]
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17460
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A statement of number contains a predication about a concept [Frege]
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17820
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If you can subdivide objects many ways for counting, you can do that to set-elements too [Yourgrau on Frege]
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16890
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Frege's problem is explaining the particularity of numbers by general laws [Frege, by Burge]
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8630
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Individual numbers are best derived from the number one, and increase by one [Frege]
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11029
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'Exactly ten gallons' may not mean ten things instantiate 'gallon' [Rumfitt on Frege]
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10013
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Numerical statements have first-order logical form, so must refer to objects [Frege, by Hodes]
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18181
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The Number for F is the extension of 'equal to F' (or maybe just F itself) [Frege]
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18103
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Numbers are objects because they partake in identity statements [Frege, by Bostock]
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9586
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In a number-statement, something is predicated of a concept [Frege]
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10553
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A number is a class of classes of the same cardinality [Frege, by Dummett]
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3331
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If '5' is the set of all sets with five members, that may be circular, and you can know a priori if the set has content [Benardete,JA on Frege]
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9949
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There is the concept, the object falling under it, and the extension (a set, which is also an object) [Frege, by George/Velleman]
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10623
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Frege defined number in terms of extensions of concepts, but needed Basic Law V to explain extensions [Frege, by Hale/Wright]
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9975
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Frege ignored Cantor's warning that a cardinal set is not just a concept-extension [Tait on Frege]
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10020
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Frege's biggest error is in not accounting for the senses of number terms [Hodes on Frege]
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10029
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Numbers need to be objects, to define the extension of the concept of each successor to n [Frege, by George/Velleman]
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9973
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The number of F's is the extension of the second level concept 'is equipollent with F' [Frege, by Tait]
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16500
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Frege showed that numbers attach to concepts, not to objects [Frege, by Wiggins]
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9990
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Frege replaced Cantor's sets as the objects of equinumerosity attributions with concepts [Frege, by Tait]
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7738
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Zero is defined using 'is not self-identical', and one by using the concept of zero [Frege, by Weiner]
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23456
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Frege said logical predication implies classes, which are arithmetical objects [Frege, by Morris,M]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
9956
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'The number of Fs' is the extension (a collection of first-level concepts) of the concept 'equinumerous with F' [Frege, by George/Velleman]
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13527
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Frege's cardinals (equivalences of one-one correspondences) is not permissible in ZFC [Frege, by Wolf,RS]
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22292
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Hume's Principle fails to implicitly define numbers, because of the Julius Caesar [Frege, by Potter]
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17442
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Frege thinks number is fundamentally bound up with one-one correspondence [Frege, by Heck]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
11030
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The words 'There are exactly Julius Caesar moons of Mars' are gibberish [Rumfitt on Frege]
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10030
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'Julius Caesar' isn't a number because numbers inherit properties of 0 and successor [Frege, by George/Velleman]
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8690
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From within logic, how can we tell whether an arbitrary object like Julius Caesar is a number? [Frege, by Friend]
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10219
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Frege said 2 is the extension of all pairs (so Julius Caesar isn't 2, because he's not an extension) [Frege, by Shapiro]
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13889
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Fregean numbers are numbers, and not 'Caesar', because they correlate 1-1 [Frege, by Wright,C]
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18142
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One-one correlations imply normal arithmetic, but don't explain our concept of a number [Frege, by Bostock]
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9046
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Our definition will not tell us whether or not Julius Caesar is a number [Frege]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
16896
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If numbers can be derived from logic, then set theory is superfluous [Frege, by Burge]
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9643
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Set theory may represent all of mathematics, without actually being mathematics [Brown,JR]
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9644
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When graphs are defined set-theoretically, that won't cover unlabelled graphs [Brown,JR]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
9625
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To see a structure in something, we must already have the idea of the structure [Brown,JR]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
8639
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If numbers are supposed to be patterns, each number can have many patterns [Frege]
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9628
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Sets seem basic to mathematics, but they don't suit structuralism [Brown,JR]
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6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
13874
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Numbers seem to be objects because they exactly fit the inference patterns for identities [Frege]
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13875
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Frege's platonism proposes that objects are what singular terms refer to [Frege, by Wright,C]
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7731
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How can numbers be external (one pair of boots is two boots), or subjective (and so relative)? [Frege, by Weiner]
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7737
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Identities refer to objects, so numbers must be objects [Frege, by Weiner]
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8635
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Numbers are not physical, and not ideas - they are objective and non-sensible [Frege]
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8652
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Numbers are objects, because they can take the definite article, and can't be plurals [Frege]
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9580
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Our concepts recognise existing relations, they don't change them [Frege]
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9589
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Numbers are not real like the sea, but (crucially) they are still objective [Frege]
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9606
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The irrationality of root-2 was achieved by intellect, not experience [Brown,JR]
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6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
9831
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Geometry appeals to intuition as the source of its axioms [Frege]
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17816
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Frege's logicism aimed at removing the reliance of arithmetic on intuition [Frege, by Yourgrau]
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
8633
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There is no physical difference between two boots and one pair of boots [Frege]
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9577
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The naïve view of number is that it is like a heap of things, or maybe a property of a heap [Frege]
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9612
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There is an infinity of mathematical objects, so they can't be physical [Brown,JR]
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9610
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Numbers are not abstracted from particulars, because each number is a particular [Brown,JR]
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6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
9951
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It appears that numbers are adjectives, but they don't apply to a single object [Frege, by George/Velleman]
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9952
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Numerical adjectives are of the same second-level type as the existential quantifier [Frege, by George/Velleman]
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11031
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'Jupiter has many moons' won't read as 'The number of Jupiter's moons equals the number many' [Rumfitt on Frege]
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8637
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The number 'one' can't be a property, if any object can be viewed as one or not one [Frege]
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9999
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For science, we can translate adjectival numbers into noun form [Frege]
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9620
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Empiricists base numbers on objects, Platonists base them on properties [Brown,JR]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
7739
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Arithmetic is analytic [Frege, by Weiner]
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9945
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Logicism shows that no empirical truths are needed to justify arithmetic [Frege, by George/Velleman]
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8782
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Frege offered a Platonist version of logicism, committed to cardinal and real numbers [Frege, by Hale/Wright]
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13608
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Mathematics has no special axioms of its own, but follows from principles of logic (with definitions) [Frege, by Bostock]
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5658
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Numbers are definable in terms of mapping items which fall under concepts [Frege, by Scruton]
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16905
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Arithmetic must be based on logic, because of its total generality [Frege, by Jeshion]
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18165
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My Basic Law V is a law of pure logic [Frege]
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8655
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Arithmetic is analytic and a priori, and thus it is part of logic [Frege]
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16880
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Frege aimed to discover the logical foundations which justify arithmetical judgements [Frege, by Burge]
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8689
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Eventually Frege tried to found arithmetic in geometry instead of in logic [Frege, by Friend]
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8487
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Arithmetic is a development of logic, so arithmetical symbolism must expand into logical symbolism [Frege]
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18166
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The loss of my Rule V seems to make foundations for arithmetic impossible [Frege]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
10607
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Frege's logic has a hierarchy of object, property, property-of-property etc. [Frege, by Smith,P]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
10831
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Frege only managed to prove that arithmetic was analytic with a logic that included set-theory [Quine on Frege]
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13864
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Frege's platonism and logicism are in conflict, if logic must dictates an infinity of objects [Wright,C on Frege]
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10033
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Why should the existence of pure logic entail the existence of objects? [George/Velleman on Frege]
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10010
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Frege's belief in logicism and in numerical objects seem uncomfortable together [Hodes on Frege]
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9545
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Late in life Frege abandoned logicism, and saw the source of arithmetic as geometrical [Frege, by Chihara]
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6. Mathematics / C. Sources of Mathematics / 7. Formalism
9629
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For nomalists there are no numbers, only numerals [Brown,JR]
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9639
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Does some mathematics depend entirely on notation? [Brown,JR]
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9631
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Formalism fails to recognise types of symbols, and also meta-games [Frege, by Brown,JR]
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9887
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Formalism misunderstands applications, metatheory, and infinity [Frege, by Dummett]
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8751
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Only applicability raises arithmetic from a game to a science [Frege]
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9630
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The most brilliant formalist was Hilbert [Brown,JR]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
9608
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There are no constructions for many highly desirable results in mathematics [Brown,JR]
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9645
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Constructivists say p has no value, if the value depends on Goldbach's Conjecture [Brown,JR]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
9875
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Frege was completing Bolzano's work, of expelling intuition from number theory and analysis [Frege, by Dummett]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
8642
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Abstraction from things produces concepts, and numbers are in the concepts [Frege]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / e. Psychologism
8621
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Mental states are irrelevant to mathematics, because they are vague and fluctuating [Frege]
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