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Single Idea 13864
[filed under theme 6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
]
Full Idea
Frege's platonism seems to be in some tension with logicism: for the thought is unprepossessing that logic should dictate the existence of infinitely many objects of some kind.
Gist of Idea
Frege's platonism and logicism are in conflict, if logic must dictates an infinity of objects
Source
comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Crispin Wright - Frege's Concept of Numbers as Objects Intro
Book Ref
Wright,Crispin: 'Frege's Conception of Numbers' [Scots Philosophical Monographs 1983], p.-1
A Reaction
Obviously Frege didn't think this, but then the crux seems to be that Frege believed that there was a multitude of logical truths awaiting discovery, while modern logic just seems to be about the logical consequences of things.
Related Ideas
Idea 13234
The view of logic as knowing a body of truths looks out-of-date [Beall/Restall]
Idea 10010
Frege's belief in logicism and in numerical objects seem uncomfortable together [Hodes on Frege]
The
40 ideas
with the same theme
[objections to the logicism view of maths]:
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12458
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Kant taught that mathematics is independent of logic, and cannot be grounded in it
[Kant, by Hilbert]
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2795
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If 7+5=12 is analytic, then an infinity of other ways to reach 12 have to be analytic
[Kant, by Dancy,J]
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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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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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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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17698
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Logic already contains some arithmetic, so the two must be developed together
[Hilbert]
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10305
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In 'Principia Mathematica', logic is exceeded in the axioms of infinity and reducibility, and in the domains
[Bernays on Russell/Whitehead]
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13426
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Formalists neglect content, but the logicists have focused on generalizations, and neglected form
[Ramsey]
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10306
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Mathematical abstraction just goes in a different direction from logic
[Bernays]
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6849
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Wittgenstein hated logicism, and described it as a cancerous growth
[Wittgenstein, by Monk]
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23509
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The logic of the world is shown by tautologies in logic, and by equations in mathematics
[Wittgenstein]
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9004
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If set theory is not actually a branch of logic, then Frege's derivation of arithmetic would not be from logic
[Quine]
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1635
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Mathematics reduces to set theory (which is a bit vague and unobvious), but not to logic proper
[Quine]
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1613
|
Logicists cheerfully accept reference to bound variables and all sorts of abstract entities
[Quine]
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8754
|
Logic is dependent on mathematics, not the other way round
[Heyting, by Shapiro]
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17808
|
Saying mathematics is logic is merely replacing one undefined term by another
[Curry]
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9876
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Set theory isn't part of logic, and why reduce to something more complex?
[Dummett]
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9904
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The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members
[Benacerraf]
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12331
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Logic is definitional, but real mathematics is axiomatic
[Badiou]
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18111
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Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality
[Bostock]
|
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18129
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Many crucial logicist definitions are in fact impredicative
[Bostock]
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18146
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If Hume's Principle is the whole story, that implies structuralism
[Bostock]
|
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12423
|
Analyticity avoids abstract entities, but can there be truth without reference?
[Kitcher]
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13863
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Logicism seemed to fail by Russell's paradox, Gödel's theorems, and non-logical axioms
[Wright,C]
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13895
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The standard objections are Russell's Paradox, non-logical axioms, and Gödel's theorems
[Wright,C]
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18215
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It seems impossible to explain the idea that the conclusion is contained in the premises
[Field,H]
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13471
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Mathematics makes existence claims, but philosophers usually say those are never analytic
[Hart,WD]
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12224
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Are neo-Fregeans 'maximalists' - that everything which can exist does exist?
[Hale/Wright]
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13625
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Mathematics and logic have no border, and logic must involve mathematics and its ontology
[Shapiro]
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8744
|
Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own
[Shapiro]
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21713
|
Did logicism fail, when Russell added three nonlogical axioms, to save mathematics?
[Linsky,B]
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21715
|
For those who abandon logicism, standard set theory is a rival option
[Linsky,B]
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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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17458
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Equinumerosity is not the same concept as one-one correspondence
[Heck]
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10006
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First-order logic captures the inferential relations of numbers, but not the semantics
[Hofweber]
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23441
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Logical truth is true in all models, so mathematical objects can't be purely logical
[Linnebo]
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11063
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Logicism struggles because there is no decent theory of analyticity
[Hanna]
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17724
|
It is not easy to show that Hume's Principle is analytic or definitive in the required sense
[Jenkins]
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