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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique

[objections to the logicism view of maths]

38 ideas
Kant taught that mathematics is independent of logic, and cannot be grounded in it [Kant, by Hilbert]
If 7+5=12 is analytic, then an infinity of other ways to reach 12 have to be analytic [Kant, by Dancy,J]
Frege's platonism and logicism are in conflict, if logic must dictates an infinity of objects [Wright,C on Frege]
Why should the existence of pure logic entail the existence of objects? [George/Velleman on Frege]
Frege's belief in logicism and in numerical objects seem uncomfortable together [Hodes on Frege]
Frege only managed to prove that arithmetic was analytic with a logic that included set-theory [Quine on Frege]
Late in life Frege abandoned logicism, and saw the source of arithmetic as geometrical [Frege, by Chihara]
Logic already contains some arithmetic, so the two must be developed together [Hilbert]
In 'Principia Mathematica', logic is exceeded in the axioms of infinity and reducibility, and in the domains [Bernays on Russell/Whitehead]
Formalists neglect content, but the logicists have focused on generalizations, and neglected form [Ramsey]
Mathematical abstraction just goes in a different direction from logic [Bernays]
Wittgenstein hated logicism, and described it as a cancerous growth [Wittgenstein, by Monk]
If set theory is not actually a branch of logic, then Frege's derivation of arithmetic would not be from logic [Quine]
Mathematics reduces to set theory (which is a bit vague and unobvious), but not to logic proper [Quine]
Logicists cheerfully accept reference to bound variables and all sorts of abstract entities [Quine]
Logic is dependent on mathematics, not the other way round [Heyting, by Shapiro]
Saying mathematics is logic is merely replacing one undefined term by another [Curry]
Set theory isn't part of logic, and why reduce to something more complex? [Dummett]
The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members [Benacerraf]
Logic is definitional, but real mathematics is axiomatic [Badiou]
Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock]
Many crucial logicist definitions are in fact impredicative [Bostock]
If Hume's Principle is the whole story, that implies structuralism [Bostock]
Analyticity avoids abstract entities, but can there be truth without reference? [Kitcher]
Logicism seemed to fail by Russell's paradox, Gödel's theorems, and non-logical axioms [Wright,C]
The standard objections are Russell's Paradox, non-logical axioms, and Gödel's theorems [Wright,C]
It seems impossible to explain the idea that the conclusion is contained in the premises [Field,H]
Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD]
Are neo-Fregeans 'maximalists' - that everything which can exist does exist? [Hale/Wright]
Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro]
Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro]
Did logicism fail, when Russell added three nonlogical axioms, to save mathematics? [Linsky,B]
For those who abandon logicism, standard set theory is a rival option [Linsky,B]
We can understand cardinality without the idea of one-one correspondence [Heck]
Equinumerosity is not the same concept as one-one correspondence [Heck]
First-order logic captures the inferential relations of numbers, but not the semantics [Hofweber]
Logicism struggles because there is no decent theory of analyticity [Hanna]
It is not easy to show that Hume's Principle is analytic or definitive in the required sense [Jenkins]