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What is the difference between logicism and formalism?

Is there a conflict in being a formalist and logicist?

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There is a big difference, and that is truth.

Formalists tend to look at mathematics as a game played by following rules to manipulate symbols--- that is, they tend to reduce mathematics to its syntax. For them, there is no question as to whether the axioms are true, they are just framework presuppositions of the "mathematics game".

Logicists, on the other hand, tend to think that the axioms are true. In fact, the claim that mathematical axioms are true in virtue of logic alone is the distinctive claim of Logicism.

(DISCLAIMER: There are subtler varieties of each view that may not be adequately captured by the above statements. This answer deals merely with the views in their most generic senses.)

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    It seems to me that logicists treat axioms as statements with meaning and not just a string with characters. Am I right about that ? – Amr Mar 26 '13 at 14:02
  • If this is true, then what is the difference between intuitionism and logicism ? – Amr Mar 26 '13 at 14:03
  • @Amr Yes to your first question. The difference between intuitionism and logicism is (at least) two-fold. Intuitionists typically don't take the axioms to be justified on the basis of logic alone (rather, on some forms of intuitionism they are given to us by (mathematical?) intuition). In addition they tend to understand "p is true" as "there is a proof of p", so they collapse truth into proof. As a result, they think that certain statements (the unproven ones like Goldbach's Conjecture) are neither true nor false. They would say something similar about all statements which are undecidable. – Dennis Mar 26 '13 at 14:14
  • @Amr As the last sentence of my previous comment might suggest, intuitionists reject the law of excluded middle (p or not-p), since there are some instances of that they will say are not true. Also, intuitionists tend to regard mathematics as an essentially mental (perhaps even "creative"/"inventive") activity and might take mathematical objects to be mind-dependent (this is true of old-school Brouwerian intuitionism, at least). – Dennis Mar 26 '13 at 14:16
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    @Amr This article does a good job of explaining the difference between these schools in a broad and accessible way. In general, I recommend the Stanford Encyclopedia of Philosophy as a first stop for these sorts of questions. The articles are all commissioned and written by experts so there is a bit more clarity and reliability than Wikipedia (generally). – Dennis Mar 26 '13 at 14:23
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I suspect that Formalism was inspired by the turn towards language inspired by Wittgenstein, and also by certain movements in mathematics; specifically Hilberts programme to formalise mathematics, in fact that is to reduce it to logic. Also the invention of model theory which allowed mathematicians to examine their own discipline through the mathematical microscope was further inspiration to that philosophy. (That is by taking as axiomatic that a theory is simply a set of self-consistent axioms with classical logic - that is a grammar).

In this sense formalism and logicism are closely implicated in each other.

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