The Synthetic a Priori
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Empiricist philosophies of mathematics - conventionalism (formalism)
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If we look a the pairs of terms synthetic/analytic and a priori/a posteriori we see that Kant's interpretation of mathematics can be represented diagrammatically as follows. ... The overlap of he synthetic with the a priori indicates that he asserts that there there are judgments (statements) that are synthetic a priori. As already indicated his main example are statements from mathematics, but he would also regard logical laws and certain fundamental principles of science as synthetic a priori. The moral law is also, in his opinion, synthetic a priori. By means of his distinctions between the a priori and a posteriori, and between the analytic and synthetic, Kant denies other philosophies of mathematics that would be consistent and compatible with empiricism. Conventionalism: This is also called formalism. In Kantian terms this is the view that mathematics is analytical a priori. In other words, that all mathematical statements are true by definition or convention. Diagrammatically, this is equivalent to the following ... In other words, conventionalists deny that two pairs of concepts are necessary. Whatever is analytic is a priori; whatever is synthetic is a posteriori. This philosophy of mathematics is quite popular with mathematicians, who feel that the systems of axioms they set up, from which they deduce their theorems, are often quite arbitrary and definitions only. The famous mathematician Hilbert was a formalist. However, not all mathematicians adopt this point-of-view. We will examine the philosophical defence of conventionalism proposed by A.J. Ayer below.
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Contents of
The Synthetic a Priori
1 Empiricism, Platonism, Innate Ideas and the A Priori
2 Analytic a priori
3 Kant and the synthetic a priori
4 Compound (molecular) and atomic sentences
5 Logically atomic sentences and the philosophy of logical atomism
6 Complex sentences and attitudes
7 Subject and predicate, individual and property
8 Synthetic and analytic, definitions offered by Kant
9 A priori and a posteriori
10 The synthetic a priori in Kant - the Critique of Pure Reason
11 Kant, The Critique of Pure Reason, the self and transcendental apperception
12 Empiricist philosophies of mathematics - conventionalism (formalism)
13 Empiricist philosophies of mathematics - the empiricism of J.S. Mill
14 Hybrid empiricist philosophies of mathematics
15 Empiricist philosophies of mathematics - Wittgenstein and non-cognitivism
16 A.J. Ayer and conventionalism - his reply to Kant
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