The Synthetic a Priori
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The synthetic a priori in Kant - the Critique of Pure Reason
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Kant breaks down his argument that mathematical knowledge is not derived from experience into two parts. First argument. Kant calls knowledge of the meanings of our language, expressed in definitions, analytic knowledge. In contrast, knowledge that is not based on definitions and meanings is synthetic knowledge. He argues that mathematical knowledge could not be true by definition and hence must be synthetic knowledge. Mathematical knowledge conveys real information about the world. He argues that mathematical knowledge is synthetic in the following passage. "Indeed, we might think that the proposition '7 + 5 = 12' is merely an analytic proposition, and that it follows by means of the principle of contradiction from the concept of the sum of 7 and 5. But on closer examination we discover that the concept of the sum of 7 and 5 does not contain anything except the idea of bringing two numbers together into one, and in this concept there is no mention of what that single number might be that is the result of the combination of both. It is by no means true that the concept of 12 is already given in the thought of the union of 7 and 5; I may analyse the concept of such a possible sum as much as I like, but I shall never find that it contains the number 12. We have to employ [mathematical] intuition to lead us outside the concept [of the union of 5 and 7] to find it…. The propositions of arithmetic are consequently always synthetic. This is even more obvious in the case of larger numbers. Then it is clear that, however much we may examine our concepts, we could never discover by analysis alone, and without the aid of [mathematical] intuition, what the number is that is the sum [of two numbers]" Second Argument" Mathematical knowledge could not be a hypothesis of science because no hypothesis of science can ever have the degree of certainty that mathematical knowledge has. All scientific knowledge is subject to the scepticism expressed by the paradox of induction, but the fundamental propositions of arithmetic and geometry are not. When knowledge is derived from particular experience, either directly or through generalization, Kant calls it a posteriori knowledge. Knowledge which is not derived from particular experience is what Kant called a priori knowledge. Kant writes, "But even if all our knowledge is awoken by experience, it does not follow that it is all based on experience. It is possible that even what we deem to be empirical knowledge is composed of both information that we receive through the senses, and information provided by our own understanding, which sense experience serves as the occasion to awaken. If our understanding [faculty of knowledge] did make such an addition, it is possible that we would not be able at first to differentiate it from the rest of the material, and would require a lot of practice in our to acquire the skill of separating it." He concludes about mathematical knowledge. "It has to be noted that all strictly mathematical propositions are always a priori judgements, and not empirical ones. This is because they contain within them a necessity that could not be derived from experience." This argument poses a fundamental challenge to empiricism. If it is true, then mathematical knowledge is substantial knowledge about the real world that is not derived from sense-experience. We must therefore have some means of discovering things about the real world that is independent of our five external senses. We know that '7 + 5 = 12' but we could not know this through sense-experience.
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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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