Kant-- Mathematics as A Set of A Priori Synthetic Propositions

[JamesDillon]

I've been recently reading Kant's Critique of Pure Reason, and some commentaries on the work including Bertrand Russell's The Problems of Philosophy. I'm a bit confused by Kant's assertion (which seems to be generally accepted, unless I've been reading the wrong commentaries) that mathematics is a system of a priori synthetic, as opposed to analytic, propositions. Kant's example is the mathematical statement 7+5=12; he argues that because the idea of "12" is not inherent in the ideas of "7" and "5", the statement is synthetic rather than analytic.

I'm probably wrong, but it seems to me that Kant is overlooking an obvious objection here. Consider the equivalent statement "12=7+5." Now consider the fact that "12" is symbolic shorthand for a more primitive idea that can be represented as "............". It seems to me that the idea of "(.......) + (.....)" is inherent in the idea "............", so that the statement "12=7+5" really is analytic. Am I missing something here?

Thanks,
James

[UndercoverElephant]

I've been recently reading Kant's Critique of Pure Reason, and some commentaries on the work including Bertrand Russell's The Problems of Philosophy. I'm a bit confused by Kant's assertion (which seems to be generally accepted, unless I've been reading the wrong commentaries) that mathematics is a system of a priori synthetic, as opposed to analytic, propositions. Kant's example is the mathematical statement 7+5=12; he argues that because the idea of "12" is not inherent in the ideas of "7" and "5", the statement is synthetic rather than analytic.


This is not generally accepted. It is generally rejected.


I'm probably wrong, but it seems to me that Kant is overlooking an obvious objection here. Consider the equivalent statement "12=7+5." Now consider the fact that "12" is symbolic shorthand for a more primitive idea that can be represented as "............". It seems to me that the idea of "(.......) + (.....)" is inherent in the idea "............", so that the statement "12=7+5" really is analytic. Am I missing something here?

Thanks,
James

You aren't missing anything. Mathematics isn't synthetic. Most people disagree with Kant on this one.

[Robin]

I've been recently reading Kant's Critique of Pure Reason, and some commentaries on the work including Bertrand Russell's The Problems of Philosophy. I'm a bit confused by Kant's assertion (which seems to be generally accepted, unless I've been reading the wrong commentaries) that mathematics is a system of a priori synthetic, as opposed to analytic, propositions. Kant's example is the mathematical statement 7+5=12; he argues that because the idea of "12" is not inherent in the ideas of "7" and "5", the statement is synthetic rather than analytic.

I'm probably wrong, but it seems to me that Kant is overlooking an obvious objection here. Consider the equivalent statement "12=7+5." Now consider the fact that "12" is symbolic shorthand for a more primitive idea that can be represented as "............". It seems to me that the idea of "(.......) + (.....)" is inherent in the idea "............", so that the statement "12=7+5" really is analytic. Am I missing something here?

Thanks,
James
I'm with you. And very nicely put by the way.