“Just to reassure him [Badiou], we will not cut off his fingers, for how would he do arithmetic then?”
— Laruelle, Anti-Badiou (via spiritandteeth)
Consider two well-known examples. In the introduction to the Critique, the ‘demonstration’ of the fact that ‘mathematical judgments are all synthetic’ relies heavily on the remark about 7 + 5 = 12, to wit that it is undoubtedly a synthetic judgment, since ‘the concept of twelve is by no means already thought merely by my thinking of that unification of 7 and 5, and no matter how long I analyze my concept of such a possible sum I will still not find 12 in it’. Following which, Kant makes a rather pitiful use of the intuition of the fingers of the hand to pass from 7 to 12. Much further on, he will also say, speaking of the concept of magnitude, that mathematics ‘looks for its consistency and meaning in number, and for the latter in the fingers, the grains of the calculating tablet, or in lines and points placed under the eyes’. This is to imprison the ‘intuition’ of mathematical structures in a naïve empiricism which it has been their obvious aim, ever since the Greeks, to overturn.
Truth be told, Kant’s crucial remark on ‘synthesis’ in 7 + 5 = 12 is totally hollow. We certainly don’t need to wait for Peano’s axiomatization of elementary arithmetic to be persuaded of its vacuity. Almost a century earlier, objecting to some Cartesians who took the judgment ‘two plus two makes four’ as an example of clear and distinct knowledge through intuition (and hence as synthetic knowledge), Leibniz argued for the purely analytical character of this equality, proposing its demonstration ‘through definitions whose possibility is recognized’, a demonstration essentially identical to that of modern logicians. What’s more, this is not the only ‘detail’ on which Kant, believing himself to be demolishing the ‘dogmatism’ of Leibniz’s metaphysics, was surpassed by the formal anticipations harboured by that supposed dogmatism.
-Badiou, Logics of Worlds 236