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Single Idea 16928

[filed under theme 6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism ]

Full Idea

Mathematical propositions are always judgements a priori, and not empirical, because they carry with them necessity, which cannot be taken from experience.

Gist of Idea

Mathematics cannot be empirical because it is necessary, and that has to be a priori

Source

Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 268)

Book Ref

Kant,Immanuel: 'Prolegomena to Any Future Metaphysic', ed/tr. Lucas,Peter G. [Manchester UP 1971], p.18

A Reaction

Presumably there are necessities in the physical world, and we might discern them by generalising about that world, so that mathematics is (by a tortuous abstract route) a posteriori necessary? Just a thought…

The 27 ideas from 'Prolegomena to Any Future Metaphysic'

16926 Analytic judgements say clearly what was in the concept of the subject [Kant]
16927 Analytic judgement rests on contradiction, since the predicate cannot be denied of the subject [Kant]
16928 Mathematics cannot be empirical because it is necessary, and that has to be a priori [Kant]
16929 7+5 = 12 is not analytic, because no analysis of 7+5 will reveal the concept of 12 [Kant]
16930 Geometry is not analytic, because a line's being 'straight' is a quality [Kant]
16931 Metaphysics is generating a priori knowledge by intuition and concepts, leading to the synthetic [Kant]
16910 Mathematics can only start from an a priori intuition which is not empirical but pure [Kant]
16913 I can't intuit a present thing in itself, because the properties can't enter my representations [Kant]
16911 Intuition is a representation that depends on the presence of the object [Kant]
16912 Some concepts can be made a priori, which are general thoughts of objects, like quantity or cause [Kant]
16916 A priori synthetic knowledge is only of appearances, not of things in themselves [Kant]
16915 A priori intuitions can only concern the objects of our senses [Kant]
16914 A priori intuition of objects is only possible by containing the form of my sensibility [Kant]
16917 All necessary mathematical judgements are based on intuitions of space and time [Kant]
16918 Mathematics cannot proceed just by the analysis of concepts [Kant]
16919 Geometry rests on our intuition of space [Kant]
16920 Numbers are formed by addition of units in time [Kant]
16921 If all empirical sensation of bodies is removed, space and time are still left [Kant]
16922 Space must have three dimensions, because only three lines can meet at right angles [Kant]
16924 I count the primary features of things (as well as the secondary ones) as mere appearances [Kant]
16923 I admit there are bodies outside us [Kant]
21447 I can make no sense of the red experience being similar to the quality in the object [Kant]
16925 Appearance gives truth, as long as it is only used within experience [Kant]
11833 The substance, once the predicates are removed, remains unknown to us [Kant]
21441 'Transcendental' is not beyond experience, but a prerequisite of experience [Kant]
21955 My dogmatic slumber was first interrupted by David Hume [Kant]
21957 'Transcendental' concerns how we know, rather than what we know [Kant]