[1435]. If anyone wished to write in mathematical fashion in metaphysics or ethics, nothing would prevent him from so doing with vigor. Some have professed to do this, and we have a promise of mathematical demonstrations outside of mathematics; but it is very rare that they have been successful. This is, I believe, because they are disgusted with the trouble it is necessary to take for a small number of readers where they would ask as in Persius: Quis leget haec, and reply: Vel duo vel nemo.—Leibnitz.
New Essay concerning Human Understanding, Langley, Bk 2, chap. 29, sect. 12.
[1436]. It is commonly asserted that mathematics and philosophy differ from one another according to their objects, the former treating of quantity, the latter of quality. All this is false. The difference between these sciences cannot depend on their object; for philosophy applies to everything, hence also to quanta, and so does mathematics in part, inasmuch as everything has magnitude. It is only the different kind of rational knowledge or application of reason in mathematics and philosophy which constitutes the specific difference between these two sciences. For philosophy is rational knowledge from mere concepts, mathematics, on the contrary, is rational knowledge from the construction of concepts.
We construct concepts when we represent them in intuition a priori, without experience, or when we represent in intuition the object which corresponds to our concept of it.—The mathematician can never apply his reason to mere concepts, nor the philosopher to the construction of concepts.—In mathematics the reason is employed in concreto, however, the intuition is not empirical, but the object of contemplation is something a priori.
In this, as we see, mathematics has an advantage over philosophy, the knowledge in the former being intuitive, in the latter, on the contrary, only discursive. But the reason why in mathematics we deal more with quantity lies in this, that magnitudes can be constructed in intuition a priori, while qualities, on the contrary, do not permit of being represented in intuition.—Kant, E.
Logik; Werke [Hartenstein], (Leipzig, 1868), Bd. 8, pp.23-24.
[1437]. Kant has divided human ideas into the two categories of quantity and quality, which, if true, would destroy the universality of Mathematics; but Descartes’ fundamental conception of the relation of the concrete to the abstract in Mathematics abolishes this division, and proves that all ideas of quality are reducible to ideas of quantity. He had in view geometrical phenomena only; but his successors have included in this generalization, first, mechanical phenomena, and, more recently, those of heat. There are now no geometers who do not consider it of universal application, and admit that every phenomenon may be as logically capable of being represented by an equation as a curve or a motion, if only we were always capable (which we are very far from being) of first discovering, and then resolving it.
The limitations of Mathematical science are not, then, in its nature. The limitations are in our intelligence: and by these we find the domain of the science remarkably restricted, in proportion as phenomena, in becoming special, become complex.—Comte, A.
Positive Philosophy [Martineau], Bk. 1, chap. 1.