Phenomenology of Spirit, Preface, paragraph 42

As to mathematical truths, we should be less inclined to consider anyone a geometer who knew Euclid’s theorems outwardly without knowing the proofs, without, if we may say so by way of contrast, knowing them inwardly. Similarly, we should not regard the knowledge acquired by measuring many right triangles, that their sides are related in the way everybody knows, as being satisfactory. All the same, the essentiality of the proof even with mathematical knowledge still does not have the significance and nature of being a moment of the result itself; rather, in this it is past and gone. As far as the result is concerned, the theorem is, no doubt, viewed as being true. But this additional circumstance does not concern its content, but only its relation to the [thinking] subject. The movement of mathematical proof does not pertain to that which is object; it is a deed external to the matter at hand. Thus, the nature of a right triangle is not broken up into factors in the manner set forth in the mathematical construction which is required to prove the proposition expressing the relation of its parts. The entire generation of the result is a procedure [Gang] and medium of knowledge. In philosophical knowledge, too, the becoming of existence as existence is distinct from the becoming of essence or the inner nature of the matter. But philosophical knowledge, for one thing, contains both of these, while mathematical knowledge only sets forth the becoming of existence, i.e., it portrays in knowledge the being of the nature of the case. For another thing, philosophical knowledge unites both these particular movements. The inward origin or the becoming of substance is an unsevered transition into outwardness or into existence, being for another; and conversely, the becoming of existence is the taking-back into the essence. The movement is thus the twofold process and becoming of the whole, such that each at the same time posits the other, and each on that account has in it both as its two aspects. Together they make the whole, through their dissolving each other, and making themselves into moments of the whole.