A Proof of the Pythagorean Theorem
"As for mathematical truths, we should be even less inclined to regard anyone as a geometer who knew Euclid's theorems outwardly by rote, without knowing their proofs, without, as we might say, to point the contrast, kinowing them inwardly. Similarly, if someone became aware, through measuring a number of right-angled triangles, that their sides do, in fact, have the well-known relation to one another, we should consider his [mere] awareness of the fact unsatisfactory.
"Yet, even in mathematical cognition, the essentiality of the proof does not have the significance and nature of being a moment of the result itself; when the latter is reached, the demonstration is over and has disappeared. It is, of course, as a result that the theorem is something seen to be true; but this added circumstance has no bearing on its content, but only on its relation to the knowing Subject.
"The movement of mathematical proof does not belong to the object, but rather is an activity external to the matter in hand. Thus the nature of the right-angled triangle does not divide itself for the proof of the proposition that expresses its ratio. The way and the means by which the result is brought forth belong entirely to the cognitive process.
"In philosophical cognition,too, the way in which the [outer ] existence [insofar as the] existence of a thing comes about, is distinct from the way in which its essence or inner nature comes to be. But to begin with, philosophical cognition includes both [existence and essence], whereas mathematical cognition sets forth only the genesis of the existence, i.e. the being of the nature of the thing in cognition as such.
"What is more, philosophical cognition also unites these two distinct processes. The inner coming-to-be or genesis of substance is an unbroken transition into outer existence, into being-for-another, and conversely, the genesis of existence is how existence is by itself taken back into essence.
"The movement is the two-fold process and the genesis of the whole, in such wise that each side simultaneously posits the other, and each therefore has both perspectives within itself; together they thus constitute the whole by dissolving themselves, and by making themselves into its moments."
["_Hegel's Phenomenology of Spirit_" translated by A.V. Miller, Oxford University Press, 1977, p.24-25]
What?
I get the sense that he is describing an intuitive problem-solving process wherein the mechanics of solving the particular problem are known well enough that the problem "answers itself" inwardly as though it were answered by an outer agent.
Wow! I bet you never thought of math as a spiritual event! Did ya?
sbk
I like the caption at the top of the picture: Pythagoras's theorem. Good ol' Pythagor and his asses. They sure know how to prove a theorem. [Kidding, LOL]
__._,_.___ "As for mathematical truths, we should be even less inclined to regard anyone as a geometer who knew Euclid's theorems outwardly by rote, without knowing their proofs, without, as we might say, to point the contrast, kinowing them inwardly. Similarly, if someone became aware, through measuring a number of right-angled triangles, that their sides do, in fact, have the well-known relation to one another, we should consider his [mere] awareness of the fact unsatisfactory.
"Yet, even in mathematical cognition, the essentiality of the proof does not have the significance and nature of being a moment of the result itself; when the latter is reached, the demonstration is over and has disappeared. It is, of course, as a result that the theorem is something seen to be true; but this added circumstance has no bearing on its content, but only on its relation to the knowing Subject.
"The movement of mathematical proof does not belong to the object, but rather is an activity external to the matter in hand. Thus the nature of the right-angled triangle does not divide itself for the proof of the proposition that expresses its ratio. The way and the means by which the result is brought forth belong entirely to the cognitive process.
"In philosophical cognition,too, the way in which the [outer ] existence [insofar as the] existence of a thing comes about, is distinct from the way in which its essence or inner nature comes to be. But to begin with, philosophical cognition includes both [existence and essence], whereas mathematical cognition sets forth only the genesis of the existence, i.e. the being of the nature of the thing in cognition as such.
"What is more, philosophical cognition also unites these two distinct processes. The inner coming-to-be or genesis of substance is an unbroken transition into outer existence, into being-for-another, and conversely, the genesis of existence is how existence is by itself taken back into essence.
"The movement is the two-fold process and the genesis of the whole, in such wise that each side simultaneously posits the other, and each therefore has both perspectives within itself; together they thus constitute the whole by dissolving themselves, and by making themselves into its moments."
["_Hegel's Phenomenology of Spirit_" translated by A.V. Miller, Oxford University Press, 1977, p.24-25]
What?
I get the sense that he is describing an intuitive problem-solving process wherein the mechanics of solving the particular problem are known well enough that the problem "answers itself" inwardly as though it were answered by an outer agent.
Wow! I bet you never thought of math as a spiritual event! Did ya?
sbk
I like the caption at the top of the picture: Pythagoras's theorem. Good ol' Pythagor and his asses. They sure know how to prove a theorem. [Kidding, LOL]
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