The nature of mathematical demonstration is totally different from all other, and the difference consists in this—that, instead of showing the contrary of the proposition asserted to be only improbable, it proves it at once to be absurd and impossible. This is done by showing that the contrary of the proposition which is asserted is in direct contradiction to some extremely evident fact, of the truth of which our eyes and hands convince us.

The second point to be considered is the method adopted by Spinoza for setting forth his philosophy; it is the demonstrative method of geometry as employed by Euclid, in which we find definitions, explanations, axioms, and theorems. Even Descartes made it his starting-point that philosophic propositions must be mathematically handled and proved, that they must have the very same evidence as mathematics. The mathematical method is considered superior to all others, on account of the nature of its evidence; and it is natural that independent knowledge in its re-awakening lighted first upon this form, of which it saw so brilliant an example. The mathematical method is, however, ill-adapted for speculative content, and finds its proper place only in the finite sciences of the understanding. In modern times Jacobi has asserted (Werke, Vol. IV. Section I. pp. 217-223) that all demonstration, all scientific knowledge leads back to Spinozism, which alone is a logical method of thought; and because it must lead thither, it is really of no service whatever, but immediate knowledge is what we must depend on. It may be conceded to Jacobi that the method of demonstration leads to Spinozism, if we understand thereby merely the method of knowledge belonging to the understanding. But the fact is that Spinoza is made a testing-point in modern philosophy, so that it may really be said: You are either a Spinozist or not a philosopher at all. This being so, the mathematical and demonstrative method of Spinoza would seem to be only a defect in the external form; but it is the fundamental defect of the whole position. In this method the nature of philosophic knowledge and the object thereof, are entirely misconceived, for mathematical knowledge and method are merely formal in character and consequently altogether unsuited for philosophy. Mathematical knowledge exhibits its proof on the existent object as such, not on the object as conceived; the Notion is lacking throughout; the content of Philosophy, however, is simply the Notion and that which is comprehended by the Notion. Therefore this Notion as the knowledge of the essence is simply one assumed, which falls within the philosophic subject; and this is what represents itself to be the method peculiar to Spinoza's philosophy.

The second point to be considered is the method adopted by Spinoza for setting forth his philosophy; it is the demonstrative method of geometry as employed by Euclid, in which we find definitions, explanations, axioms, and theorems. Even Descartes made it his starting-point that philosophic propositions must be mathematically handled and proved, that they must have the very same evidence as mathematics. The mathematical method is considered superior to all others, on account of the nature of its evidence; and it is natural that independent knowledge in its re-awakening lighted first upon this form, of which it saw so brilliant an example. The mathematical method is, however, ill-adapted for speculative content, and finds its proper place only in the finite sciences of the understanding. In modern times Jacobi has asserted (Werke, Vol. IV. Section I. pp. 217-223) that all demonstration, all scientific knowledge leads back to Spinozism, which alone is a logical method of thought; and because it must lead thither, it is really of no service whatever, but immediate knowledge is what we must depend on. It may be conceded to Jacobi that the method of demonstration leads to Spinozism, if we understand thereby merely the method of knowledge belonging to the understanding. But the fact is that Spinoza is made a testing-point in modern philosophy, so that it may really be said: You are either a Spinozist or not a philosopher at all. This being so, the mathematical and demonstrative method of Spinoza would seem to be only a defect in the external form; but it is the fundamental defect of the whole position. In this method the nature of philosophic knowledge and the object thereof, are entirely misconceived, for mathematical knowledge and method are merely formal in character and consequently altogether unsuited for philosophy. Mathematical knowledge exhibits its proof on the existent object as such, not on the object as conceived; the Notion is lacking throughout; the content of Philosophy, however, is simply the Notion and that which is comprehended by the Notion. Therefore this Notion as the knowledge of the essence is simply one assumed, which falls within the philosophic subject; and this is what represents itself to be the method peculiar to Spinoza's philosophy.

I thought fit to... explain in detail in the same book the peculiarity of a certain method, by which it will be possible... to investigate some of the problems in mathematics by means of mechanics. This procedure is... no less useful even for the proof of the theorems themselves; for certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards... But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.

[C]onsider the steps by which the antients were able... from the mensuration of right-lined figures, to judge of such as were bounded by curve lines; for as they did not allow themselves to resolve curvilineal figures into rectilineal elements, it is worth examin[ing] by what art they could make a transition from the one to the other: and as they... finish their demonstrations in the most perfect manner... by following their example... in demonstrating a method so much more general than their's, we may best guard against exceptions and cavils, and vary less from the old foundations of geometry.

The first demonstration (Disq. Arith., Arts. 125-145) which is presented by Gauss in a form very repulsive to any but the most laborious students, has been resumed by Lejeune Dirichlet in a memoir in Crelle's Journal... and has been developed by him with that luminous perspicuity by which his mathematical writings are distinguished.

Rules for Demonstrations. I. Not to undertake to demonstrate any thing that is so evident of itself that nothing can be given that is clearer to prove it. II. To prove all propositions at all obscure, and to employ in their proof only very evident maxims or propositions already admitted or demonstrated. III. To always mentally substitute definitions in the place of things defined, in order not to be misled by the ambiguity of terms which have been restricted by definitions.

There is another point... and that is the method of demonstration by geometrical figures. In the first solution of Isoperimetrical problems, the Bernoullis use diagrams and their properties. Euler, in his early essays, does the same; then, as he improves the calculus he gets rid of constructions. In his Treatise [footnote: Methodus inveniendi, &c.], he introduces geometrical figures, but almost entirely, for the purpose of illustration: and finally, in the tenth volume of the Novi Comm. Petrop. as Lagrange had done in the Miscellanea Taurinensea, he expounds the calculus, in its most refined state, entirely without the aid of diagrams and their properties. A similar history will belong to every other method of calculation, that has been advanced to any degree of perfection.

Experience has convinced me that the proper way of teaching is to bring together that which is simple from all quarters, and, if I may use such a phrase, to draw upon the surface of the subject a proper mean between the line of closest connexion and the line of easiest deduction. This was the method followed by Euclid, who, fortunately for us, never dreamed of a geometry of triangles, as distinguished from a geometry of circles, or a separate application of the arithmetics of addition and subtraction; but made one help out the other as he best could.

The primary method of mathematics is deduction; the primary method of philosophy is descrip- [16] tive generalization. Under the influence of mathematics, deduction has been foisted onto philosophy as its standard method, instead of taking its true place as an essential auxiliary mode of verification whereby to test the scope of generalities. This misapprehension of philosophic method has veiled the very considerable success of philosophy in providing generic notions which add lucidity to our apprehension of the facts of experience. The depositions of Plato, Aristotle, Thomas Aquinas, Descartes, Spinoza, Leibniz,† Locke, Berkeley, Hume, Kant, Hegel, merely mean that ideas which these men introduced into the philosophic tradition must be construed with limitations, adaptations, and inversions, either unknown to them, or even explicitly repudiated by them. A new idea introduces a new alternative; and we are not less indebted to a thinker when we adopt the alternative which he discarded. Philosophy never reverts to its old position after the shock of a great philosopher.

One finds in this subject a kind of demonstration which does not carry with it so high a degree of certainty as that employed in geometry; and which differs distinctly from the method employed by geometers in that they prove their propositions by well-established and incontrovertible principles, while here principles are tested by inferences which are derivable from them. The nature of the subject permits of no other treatment. It is possible, however, in this way to establish a probability which is little short of certainty. This is the case when the consequences of the assumed principles are in perfect accord with the observed phenomena, and especially when these verifications are numerous; but above all when one employs the hypothesis to predict new phenomena and finds his expectations realized.

He considered magnitudes as generated by a or motion, and showed how the velocities of the generating motions were to be compared together. There was nothing in this doctrine but what seemed to be natural and agreeable to the antient geometry. But what he has given us on this subject being very short, his conciseness maybe supposed to have given some occasion to the objections which have been raised against his method.

. . .it is not my design to teach the method that everyone must follow in order to use his reason properly, but only to show the way in which I have tried to use my own.

The reason for this method of Revelation is not far to seek; it is the only way in which one teaching can be made available for minds at different stages of evolution, and thus train not only those to whom it is immediately given, but also those who, later in time, shall have progressed beyond those to whom the Revelation was first made. p. 373