are separated from them and set over against them, as something independent of them, as laws, which apparently come from the outside, in accordance with which the material world must necessarily conduct itself. So, it has happened in society and the state, so, and not otherwise, pure mathematics though borrowed from the world is applied to the world, and though it only shows a portion of its component factors is all the better applicable on that account.
But as Herr Duehring imagines that the whole of pure mathematics can be derived from the mathematical axioms, "which according to purely logical concepts are neither capable of proof nor in need of any, and without empirical ingredients anywhere and that these can be applied to the universe, he likewise imagines, in the first place, the foundation forms of being, the single ingredients of all knowledge, the axioms of philosophy, to be produced by the intellect of man; he imagines also that he can derive the whole of philosophy or plan of the universe from these, and that his sublime genius can compel us to accept this, his conception of nature and humanity." Unfortunately nature and humanity are not constituted like the Prussians of the Manteuffel regime of 1850.
The axioms of mathematics are expressions of the most elementary ideas which mathematics must borrow from logic. They may be reduced to two.
(1) The whole is greater than its part; this statement is mere tautology, since the quantitatively limited concept, "part," necessarily refers to the concept, "whole," – in that "part" signifies no more than that the quantitative "whole" is made up of quantitative "parts." Since the so-called axiom merely asserts this much we are not a step further. This can be shown to be a tautology if we say "The whole is that which consists of several parts – a part is that several of which make up a whole, therefore the part is less than the whole." Where the barrenness of the repetition shows the lack of content all the more strongly.
(2) If two magnitudes are equal to a third they are equal to one another; this statement is, as Hegel has shown, a conclusion, upon the correctness of which all logic depends, and which is demonstrated therefore outside of pure mathematics. The remaining axioms with regard to equality and inequality are merely logical extensions of this conclusion. Such barren statements are not enticing either in mathematics or anywhere else. To proceed we must have realities, conditions and forms taken from real material things; representations of lines, planes, angles, polygons, spheres, etc., are all borrowed from reality, and it is just naive ideology to believe the mathematicians, who assert that the first line was made by causing a point to progress through space, the first plane by means of the movement of a line, and the first solid by revolving a plane, etc. Even speech rebels against this idea. A mathematical figure of three dimensions is called a solid – corpus solidum – and hence, according to the Latin, a body capable of being handled. It has a name derived, therefore, by no means from the independent play of imagination but from solid reality.
But to what purpose is all this prolixity? After Herr Duehring has enthusiastically proclaimed the independence of pure mathematics of the world of experience, their apriorism, their connection with free creation and imagination, he says "it will be readily seen that these mathematical elements (number, magnitude, time, space, geometric progression), are therefore ideal forms with relation to absolute magnitudes and therefore something quite empiric, no matter to what species they belong." But "mathematical general notions are, apart from experience, nevertheless capable of sufficient characterization," which latter proceeds, more or less, from each abstraction, but does not by any means prove that it is not deprived from the actual. In the scheme of the universe of our author pure mathematics originated in pure thought, in his philosophy of nature it is derived from the external world and then set apart from it. What are we then to believe?
"All-comprehending existence is sole. It is sufficient to itself and has nothing above or below it. To associate a second existence with it would be to make it just what it is not, a part of a constituent or all-embracing whole. When we conceive of our idea of soleness as a frame there is nothing which can enter into this, nothing which retains twofoldness can enter into this concept of unity. But nothing can alienate itself from this concept of unity. The essence of all thought consists in uniting the elements of consciousness in a unity. The indivisible concept of the universe has arisen by comprehending everything, and the universe, as the word signifies, is recognised as something in which everything is united into one unity."
So far Herr Duehring is quoted. The mathematical method, "Everything must be decided on simple axiomatic foundation principles, just as if it were concerned with the simple principles of mathematics," this method is for the first time here applied.
"The all-embracing existence is sole." If tautology, simple repetition in the predicate of what has been stated in the subject, if this constitutes an axiom, then we have a splendid specimen. In the subject Herr Duehring tells us that existence comprehends everything, in the predicate he explains intrepidly that there is nothing outside it. What a system-shaping thought. It is indeed system-shaping until we find six lines further down that Herr Duehring has transformed the soleness of being by means of our idea of unity into its one-ness. As the work of all thought consists in the bringing together of all thought into a unity so is existence, as soon as it is conceived, thought of as a unity, an indivisible concept of the universe, and because existence so conceived is the sole universal concept, so is real existence, the real universe, just as much an indivisible unity, and consequently "the beings in the beyond have no further place as soon as the mind has learned to comprehend existence in the homogeneous universality."
That is a campaign with which in comparison Austerlitz and Jena, Koeniggratz and Sedan sink in insignificance. In a couple of expressions after we have set the first axiom moving we have abolished, put away, and destroyed all the inhabitants of the spirit-world, God, the heavenly hierarchies, heaven, hell and purgatory as well as the immortality of the soul.
How do we arrive at the idea of the unity of existence from that of its soleness? As a matter of fact, we generally conceive it. As we spread out our idea of unity as a frame around it the concept of existence becomes the concept of unity, for the existence of all thought consists in the bringing of elements of consciousness into unity.
This last statement is simply false. In the first place thought consists in the decomposition of objects of consciousness into their elements as well as in the uniting of mutually connected elements into a unity. There can be no synthesis without analysis. In the second place, thought can, without error, only bring those elements of consciousness into a unity in which or in the actual prototypes of which this unity already existed beforehand. If I comprehend a shoebrush under the class mammal, it does not thereupon become a milk-giver. The unity of existence is therefore just the thing which had to be proved in order to justify his concept of thought as a unity, and if Herr Duehring assures us that he regards existence as a unity and not as twofold he tells us nothing more than that he himself personally thinks so.
To give a clear explanation of his method of reasoning, it is as follows, "I begin with existence. Therefore I think of existence. The idea of existence is an idea of unity. Thought and existence must therefore belong together, they answer one another, they mutually cover each other. Therefore existence is in reality a unity and there are no beings beyond." But if Herr Duehring had spoken thus plainly instead of entertaining us with oracular statements, the ideology of his argument would have been completely exposed. To attempt to undertake to prove from the identity of thought and existence the reality of the result of thought, that indeed were one of the fever-phantoms of a Hegel.
If his entire method of proof were really correct Herr Duehring would not have gained a single point over the spiritists. The spiritists would curtly reply, "The universe is simple from our standpoint also. The division into the hither and the beyond only exists from our special earthly original sin standpoint. In its essence, that is God, the entire universe is a unity." And they will take Herr Duehring with them to his beloved heavenly bodies, and will show him one or more where no original sin can be found, and where there is therefore no antagonism between the hither and the beyond, and the oneness of the universe is a demand of faith.
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