(b) Mathematical definitions cannot be erroneous.For the conception is given only in and through the definition, and thus it contains only what has been cogitated in the definition.But although a definition cannot be incorrect, as regards its content, an error may sometimes, although seldom, creep into the form.This error consists in a want of precision.Thus the common definition of a circle- that it is a curved line, every point in which is equally distant from another point called the centre- is faulty, from the fact that the determination indicated by the word curved is superfluous.For there ought to be a particular theorem, which may be easily proved from the definition, to the effect that every line, which has all its points at equal distances from another point, must be a curved line- that is, that not even the smallest part of it can be straight.Analytical definitions, on the other hand, may be erroneous in many respects, either by the introduction of signs which do not actually exist in the conception, or by wanting in that completeness which forms the essential of a definition.In the latter case, the definition is necessarily defective, because we can never be fully certain of the completeness of our analysis.For these reasons, the method of definition employed in mathematics cannot be imitated in philosophy.
2.Of Axioms.These, in so far as they are immediately certain, are a priori synthetical principles.Now, one conception cannot be connected synthetically and yet immediately with another; because, if we wish to proceed out of and beyond a conception, a third mediating cognition is necessary.And, as philosophy is a cognition of reason by the aid of conceptions alone, there is to be found in it no principle which deserves to be called an axiom.Mathematics, on the other hand, may possess axioms, because it can always connect the predicates of an object a priori, and without any mediating term, by means of the construction of conceptions in intuition.Such is the case with the proposition: Three points can always lie in a plane.
On the other hand, no synthetical principle which is based upon conceptions, can ever be immediately certain (for example, the proposition: Everything that happens has a cause), because I require a mediating term to connect the two conceptions of event and cause-namely, the condition of time-determination in an experience, and Icannot cognize any such principle immediately and from conceptions alone.Discursive principles are, accordingly, very different from intuitive principles or axioms.The former always require deduction, which in the case of the latter may be altogether dispensed with.
Axioms are, for this reason, always self-evident, while philosophical principles, whatever may be the degree of certainty they possess, cannot lay any claim to such a distinction.No synthetical proposition of pure transcendental reason can be so evident, as is often rashly enough declared, as the statement, twice two are four.It is true that in the Analytic I introduced into the list of principles of the pure understanding, certain axioms of intuition; but the principle there discussed was not itself an axiom, but served merely to present the principle of the possibility of axioms in general, while it was really nothing more than a principle based upon conceptions.For it is one part of the duty of transcendental philosophy to establish the possibility of mathematics itself.
Philosophy possesses, then, no axioms, and has no right to impose its a priori principles upon thought, until it has established their authority and validity by a thoroughgoing deduction.
3.Of Demonstrations.Only an apodeictic proof, based upon intuition, can be termed a demonstration.Experience teaches us what is, but it cannot convince us that it might not have been otherwise.
Hence a proof upon empirical grounds cannot be apodeictic.A priori conceptions, in discursive cognition, can never produce intuitive certainty or evidence, however certain the judgement they present may be.Mathematics alone, therefore, contains demonstrations, because it does not deduce its cognition from conceptions, but from the construction of conceptions, that is, from intuition, which can be given a priori in accordance with conceptions.The method of algebra, in equations, from which the correct answer is deduced by reduction, is a kind of construction- not geometrical, but by symbols-in which all conceptions, especially those of the relations of quantities, are represented in intuition by signs; and thus the conclusions in that science are secured from errors by the fact that every proof is submitted to ocular evidence.Philosophical cognition does not possess this advantage, it being required to consider the general always in abstracto (by means of conceptions), while mathematics can always consider it in concreto (in an individual intuition), and at the same time by means of a priori representation, whereby all errors are rendered manifest to the senses.The former- discursive proofs- ought to be termed acroamatic proofs, rather than demonstrations, as only words are employed in them, while demonstrations proper, as the term itself indicates, always require a reference to the intuition of the object.
It follows from all these considerations that it is not consonant with the nature of philosophy, especially in the sphere of pure reason, to employ the dogmatical method, and to adorn itself with the titles and insignia of mathematical science.It does not belong to that order, and can only hope for a fraternal union with that science.
