The Discipline of Pure Reason in Dogmatism

SECTION I. The Discipline of Pure Reason in the Sphere of Dogmatism

The science of mathematics is the most brilliant example of the extension of pure reason without the aid of experience.

Pure reason hopes to extend its empire in the transcendental sphere with equal success and security, especially when it applies the same method which was attended with such brilliant results in the science of mathematics.

Demonstrative certainty is called mathematical.

Philosophy to mathematics is dogmatical.

Is the method of arriving at mathematical certainty identical with philosophical certainty?

Philosophical cognition is the cognition of reason through ideas.

Mathematical cognition is cognition through the construction of ideas.

The construction of an idea is the within-the-mind presentation of the intuition which corresponds to the idea.

For this purpose, a non-empirical intuition is needed. As an intuition it is an individual object.

The construction of an idea (as a general representation) should be universally valid for all the possible intuitions of that idea.

I draw a triangle either:

• in my mind, in pure intuition or
• on paper, in empirical intuition

In both cases, these are completely within-the-mind [since even the drawing on paper is from the mind].

Philosophical cognition regards the particular only in the general.

Mathematical cogntion regards the general in the individual.

The essential difference of these 2 modes of cognition is in this formal quality it does not regard the difference of the matter or objects of both.

Some thinkers distinguish philosophy from mathematics by asserting that:

• philosophy has to do only with quality
• mathematics hs to do only with quantity

They have mistaken the effect for the cause.

Mathematical cognition relates only to quantity is only because of its form.

• This is because mathematical cognition can only be constructed through the idea of numbers

But qualities need an empirical intuition.

Hence the cognition of qualities by reason is possible only through ideas.

No one can find an intuition which shall correspond to the conception of reality, except in experience.

It cannot be presented to the mind a priori and antecedently to the empirical consciousness of a reality.

We can form an intuition, by means of the mere conception of it, of a cone, without the aid of experience; but the colour of the cone we cannot know except from experience. I cannot present an intuition of a cause, except in an example which experience offers to me. Besides, philosophy, as well as mathematics, treats of quantities; as, for example, of totality, infinity, and so on. Mathematics, too, treats of the difference of lines and surfaces—as spaces of different quality, of the continuity of extension—as a quality thereof.

But, although in such cases they have a common object, the mode in which reason considers that object is very different in philosophy from what it is in mathematics. The former confines itself to the general conceptions; the latter can do nothing with a mere conception, it hastens to intuition.

In this intuition it regards the conception in concreto, not empirically, but in an a priori intuition, which it has constructed; and in which, all the results which follow from the general conditions of the construction of the conception are in all cases valid for the object of the constructed conception.

Let us give the idea a triangle to a philosopher. We make him discover, by the philosophical method, what relation the sum of its angles bears to a right angle.

He may analyse the idea of a right line, of an angle, or of the number 3 as long as he pleases. But he will not discover any properties not contained in these ideas.*

But, if this question is asked to a geometrician, he at once begins by constructing a triangle.

He knows that 2 right angles are equal to the sum of all the contiguous angles which proceed from one point in a straight line.

He goes on to produce one side of his triangle, thus forming two adjacent angles which are together equal to two right angles.

He then divides the exterior of these angles, by drawing a line parallel with the opposite side of the triangle, and immediately perceives that he has thus got an exterior adjacent angle which is equal to the interior.

Proceeding in this way, through a chain of inferences, and always on the ground of intuition, he arrives at a clear and universally valid solution of the question.

But mathematics does not confine itself to the construction of quantities (quanta), as in the case of geometry; it occupies itself with pure quantity also (quantitas), as in the case of algebra, where complete abstraction is made of the properties of the object indicated by the conception of quantity.

In algebra, a certain method of notation by signs is adopted, and these indicate the different possible constructions of quantities, the extraction of roots, and so on. After having thus denoted the general conception of quantities, according to their different relations, the different operations by which quantity or number is increased or diminished are presented in intuition in accordance with general rules.

Thus, when one quantity is to be divided by another, the signs which denote both are placed in the form peculiar to the operation of division; and thus algebra, by means of a symbolical construction of quantity, just as geometry, with its ostensive or geometrical construction (a construction of the objects themselves), arrives at results which discursive cognition cannot hope to reach by the aid of mere conceptions.

What is the cause of this difference in the fortune of the philosopher and the mathematician?

The philosoper follows the path of ideas.

The mathematician folows that of intuitions which he represents within-the-mind in correspondence with his ideas?

I do not want to discover passive-thinking propositions which may be produced merely by analysing our ideas.

• In this, the philosopher would have the advantage over his rival mathematician.

I want to discover active-thinking propositions that can be created within-the-mind.

I must not confine myself to that which I actually cogitate in my conception of a triangle, for this is nothing more than the mere definition;

I must try to go beyond that, and to arrive at properties which are not contained in, although they belong to, the conception.

This is impossible, unless I determine the object present to my mind according to the conditions, either of empirical, or of pure, intuition.

In the former case, I should have an empirical proposition (arrived at by actual measurement of the angles of the triangle), which would possess neither universality nor necessity; but that would be of no value. In the latter, I proceed by geometrical construction, by means of which I collect, in a pure intuition, just as I would in an empirical intuition, all the various properties which belong to the schema of a triangle in general, and consequently to its conception, and thus construct synthetical propositions which possess the attribute of universality.

It would be vain to philosophize upon the triangle, that is, to reflect on it discursively; I should get no further than the definition with which I had been obliged to set out. There are certainly transcendental synthetical propositions which are framed by means of pure conceptions, and which form the peculiar distinction of philosophy; but these do not relate to any particular thing, but to a thing in general, and enounce the conditions under which the perception of it may become a part of possible experience. But the science of mathematics has nothing to do with such questions, nor with the question of existence in any fashion; it is concerned merely with the properties of objects in themselves, only in so far as these are connected with the conception of the objects.

In the above example, we merely attempted to show the great difference which exists between the discursive employment of reason in the sphere of conceptions, and its intuitive exercise by means of the construction of conceptions. The question naturally arises: What is the cause which necessitates this twofold exercise of reason, and how are we to discover whether it is the philosophical or the mathematical method which reason is pursuing in an argument?

All our knowledge relates, finally, to possible intuitions, for it is these alone that present objects to the mind. An a priori or non-empirical conception contains either a pure intuition—and in this case it can be constructed; or it contains nothing but the synthesis of possible intuitions, which are not given a priori. In this latter case, it may help us to form synthetical a priori judgements, but only in the discursive method, by conceptions, not in the intuitive, by means of the construction of conceptions.