Demonstrations

3. Demonstrations

Only an apodeictic proof, based on 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 on 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.

Therefore, mathematics alone contains demonstrations.

This is because it does not deduce its ideas from active-knowing, but from the construction of ideas or from passive-knowing which can be given within-the-mind in accordance with active-knowing.

The method of algebra deduced by reduction to get the correct answer.

It is a kind of nom-geometrical construction by symbols.

It uses signs represented by passive-knowing for all its ideas, especially those of the relations of quantities.

Thus, the conclusions in mathematics are secured from errors by every proof being submitted to ocular evidence.*

Philosophical cognition does not possess this advantage.

It must consider the general always in abstract.

Mathematics can always consider it in concrete.

Discursive proofs should be called acroamatic proofs rather than demonstrations because:

• demonstrations always require a reference to the passive-knowing of the object.
• discursive proofs use only words

Therefore, the dogmatical method is not consistent with the nature of abstract philosophy.

Abstract philosophy should not use the titles and insignia of mathematical science.

Its attempts at mathematical evidence are vain pretensions.

The true aim of abstract philosophy is:

• to detect the illusory procedure of reason when transgressing its proper limits
• to fully explain and analyse our conceptions
• to divert us from the dim regions of speculation to the clear region of modest self-knowledge.

In its transcendental endeavours, reason must not look forward with such mathematical confidence.

All apodeictic propositions are either:

1. Dogmata

A dogma is a direct active-thinking proposition, based on conceptions.

2. Mathemata

A mathema is direct active-thinking proposition based on the construction of conceptions.

Passive-thiking judgements do not teach us any more about an object than what was in our conception of it.

This is because they do not extend our cognition beyond our conception of an object.

• They merely elucidate the conception.

They cannot therefore be called dogmas.

Only the propositions employed in philosophy can be called dogmas.

The propositions of arithmetic or geometry cannot be dogmas.*

Thus, pure reason, in the sphere of speculation, does not contain a single direct synthetical judgement based upon conceptions.

By means of ideas, it is, as we have shown, incapable of producing synthetical judgements, which are objectively valid.

By means of the conceptions of the understanding, it establishes certain indubitable principles, not, however, directly on the basis of conceptions, but only indirectly by means of the relation of these conceptions to something of a purely contingent nature, namely, possible experience.

When experience is presupposed, these principles are apodeictically certain, but in themselves, and directly, they cannot even be cognized a priori.

Thus the given conceptions of cause and event will not be sufficient for the demonstration of the proposition: Every event has a cause. For this reason, it is not a dogma; although from another point of view, that of experience, it is capable of being proved to demonstration.

The proper term for such a proposition is principle, and not theorem (although it does require to be proved), because it possesses the remarkable peculiarity of being the condition of the possibility of its own ground of proof, that is, experience, and of forming a necessary presupposition in all empirical observation.

If then, in the speculative sphere of pure reason, no dogmata are to be found; all dogmatical methods, whether borrowed from mathematics, or invented by philosophical thinkers, are alike inappropriate and inefficient. They only serve to conceal errors and fallacies, and to deceive philosophy, whose duty it is to see that reason pursues a safe and straight path. A philosophical method may, however, be systematical. For our reason is, subjectively considered, itself a system, and, in the sphere of mere conceptions, a system of investigation according to principles of unity, the material being supplied by experience alone. But this is not the proper place for discussing the peculiar method of transcendental philosophy, as our present task is simply to examine whether our faculties are capable of erecting an edifice on the basis of pure reason, and how far they may proceed with the materials at their command.