The Twofold Exercise of Reason

There is thus a twofold exercise of reason.

Both modes have the properties of imposition and a confined-to-mind origin.

But procedurally, they very different.

This is because the real world has 2 main elements:

1. The form of intuition (space and time)

This can be cognized and determined completely within-the-mind

2. The matter or content

This is presented in space and time.

Real things have no within-the-mind notions which relate to them, except the undetermined active-knowings of the active-thinking of possible sensations, in so far as these belong (in a possible experience) to the unity of consciousness.

We create the objects of the active-knowings in space and time.

• We regard these objects simply as quanta.

In the one case, reason proceeds according to conceptions and can do nothing more than subject phenomena to these—which can only be determined empirically, that is, a posteriori—in conformity, however, with those conceptions as the rules of all empirical synthesis. In the other case, reason proceeds by the construction of conceptions;

These active-knowings relate to a passive-knowing within-the-mind.

They may be determined purely passively within-the-mind.

Philosophical cognition is the examination of everything in space or time which includes:

• the possibility of its existence
• its reality and necessity or opposites
• whether it is primary or the effect of something else

Mathematical cognition includes:

• determining within-the-mind a thought of shape
• determining, by number, a thought from space and time
• dividing time into periods

Mathematical cognition is an operation of reason through the construction of conceptions.

Pure reason succeeds in the sphere of mathematics. It leads to the expectation that the mathematical method can be applied to other regions of mental endeavour. In this way, it can make itself a master over nature.

On the other hand, pure philosophy, with its within-the-mind discursive active-knowings, bungles around in the world of nature.

• It cannot show any within-the-mind evidence of the reality of these active-knowings.

Masters in the science of mathematics are confident of the success of this method.

• It is a common persuasion that mathematics can be applied to any subject of human thought.

They have hardly ever reflected or philosophized on their favourite science—a task of great difficulty;

The specific difference between the two modes of employing the faculty of reason has never entered their thoughts.

Rules current in the field of common experience, and which common sense stamps everywhere with its approval, are regarded by them as axiomatic.

From what source the conceptions of space and time, with which (as the only primitive quanta) they have to deal, enter their minds, is a question which they do not trouble themselves to answer; and they think it just as unnecessary to examine into the origin of the pure conceptions of the understanding and the extent of their validity.

All they have to do with them is to employ them.

In all this they are perfectly right, if they do not overstep the limits of the sphere of nature.

But they pass, unconsciously, from the world of sense to the insecure ground of pure transcendental conceptions (instabilis tellus, innabilis unda), where they can neither stand nor swim, and where the tracks of their footsteps are obliterated by time; while the march of mathematics is pursued on a broad and magnificent highway, which the latest posterity shall frequent without fear of danger or impediment.

As we have taken upon us the task of determining, clearly and certainly,

the limits of pure reason in the sphere of transcendentalism,

the efforts of reason in this direction are persisted in, even after the plainest and most expressive warnings,

hope still beckoning us past the limits of experience into the splendours of the intellectual world—it becomes necessary to cut away the last anchor of this fallacious and fantastic hope.

The mathematical method is unattended in the sphere of philosophy by the least advantage—except, perhaps, that it more plainly exhibits its own inadequacy

Geometry and philosophy are 2 different things.

They go hand in hand in the field of natural science.

Consequently, that the procedure of the one can never be imitated by the other.

The evidence of mathematics rests on definitions, axioms, and demonstrations.

None of these forms can be employed or imitated in philosophy in the sense in which they are understood by mathematicians.

The geometrician, if he employs his method in philosophy, will succeed only in building card-castles.

The employment of the philosophical method in mathematics can result only mere verbiage.

The essential business of philosophy is to mark out the limits of the science.