Immanuel Kant – The Critique of Pure Reason; Part 26

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TRANSCENDENTAL DOCTRINE OF METHOD

If we regard the sum of the cognition of pure speculative reason as an
Edifice, the idea of which, at least, exists in the human mind, it may
Be said that we have in the Transcendental Doctrine of Elements examined
The materials and determined to what edifice these belong, and what
Its height and stability. We have found, indeed, that, although we had
Purposed to build for ourselves a tower which should reach to Heaven
The supply of materials sufficed merely for a habitation, which was
Spacious enough for all terrestrial purposes, and high enough to
Enable us to survey the level plain of experience, but that the bold
Undertaking designed necessarily failed for want of materials--not to
Mention the confusion of tongues, which gave rise to endless disputes
Among the labourers on the plan of the edifice, and at last scattered
Them over all the world, each to erect a separate building for himself
According to his own plans and his own inclinations. Our present task
Relates not to the materials, but to the plan of an edifice; and, as we
Have had sufficient warning not to venture blindly upon a design which
May be found to transcend our natural powers, while, at the same time
We cannot give up the intention of erecting a secure abode for the mind
We must proportion our design to the material which is presented to us
And which is, at the same time, sufficient for all our wants

I understand, then, by the transcendental doctrine of method, the
Determination of the formal conditions of a complete system of pure
Reason. We shall accordingly have to treat of the discipline, the canon
The architectonic, and, finally, the history of pure reason. This part
Of our Critique will accomplish, from the transcendental point of view
What has been usually attempted, but miserably executed, under the name
Of practical logic. It has been badly executed, I say, because general
Logic, not being limited to any particular kind of cognition (not
Even to the pure cognition of the understanding) nor to any particular
Objects, it cannot, without borrowing from other sciences, do more than
Present merely the titles or signs of possible methods and the technical
Expressions, which are employed in the systematic parts of all sciences;
And thus the pupil is made acquainted with names, the meaning and
Application of which he is to learn only at some future time

CHAPTER I. The Discipline of Pure Reason

Negative judgements--those which are so not merely as regards their
Logical form, but in respect of their content--are not commonly held in
Especial respect. They are, on the contrary, regarded as jealous enemies
Of our insatiable desire for knowledge; and it almost requires an
Apology to induce us to tolerate, much less to prize and to respect
Them

All propositions, indeed, may be logically expressed in a negative form;
But, in relation to the content of our cognition, the peculiar province
Of negative judgements is solely to prevent error. For this reason, too
Negative propositions, which are framed for the purpose of correcting
False cognitions where error is absolutely impossible, are undoubtedly
True, but inane and senseless; that is, they are in reality purposeless
And, for this reason, often very ridiculous. Such is the proposition
Of the schoolman that Alexander could not have subdued any countries
Without an army

But where the limits of our possible cognition are very much contracted
The attraction to new fields of knowledge great, the illusions to
Which the mind is subject of the most deceptive character, and the
Evil consequences of error of no inconsiderable magnitude--the negative
Element in knowledge, which is useful only to guard us against error
Is of far more importance than much of that positive instruction which
Makes additions to the sum of our knowledge. The restraint which is
Employed to repress, and finally to extirpate the constant inclination
To depart from certain rules, is termed discipline. It is distinguished
From culture, which aims at the formation of a certain degree of skill
Without attempting to repress or to destroy any other mental power
Already existing. In the cultivation of a talent, which has given
Evidence of an impulse towards self-development, discipline takes a
Negative, culture and doctrine a positive, part

That natural dispositions and talents (such as imagination and wit)
Which ask a free and unlimited development, require in many respects the
Corrective influence of discipline, every one will readily grant. But
It may well appear strange that reason, whose proper duty it is to
Prescribe rules of discipline to all the other powers of the mind
Should itself require this corrective. It has, in fact, hitherto
Escaped this humiliation, only because, in presence of its magnificent
Pretensions and high position, no one could readily suspect it to be
Capable of substituting fancies for conceptions, and words for things

Reason, when employed in the field of experience, does not stand in need
Of criticism, because its principles are subjected to the continual test
Of empirical observations. Nor is criticism requisite in the sphere of
Mathematics, where the conceptions of reason must always be presented
In concreto in pure intuition, and baseless or arbitrary assertions are
Discovered without difficulty. But where reason is not held in a plain
Track by the influence of empirical or of pure intuition, that is, when
It is employed in the transcendental sphere of pure conceptions, it
Stands in great need of discipline, to restrain its propensity to
Overstep the limits of possible experience and to keep it from wandering
Into error. In fact, the utility of the philosophy of pure reason is
Entirely of this negative character. Particular errors may be corrected
By particular animadversions, and the causes of these errors may be
Eradicated by criticism. But where we find, as in the case of pure
Reason, a complete system of illusions and fallacies, closely connected
With each other and depending upon grand general principles, there
Seems to be required a peculiar and negative code of mental legislation
Which, under the denomination of a discipline, and founded upon the
Nature of reason and the objects of its exercise, shall constitute a
System of thorough examination and testing, which no fallacy will be
Able to withstand or escape from, under whatever disguise or concealment
It may lurk

But the reader must remark that, in this the second division of our
Transcendental Critique the discipline of pure reason is not directed
To the content, but to the method of the cognition of pure reason. The
Former task has been completed in the doctrine of elements. But there
Is so much similarity in the mode of employing the faculty of reason
Whatever be the object to which it is applied, while, at the same time
Its employment in the transcendental sphere is so essentially different
In kind from every other, that, without the warning negative influence
Of a discipline specially directed to that end, the errors are
Unavoidable which spring from the unskillful employment of the methods
Which are originated by reason but which are out of place in this
Sphere

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

The science of mathematics presents the most brilliant example of the
Extension of the sphere of pure reason without the aid of experience
Examples are always contagious; and they exert an especial influence on
The same faculty, which naturally flatters itself that it will have
The same good fortune in other case as fell to its lot in one fortunate
Instance. Hence pure reason hopes to be able 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. It is, therefore, of the
Highest importance for us to know whether the method of arriving at
Demonstrative certainty, which is termed mathematical, be identical with
That by which we endeavour to attain the same degree of certainty in
Philosophy, and which is termed in that science dogmatical

Philosophical cognition is the cognition of reason by means of
Conceptions; mathematical cognition is cognition by means of the
Construction of conceptions. The construction of a conception is
The presentation a priori of the intuition which corresponds to the
Conception. For this purpose a non-empirical intuition is requisite
Which, as an intuition, is an individual object; while, as the
Construction of a conception (a general representation), it must be seen
To be universally valid for all the possible intuitions which rank under
That conception. Thus I construct a triangle, by the presentation of the
Object which corresponds to this conception, either by mere imagination
In pure intuition, or upon paper, in empirical intuition, in both cases
Completely a priori, without borrowing the type of that figure from any
Experience. The individual figure drawn upon paper is empirical; but
It serves, notwithstanding, to indicate the conception, even in its
Universality, because in this empirical intuition we keep our eye merely
On the act of the construction of the conception, and pay no attention
To the various modes of determining it, for example, its size, the
Length of its sides, the size of its angles, these not in the least
Affecting the essential character of the conception

Philosophical cognition, accordingly, regards the particular only in
The general; mathematical the general in the particular, nay, in the
Individual. This is done, however, entirely a priori and by means of
Pure reason, so that, as this individual figure is determined under
Certain universal conditions of construction, the object of the
Conception, to which this individual figure corresponds as its schema
Must be cogitated as universally determined

The essential difference of these two modes of cognition consists
Therefore, in this formal quality; it does not regard the difference of
The matter or objects of both. Those thinkers who aim at distinguishing
Philosophy from mathematics by asserting that the former has to do with
Quality merely, and the latter with quantity, have mistaken the effect
For the cause. The reason why mathematical cognition can relate only to
Quantity is to be found in its form alone. For it is the conception of
Quantities only that is capable of being constructed, that is, presented
A priori in intuition; while qualities cannot be given in any other than
An empirical intuition. Hence the cognition of qualities by reason is
Possible only through conceptions. 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

Suppose that the conception of a triangle is given to a philosopher
And that he is required to discover, by the philosophical method, what
Relation the sum of its angles bears to a right angle. He has nothing
Before him but the conception of a figure enclosed within three right
Lines, and, consequently, with the same number of angles. He may analyse
The conception of a right line, of an angle, or of the number three
As long as he pleases, but he will not discover any properties not
Contained in these conceptions. But, if this question is proposed to
A geometrician, he at once begins by constructing a triangle. He knows
That two right angles are equal to the sum of all the contiguous angles
Which proceed from one point in a straight line; and 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

Now, what is the cause of this difference in the fortune of the
Philosopher and the mathematician, the former of whom follows the path
Of conceptions, while the latter pursues that of intuitions, which he
Represents, a priori, in correspondence with his conceptions? The cause
Is evident from what has been already demonstrated in the introduction
To this Critique. We do not, in the present case, want to discover
Analytical propositions, which may be produced merely by analysing our
Conceptions--for in this the philosopher would have the advantage over
His rival; we aim at the discovery of synthetical propositions--such
Synthetical propositions, moreover, as can be cognized a priori. 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. Now, 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

The only a priori intuition is that of the pure form of phenomena--space
And time. A conception of space and time as quanta may be presented
A priori in intuition, that is, constructed, either alone with their
Quality (figure), or as pure quantity (the mere synthesis of the
Homogeneous), by means of number. But the matter of phenomena, by which
Things are given in space and time, can be presented only in perception
A posteriori. The only conception which represents a priori this
Empirical content of phenomena is the conception of a thing in general;
And the a priori synthetical cognition of this conception can give
Us nothing more than the rule for the synthesis of that which may be
Contained in the corresponding a posteriori perception; it is utterly
Inadequate to present an a priori intuition of the real object, which
Must necessarily be empirical

Synthetical propositions, which relate to things in general, an a priori
Intuition of which is impossible, are transcendental. For this
Reason transcendental propositions cannot be framed by means of the
Construction of conceptions; they are a priori, and based entirely on
Conceptions themselves. They contain merely the rule, by which we are to
Seek in the world of perception or experience the synthetical unity
Of that which cannot be intuited a priori. But they are incompetent
To present any of the conceptions which appear in them in an a priori
Intuition; these can be given only a posteriori, in experience, which
However, is itself possible only through these synthetical principles

If we are to form a synthetical judgement regarding a conception, we
Must go beyond it, to the intuition in which it is given. If we keep
To what is contained in the conception, the judgement is merely
Analytical--it is merely an explanation of what we have cogitated in the
Conception. But I can pass from the conception to the pure or empirical
Intuition which corresponds to it. I can proceed to examine my
Conception in concreto, and to cognize, either a priori or a posterio
What I find in the object of the conception. The former--a priori
Cognition--is rational-mathematical cognition by means of the
Construction of the conception; the latter--a posteriori cognition--is
Purely empirical cognition, which does not possess the attributes of
Necessity and universality. Thus I may analyse the conception I have
Of gold; but I gain no new information from this analysis, I merely
Enumerate the different properties which I had connected with the notion
Indicated by the word. My knowledge has gained in logical clearness
And arrangement, but no addition has been made to it. But if I take the
Matter which is indicated by this name, and submit it to the examination
Of my senses, I am enabled to form several synthetical--although still
Empirical--propositions. The mathematical conception of a triangle I
Should construct, that is, present a priori in intuition, and in
This way attain to rational-synthetical cognition. But when the
Transcendental conception of reality, or substance, or power is
Presented to my mind, I find that it does not relate to or indicate
Either an empirical or pure intuition, but that it indicates merely the
Synthesis of empirical intuitions, which cannot of course be given
A priori. The synthesis in such a conception cannot proceed a
Priori--without the aid of experience--to the intuition which
Corresponds to the conception; and, for this reason, none of these
Conceptions can produce a determinative synthetical proposition, they
Can never present more than a principle of the synthesis* of possible
Empirical intuitions. A transcendental proposition is, therefore, a
Synthetical cognition of reason by means of pure conceptions and the
Discursive method, and it renders possible all synthetical unity in
Empirical cognition, though it cannot present us with any intuition a
Priori

There is thus a twofold exercise of reason. Both modes have the
Properties of universality and an a priori origin in common, but are
In their procedure, of widely different character. The reason of this is
That in the world of phenomena, in which alone objects are presented to
Our minds, there are two main elements--the form of intuition (space and
Time), which can be cognized and determined completely a priori, and the
Matter or content--that which is presented in space and time, and which
Consequently, contains a something--an existence corresponding to our
Powers of sensation. As regards the latter, which can never be given in
A determinate mode except by experience, there are no a priori notions
Which relate to it, except the undetermined conceptions of the synthesis
Of possible sensations, in so far as these belong (in a possible
Experience) to the unity of consciousness. As regards the former, we
Can determine our conceptions a priori in intuition, inasmuch as we are
Ourselves the creators of the objects of the conceptions in space and
Time--these objects being regarded 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; and, as these conceptions relate
To an a priori intuition, they may be given and determined in pure
Intuition a priori, and without the aid of empirical data. The
Examination and consideration of everything that exists in space
Or time--whether it is a quantum or not, in how far the particular
Something (which fills space or time) is a primary substratum, or a mere
Determination of some other existence, whether it relates to anything
Else--either as cause or effect, whether its existence is isolated or in
Reciprocal connection with and dependence upon others, the possibility
Of this existence, its reality and necessity or opposites--all these
Form part of the cognition of reason on the ground of conceptions, and
This cognition is termed philosophical. But to determine a priori an
Intuition in space (its figure), to divide time into periods, or merely
To cognize the quantity of an intuition in space and time, and to
Determine it by number--all this is an operation of reason by means of
The construction of conceptions, and is called mathematical

The success which attends the efforts of reason in the sphere of
Mathematics naturally fosters the expectation that the same good fortune
Will be its lot, if it applies the mathematical method in other regions
Of mental endeavour besides that of quantities. Its success is thus
Great, because it can support all its conceptions by a priori intuitions
And, in this way, make itself a master, as it were, over nature; while
Pure philosophy, with its a priori discursive conceptions, bungles
About in the world of nature, and cannot accredit or show any a priori
Evidence of the reality of these conceptions. Masters in the science of
Mathematics are confident of the success of this method; indeed, it is a
Common persuasion that it is capable of being applied to any subject of
Human thought. They have hardly ever reflected or philosophized on
Their favourite science--a task of great difficulty; and 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, and as
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. We shall, accordingly, show that the mathematical
Method is unattended in the sphere of philosophy by the least
Advantage--except, perhaps, that it more plainly exhibits its own
Inadequacy--that geometry and philosophy are two quite different things
Although they go band in hand in hand in the field of natural science
And, consequently, that the procedure of the one can never be imitated
By the other

The evidence of mathematics rests upon definitions, axioms, and
Demonstrations. I shall be satisfied with showing that none of these
Forms can be employed or imitated in philosophy in the sense in which
They are understood by mathematicians; and that the geometrician, if
He employs his method in philosophy, will succeed only in building
Card-castles, while the employment of the philosophical method in
Mathematics can result in nothing but mere verbiage. The essential
Business of philosophy, indeed, is to mark out the limits of the
Science; and even the mathematician, unless his talent is naturally
Circumscribed and limited to this particular department of knowledge
Cannot turn a deaf ear to the warnings of philosophy, or set himself
Above its direction

I. Of Definitions. A definition is, as the term itself indicates, the
Representation, upon primary grounds, of the complete conception of
A thing within its own limits.* Accordingly, an empirical conception
Cannot be defined, it can only be explained. For, as there are in such
A conception only a certain number of marks or signs, which denote a
Certain class of sensuous objects, we can never be sure that we do not
Cogitate under the word which indicates the same object, at one time
A greater, at another a smaller number of signs. Thus, one person may
Cogitate in his conception of gold, in addition to its properties of
Weight, colour, malleability, that of resisting rust, while another
Person may be ignorant of this quality. We employ certain signs only so
Long as we require them for the sake of distinction; new observations
Abstract some and add new ones, so that an empirical conception never
Remains within permanent limits. It is, in fact, useless to define a
Conception of this kind. If, for example, we are speaking of water and
Its properties, we do not stop at what we actually think by the word
Water, but proceed to observation and experiment; and the word, with
The few signs attached to it, is more properly a designation than a
Conception of the thing. A definition in this case would evidently be
Nothing more than a determination of the word. In the second place, no
A priori conception, such as those of substance, cause, right, fitness
And so on, can be defined. For I can never be sure, that the clear
Representation of a given conception (which is given in a confused
State) has been fully developed, until I know that the representation
Is adequate with its object. But, inasmuch as the conception, as it is
Presented to the mind, may contain a number of obscure representations
Which we do not observe in our analysis, although we employ them in our
Application of the conception, I can never be sure that my analysis is
Complete, while examples may make this probable, although they can never
Demonstrate the fact. Instead of the word definition, I should rather
Employ the term exposition--a more modest expression, which the critic
May accept without surrendering his doubts as to the completeness of the
Analysis of any such conception. As, therefore, neither empirical nor a
Priori conceptions are capable of definition, we have to see whether the
Only other kind of conceptions--arbitrary conceptions--can be subjected
To this mental operation. Such a conception can always be defined; for
I must know thoroughly what I wished to cogitate in it, as it was I who
Created it, and it was not given to my mind either by the nature of my
Understanding or by experience. At the same time, I cannot say that, by
Such a definition, I have defined a real object. If the conception is
Based upon empirical conditions, if, for example, I have a conception of
A clock for a ship, this arbitrary conception does not assure me of the
Existence or even of the possibility of the object. My definition of
Such a conception would with more propriety be termed a declaration of
A project than a definition of an object. There are no other conceptions
Which can bear definition, except those which contain an arbitrary
Synthesis, which can be constructed a priori. Consequently, the science
Of mathematics alone possesses definitions. For the object here thought
Is presented a priori in intuition; and thus it can never contain more
Or less than the conception, because the conception of the object has
Been given by the definition--and primarily, that is, without deriving
The definition from any other source. Philosophical definitions are
Therefore, merely expositions of given conceptions, while mathematical
Definitions are constructions of conceptions originally formed by the
Mind itself; the former are produced by analysis, the completeness of
Which is never demonstratively certain, the latter by a synthesis. In
A mathematical definition the conception is formed, in a philosophical
Definition it is only explained. From this it follows:

(a) That we must not imitate, in philosophy, the mathematical usage of
Commencing with definitions--except by way of hypothesis or experiment
For, as all so-called philosophical definitions are merely analyses of
Given conceptions, these conceptions, although only in a confused form
Must precede the analysis; and the incomplete exposition must precede
The complete, so that we may be able to draw certain inferences from the
Characteristics which an incomplete analysis has enabled us to discover
Before we attain to the complete exposition or definition of the
Conception. In one word, a full and clear definition ought, in
Philosophy, rather to form the conclusion than the commencement of our
Labours.* In mathematics, on the contrary, we cannot have a conception
Prior to the definition; it is the definition which gives us the
Conception, and it must for this reason form the commencement of every
Chain of mathematical reasoning

(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 I cannot 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. Its attempts at
Mathematical evidence are vain pretensions, which can only keep it back
From its true aim, which is to detect the illusory procedure of reason
When transgressing its proper limits, and by fully explaining and
Analysing our conceptions, to conduct us from the dim regions of
Speculation to the clear region of modest self-knowledge. Reason must
Not, therefore, in its transcendental endeavours, look forward with such
Confidence, as if the path it is pursuing led straight to its aim
Nor reckon with such security upon its premisses, as to consider it
Unnecessary to take a step back, or to keep a strict watch for errors
Which, overlooked in the principles, may be detected in the arguments
Themselves--in which case it may be requisite either to determine these
Principles with greater strictness, or to change them entirely

I divide all apodeictic propositions, whether demonstrable or
Immediately certain, into dogmata and mathemata. A direct synthetical
Proposition, based on conceptions, is a dogma; a proposition of the same
Kind, based on the construction of conceptions, is a mathema. Analytical
Judgements do not teach us any more about an object than what was
Contained in the conception we had of it; because they do not extend our
Cognition beyond our conception of an object, they merely elucidate the
Conception. They cannot therefore be with propriety termed dogmas. Of
The two kinds of a priori synthetical propositions above mentioned, only
Those which are employed in philosophy can, according to the general
Mode of speech, bear this name; those of arithmetic or geometry would
Not be rightly so denominated. Thus the customary mode of speaking
Confirms the explanation given above, and the conclusion arrived at
That only those judgements which are based upon conceptions, not on the
Construction of conceptions, can be termed dogmatical

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

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