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contain some of the elements requisite to form a complete
definition。 If a conception could not be employed in reasoning
before it had been defined； it would fare ill with all philosophical
thought。 But； as incompletely defined conceptions may always be
employed without detriment to truth； so far as our analysis of the
elements contained in them proceeds； imperfect definitions； that is；
propositions which are properly not definitions； but merely
approximations thereto； may be used with great advantage。 In
mathematics； definition belongs ad esse； in philosophy ad melius esse。
It is a difficult task to construct a proper definition。 Jurists are
still without a complete definition of the idea of right。
（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