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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：
*The definition must describe the conception completely that is；
omit none of the marks or signs of which it composed； within its own
limits； that is； it must be precise； and enumerate no more signs
than belong to the conception； and on primary grounds； that is to say；
the limitations of the bounds of the conception must not be deduced
from other conceptions； as in this case a proof would be necessary；
and the so…called definition would be incapable of taking its place at
the bead of all the judgements we have to form regarding an object。
（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。
*Philosophy abounds in faulty definitions； especially such as
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 prio