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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 pleteness
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 pletely that is；
omit none of the marks or signs of which it posed； 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 mencing 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 inplete
exposition must precede the plete； so that we may be able to draw
certain inferences from the characteristics which an inplete
analysis has enabled us to discover； before we attain to the
plete exposition or definition of the conception。 In one word； a
full and clear definition ought； in philosophy； rather to form the
conclusion than the mencement 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 mencement of every chain of mathematical
reasoning。
*Philosophy abounds in faulty definitions； especially such as
contain some of the elements requisite to form a plete
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 inpletely 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 plete 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 mon 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 pleteness 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
pleteness 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 t