For the construction of a concept we need a non-empirical intuition.
This non-empirical intuition must be a single object.
It must also express, in its representation, universal validity for all possible intuitions which fall under the concept.
Philosophical knowledge considers the particular only in the universal; mathematical knowledge the universal in the particular a priori by means of reason.
This is the essential difference between these two kinds of knowledge and not the difference of their material or objects.
The concept of quantities with which mathematics deals, is the only one that allows of being constructed, that is, exhibited a priori in intuition; whereas qualities cannot be presented in any intuition that is not empirical.
i.e. The shape of a cone we can form in intuition, unassisted by any experience, according to the concept alone, but the color of this cone must be given in some experience or other.
The concepts constructed by the mathematical method are universal synthetic propositions.
There is thus a twofold employment of reason; and while the two modes of employment resemble each other in the universality and a priori origin of their knowledge, in outcome they are very different.
A twofold employment of reason because in appearance there are two elements:
The form of intuition (space and time), which can be known and determined a priori.
We can determine our concepts in a prioriintuition, inasmuch as we create for ourselves the objects themselves through a homogeneous synthesis of space and time.
This is the employment of reason through the construction of concepts.
This is the mathematical use of knowledge.
Matter (the physical element) or content, which is met with in space and time and therefore contains an existent corresponding to sensation.
This can only be determined empirically, we can have nothing a priori except indeterminate concepts of the synthesis of possible sensations.
This is the employment of reason is accordance with concepts.
The questions of possibility of existence, its actuality and necessity of these objects is the philosophical use of reason.
Transcendental propositions can never be given through construction of concepts, but only in accordance with concepts that are a priori.
Synthetic propositions in regard to things in general are transcendental.
But these synthetic principles cannot exhibit a priori any one of their concepts in a specific instance.
The exactness of mathematics rests upon definitions, axioms and demonstrations. None of these, in the sense understood by the mathematician, can be achieved or imitated by the philosopher.
Definitions: To define means to present the complete, original concept of a thing within the limits of its concept.
Completeness: clearness and sufficiency of characteristics.
Limits: the precision shown in there not being more of these characteristics than belong to the complete concept.
Original: the determination of the limits is not derived from anything else and therefore does not require proof.
An empirical concept, therefore, can not be defined, it can only be made explicit.
It is never certain that we are not using the word, in denoting one and same object, sometimes so as to stand for more, sometimes to stand for fewer characteristics.
What purpose could be served by defining an empirical concept such as water?
No concept given a priori can be defined; i.e. concepts such as cause, right, equity, etc.
The completeness of the concept is always in doubt.
Instead of definition, the term 'exposition' applies.
The only kind of concept that allows of definition are arbitrarily invented concepts.
If I invented it, I can always define it.
Therefore, mathematics is the only science that has definitions.
Philosophical definitions are never more that expositions.
Since mathematical definitions make their concepts and in philosophical definitions the concepts are only explained, it follows that:
In philosophy we must not imitate mathematics by beginning with definitions, since the definitions are analyses of given concepts, they presuppose the prior presence of the concepts. The definition ought to come at the end of our enquiries. In mathematics we have no concept prior to the definition.
Mathematical definitions can never be in error.
Axioms: These, in so far as they are immediately certain, are synthetic a priori principles.
Mathematics can have axioms since by means of the construction of concepts in the intuition of the object it can combine predicates of the object both a priori and immediately.
Philosophy has no axioms since it is simply what reason knows by means of concepts.
The axioms of intuition in the Table of Principles are not axioms in the mathematical sense, they are the principles that specify the possibility of axioms in general.
Demonstrations: an apodeictic proof only in so far as it is intuitive.
No empirical grounds of proof can ever amount to apodeictic proof.
Mathematics alone contains demonstrations since it derives its knowledge not from concepts but from the construction of them, that is, from intuition, which can be given a priori in accordance with the concepts.
Philosophy does not have demonstrations because it has always to consider the universal in abstracto (by means of concepts), while mathematics can consider the universal in concreto (in the single intuition).