48. Indeterminate ideas, in order to represent any thing, must be applied to some property which is the condition of their actual or possible realization. Until this application is made, they are pure intellectual forms, which represent nothing determinate. I do not mean by this, that these ideas are empty conceptions, which cannot be applied outside of the sensible order, as was maintained by Kant;[37] but only that granting them an universal value, I deny that they have by themselves alone a value representative of any thing that can be realized, beyond the property which they express. The idea of a _pure_ triangle can not be realized, for every _real_ triangle would contain something more than is in the idea: it would be a right-angled or oblique-angled, etc., all which, the pure idea abstracts. The object will be indeterminate, in proportion to the indeterminateness of the properties contained in the conception; consequently, that which is presented to the understanding will also be more vague, and the applications which may be made of the idea, will be more varied and numerous, as is the case in the ideas of being, not-being, limit, and the like. [37] See Book V., Chapters XIV., XV., and XVI. CHAPTER VII. FUNDAMENTAL EXPLANATION OF THE ABSTRACT IDEA OF THE INFINITE. 49. Supposing that our idea of the infinite is not intuitive but abstract, let us see how its true nature may be explained. We have the ideas of being and of its opposite, not-being; these ideas considered in themselves are general, indeterminate, and may be applied to every thing which is subjected to our experience. We may affirm and deny something of every limited being: we may affirm what it is: we may deny what it is not: the limit is only conceived as such when something is denied of it. 50. The activity of our being is unceasing, but it is limited by the absence or the resistance of objects; the external world is an assemblage of beings presenting a great variety of limitations. Therefore both internal and external experience give us the idea of the finite, that is, of a being which involves some not-being. The brute has sensible perception, but no understanding: it _is_ sensitive, and herein it has being; it _is not_ intelligent, and herein it is limited. Man is sensitive and intelligent; the limit of the brute is not the limit of man. Among intelligent beings some understand more than others; therefore the limit of all is not the same. 51. Since we find a limit in both internal and external experience, it is evident that we can form the general idea of limit, that is, of a negation applied to an object.