Front Page Titles (by Subject) § 62.: Of the objective purposiveness which is merely formal as distinguished from that which is material - The Critique of Judgement
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§ 62.: Of the objective purposiveness which is merely formal as distinguished from that which is material - Immanuel Kant, The Critique of Judgement 
Kant’s Critique of Judgement, translated with Introduction and Notes by J.H. Bernard (2nd ed. revised) (London: Macmillan, 1914).
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Of the objective purposiveness which is merely formal as distinguished from that which is material
All geometrical figures drawn on a principle display a manifold, oft admired, objective purposiveness; i.e. in reference to their usefulness for the solution of several problems by a single principle, or of the same problem in an infinite variety of ways. The purposiveness is here obviously objective and intellectual, not merely subjective and aesthetical. For it expresses the suitability of the figure for the production of many intended figures, and is cognised through Reason. But this purposiveness does not make the concept of the object itself possible, i.e. it is not regarded as possible merely with reference to this use.
In so simple a figure as the circle lies the key to the solution of a multitude of problems, each of which would demand various appliances; whereas the solution results of itself, as it were, as one of the infinite number of elegant properties of this figure. Are we, for example, asked to construct a triangle, being given the base and vertical angle? The problem is indeterminate, i.e. it can be solved in an infinite number of ways. But the circle embraces them altogether as the geometrical locus of the vertices of triangles satisfying the given conditions. Again, suppose that two lines are to cut one another so that the rectangle under the segments of the one should be equal to the rectangle under the segments of the other; the solution of the problem from this point of view presents much difficulty. But all chords intersecting inside a circle divide one another in this proportion. Other curved lines suggest other purposive solutions of which nothing was thought in the rule that furnished their construction. All conic sections in themselves and when compared with one another are fruitful in principles for the solution of a number of possible problems, however simple is the definition which determines their concept.— It is a true joy to see the zeal with which the old geometers investigated the properties of lines of this class, without allowing themselves to be led astray by the questions of narrow-minded persons, as to what use this knowledge would be. Thus they worked out the properties of the parabola without knowing the law of gravitation, which would have suggested to them its application to the trajectory of heavy bodies (for the motion of a heavy body can be seen to be parallel to the curve of a parabola). Again, they found out the properties of an ellipse without surmising that any of the heavenly bodies had weight, and without knowing the law of force at different distances from the point of attraction, which causes it to describe this curve in free motion. While they thus unconsciously worked for the science of the future, they delighted themselves with a purposiveness in the [essential] being of things which yet they were able to present completely a priori in its necessity. Plato, himself master of this science, hinted at such an original constitution of things in the discovery of which we can dispense with all experience, and at the power of the mind to produce from its supersensible principle the harmony of beings (where the properties of number come in, with which the mind plays in music). This [he touches upon] in the inspiration that raised him above the concepts of experience to Ideas, which seem to him to be explicable only through an intellectual affinity with the origin of all beings. No wonder that he banished from his school the man who was ignorant of geometry, since he thought he could derive from pure intuition, which has its home in the human spirit, that which Anaxagoras drew from empirical objects and their purposive combination. For in the very necessity of that which is purposive, and is constituted just as if it were designedly intended for our use,—but at the same time seems to belong originally to the being of things without any reference to our use—lies the ground of our great admiration of nature, and that not so much external as in our own Reason. It is surely excusable that this admiration should through misunderstanding gradually rise to the height of fanaticism.
But this intellectual purposiveness, although no doubt objective (not subjective like aesthetical purposiveness), is in reference to its possibility merely formal (not real). It can only be conceived as purposiveness in general without any [definite] purpose being assumed as its basis, and consequently without teleology being needed for it. The figure of a circle is an intuition which is determined by means of the Understanding according to a principle. The unity of this principle which I arbitrarily assume and use as fundamental concept, applied to a form of intuition (space) which is met with in myself as a representation and yet a priori, renders intelligible the unity of many rules resulting from the construction of that concept, which are purposive for many possible designs. But this purposiveness does not imply a purpose or any other ground whatever. It is quite different if I meet with order and regularity in complexes of things, external to myself, enclosed within certain boundaries; as, e.g. in a garden, the order and regularity of the trees, flower-beds, and walks. These I cannot expect to derive a priori from my bounding of space made after a rule of my own; for this order and regularity are existing things which must be given empirically in order to be known, and not a mere representation in myself determined a priori according to a principle. So then the latter (empirical) purposiveness, as real, is dependent on the concept of a purpose.
But the ground of admiration for a perceived purposiveness, although it be in the being of things (so far as their concepts can be constructed), may very well be seen, and seen to be legitimate. The manifold rules whose unity (derived from a principle) excites admiration, are all synthetical and do not follow from the concept of the Object, e.g. of a circle; but require this Object to be given in intuition. Hence this unity gets the appearance of having empirically an external basis of rules distinct from our representative faculty; as if therefore the correspondence of the Object to that need of rules which is proper to the Understanding were contingent in itself, and therefore only possible by means of a purpose expressly directed thereto. Now because this harmony, notwithstanding all this purposiveness, is not cognised empirically but a priori, it should bring us of itself to this point—that space, through whose determination (by means of the Imagination, in accordance with a concept) the Object is alone possible, is not a characteristic of things external to me, but a mere mode of representation in myself. Hence, in the figure which I draw in conformity with a concept, i.e. in my own mode of representing that which is given to me externally, whatever it may be in itself, it is I that introduce the purposiveness; I get no empirical instruction from the Object about the purposiveness, and so I require in it no particular purpose external to myself. But because this consideration already calls for a critical employment of Reason, and consequently cannot be involved in the judging of the Object according to its properties; so this latter [judging] suggests to me immediately nothing but the unification of heterogeneous rules (even according to their very diversity) in a principle. This principle, without requiring any particular a priori basis external to my concept, or indeed, generally speaking, to my representation, is yet cognised a priori by me as true. Now wonder is a shock of the mind arising from the incompatibility of a representation, and the rule given by its means, with the principles already lying at its basis; which provokes a doubt as to whether we have rightly seen or rightly judged. Admiration, however, is wonder which ever recurs, despite the disappearance of this doubt. Consequently the latter is a quite natural effect of that observed purposiveness in the being of things (as phenomena). It cannot indeed be censured, whilst the unification of the form of sensible intuition (space)—with the faculty of concepts (the Understanding)—is inexplicable to us; and that not only on account of the union being just of the kind that it is, but because it is enlarging for the mind to surmise [the existence of] something lying outside our sensible representations in which, although unknown to us, the ultimate ground of that agreement may be met with. We are, it is true, not necessitated to cognise this if we have only to do a priori with the formal purposiveness of our representations; but the fact that we are compelled to look out beyond it inspires at the same time an admiration for the object that impels us thereto.
We are accustomed to speak of the above mentioned properties of geometrical figures or of numbers as beautiful, on account of a certain a priori purposiveness they have for all kinds of cognitive uses, this purposiveness being quite unexpected on account of the simplicity of the construction. We speak, e.g. of this or that beautiful property of the circle, which was discovered in this or that way. But there is no aesthetical act of judgement through which we find it purposive, no act of judgement without a concept which renders noticeable a mere subjective purposiveness in the free play of our cognitive faculties; but an intellectual act according to concepts which enables us clearly to cognise an objective purposiveness, i.e. availableness for all kinds of (infinitely manifold) purposes. We must rather call this relative perfection than a beauty of the mathematical figure. To speak thus of an intellectual beauty cannot in general be permissible; for otherwise the word beauty would lose all determinate significance, or the intellectual satisfaction all superiority over the sensible. We should rather call a demonstration of such properties beautiful, because through it the Understanding as the faculty of concepts, and the Imagination as the faculty of presenting them, feel themselves strengthened a priori. (This, when viewed in connexion with the precision introduced by Reason, is spoken of as elegant.) Here, however, the satisfaction, although it is based on concepts, is subjective; while perfection brings with itself an objective satisfaction.