Anyone care to read Kant's "Critique of Pure Reason"? On the justification for, but not the deduction of, the pure a priori conceptions of the understanding:
“…. General logic, as has been repeatedly said, makes abstraction of all content of cognition, and expects to receive representations from some other quarter, in order, by means of analysis, to convert them into conceptions. On the contrary, transcendental logic has lying before it the manifold content of à priori sensibility, which transcendental æsthetic presents to it in order to give matter to the pure conceptions of the understanding, without which transcendental logic would have no content, and be therefore utterly void. Now space and time contain an infinite diversity of determinations of pure à priori intuition, but are nevertheless the condition of the mind’s receptivity, under which alone it can obtain representations of objects, and which, consequently, must always affect the conception of these objects. But the spontaneity of thought requires that this diversity be examined after a certain manner, received into the mind, and connected, in order afterwards to form a cognition out of it. This Process I call synthesis.
By the word synthesis, in its most general signification, I understand the process of joining different representations to each other and of comprehending their diversity in one cognition. This synthesis is pure when the diversity is not given empirically but à priori (as that in space and time). Our representations must be given previously to any analysis of them; and no conceptions can arise regarding its content analytically. But the synthesis of a diversity (be it given à priori or empirically) is the first requisite for the production of a cognition, which in its beginning, indeed, may be crude and confused, and therefore in need of analysis—still, synthesis is that by which alone the elements of our cognitions are collected and united into a certain content, consequently it is the first thing on which we must fix our attention, if we wish to investigate the origin of our knowledge.
Synthesis, generally speaking, is, as we shall afterwards see, the mere operation of the imagination—a blind but indispensable function of the soul, without which we should have no cognition whatever, but of the working of which we are seldom even conscious. But to reduce this synthesis to conceptions is a function of the understanding….
….. Pure synthesis, represented generally, gives us the pure conception of the understanding. But by this pure synthesis, I mean that which rests upon a basis of à priori synthetical unity…..
….. By means of analysis different representations are brought under one conception—an operation of which general logic treats**. On the other hand, the duty of transcendental logic is to reduce to conceptions, not representations, but the pure synthesis of representations***. The first thing which must be given to us for the sake of the à priori cognition of all objects, is the diversity of the pure intuition; the synthesis of this diversity by means of the imagination is the second; but this gives, as yet, no cognition. The conceptions which give unity to this pure synthesis, and which consist solely in the representation of this necessary synthetical unity, furnish the third requisite for the cognition of an object, and these conceptions are given by the understanding. The same function which gives unity to the different representation in a judgement, gives also unity to the mere synthesis of different representations in an intuition; and this unity we call the pure conception of the understanding.
Thus, the same understanding, and by the same operations, whereby in conceptions, by means of analytical unity, it produced the logical form of a judgement, introduces, by means of the synthetical unity of the manifold in intuition, a transcendental content into its representations****, on which account they are called pure conceptions of the understanding, and they apply à priori to objects, a result not within the power of general logic.….”
(**and here is where the undetermined object of intuition becomes “apple”)
(***whatever any undetermined object becomes, it does so in accordance with a rule)
(****the rule)
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On the justification for a priori pure intuitions:
“….. It must be admitted that the Leibnitz-Wolfian philosophy has assigned an entirely erroneous point of view to all investigations into the nature and origin of our cognitions, inasmuch as it regards the distinction between the sensuous and the intellectual as merely logical, whereas it is plainly transcendental, and concerns not merely the clearness or obscurity, but the content and origin of both. For the faculty of sensibility not only does not present us with an indistinct and confused cognition of objects as things in themselves, but, in fact, gives us no knowledge of these at all. On the contrary, so soon as we abstract in thought our own subjective nature, the object represented, with the properties ascribed to it by sensuous intuition, entirely disappears, because it was only this subjective nature that determined the form of the object as a phenomenon.
In phenomena, we commonly, indeed, distinguish that which essentially belongs to the intuition of them, and is valid for the sensuous faculty of every human being, from that which belongs to the same intuition accidentally, as valid not for the sensuous faculty in general, but for a particular state or organization of this or that sense. Accordingly, we are accustomed to say that the former is a cognition which represents the object itself, whilst the latter presents only a particular appearance or phenomenon thereof. This distinction, however, is only empirical. If we stop here (as is usual), and do not regard the empirical intuition as itself a mere phenomenon (as we ought to do), in which nothing that can appertain to a thing in itself is to be found, our transcendental distinction is lost, and we believe that we cognize objects as things in themselves, although in the whole range of the sensuous world, investigate the nature of its objects as profoundly as we may, we have to do with nothing but phenomena.
Suppose, then, that space and time are in themselves objective, and conditions of the possibility of objects as things in themselves. In the first place, it is evident that both present us, with very many apodeictic and synthetic propositions à priori, but especially space—and for this reason we shall prefer it for investigation at present. As the propositions of geometry are cognized synthetically à priori, and with apodeictic certainty, I inquire: Whence do you obtain propositions of this kind, and on what basis does the understanding rest, in order to arrive at such absolutely necessary and universally valid truths? There is no other way than through intuitions or conceptions, as such; and these are given either à priori or à posteriori. The latter, namely, empirical conceptions, together with the empirical intuition on which they are founded, cannot afford any synthetical proposition, except such as is itself also empirical, that is, a proposition of experience.
But an empirical proposition cannot possess the qualities of necessity and absolute universality, which, nevertheless, are the characteristics of all geometrical propositions. As to the first and only means to arrive at such cognitions, namely, through mere conceptions or intuitions à priori, it is quite clear that from mere conceptions no synthetical cognitions, but only analytical ones, can be obtained. Take, for example, the proposition: “Two straight lines cannot enclose a space, and with these alone no figure is possible,” and try to deduce it from the conception of a straight line and the number two; or take the proposition: “It is possible to construct a figure with three straight lines,” and endeavour, in like manner, to deduce it from the mere conception of a straight line and the number three. All your endeavours are in vain, and you find yourself forced to have recourse to intuition, as, in fact, geometry always does. You therefore give yourself an object in intuition. But of what kind is this intuition? Is it a pure à priori, or is it an empirical intuition? If the latter, then neither an universally valid, much less an apodeictic proposition can arise from it, for experience never can give us any such proposition. You must, therefore, give yourself an object à priori in intuition, and upon that ground your synthetical proposition.
Now if there did not exist within you a faculty of intuition à priori; if this subjective condition were not in respect to its form also the universal condition à priori under which alone the object of this external intuition is itself possible; if the object (that is, the triangle) were something in itself, without relation to you the subject; how could you affirm that that which lies necessarily in your subjective conditions in order to construct a triangle, must also necessarily belong to the triangle in itself? For to your conceptions of three lines, you could not add anything new (that is, the figure); which, therefore, must necessarily be found in the object, because the object is given before your cognition, and not by means of it.
If, therefore, space (and time also) were not a mere form of your intuition, which contains conditions à priori, under which alone things can become external objects for you, and without which subjective conditions the objects are in themselves nothing, you could not construct any synthetical proposition whatsoever regarding external objects. It is therefore not merely possible or probable, but indubitably certain, that space and time, as the necessary conditions of all our external and internal experience, are merely subjective conditions of all our intuitions, in relation to which all objects are therefore mere phenomena, and not things in themselves, presented to us in this particular manner. And for this reason, in respect to the form of phenomena, much may be said à priori, whilst of the thing in itself, which may lie at the foundation of these phenomena, it is impossible to say anything.…”
The claim, it follows, that these justifications were not given in the body of the text, is catastrophically false.