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lf, and so a kind of meta-idea. This generates an infinite regress - no matter what is our standard of comparison (idea, meta-idea, meta-meta-idea), there always is a tertius homo or third man, a higher criterion which enables one to establish that the given two objects are similar. The same problem of similarity is pervasive in the context of rules. Indeed, Platos question may be considered as the rule-following puzzle in (Ancient) disguise. One should use the word great only in relation to great objects. But how do we know that any two objects share the feature of greatness, or are similar in this respect? Another example comes from Kants great critical project. In Critique of Pure Reason he attempts, inter alia, to explain why mathematical judgments are synthetic (they expand our knowledge), yet a priori (independent of sensual experience). Kants famous answer is that mathematical judgments are based on the intuitions of space and time, which, in turn, are not aspects of reality, but of the knowing subject. We are so constructed that our perception is always a perception in time and space, as both are elements of our cognitive structure. In this way, mathematical theorems are independent of experience, but not merely analytical or devoid of meaning. [3] For Kant, the key problem of this account lies in the fact that any representation in intuition is a concrete object. When you see or merely imagine a triangle, it is always a concrete triangle: right-angled, acute-angled or obtuse-angled. Yet, mathematical knowledge is expressed in concepts which are universally applicable: that the angles of a triangle amount to 180 degrees is true of any triangle, not just a particular one represented in an intuition. So, the question reads, how is it possible that intuitive representations of particular mathematical objects may serve to justify universal mathematical knowledge. Kants answer in Critique of Pure Reason is as follows: The individual drawn figure is empirical, and nevertheless serves to express the concept without damage to its universality, for in the case of this empirical intuition we have taken account only of the action of constructing the concept, to which many determinations, , those of the magnitude of the sides and the angles, are entirely indifferent. [4] He believes that the tertius homo, or the intermediary between concrete representations and general concepts, is the procedure of construction, which he calls a transcendental schema: Now it is clear that there must be a third thing, which must stand in homogeneity with the category on the one hand and the appearance on the other, and makes possible the application of the former to the latter. This mediating representation must be pure (without anything empirical) and yet intellectual on the one hand, and sensible on the other. Such a representation is the transcendental schema. [5] A transcendental schema of a triangle is, therefore, a procedure or a rule that enables one to construct a particular triangle in intuition, and the same holds for all mathematical objects. A schema "can never exist anywhere but in thought, and it signifies a rule of synthesis of the imagination with respect to pure figures in space", [6] making it possible to exhibit universal features inherent in particular instantiations of mathematical objects: We say that we cognize the object if we have effected synthetic unity in the manifold of intuition. But this is impossible if the intuition could not have been produced through a function of synthesis in accordance with a rule that makes the reproduction of the manifold necessary a priori and a concept in which this manifold is unified possible. Thus we think of a triangle as an object by being conscious of the composition of three straight lines in accordance with a rule according to which such an intuition can always be exhibited. [7] This is Kants solution to the problem of the relationship between particular mathematical objects and general mathematical concepts. It is noteworthy that Kant is dealing here with all three aspects of the pattern condition: similarity, infinity and projectibility. He explains how we can perceive similarities between concrete triangles, as all of them are constructed with the use of the same transcendental schema; furthermore, his theory also accounts for the fact that one concept refers to potentially infinitely many instantiations, for transcendental schemata are at work in each and every act of cognition; finally, mathematical knowledge is stable and mathematical concepts projectible, again because of the transcendental character of the schemata. It is easy to see that the problem Kant addresses, namely the nature of mathematical knowledge and mathematical concepts, may be rephrased so as to concern mathematical rules. Instead of asking how the concept of a triangle corresponds to infinitely many intuitive representations of triangles, one may ask, how does a mathematical rule