 While in philosophical questions related to mathematics, Proclus and Plato were Kepler’s most important inspirational sources, he did not always see Plato and Aristotle as completely opposed, for the latter—in Kepler’s interpretation—also accepted “a certain existence of the mathematical entities” (KGW 14, let. N° 226, p. 265; see Peters, p. 130). To a great extent Kepler understood his mathematical investigations of HM as a continuation of Euclid’s Elements, especially of the analysis of irrationalities in Book 10. The central notion that he works out here is that of “constructability”. According to Kepler, each branch of knowledge must, in principle, be reducible to geometry if it is to be accepted as knowledge in the strong sense (although, in the case of the physics, this condition is, as the AN emphasizes, only a necessary and not a sufficient condition). Thus, the new principles he was elaborating over the years in astrology were geometrical ones. A similar case occurs with the basic notions of harmony, which, after Kepler, could be reduced to geometry. Of course, not every geometrical statement is equally relevant and equally fundamental. For Kepler, the geometrical entities, principles and propositions which are especially fundamental are those that can be constructed in the classical sense, i.e., using only ruler (without measurement units) and compass. On this are based further notions according to different degrees of “knowability” (scibilitas), which begins with the circle and its diameter. Once again, Kepler understood this within the framework of his cosmological and theological philosophy: geometry, and especially geometrically constructible entities, have a higher meaning than other kinds of knowledge because God has used them to delineate and to create this perfect harmonic world. From this point of view, it is clear that Kepler defends a Platonist conception of mathematics, that he cannot assume the Aristotelian theory of abstraction and that he is not able to accept algebra, at least in the way he understood it. So, for instance, there are figures that cannot be constructed “geometrically”, although they are often assumed as safe geometrical knowledge. The best example of this is perhaps the heptagon. This figure cannot be described outside of the circle, and in the circle its sides have, of course, a determinate magnitude, but this is not knowable. Kepler himself says that this is important because here he finds the explanation for why God did not use such figures to structure the world. Consequently, he devotes many pages to discussing the issue (KGW 6, Prop. 45, pp. 47–56, see also KGW 9, p. 147). Certainly for a geometer like Kepler, approximations constitute – as mathematical theory—a painful and precarious way to progress. The philosophical background for his rejection of algebra seems to be, at least partially, Aristotelian in some of its basic suppositions: geometrical quantities are continuous quantities which therefore cannot be treated with numbers that are, in the inverse, discrete quantities. But the difference from the Aristotelian ideal of science remains an important one: for Aristotle, a crossover between arithmetic and geometry is allowed only in the case of the “middle sciences”, while for Kepler all knowledge must be reduced to its geometrical foundations. Recent research has tried to show that Kepler has followed the criteria of a “Pythagorean methodology” when studying the phenomena of light refraction. This would be evident in Kepler’s application of the principle of “sameness is knowable by sameness” and, additionally, in his tendency to avoid the infinite or, rather, to set limits to infinite magnitudes (Cardona Suárez 2016).