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Overview of Scheda (mid–1990s)

This “Scheda” from the mid-1990s, which Leibniz himself characterized as “good” amidst many other writings of the same period, provides a good overview of the foundations of analysis situs in this period. Leibniz exposes in it his own thought in deductive form (which is uncommon, but not altogether rare in geometry writings), thus showing which propositions he considers more elementary and which ones derivable from the previous ones.

In particular, it is relevant that Leibniz assumes here as the initial term of the whole essay the notion of an extensum, which is defined as a coexistent continuum (a definition that had already appeared in the first writings on Characteristica geometrica). The notion of a point, however, which in the 1680s had been coordinated as a second primitive to that of an extensum is seen here as derived from it.

In the text appear some fundamental ideas that return often in the writings of these years. The main one is probably that all points have situational relations to each other, and therefore they all belong to the same all-encompassing extensum, which Leibniz here calls “Universal Space.” Leibniz also develops there some fundamental ideas for continuity, and mainly that in any continuous extensum it is always possible to take another continuous extensum as its own part (today this is an important axiom of the open sets of a topological space). Finally, the distinction between homogeneity and congeneity recurs in the text, which is a topic particularly studied by Leibniz in these years. Congeneity is a more general relation than classical homogeneity (which has various definitions, but generally denotes geometric objects of the same dimension), and Leibniz calls congeneous those figures that can be transformed into each other through continuous motion (a point and a line, for example). With such a notion he wants to give a characterization of what is properly spatial (an instant is not congeneous to a geometric line, whereas points, surfaces and solid bodies are), and at the same time to have a notion useful to discuss some classical foundational difficulties (a curvilinear angle is not congeneous to a rectilinear angle). The concept recurs with the name “homogunum” or with the Greek συγγενές in other essays of the period.

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