Reverse analyses of three proofs of the Sylvester-Gallai theorem lead to three different and incompatible axiom systems. In particular, we show that proofs respecting the purity of the method, using only notions considered to be part of the statement of the theorem to be proved, are not always the simplest, as they may require axioms which proofs using extraneous predicates do not rely upon.

The task of this paper is to give a new, catholic interpretation of Leibniz’s concept of characteristica universalis. In § 2 we shall see that in different periods of his development, Leibniz defined this concept differently. He introduced it as “philosophical characteristic” in 1675, elaborated it further as characteristica universalis in 1679, and worked on it at least until 1690. Secondly, we shall see (in § 3) that in the last 130 years or so, different philosophers have advanced projects similar (...) to that of Leibniz, not always referring to him, at that. These very projects, we claim, threw some light on what Leibniz idea of characteristica universalis could be; or, in more positive sense, on how could it be reconstructed in a more workable way. Unfortunately, they failed to answer the question what exactly Leibniz’s philosophy was. Finally, despite the fact that Leibniz’s concept of characteristica universalis impressed generations of philosophers who tried to make sense of it, the result is that in the more than 300 years after it was introduced, it was never used in the scale its author dreamed of. This fact sets out the next objective which we are going to pursue in this paper: we shall try to find out (in §§ 4 and 5) how this concept can be interpreted in more practical form. (shrink)

This paper provides a definitive answer, based on considerations derived from first-order logic, to the question regarding the status of elementary geometry, whether elementary geometry can be reduced to algebra. The answer we arrive at is negative, and is based on a series of structural questions that can be asked only inside the geometric formal theory, as well as the consideration of reverse geometry, which is the art of finding minimal axiom systems strong enough to prove certain geometrical theorems, given (...) that there are no algebraic structures that one could associate with those minimal axiom systems. (shrink)

ABSTRACT We present and discuss a change in the introduction of Hilbert’s Grundlagen der Geometrie between the first and the subsequent editions: the disappearance of the reference to the independence of the axioms. We briefly outline the theoretical relevance of the notion of independence in Hilbert’s work and we suggest that a possible reason for this disappearance is the discovery that Hilbert’s axioms were not, in fact, independent. In the end we show how this change gives textual evidence for the (...) connection between the notions of independence and simplicity. (shrink)

Hilbert’s methodological reflection has certainly shaped a new image of the axiomatic method. However, the discussion on the procedural character of the method is still open, with commentators subs...