conceptual

What I saw at Marco Panza and collaborators’ Euclid session at Chapman University this Friday October 13th: Looking, right now at mathematicians workshop looking for a non-standard model to a formal system of axioms, it is very interesting to see how the whiteboard is used. For example, by explicitly writing out the formulas, it raises the salience of the operations involved. Thus, once it was shown, by explicitly writing the relevant formula, that a certain parameter was being calculated by a quadratic equation, this immediately suggested the use of irrationals to find the desired non-standard model.  Similarly, using ad-hoc formulas i.e. displays that belong to no actual standardized formal language but share a superficial grammar with them, for example, at some time Marco Panza wrote “SAS : Ang → Sea” to represent the fact that, in Euclid, the so-called Segment-Angle-Segment axiom (which they never called that way, always using the acronym “SAS”) states that, given c...

One of the basic problem of grounding  theory originates from an assumed fundamental asymmetry between the ultimate explanans and explananda of metaphysical explanation. While explananda can be composed of other facts in several ways, by disjunction, negation, conjunction, quantification, etc., the fundamental facts that constitute the explanans are supposed to be simple in themselves and total explanans can only be composed in of these simple facts in a straightforward aggregative way. In other words, while we need to find a metaphysical explanation to disjunctive, negative, quantified truth-bearers, etc., Neo-Aristotelian metaphysics resists the acceptance of fundamental negative, disjunctive or quantified facts. Thus, it faces the significant challenge of having to find adequate non-disjunctive grounds for disjunctive facts, positive grounds for negative facts, non-quantified grounds for quantified fact, non-modal grounds for modal facts, etc. So far, this task had proved elusiv...

In Euclid, parallelism is defined negatively by the absence of an intersection. But, even if they are not explicitly mentioned in this definition, the phenomenon of parallelism is ultimately about angles, because line intersection is also ultimately about angles (in particular, about the angle thus determined) – this is common sense, I think –; thus, in the end, parallelism is not about the angle between the parallel lines themselves (which does not exist) but about the angles produced by other, non-parallel, lines (otherwise, as I just mentioned, there would be no angles). In this context, we can think of axiom SAS as an extreme, simpler case of how to answer a more general question: how are angles are parallels related?  The other extreme is to say that they are completely independent, which is what José Gil is currently exploring. But there is an ample space between one extreme and the other. Remember that, according to SAS, if two straight lines (well, to be historically accura...