conceptual

Arguments in favor of the existence of mathematical entities like numbers can be divided into two broad kinds: on the one hand, we have those that start from mathematical practice as a given and postulate the existence of mathematical objects as part of the best explanation for mathematical practice being as its, usually, either because of its success as a scientific enterprise or because of its importance for other successful scientific and technological practices. On the other hand, there are those that, instead, take everyday linguistic practice as starting point and postulate the existence of mathematical objects as part of the best explanation of our usual linguistic practices being as they are. The overall general strategy in both cases is to argue that without numbers, it would be very hard to explain why things that we accept to be true – or at least, to be successful as claims about the world, like simple arithmetical truths like seventeen being prime, complex physical laws li...

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...

Judges use  photo finish to determine who won a race, a driver stops at a corner to ask a passer by for directions, a radiologist examines a patient’s x-ray before giving diagnosis, a traveller checks the screen at the airport to get information about her flight, a scientist checks the reading on her nanometer to determine the length of her samples, a mathematician looks at a diagram to gain insight into a new conjecture, etc. What all these cases have in common is that in all of them a person tries to get information about the world not by direct observation but by the use of representations of different sorts. Just as the radiologist need not have any direct contact with the patient, we usually do not need direct contact with whatever aspect of the world we want to know about. In every case, the information might be more or less accurate, the method we use more or less reliable, but in all of them the information is mediated by a representation: a photograph, some words, an x-ray...