Introduction

If I were to show you a hammer and ask you whether this hammer was true, you would probably consider this as an ill-posed question. Furthermore, if I were to show you a hammer and a screwdriver, and ask you which of these two is truer than the other, you could reasonably consider the question as complete nonsense.

Hammers and screwdrivers are tools, and as such we don’t ask for their “truth”, let alone in a comparative manner. We would rather ask whether they are useful; and as such, for which purpose: a hammer is useful to drive a nail into the wall, a screwdriver to do the same with a screw (and better not to do it the other way around).

We will argue that it is similarly odd to ask for truth (simpliciter) in Mathematics. This is, in fact, in accordance with modern mathematical self-conception, resulting from the shift of understanding caused by the discovery of non-Euclidean Geome­tries. The aim of the paper is to show that there is, however, still a rationale for truth in Mathematics which, however, has to be relativized to certain structures—including the possibility of non-standard structures.

Work partially supported by the Portuguese Science Foundation, FCT, through the projects The Notion of Mathematical Proof, PTDC/MHC-FIL/5363/2012, Hilbert’s 24th Problem, PTDC/MHC-FIL/2583/2014, and the Centro de Matemdtica e Aplicapdes, UID/MAT/00297/2013.

R. Kahle (El)

CMA & DM, FCT, Universidade Nova de Lisboa, Lisbon, Portugal

e-mail: kahle@mat.uc.pt

© Springer International Publishing AG 2017 395

E. Agazzi (ed.), Varieties of Scientific Realism,

DOI 10.1007/978-3-319-51608-0_22

2