Mathematical abstraction is the process of considering and ma­nipulating operations, rules, methods and concepts divested from their refe­rence to real world phenomena and circumstances, and also deprived from the content connected to particular applications. There is no one single way of per­forming mathematical abstraction. The term “abstraction” does not name a unique procedure but a general process, which goes many ways that are mostly simultaneous and intertwined; in particular, the process does not amount only to logical subsumption. I will consider (...) comparatively how philosophers consi­der abstraction and how mathematicians perform it, with the aim to bring to light the fundamental thinking processes at play, and to illustrate by signifi­cant examples how much intricate and multi-leveled may be the combination of typical mathematical techniques which include axiomatic method, invariance principles, equivalence relations and functional correspondences. (shrink)

Neurosciences and cognitive sciences provide us with myriad empirical findings that shed light on hypothesised primitive numerical processes in the brain and in the mind. Yet, the hypotheses on which the experiments are based, and hence the results, depend strongly on sophisticated abstract models used to describe and explain neural data or cognitive representations that supposedly are the empirical roots of primary arithmetical activity. I will question the foundational role of such models. I will even cast doubt upon the search (...) for a general and unified philosophical foundation of an empirical science. First, it seems to me hard to draw a global and coherent view from the innumerable and piecemeal neuropsychological experiments and their variable, and sometimes uneasily compatible or fully divergent interpretations. Secondly, I think that the aim of empirical research is to describe dynamical processes, establishing correlations between different sets of data, without meaning to fix an origin or to point to a cause, let alone to a ground. From the very scientific and philosophical point of view it is essential to distinguish between explanations, which provide correlations or, at best, causal mechanisms, and grounding, which involves a claim to some form of determinism. (shrink)