Top-Down and Bottom-Up Philosophy of Mathematics

Abstract

Abstract. The philosophy of mathematics of the last few decades is commonly distinguished into mainstream and maverick, to which a 'third way' has been recently added, the philosophy of mathematical practice. In this paper the limitations of these trends in the philosophy of mathematics are pointed out, and it is argued that they are due to the fact that all of them are based on a top-down approach, that is, an approach which explains the nature of mathematics in terms of some general unproven assumption. As an alternative, a bottom-up approach is proposed, which explains the nature of mathematics in terms of the activity of real individuals and interactions between them. This involves distinguishing between mathematics as a discipline and the mathematics embodied in organisms as a result of biological evolution, which however, while being distinguished, are not opposed. Moreover, it requires a view of mathematical proof, mathematical definition and mathematical objects which is alternative to the top-down approach.

Matematikaren filosofia fundamentista vs. ez-fundamentista

“…Lakatos hartu izan da matematika ez-fundamentistaren abiapuntu bezala (Aspray & Kitcher, 1988;Hersh, 1997;Cellucci, 2013) eta bere idatzietatik edan izan dute gerora bide berriak proposatu dituzten hainbat autorek. Ezin daiteke esan Lakatosek bere matematikaren filosofia modu sistematikoan garatu zuenik.…”

“…Kitcher da matematikaren filosofia ez-fundamentistaren tradizioan eragin handiko beste autore bat (Hersh, 1997;Cellucci, 2013). The Nature of Mathematical Knowledge (1983) liburuan azaldu zuen Kitcherrek bere posizioa matematikaren filosofian.…”

“…Harrez gero ekarpen desberdinak izan dira. Kitcherrena da horien artean garrantzitsuenetakoa Lakatosenarekin batera beste batzuengan izan duen eraginagatik, hauengan, besteak beste: Davis, Hersh, Tymoczko, Rota, Grosholz, Rav eta van Bengedem (Cellucci, 2013).…”

Formalizing darwinism, naturalizing mathematics

“…This view is at odds with a naturalist stance, because the way in which it defines mathematics and logic makes impossible to naturalistically account for our ability in producing and using such abstract devices (Cellucci, 2013a(Cellucci, , 2013b. Indeed, Dutilh Novaes also considers formal languages a "technology", i.e.…”

How can expert knowledge increase the realism of conceptual hydrological models? A case study based on the concept of dominant runoff process in the Swiss Pre-Alps

“…These two approaches have been applied in nearly all scientific disciplines, e.g. in mathematics (Cellucci, 2013), economics (Böhringer and Rutherford, 2008), and neuroscience (Gilbert and Li, 2013), as well as hydrology. Thus one type of hydrological scientist, the experimentalist or "wet" hydrologist, tries to understand catchment functioning through extended field investigations, whereas the conceptual modeller or "dry" hydrologist tends to develop theories at the catchment scale and successively tries to validate them against measurements (Seibert and McDonnell, 2002).…”