The formal sciences - mathematical as opposed to natural sciences, such as operations research, statistics, theoretical computer science, systems engineering - appear to have achieved mathematically provable knowledge directly about the real world. It is argued that this appearance is correct.

Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem--realism--is examined and rejected in favor of another approach--naturalism. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.

James Franklin has argued that the formal, mathematical sciences of complexity — network theory, information theory, game theory, control theory, etc. — have a methodology that is different from the methodology of the natural sciences, and which can result in a knowledge of physical systems that has the epistemic character of deductive mathematical knowledge. I evaluate Franklin’s arguments in light of realistic examples of mathematical modelling and conclude that, in general, the formal sciences are no more able to guarantee certainty (...) than the natural sciences. Yet the formal sciences are characterized by a ‘domain-independence’ that is philosophically interesting, and I argue that it is this property that Franklin actually employs to distinguish the formal from the natural sciences. I use Einstein’s ‘principle’/‘constructive’ theory distinction to contrast the domain-independence of physical theories with the domain-independence of formal mathematical theories, and show how both kinds of domain-independence function to generate the domain-independence that is observed in the complex systems sciences. © 1999 Elsevier Science Ltd. All rights reserved. (shrink)