An Aristotelian Realist Philosophy of Mathematics

Most of the traditional problems in the philosophy of mathematics arise, in 6 James Franklin’s words, out of the ‘oscillation between Platonism and nomi7 nalism, as if those were the only alternatives’ (p. 11). In An Aristotelian Re8 alist Philosophy of Mathematics Franklin develops a tantalizing alternative to 9 these approaches by arguing that at least some mathematical universals exist 10 in the physical realm and are knowable through ordinary methods of access to 11 physical reality. By offering a third option that lies between these extreme all12 or-nothing approaches and by rejecting the ‘dichotomy of objects into abstract 13 and concrete’, Franklin provides potential solutions to many of these tradi14 tional problems and opens up a whole new terrain for debate in the philosophy 15 of mathematics (p 15). The acknowledgement of this by no means new but 16 oft neglected Aristotelian position sheds refreshing new light on debates that 17 have become somewhat stagnant in recent times. Furthermore, by drawing at18 tention to the possibility of an Aristotelian alternative, Franklin opens the way 19 for a whole host of new debates to emerge regarding the correct Aristotelian 20 approach. The scope of the book is ambitious and the overall position defended 21 is controversial in a number of ways. As such, it gives rise to as many new ques22 tions as it provides answers. However, this should be seen as a positive, since 23 the many questions that arise are deeply significant and have been neglected by 24 philosophers of mathematics for far too long. 25 One of the beauties of the Aristotelian position introduced is that it has far 26 more scope for internal variation than either Platonist or nominalist alternatives. 27 Once one takes the step of acknowledging that some mathematical objects reside 28 in our universe, this opens up a whole new range of debates regarding how much 29 of mathematics can be understood in physical terms and in what way. Even if 30 Franklin’s is not the ultimate Aristotelian approach, it is a pioneering step that 31 could lead to many more interesting developments. Furthermore, by altering the 32 terms of debate, Franklin paves the way for novel anti-realist alternatives that 33 follow his lead by taking actual mathematical practice and empirical research 34 into the study of mathematical cognition more seriously. 35 The author’s wide-ranging understanding of the actual practice of mathe36 matics stands out throughout and it is refreshing to read a book on the phi37 losophy of mathematics by a practising mathematician. This fresh perspective 38 allows the author to develop a truly novel approach, with much room for fur39 ther expansion and refinement, without being burdened by some of the stale 40