Philosophy of Mathematics Seminar


Philosophy of Mathematics seminar


Mary Leng, York University

To Be (Thin) Or Not To Be? Are Thin Objects Fictional Or Are Fictional Objects Thin?  

Øystein Linnebo offers an account of mathematical objects that promises to bypass some of the features that can make standard Platonism seem unpalatable. Although Linnebo’s mathematical objects are abstract and indeed exist of necessity, their existence is lightweight: they make little by way of demands on the world. Given the quite bearable lightness of being, on Linnebo’s account, why persist with fictionalist qualms about the existence of mathematical objects? This paper has two main aims. First, by considering some key features of Linnebo’s account of mathematical objects as ‘thin’, it raises the question, ‘Are thin objects really just fictional objects?’. For all Linnebo’s insistence on the necessary existence of mathematical objects, it will be argued that there are features of the ‘objects’ he introduces that suggest that they fall short of really existing and should instead be thought of as merely fictional. Second, it considers the rejoinder that, rather than thin objects being merely fictional (and so not existing at all), the bar to existence has been lowered so that fictional objects are shown to exist and to be thin (so that fictionalism turns out not to be a nominalist position at all, with fictional objects turning out to be thin objects). Conceding that the difference between Linnebo’s thin realist perspective and the fictionalist perspective might be more a matter of taste (concerning how to use the word ‘exists’) than of substance, it will finally be noted that Linnebo’s account is likely to be vulnerable to similar challenges faced by the fictionalist, particularly around its account of modality, and its account of the explanatory role played by mathematical objects in empirical science.