Logical pluralism is the view that different logics are equally appropriate, or equally correct. Logical relativism is a pluralism according to which validity and logical consequence are relative to something. Stewart Shapiro explores various such views. He argues that the question of meaning shift is itself context-sensitive and interest-relative.

Mathematical fictionalism (or as I'll call it, fictionalism) is best thought of as a reaction to mathematical platonism. Platonism is the view that (a) there exist abstract mathematical objects (i.e., nonspatiotemporal mathematical objects), and (b) our mathematical sentences and theories provide true descriptions of such objects. So, for instance, on the platonist view, the sentence ‘3 is prime’ provides a straightforward description of a certain object—namely, the number 3—in much the same way that the sentence ‘Mars is red’ provides a (...) description of Mars. But whereas Mars is a physical object, the number 3 is (according to platonism) an abstract object. And abstract objects, platonists tell us, are wholly nonphysical, nonmental, nonspatial, nontemporal, and noncausal. Thus, on this view, the number 3 exists independently of us and our thinking, but it does not exist in space or time, it is not a physical or mental object, and it does not enter into causal relations with other objects. This view has been endorsed by Plato, Frege (1884, 1893-1903, 1919), Gödel (1964), and in some of their writings, Russell (1912) and Quine (1948, 1951), not to mention numerous more recent philosophers of mathematics, e.g., Putnam (1971), Parsons (1971), Steiner (1975), Resnik (1997), Shapiro (1997), Hale (1987), Wright (1983), Katz (1998), Zalta (1988), and Colyvan (2001). (shrink)

In this paper we introduce a paraconsistent reasoning strategy, Chunk and Permeate. In this, information is broken up into chunks, and a limited amount of information is allowed to flow between chunks. We start by giving an abstract characterisation of the strategy. It is then applied to model the reasoning employed in the original infinitesimal calculus. The paper next establishes some results concerning the legitimacy of reasoning of this kind - specifically concerning the preservation of the consistency of each chunk (...) and concludes with some other possible applications and technical questions. (shrink)

Many advocates of non-classical logic for reasons external to mathematics claim that their proposed revisions are consistent with the use of classical logic within pure mathematics. Doubts are raised about such claims, concerning the applicability of pure mathematics to natural and social science. -/- .

In the paper of Brown and Priest 2004, the authors developed the chunk and permeate method, which they described as a ?paraconsistent reasoning strategy?. There it is suggested that the method of chunk and permeate could apply to the historical infinitesimal calculus. However, no attempt was made to look at actual historical examples. In this paper, I show that the method of chunk and permeate can indeed apply, as a rational reconstruction, to certain of Isaac Newton's arguments that use infinitesimals. (...) This rational reconstruction maintains and uses, rather than sidesteps, the apparent contradictions in Newton's arguments. The applicability of chunk and permeate to other historical arguments, e.g. of Leibniz/L'Hospital and Fermat, has also been investigated and will be communicated in future publications. (shrink)

In this paper I will review the developments in the foundations of mathematics in the last 150 years in such a way as to show that they have delivered something of a rather different kind: mathematical pluralism.