The main goal of my current research is to discover a philosophy of language suitable for mathematical language, in order to better understand how we acquire mathematical knowledge. This leads me to issues concerning the nature of concepts and conceptual change more generally. Although my work draws on insights from the analytic tradition (of Frege, Russell, Church, and Kripke), I also use empirical work on the cognitive capacities that are associated with the relevant aspects of thought and language, as well as work on the history and anthropology of mathematics, to the extent that this work augments the lessons of the analytic tradition.

My latest project begins with a critical evaluation of Frege’s view that the history of conceptual change is really the history of human beings gaining an increasingly clear and complete understanding of *the same* stock of objective and immutable notions, axioms and theorems. I evaluate the plausibility of this view in the light of the various kinds of partial mathematical understanding that have been exhibited throughout history, especially those that involve not merely a failure to reflect on knowledge and abilities that one already has, but also a lack of theoretical knowledge. I will now discuss an example of this, one that also reveals a flaw in the doctrine —suggested by Frege’s examples— that names (like ‘Aristotle’) are defined in terms of definite descriptions (like ‘the greatest student of Plato’).

Turning to mathematical names (like ‘π’ and ‘*e*’), Frege is committed to the view that the meaning of ‘π’ is as eternal and immutable as the meaning of any other mathematical expression, and that the familiar geometric definition of it —as the ratio of the circumference of any circle to its diameter— represents a partial understanding that can only be completed holistically, by understanding how π behaves in many different mathematical settings outside of geometry. This example of partial understanding exemplifies understanding that involves a lack of substantive theoretical knowledge, in this case a lack of non-geometrical mathematical knowledge. Further, it also exemplifies how partial understanding is answerable to objective facts: in this case, facts about how π relates to other entities in non-geometrical settings. This is a claim that I will develop using the history of π as a case study. I will argue that, for Frege, the objectively correct definition of ‘π’ —the one that represents complete understanding— must coincide with each of the older and more specialized definitions, when the newer definition is applied to the respective subject matters of the older definitions.

My study of ‘π’ will form part of a broader critical study of the role of mathematical names in Frege’s philosophy. From a Fregean perspective, the role of such names seems to be psychological, in that they are introduced to help us overcome our intellectual limitations. For example, by introducing names and common nouns, we can draw attention to tacit knowledge that was previously unrecognized; we can also abbreviate the relevant descriptive definitions within calculations and proofs, which would otherwise be beyond our capacity to parse and, consequently, psychologically impossible for us to understand. I am currently investigating whether this Fregean view does justice to the role of mathematical names such as ‘π’, or whether they also allow us to represent mathematical entities in a way that is singular (or *de re*).

This issue is closely related to Kripke’s distinction between defining a term and fixing its reference (as well as to his much disputed claim that there are truths that are contingent but known *a priori*). Armed with this distinction, it becomes easier to see why one might think that mathematical names refer in a way that is singular (or *de re*): it is because they always refer to the same thing despite changing definitions. Crucially, as mathematics evolves over time, so does the cluster of descriptive definitions that are associated with a name. As I have already suggested, in the case of ‘π’, these definitions describe how π occurs in new settings – ones outside of geometry. Thus, as Kripke claims, ‘π’ does not simply abbreviate any particular descriptive definition, since no such definition is “the last word” about π. For while the sophisticated, general definitions that were discussed previously are a major achievement, the claim that there is one such definition of ‘π’ that is, objectively, the correct definition, implies the false claim that of all the many proofs of the various theorems about π, there is, in each case, only one proof that is objectively the right proof. This does not accurately describe the situation in mathematics.

Even if one accepts this argument that ‘π’ does not abbreviate any particular definition, one must still show that it always refers to the same thing despite changing definitions. The problem is that although mathematical hindsight suggests that changes in the meaning of ‘π’ are determined by the aforementioned facts about how π relates to other entities in non-geometrical settings, this kind of determination does not, on its own, suffice to justify the claim that sameness of reference is preserved. After all, in ancient Greek mathematics, π is a constant ratio of geometric magnitudes, whereas, in current mathematics, π is a real number and so not a geometric entity. The natural thing to say, from a Kripkean point of view, is that π is a real number to which prior mathematicians were able to refer, despite the fact that they described it purely geometrically. A key goal of my project will be to see if this claim can be justified by studying the history of π —and thus the history of the concept of the limit of an infinite series— through the lens of Kripke’s work in *Naming and Necessity*, his famous distinction between speaker’s and semantic reference, as well as his unpublished lectures and seminars on *de re *attitudes towards numbers.