# Richard Lawrence

## Research

(For a printable PDF version of my research statement, click here.)

My research engages with contemporary and historical issues in the
philosophy of mathematics, philosophy of language, logic, and
linguistics. My research in these areas stems from an interest in a
foundational question: how does *meaning* relate to our practices of
*inquiry*? How is what we say about the world shaped by the ways we
ask and answer questions about it?

This question is important for anyone who holds that we can use language to communicate knowledge. For a sentence to express knowledge about something, its meaning must be related to what we inquire into when we want to show that it is true. The sentence “Jupiter is a gaseous planet”, for example, can communicate knowledge about astronomy. But if we could show, without undertaking any astronomical inquiry, that this sentence was true, then no astronomical knowledge would be required to justify asserting it; and so no such knowledge could be communicated from someone who asserts it to someone who understands its meaning. The idea that a particular sentence can communicate a particular bit of knowledge thus presupposes a relationship between what the sentence means and how we inquire into its truth.

I am interested in many issues in logic, mathematics, and natural language semantics that address this relation of meaning to inquiry. In the philosophy of logic, I am interested in alternative logics and styles of semantics which connect meaning directly with conditions of inquiry, such as game-theoretic and proof-theoretic semantics, dependence logics, and erotetic logic.

My work focuses in particular on the logical distinction between complete expressions (e.g., terms and sentences) and incomplete expressions (e.g. predicates and functions) from a semantic perspective. This encompasses historical questions, such as how Frege understood his distinction between objects and concepts, as well as more systematic questions, such as how we understand first- and second-order variables. I have argued in my dissertation and subsequent work that we can best understand this semantic distinction in terms of two different roles in inquiry.

The distinction is directly relevant for issues in contemporary philosophy of mathematics…

##### “Giving the value of a variable”

In this article, which is based on my dissertation work, I examine the
connection between inquiry and meaning by taking problems in
elementary algebra as a case study. In elementary algebra, an equation
like \(x^2 - 6x + 9 = 0\) gives a problem: what is the value of \(x\)?
We answer this question with another equation that gives the value of
\(x\): \(x = 3\). Since answers to many other kinds of questions can
be conceived as giving the values of variables, the process of
answering an algebraic question serves as a model for inquiries more
generally. But a puzzle arises about what these equations mean. These
two equations are truth-conditionally equivalent; each is true if, and
only if, \(x\) takes the number \(3\) as its value. So in what sense
could the second equation be an informative answer? Why does it count
as *solving* the problem that the first equation poses? I argue that
four different features are required for an equation to give the value
of a variable and thereby answer an algebraic question: the variable
must be in the *scope* of the problem statement; the values given must
be in the *range* of the variable; the statement giving the values
must represent a *complete* solution; and it must be in a *canonical
form*. These features are a guide to the structure of inquiries more
generally: to answer a question is to find a statement that has these
four features.

##### “Frege, Hankel, and formalism in the *Foundations*”

Since completing my dissertation, my work has focused on historical
research about Frege’s distinction between objects and concepts. In
the *Foundations of Arithmetic*, Frege explicitly introduces this
distinction in order to offer a critique of formalism in mathematics.
In this article, I examine Frege’s early engagement with formalism in
the *Foundations*, where his main formalist interlocutor is Hermann
Hankel. Hankel’s text is not well-known among Frege scholars, but I
argue that it had an important influence on Frege. Hankel, like Frege,
wants to show against Kant that arithmetic is analytic, and he does so
by arguing that basic arithmetic truths like \(7 + 5 = 12\) can be
deduced from purely analytic axioms. Hankel’s formalism thus
anticipates Frege’s logicism to a significant extent. This undercuts
Frege’s claim that his logicism is “completely
different” from Hankel’s formalism, and raises the question of
where the differences really lie. I argue that Hankel and Frege both
recognize concepts as a kind of content which may be freely postulated
or defined. But Frege differs from Hankel in recognizing that
arithmetical terms have a different kind of content which we cannot
merely postulate; before we can use arithmetical terms, we first need
to *prove* that there are objects which serve as their contents. Frege
thus aligns the distinction between concepts and objects with a
practical distinction in mathematical inquiry, between what can be
postulated, and what must be proven.

##### Book project: *Frege among the formalists*

There is additional historical work to be done on Frege’s engagement
with formalism. While Frege’s opposition to formalism is well known,
the views of the formalist figures he engages with, and the details of
Frege’s arguments against them, are under-explored. Moreover, his
criticisms of formalism are important textual sources for his thinking
about objects, functions, and the category of *Bedeutung* in general.
Frege continues to stress in his later work that formalism fails
because it postulates, rather than demonstrates, the existence of
mathematical objects, which results from a failure to distinguish
objects from concepts. He also sometimes charges formalism with
failing to distinguish signs from their *Bedeutung* entirely, or with
assigning them a kind of *Bedeutung* that makes them unsuitable for
use in science. To make sense of these claims and the consequences
they have for Frege’s overall theory, we need a better picture of his
other formalist interlocutors, including Heine and Thomae (whose views
stem from Weierstrass), Hilbert, and Wittgenstein. My plan for the
next three years is to write a book on this topic. The book will
investigate the views of these interlocutors on their own terms, in
order to bring unknown details to light, to describe their impact on
Frege’s views, and to explain the influence that they had—both
through Frege and in spite of him—on the later development of
analytic philosophy. I have developed this research into a grant
proposal; the proposal is currently under review at the Deutsche
Forschungsgemeinschaft.