In the twentieth century investigations within the analytic tradition into the development of scientific theories can be classified into two broad strands: one that emphasizes the linguistic character of theories, but considers the notion of discovery as not amenable to philosophical inquiry (e.g., Popper, Hempel), and one that focuses on discovery processes, but vehemently rejects considering theories as linguistic entities (e.g., Hanson, van Fraassen). The purpose of my dissertation is to bridge this unfortunate divide by elucidating a particular aspect of the methodology of theoretical science, with special emphasis on mathematics, namely the practice of developing, employing, or studying systems of axioms, for short, axiomatics.

More specifically, my thesis is guided by the following two questions:

• Why do mathematicians, and theoretical scientists in general, axiomatize the body of knowledge they are investigating?
• How does the fact that they do so influence further research, inventions, and discoveries?

The standard answers to these two questions are (1) that they axiomatize theories purely for expository purposes, and (2) that as such it has little or no influence on further research and discoveries. In contrast, I claim that axiomatics is an engine for driving the development of mathematics and science; in particular, it facilitates finding and expressing analogies and it provides a powerful handle for grasping and using abstract notions.

In my thesis I present a characterization of axiomatic theories that is broad enough to encompass such works as Euclid's Elements, Newton's Principia, and Hilbert's Foundations of Geometry. Formulating axiomatic theories, as well as employing and investigating them, are all parts of axiomatics. This broad conception of axiomatics allows us to sidestep particular views that have been held regarding the proper form and function of axiom systems, their epistemological status, and the aims of particular axiomatizations, and concentrate on those features of scientific and mathematical theories that result in virtue of the axiomatic formulation. This is very much in the spirit of Patrick Suppes's comment that the classical philosophical discussions of the nature of abstract mathematical objects may fruitfully be replaced by concentration, not on mathematical objects, but on the character of mathematical thinking.

Naturally there is a close connection between axiomatics and the notion of logical consequence, since the latter is employed for drawing conclusions from the axioms. In fact a clean separation between assertions of a theory and inferences between them is a consequence of the attempt to explicitly state all assumptions being made, which is one of the main motivations behind axiomatics. However, in my discussion the issues regarding axiomatics and logical consequences have been separated as far as possible, so that the main aspects of axiomatics can be presented without presupposing a particular notion of logical consequence. This allows us to concentrate on those issues that depend on the fact that a theory is presented axiomatically and avoid the popular conflation of axiom systems and formalizations (I take this conflation to be the basis for the rejection of axiomatics by the second strand mentioned above).

Because of the dual role of, on the one hand, regimenting the language of discourse and, on the other hand, of determining a class of models, axiom systems constitute a link between sentences and abstract notions. By providing access to a theory via a small number of axioms, the attention of the theoretician is focused on a limited set of statements, which can be investigated and manipulated in a systematic manner. This use of axiomatics goes back to the ancient Greeks and many philosophers since then have taken Euclid's presentation as model for their own investigations. Thus, it is all the more surprising that the benefits of axiomatics in theory development have not yet received a systematic treatment in the philosophical literature. Such benefits include that the formulation of axioms can bring out hidden assumptions, explicate informal concepts, or reveal gaps in the argumentations; once a theory is axiomatized it can be studied through the axioms, and relations to other theories can thereby be established; manipulations of axioms, which can be motivated by empirical findings that contradict some theorem (e.g., the psychologist Clark Hull's theory of rote learning) or by attempts to prove the independence of the axioms can suggest new theories (non-Euclidean geometries are the most famous outcome of the latter, but there are many others).

In addition, I argue that axiomatics has a considerable effect on the perception and formulation of analogies between scientific theories or mathematical notions. This has been mentioned and put into use by researchers in the late nineteenth century (e.g., Maxwell, Dedekind, Hilbert), but has been neglected almost completely in the discussion of analogical reasoning in philosophy of science and cognitive science, where it has become standard to explicate analogies purely in terms of structure preserving mappings (Gentner). According to this view two domains are analogous, if some objects and relations of one domain can be put into a one to one correspondence to objects and relations of the other. This correspondence then justifies the transfer of knowledge from one domain to the other. Although this characterization works for simple examples, I show that it is inadequate for explicating analogies within a large number of domains that frequently occur in mathematics, where an axiomatic approach is successful.

Moreover, I argue that abstract notions are a rich source for mathematical innovations, which gives a reason for the fact that mathematics has become "more abstract" over the past centuries. Although the claim is often made that we can have some kind of intuitive grasp of particular abstract notions like natural numbers, this is highly implausible for most abstract structures that occur in the sciences (non-Euclidean geometries, abstract spaces, etc.). Rather, our abilities to access, exert control over, and understand such notions make essential use of axiomatics (this is acknowledged in the fine print even by proponents of the "semantic view of theories" and "structuralists" with respect to the ontology of mathematics).

In sum, I show how the role of axiomatics goes beyond purely aesthetic considerations, and also beyond providing some form of justification for the axiomatized theory. The systematic investigations in my dissertation into the role of axiomatics in driving discoveries in the theoretical sciences are augmented by a number of case studies from the history of mathematics and science, which provide evidence that axiomatics has in fact been used in the way I argue for.

The use of axiomatics in scientific research is part of my general interest in the questions

• How we can systematically increase our knowledge?
• How do the means we thereby employ (e.g., axiomatics) influence the kind of knowledge we obtain?

The domains of knowledge that I am most curious about are mathematics and science. These questions touch upon a wide variety of problems from different areas that I intend to pursue in my future research. These include further historical studies of scientific episodes that reveal the impact of axiomatics for attaining progress, as well as the careful study of such views that deny such a role (e.g., Brouwer). At the same time I am also interested in systematic investigations regarding analogical reasoning, concept formation, and theory development, and I'd like to continue pursing attempts to clarify the nature of the notions that arise in theoretical investigations in a way that takes the actual practice of these investigations seriously (e.g., Dedekind, Bernays). Here computational and cognitive aspects have to be taken in consideration. Thus, my research interests fall into the areas of history and philosophy of mathematics and science, epistemology, and cognitive science.