# Strategic Research Project on Logic and Philosophy of Mathematical Practices

#### Core Aims

The CLWF has been internationally recognized for its contribution to the development of the study of (the philosophy of) mathematical practices, and especially its role within the APMP. Research within this relatively new domain has already dealt with a number of questions, but a number of clear challenges for the future have arisen as well. These have primarily to do with bridging divides that exist at the moment. Of these, the most important ones are the following two:

**The internal-external divide (IED):**up to now the research focus has been largely on the internal development and growth of mathematics (a line of research that was initiated by Imre Lakatos in his seminal Proofs and Refutations book of 1976, where the “history” of a mathematical proof is sketched). This internal focus has resulted in an uneasy relationship with the sociology of mathematics (and related topics, such as ethnomathematics and mathematics education although these issues will not be addressed in this application) that pays more attention to the external development, an approach that was most strongly developed by such researchers as Sal Restivo (1992, 1993), David Bloor (1976), and, for ethnomathematics, Alan J. Bishop (1988) and Ubiratan D’Ambrosio (1990, 2007).**Foundations and practice (FP):**the relation with the still dominant foundational approach to mathematics and its philosophy remains problematic, and is at times even antagonistic. Of course, one could claim that these are indeed two different approaches, complementary at best but definitely without overlap. On the other hand, it seems clear that foundational theories in mathematics, such as Zermelo-Fraenkel set theory, can themselves become the topic of a case-study in the philosophy of mathematical practice. Conversely, a better understanding of mathematical practices may (if not should) lead to a better insight into what theories can claim to be foundational. Furthermore, as Hersh (1997) argues, there is also a connection between the foundational approach to mathematics and the teachability of mathematics.

#### Research questions

The specific research questions that follow deal with selected aspects of these two challenges and are therefore not meant to be exhaustive. They are rather determined by the present, available expertise in the CLWF. Of greater importance is the fact that they are to a certain extent relatively independent from one another. Consequently, the five themes proposed below can run perfectly in parallel and this is the scheme we have in mind. They all start simultaneously and provide input for each other on a continuous basis.

##### IED1: Explanatory value of proofs and the power of conviction.

Explanatory value of proofs and the power of conviction. It is clear that explanation plays an important role for mathematicians. A mathematical proof that has explanatory power is usually preferred over a proof that lacks it. Any theory of mathematical practice should deal with this problem. So far a few approaches have been outlined but a lot of work remains to be done (see, e.g., Mancosu (2008)). The guiding hypothesis here is that, in contrast with the existing models that refer to purely internal characteristics, a broader idea of what explanation is, plays a central role. The task of this subproject is to go through existing theories of explanation and see whether these are applicable to mathematics. If none of these proves to be satisfactory, a bottom-up strategy will be followed: case studies will have to be gathered to arrive at insights and (a) proposal(s) for mathematical explanation.

##### IED2: The nature of diagrammatic reasoning.

We tend to think of mathematical proofs as written texts, mixtures of ordinary language (using expressions, such as “Therefore”, “Let this or that be given”, …) and mathematical symbols. In the ideal situation all use of ordinary language can be eliminated. This view is contradicted by actual mathematical practice where drawings, figures, diagrams, …, play a fundamentally important role. Hence it is important to understand how these diagrams function. As said, work has been done (see, e.g., Carter (2010)), but a lot of bottom-up work will be necessary, so in the first phase materials have to be collected to come to a theory of diagrams in the second phase. It must be mentioned here that there are logical approaches to the problem, see, e.g., Shin (1995) and Mumma (2010), which establishes a relation with the FP subprojects.

##### IED3: Mathematics and its social organization.

The recent development of the Polymath project, see Van Bendegem et al. (to appear 2013), where the search for a proof of a theorem was posted on the web thereby inviting everyone to participate, made clear that the way a scientific community is organized influences the results that community can obtain. Although this is a well-known idea in the philosophy of science, this is less so, if existent at all, in mathematics. As there are several good instruments for modelling a research community, for which the extensive literature in the sociology of science is a guarantee, the task of this subproject is to study (a) whether social organization has an impact on the results that are obtained and (b) if so, what precise mechanisms are at work. See, e.g., Geist et al. (2010) for a case study concerning peer review. It is not to be excluded that the results of this subproject could in principle be tested by, e.g., setting up experiments similar to the Polymath project.

##### IED4: Ethnomathematics: how mathematics is embedded in a culture.

Although it seems obvious that mathematical practices in the Western world, i.e., mathematics as it is mostly practised in the academic setting, combined with a very peculiar and particular model of transmitting mathematical knowledge to society at large, both to young people and adults, can and should be compared to mathematical practices in other cultures, there is hardly any material to be found. Either it is considered to be part of anthropology, sociology or cultural studies but not of philosophy, let alone philosophy of mathematics. The most fascinating aspect is the way different mathematical practices (including Western mathematical practices) are embedded in society at large. Of course, the ethnomathematical literature is quite extensive, so little or no material needs to be collected by ourselves, but a lot of work remains to be done to characterize sociologically Western mathematics to make a cross-cultural comparison possible and meaningful (see, e.g., Pinxten & François, 2011).

##### FP1: Development of the logics required for modelling mathematical practice.

To develop a formal approach to the study of mathematical practice, needed to make comparisons possible with foundational studies, we shall focus on the development of two separate features of the formalisms we believe are needed. One aspect is fairly general, and focuses on the need to model the processes whereby we share and communicate mathematical knowledge. This approach is continuous with recent research in dynamic epistemic and dynamic doxastic logic (including belief revision), and should allow us to model the information flow within as well as between different mathematical communities, see, e.g., Van Benthem (2007); Van Ditmarsch et al. (2007). A second aspect is more specific, as it relates to the specificity of what mathematicians really communicate. This means that we should be interested in “real” rather than in idealized proofs. To incorporate these features, we shall have to go beyond existing formal approaches.

##### FP2: Argumentation and problem-solving networks.

Mathematical knowledge is not only shared and communicated by passing on proofs, but proofs are also used to convince that a certain theorem is true. To model this aspect, we propose to explore the so-called argumentation networks and problem-solving networks that were mainly developed by researchers in Artificial Intelligence (see Dung (1995)). Groundbreaking work has already been done very recently, see Pease & Aberdein (2011), and it shows clearly that the incorporation of insights from argumentation theory is a natural move, and the adoption of formal models of that allow us to represent and study the interaction between arguments and counter-arguments (arguments can be stronger or weaker, can reinforce each other, or oppose each other) follows immediately.

#### Selected references

- Bishop, A. (1988). Mathematical enculturation. Dordrecht: Kluwer.
- Bloor, David (1976). Knowledge and social imagery. London; Boston: Routledge & K. Paul.
- Carter, Jessica (2010). “Diagrams and Proofs in Analysis”. International Studies in the Philosophy of Science, vol. 24, 1, 1-14.
- D’Ambrosio, Ubiratan (1990). “The History of Mathematics and Ethnomathematics. How a Native Culture Intervenes in the Process of Learning Science”. Impact of Science on Society. 40(4), 369-377.
- D’Ambrosio, Ubiratan (2007). “Peace, Social Justice and Ethnomathematics”. The Montana Mathematics Enthusiast. Monograph 1, 25-34.
- François, Karen & Van Bendegem, Jean Paul (Eds.). (2007). Philosophical Dimensions in Mathematics Education. New York: Springer.
- Geist, Christian, Benedikt Löwe & Bart Van Kerkhove (2010), “Peer Review and Knowledge by Testimony in Mathematics”, in: Benedikt Löwe & Thomas Müller (eds.), Philosophy of Mathematics: Sociological Aspects and Mathematical Practice. Texts in Philosophy 11, College Publications, London: 155-78.
- Hersh, R. (1997). What is Mathematics, Really? London: Jonathan Cape.
- Lakatos, Imre (1976). Proofs and Refutations. Cambridge: Cambridge University Press.
- Mancosu, Paolo (ed.) (2008): The Philosophy of Mathematical Practice. Oxford: Oxford University Press.
- Mumma, John (2010). “Proofs, Pictures and Euclid,” Synthese, 175(2): 255-287.
- Pease, Alison & Aberdein, Andrew (2011). “Five theories of reasoning: interconnections and applications to mathematics”, Logic and Logical Philosophy, 20(1-2): 7-57.
- Restivo, Sal (1992). Mathematics in Society and History. Sociological Inquiries. Series Episteme. Volume 20. Kluwer Academic Publishers, Dordrecht.
- Restivo, Sal; van Bendegem, Jean Paul & Fischer. Roland (Eds.). (1993). Math worlds: philosophical and social studies of mathematics and mathematics education. Albany : State University of New York Press.
- Shin Sun-Joo (1995). The logical status of diagrams. Cambridge: Cambridge University Press.
- Van Bendegem, Jean Paul, Bart Van Kerkhove & Patrick Allo (to appear 2013). “Mathematical Arguments and Distributed Knowledge”, in: Andrew Aberdein & Ian J. Dove (eds.), The Argument of Mathematics. New York: Springer.
- Van Benthem, J. (2007). “Dynamic logic for belief revision.” Journal of Applied Non-classical Logics 17(2): 129-55.
- Van Ditmarsch, H., W. van der Hoek, et al. (2007). Dynamic Epistemic Logic. Dordrecht, Springer.