Mathematical knowledge has traditionally been taken to be absolutely objective, i.e. completely independent of contingent facts about the agents who discover the results. Today, this absolutistic view of mathematics has been challenged by a number of different theories. Most noticeably, social constructivists such as David Bloor (1981, 2011) and Donald MacKenzie (1979) have stress the influence social factors have had on the development of mathematics, and Bloor simply describes mathematics as a social institution. Other theorists such as Rafael Núñez and George Lakoff (2000) have claimed mathematics to be embodied and fundamentally shaped by the practitioners' sensory-motor experience. In this paper will report from a qualitative study of the practice of working mathematicians (Johansen and Misfeldt, 2014). The study shows that the production of mathematical knowledge is conditioned both by social factors and by our experience of the material world. Thus, the study confirms some of the basic ideas of the two approaches mentioned above. However, the study also shows that mathematicians actively use and shape the material world as part of their work process, and thus the material world conditions mathematics not only through sensory-motor experiences but also thorough the affordances is offers especially concerning the creation and manipulation of representations. Furthermore, our study gives reason to questions the reductionism inherent in both the social constructivist and the embodiment approach. Mathematics cannot be reduced either to the social or to the material. On the contrary we will show how the interplay between these two types of conditions is clearly visible and shapes the development of mathematics.
How do the practices of doing research level mathematics generate knowledge? In this talk we look at one specific case of current research in pure mathematics, and analyze its minute details in order to gain some insight into the kinds of resources used to put in place a new piece of mathematical knowledge. The case we look at is a result in Field Theory and Algebraic Geometry. The result was discussed, proved, written as a paper, and published between 2012 and 2014 by a group of three mathematicians working in German and Israeli universities. The talk is based on observations and data (mainly drafts and email correspondence) collected in real time, and on multiple interviews with the participants.
The talk takes as a starting point the assumption that mathematical knowledge creation and use are human collective practices, and should be described and analyzed as such. It traces some minute details of the process of this mathematical research, and uses this story to point out some of the practices in research level mathematics which make possible the creation of new mathematical knowledge. Compared to the knowledge produced in other disciplines, mathematical knowledge is commonly granted a distinctive type of certainty. This certainty, as an empirical phenomenon, is usually associated with consensus and lack of even a possibility of disagreement. The talk therefore pays extra attention to the many types of disagreement that emerged during this production of new knowledge, and, more importantly – to the diverse ways of solving (and at many occasions- dissolving) those disagreements. The different resources used to deal with such problems are considered. These include textual resources, personal and social resources, and material resources. We focus on a few specific exchanges (from e-mail correspondences of the researchers) and see how the separation of “internal” issues of proof (“pure” mathematical technical context) from “external” issues (applications of the theorems, presentation of the ideas, format and wording, expected audiences of the paper, etc.) collapses when looking at mathematical research practices. I will claim that mathematical certainty (in this case) is achieved precisely through such a contingent assemblage of practices, some of which are later removed from the published product.
Finally, the case study will be used as a basis for a general consideration of the differences between thinking of mathematical certainty as a norm and abstract ideal, and thinking of it as a practical achievement.
Model theory is often related to philosophy of science in the context of the reductionist project promoted by Patrick Suppes, which, roughly speaking, revolved around the identification of models in the sense of the working scientist with mathematical models definable by a set-theoretical predicate. The main theme of this talk is that the rejection of this identification as implausible (as suggested more or less explicitly in the literature) should not come with a rejection of all interaction between model-theory and scientific modelling as irrelevant. A local use of model-theoretical notions or techniques can shed significant light on scientific practice (at times, hardly to be had in alternative ways), offer a sophisticated analysis of its relation to mathematics and afford subtle ways of understanding the conceptual dynamics of mathematical modeling itself. In support of this claim I discuss two examples involving a very small amount of model-theory, in essence only the concept of satisfiability, both summarized below.
Measurement models in empirical research rely on the idea that numbers measure certain empirical attributes. The foundational question concerning the nature of measuring numbers is also a practical question concerning the meaningfulness of numerical practices wherever it is sought to introduce them as instruments of investigation (psychology is a notable example). Numerical measures can be seen as model-theoretical objects and this point of view literally allows one to conceptualize the construction of measuring numbers from experimental practice. For measurement with a unit $u$ (and an absolute zero), a complete list of experimental records $M(x, u)$ (these are formulae) on an environment $E$ determines the measure of an object $a$ relative to $u$ as the subset of $M(x, u)$ satisfied by $a$ in $E$. This makes it apparent that numbers codify experimental interactions and can be seen as compressions of experimental information whose structure is directly induced by experimental operations. Whenever this account can be given for a type of experimental practice, measurement is meaningful for it.
Social scientists (especially economists) often confront descriptions of types of design that admit of no solution (an example is provided by Arrovian aggregation rules). These descriptions are linguistic but depend on set-theoretical parameters and they can be written as formulae with one free-variable in a sufficiently rich first-order language. Thus, an impossibility theorem amounts to the fact that, in a modelling universe $U$, a certain formula $F(x(1),\dots,x(n), P(1),\dots, P(n))$ with parameters $P(i)$ is not satisfiable. It is often thought that the only possible way of avoiding an impossibility consists in replacing $F$ with $G$, which expresses a weaker description of the original design (in the sense that it is strictly entailed by the original description). The model-theoretical formulation of the problem, however, shows very clearly that impossibilities may also be removed by a change of parameters, i.e., by replacing $F(x(1), \dots,x(n), P(1), \dots, P(n))$ with $F(x(1), \dots, x(n), Q(1), \dots, Q(n))$ (for at least one $i$ $P(i)$ is different from $Q(i)$). In particular, one may remove an impossibility while adopting a description strictly stronger than $F$. Several examples of occur in aggregation theory and utility theory.
I briefly conclude by pointing to areas in which only a little work has been done but significant progress is likely to come from the adoption of a model-theoretical framework (notably, the reconstruction of aggregation procedures as amalgamations of structures and the use of model-theoretical mixtures to build and study probabilistic models).