# 097 - Modelling in Mathematics: a common interest for Philosophy and History?

## Abstract

Clearly mathematics offers the technical tools for other disciplines to make models of phenomena of interest. For example, mathematical models in astronomy, physics, biology and economics are familiar – if not in detail, at least for their important role in establishing some predictive power, and thereby credibility and status, for the discipline. Often such models are stepping stones, or mediators [1] between the world and theory. Perhaps such models help our understanding. But predictive power is not necessarily explanatory power [2]. The legitimation and adoption of such models (whether mathematical or not) is often accompanied by the telling of an informal ‘story’ concerning the context, mechanisms and consequences of the model in question (cf. Hartmann in [1]). How do the story-composition and the story-telling relate to the model-making and why are they so effective? How does the link between story and model operate in relation to understanding for the first time (discovery/invention) and to understanding by others (learning)?

What is so useful about mathematical tools? Mathematical language offers a precise, clear representation of a problem situation relating to phenomena in the world; symbolic methods facilitate the manipulation of such representations for the sake of proofs and application of theories; well-established theories at many levels of abstraction allow for the application of powerful methods and results. These are generally regarded as advantages of mathematical methods over those that rely wholly on natural language or physical methods. However, these advantages are properties of the mathematical method once it is underway – they belie the cognitive processes of identification, experience, interpretation and attention that are essential preliminaries.

But what of mathematics itself? Can the practice of modelling also be applied to mediate between mathematical domains and their theory? Philosophically, does the difference between a phenomenal and an abstract ‘world’ matter here? Historically, do we not already do such modelling in mathematical practice? What does modelling in such a context really mean?

The construction, and use of mathematical models has been transformed in recent decades by the translation of their development and use to computer-based environments. This has opened the way to large scale, automated data collection and mining, and to the use of libraries, tools and processing far beyond the human scale. It has given rise to so-called ‘computational science’ [3]. It has also, perhaps unwittingly, been both constrained and promoted by conventional paradigms for the programming and development of systems that have given tacit priority to language, logic and function at the expense of more human, situated factors such as context, experience and interpretation.

An unconventional computing paradigm – Empirical Modelling [4] as developed at Warwick – supports a much broader style of modelling that embraces both the symbolic and the experiential elements of our thinking. At the heart of this modelling is the notion of construal as defined by Gooding, ‘…a representation which combines images and words as a provisional or tentative interpretation of novel experience’ [5]. This is a kind of modelling that is much closer to human cognitive modelling than the ‘purely’ mathematical, but also allows access to some of the benefits of computing power. We suggest this kind of modelling can capture the combination of the informal and the formal, the intuitive and the logical, that is the essential pre-requisite to the development and expression of mathematics. We shall give a brief introduction to Empirical Modelling and relate it to a mathematical example. We shall also illustrate the role of a story in relation to a mathematical model from early work of Bolzano and a current first year Mathematics for Computer Scientists module at Warwick.

[2] Max Black, Chapter 13 of Models and Metaphors, Cornell University Press, 1962

[3] Paul Humphreys, Chapter 1 of Extending Ourselves: Computational Science, Empiricism and Scientific Method, Oxford University Press, 2004

[4] Links to information, publications and software are at: http://www.dcs.warwick.ac.uk/modelling

[5] David Gooding, Experiment and the Making of Meaning, Kluwer, 1990

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Full paper still in preparation.