I recently attended the Second International Conference on the Philosophy of Mathematics, in Moscow, 28-30 May http://vfc.org.ru/eng/events/conferences/philmath2009/. For some time I have been interested in the ways that engineers and technologists use maths in a very different way from scientists and mathematicians themselves. Here’s a (rather long) abstract of my paper.
Models of technology
Engineeering, like science, uses mathematical descriptions of the world. Because engineering models use many of the same mathematical techniques as scientific models (differential equations, Fourier and Laplace transformations, vectors, tensors, for example) it is easy to assume that they are one and the same in essence. Yet in the case of engineering (and technology in general) such models are likely to be used more for design than for understanding the natural world. This means that historically there has been just as great – if not greater – emphasis on rules-of-thumb, charts, and empirical models as there has been on analytical models (although the latter have also been vitally important in areas such as electronics, mechanics, chemical and civil engineering, and so on).
Engineering models of the type discussed in this paper are not always highly valued in formal engineering education at university level, which often takes an “applied science” approach close to that of the natural sciences (something that can result in disaffection on the part of students). Yet in an informal context, such as laboratories, industrial placements, and so on, a very different situation obtains. The paper will consider such epistemological aspects, as well as the status of different types of models within the engineering education community.
The professional training of engineers includes a great deal of mathematics, conventionally said to ‘underpin’ the various engineering disciplines. Yet practising engineers often claim never to have used the majority of the mathematics they were taught. If by this they mean, for example, that they rarely if ever solve differential equations, invert matrices, or use vector calculus then – unless they are working in highly specialised research and development – they are almost certainly correct.
This apparent paradox is best understood by examining the social context of the use of mathematics by the vast majority of professional engineers. This paper will draw upon examples from information engineering – disciplines such as telecommunications, control engineering and signal processing. In particular, such engineers have developed visual or pictorial ways of representing systems that not only avoid the use of complex mathematics (although the techniques may well be isomorphic with the conventional formulations taught in universities), but have enabled ways of seeing and talking about systems that draw on the graphical features of the models. This approach develops within a community of engineering practice where the interpretation and understanding of these visual representations of systems behaviour are learnt, shared and become part of the normal way of talking. Engineers put models to work by using them as the focal point for a story or conversation about how a system behaves and how that behaviour can be changed. It is by mediating in this process – acting to focus language by stressing some features of the real system while ignoring others – that models contribute to new shared understandings in a community of engineering practice. Interestingly, modern computer tools continue to exploit many much earlier information engineering techniques – techniques originally designed to eliminate computation but now used primarily to facilitate communication and human-machine interaction.
This paper will thus examine some of the characteristics of technological/engineering models that are likely to be unfamiliar to those who are interested primarily in the history and philosophy of mathematics, and which differentiate technological models from scientific and mathematical ones. Themes that will be highlighted include:
• the role of language: the models developed for engineering design have resulted in new ways of talking about technological systems
• communities of practice: related to the previous point, particular engineering communities have particular ways of sharing and developing knowledge
• graphical (re)presentation: engineers have developed many ways of reducing quite complex mathematical models to more simple representations
• reification: highly abstract mathematical models are turned into ‘objects’ that can be manipulated almost like components of a physical system
• machines: not only the currently ubiquitous digital computer, but also older analogue devices – slide rules, physical models, wind tunnels and other small-scale simulators, as well as mechanical, electrical and electronic analogue computers