The Mathematical Components of Engineering



Division of mathematical labour and mathematical interfaces

The fact that the majority of design engineers can work without having to do advanced
mathematics is due to the sophisticated distribution of expertise in engineering practice.
We identified and studied three main areas of divided mathematical labour:

1. Ubiquitous computer programs as design tools

2. Codes of Practice which distil knowledge and expertise into a form usable for design

3. Analytical specialists acting as consultants to design practice

A design engineer at work on a project must interact with the first two of these areas, and
in some cases, the third. As research progressed, we began to recognise the
mathematical
interfaces
that exist between specialist domains of knowledge, and between the experts in
those domains.

We have tried to characterise what kind of knowledge is necessary to make sense of a
“hidden” calculation through an interface. There are situations where a visualisation of
the inside workings of a mathematical calculation is
not required to make an informed
judgement about it: the judgement can come directly from engineering understanding. On
the other hand, even in a multi-disciplinary design team with its own mathematical
specialists, mathematical analysis cannot be a totally black box for any engineer who has
to use a mathematical result and take responsibility for its use.

Finding 3: The team of design engineers is comprised of many specialisms,
and only a minority of them are mathematics-based. Wherever there is a
division of labour there is a need to communicate information across those
divisions. Each division entails its own “interface”: that is, the forms and
language of communication between individuals across that division.

Characterising the embedding of mathematics in expert knowledge

An obvious question that arises is where does a design engineer’s understanding of
hidden calculations come from? Although the scope of this project does not allow a
definitive answer, there are clues to be found in the expert structural engineer’s repertoire
of qualitative and quantitative ways of thinking about structures.

Approaches to analysing structures vary along a quantitative-qualitative spectrum,
running from exact, explicitly mathematical or computer-based methods, through “rough
calculations”, to qualitative approaches based on “structural feel”. Understanding comes
through connecting across the different approaches, in the simplest instances cross-
checking of, say, an exact answer against a rough calculation.

The type of qualitative thinking that characterises the use of “feel” in design is
exemplified by the concept of
load path, the notion that the loads acting on a structure
have to “flow down into the ground” like a kind of fluid. It is an extremely useful concept
because it provides a way of thinking about a structure before any quantitative analysis is
done.

Load path is closely-related to structural geometry: engineers use mathematics to carry
around in a very compact form the shapes and magnitudes of the deformations of
structural elements when loads are applied: understanding is “situated” in the sense that
structural engineers think about geometrical and algebraic forms for what they mean in
structural terms. From the outside observer’s point of view, this can — we think
mistakenly — be interpreted as an absence of mathematical knowledge.



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