Michael Mahoney, excerpt from “The Structures of Computation and the Mathematical Structure of Nature” in Histories of Computing
Is mathematics a science of the natural world, or can it be? Computational science puts the question in a new light. Science is not about nature, but about how we represent nature to ourselves. We know about nature through the models we build of it, constructing them by abstraction from our experience, manipulating them physically or conceptually, and testing their implications back against nature. How we understand nature reflects, first, the mapping between the observable or measurable parameters of a physical system and the operative elements of the model and, second, our capacity to analyze the structure of the model and the transformations of which is capable. In the physical sciences, the elements have been particles in motion or distributions of force in fields described in differential equations; in the life sciences, they have been organisms arranged in a taxonomy or gathered in statistically distributed populations. The scope and power of these models have depended heavily on the development of mathematical techniques to analyze them and to derive new structures and relations within them. Developed initially as a tool for solving systems of equations that were otherwise intractable, computers have evolved to provide a means for a new kind of modeling and thus a new kind of understanding of the world, naming dynamic simulation, which is what we generally mean when we talk of ‘computational science’. But with the new power come new problems, first, to define the mapping that relates the operative elements of the simulation to what we take to be those of the natural system and, second, to develop the mathematical tools to analyze the dynamic behavior of computational processes and to relate their structures to one another.
In short, understanding nature appears to be coming down to understanding computers and computation. Whether computer science, broadly conceived, should be or even could be a wholly mathematical science remains a matter of debate among practitioners, on which historians must be agnostic. What matters to this historian is that from the earliest days, leading members of the community of computer scientists, as measured by citations, awards, and honors, have turned to mathematics as the foundation of computer science. I am not stating a position but reporting a position taken by figures who would seem to qualify as authoritative. A look at what current computing curricula take to be fundamental theory confirms that position, and developments in computational science reinforce it further.
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