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What about origination in mathematics? This is also a linking, but this time of what needs to be demonstrated-usually a theorem-to certain conceptual forms or principles that will together construct the demonstration. Think of a theorem as a carefully constructed logical argument. It is valid if it can be constructed under accepted logical rules from other valid components of mathematics-other theorems, definitions, and lemmas that form the available parts and assemblies in mathematics.
Typically the mathematcian "sees" or struggles to see one or two overarching principles: conceptual ideas that if provable provide the overall route to a solution. To be proved, these must be constructed from other accepted subprinciples or theorems. Each part moves the argument part of the way. Andrew Wiles' proof of Fermat's theorem uses as its base principle a conjecture by the Japanese mathematicians Taniyama and Shimura that connects two main structures he needs, modular forms and elliptic equations.
To prove this conjecture and link the components of the argument, Wiles uses many subprinciples. "You turn to a page and there's a brief appearance of some fundamental theorem by Deligne," says mathematician Kenneth Ribet, "and then you turn to another page and in some incidental way there's a theorem by Hellegouarch-all of these things are just called into play and used for a moment before going on to the next idea." The whole is a concatenation of principles-conceptual ideas-architected together to achieve the purpose. And each component principle, or theorem, derives from some earlier concatenation. Each, as with technology, provides some generic functionality-some key piece of the argument-used in the overall structure.
That origination in science or in mathematics is not fundamentally different from that in technology should not be surprising. The correspondences exist not because science and mathematics are the same as technology. They exist because all three are purposed systems-means to purposes, broadly interpreted-and therefore must follow the same logic. All three are constructed from forms or principles: in the case of technology, conceptual methods; in the case of science, explanatory structures; in the case of mathematics, truth structures consistent with basic axioms. Technology, scientific explanation, and mathematics therefore come into being via similar types of heuristic process-fundamentally a linking between a problem and the forms that will satisfy it.
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W. Brian Arthur (The Nature of Technology: What It Is and How It Evolves)