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I'm interested in the development of mathematical definitions for the sake of engineering, and what makes a particular definition better suited for a problem than another in any particular context.

Ideally, I aim to compile a list of examples of definitions, their historical context and a list of inadequate definitions which were proposed, along with a rationale of what was missing from the inadequate definition.

A concrete example is Bell Labs being contracted to create secure military communications, and Claude Shannon developing his theory of mutual information as a result. A failed attempt would be Karl Pearson's suggestion of mutual information as P(x,y)^2.

Here's a blog post that goes a little bit into that history, and points at the kind of comparitive research I'm interested in: https://www.lesswrong.com/posts/GhFoAxG49RXFzze5Y/what-s-so-bad-about-ad-hoc-mathematical-definitions

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    $\begingroup$ One place to investigate is Charles Proteus Steinmetz's influence on getting complex numbers and aspects of complex function theory as mathematical tools for electrical engineers. Another is Josiah Willard Gibbs's influence on getting vector analysis as a mathematical tool for physicists and engineers. Still another is Paul Dirac's introduction of the "Dirac delta function", whose later mathematical formulation gave rise to the theory of distributions. $\endgroup$ Jul 8, 2022 at 12:04
  • $\begingroup$ A pretty advanced area of abstract algebra, so-called clone theory, grew out of the problem of designing switching circuits for electronic devices. The idea was to find all collections of ($k$-valued) logic gates that are sufficient for encoding any function, and to optimize their set, see Rosenberg, Completeness properties of multiple-valued logic algebras for historical remarks. $\endgroup$
    – Conifold
    Jul 9, 2022 at 6:44
  • $\begingroup$ For another recent example see Zimmerman, Hückel Energy of a Graph: Its Evolution From Quantum Chemistry to Mathematics. $\endgroup$
    – Conifold
    Jul 9, 2022 at 6:47
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    $\begingroup$ I am not sure how far back you wish to go, but Archimedes's mass point geometry was clearly motivated by the law of the lever, and the machinery of On Floating Bodies by navigation. $\endgroup$
    – Conifold
    Jul 9, 2022 at 6:58
  • $\begingroup$ Chebyshev polynomials were motivated by a mechanical engineering problem. $\endgroup$ Jul 9, 2022 at 21:41

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Stable polynomials. A polynomial is called stable if its zeros lie in the left half-plane. This is a substantial research topic for the last 150 years. The problem arose as a pure engineering problem of stabilizing various automatically controlled machines (First Airy was investigating stabilization of his telescopes, then Stodola investigated stabilization of steam engines etc.) Important contributions were made in 19 century by Maxwell, then Routh and Hurwitz, but research on stable polynomials continues to this day.

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