Applications of Complex Numbers and Quaternions: Historical Remarks, with a Note on Clifford Algebra
Applications of Complex Numbers and Quaternions: Historical Remarks, with a Note on Clifford Algebra
Frege required his logicist definitions of numbers to obey the applicability constraint, that all potential applications somehow be prefigured in the definitions. Can this constraint hold for neo-logicist definitions of further mathematical kinds? Two natural extensions of Frege’s interests are to complex numbers and quaternions. The former were introduced for purely mathematical reasons, and did not find extra-mathematical application until the end of the 19th century, calculating electrical impedance in AC circuits. They are now seemingly indispensable in quantum theory. Quaternions were developed with an eye to geometry, but lost out to vector analysis for those seeking geometric applications. Despite recent use in representing rotations, they have never been employed in their full structure. Ironically, vector theory now has a serious rival in geometric or Clifford algebras. The moral is that neo-logicism should examine more of the mathematics that is useful in applications.
Keywords: complex numbers, quaternions, Clifford algebras, neo-logicism, abstraction, applicability
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