Proposal for The Common Denominator Conference, Universität Leipzig, March 2014.
Keywords: Critical Reasoning; Symbolic Reasoning; Amphiboly, Reflection and Projection; Computability
In 1898 Alfred North Whitehead published A Treatise on Universal Algebra in order to present “a comparative study of the various Systems of Symbolic Reasoning“ that had been allied to ordinary Algebra since mid 19th century. The background to this treatise were developments in number theory: calculating with the so-called imaginary unit, a symbol for the “impossible” quantity of the square root of negative one. At stake with legitimating the domain of the complex numbers in mathematics was nothing less than a profound challenge to the enlightenment secularization of science and philosophy. Intellectuals like Gottlob Frege, Bertrand Russell, Karl Weierstrass did not fail to see that the understanding of mathematics as guaranteeing critical foundations to knowledge was severely unsettled by these new practices in algebraic computation, and they strived to hold on to the Kantian perspective of banning amphiboly from all concepts of reflection; at the same time, intellectuals like Richard Dedekind, George Boole, Hermann Grassmann and Bernhard Riemann affirmed the imaginary roots of calculations and constructions, yet also not without awareness of the unsettling implications. But instead of aiming to purify the reflective faculties of reason, they set out to develop novel systems of how the symbolic constitutions for reflective projections might be engendered such that they allow for critical reasoning. While the former linked the dream of a universal method (mathesis universalis) to a mechanization of objective thought, the latter insisted on a creativity of subjective acts of thinking, and linked their dream of a universal method to the adventure of exploring systematically the open totality of the thinkable. This paper will recount a series of short stories around how the developments in symbolic reasoning were received at the time, and how they continue to live on in contemporary theoretical stances vis-à-vis computability.
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