“Bombelli [(1526-1572)] had given meaning to the “meaningless” by thinking the “unthinkable,” namely that square roots of negative numbers could be manipulated in a meaningful way to yield significant results. This was a very bold move on his part. As he put it: ‘it was a wild thought in the judgment of many; and I too was for a long time of the same opinion. The whole matter seemed to rest on sophistry rather than on truth. Yet I sought so long until I actually proved this to be the case.’ Bombelli developed a “calculus” for complex numbers, stating such rules as (+√−1)(+√−1) = −1 and (+√−1)(−√−1) = 1, and defined addition and multiplication of specific complex numbers. This was the birth of complex numbers. But birth did not entail legitimacy.”
Israel Kleiner, A History of Abstract Algebra, p. 8
In his appendix entitled THE AMPHIBOLY OF CONCEPTS OF REFLECTION Kant criticized that Leibniz, in his thoughts on a universal characteristics, departed from an intellectual notion of intuition instead of a sensible one; he rightly observed that in consequence of this, judgements about a thing in general – i.e. about an object – can never be possible in an unproblematical manner. We can easily understand the confusion Kant was pointing out when we remember that Leibniz did not hesitate to use quantities in his calculations whose nature was dubious – he called his infinitesimals useful fictions. By the time of the 19th century, numbers counting quantities of magnitudes whose nature was unclear – i.e. symbolic – had literally proliferated. Bombelli’s dubious domain of imaginary numbers had been firmly established in practice and not only for conceptual considerations – their application engendered sensible phenomena like electricity, which cannot, mathematically, be controlled without applying synthetic procedures rooted in the realm of imaginary numerosity. The complex number plane allowed to engender purely symbolic numerical ‘species’ for which the Kantian distinction between the transcendental and the phenomenal is stubbornly blind. Within algebraic formula, terms can live like the amphibians which have their name because they can exist in different milieus, in water as well as in air.
Polynomial terms too can live lives in multiple numerical worlds.
Every term of a formula can be written as a polynomial, involving variable values and constant values, of which the latter can be ‘spelt’ by attaching them to designate-able and balancable constellations of Coefficients. We can take the meaning of Coefficients quite literally, from efficient for capable of producing a desired effect, marked with the prefix com which indicates that the capability depends upon a the actuality of a distribution. The coefficients need to be balanced – it is this balancing which synthetic procedures can govern in a myriad of ways. Since the invention of the group concept, computing artists not only manipulate magnitudes as values that can actualize in variable intensities; the manipulation of coefficients allows them to modulate the solvability of formulaic constellations generically. It allows for breeding solutions. In other words, the computing scientists and artists can equip that which counts, the multitudinality, with a logistic setup. To this, we owe the sheer explosion of technical feasibility and synthetically real possibility in which we live today. Every artefact incorportes such a ‘doped’ domain.
Yet firm establishment in practice and in method did not settle the question of philosophical legitimacy of these ‘numerical species’.
Or asked differently, what is one grasping when articulating quantities in polynomial terms ?
Polynomials name terms that comprehend ever so much as the term is capable of bounding within a constellation of terms as incorporated by a formulaic system. the determinability of this so much is added separately, by the decision regarding which numerical domain is being put at the basis of the solution space.