Medieval Arabic Algebra as an Artificial Language

Abstract

Medieval Arabic algebra is a good example of an artificial language. Yet despite its abstract, formal structure, its utility was restricted to problem solving.

The Unattainable Attempt to Avoid the Casus Irreducibilis for Cubic Equations

“…81 Summing up, we remark again that the hypothesis that the Ars Magna comes from a further development of the Ars Magna Arithmeticae is not at all unlikely. Since there are no definitively valid arguments for that, I prefer to trust Chapters XX and XXIV and consider that only x 3 + a 2 x 2 + a 1 x = a 0 and x 3 + a 1 x = a 2 x 2 + a 0 have a "general" rule.…”

“…10 I rather agree with [81] and 9 Under the same viewpoint and for the sake of brevity, I will sometimes use notations concerning equations, especially the 'discriminant Δ3', referring to Cardano's equations. In particular, by writing the equations as in the above table, I do not obviously mean that Cardano is interested in the zeros of the function associated to the appropriated polynomial (intended, in modern terms, as an infinite, eventually zero sequence of coefficients) -it would simply sound ridiculous.…”

“…For instance, when we want to draw a solution of x 3 + 6x = 40 using the cubic formula, 3 81 Indeed, in Ars Magna, Chapter XII Cardano does not know that the terms under the cubic roots are two cubes. For instance, when we want to draw a solution of x 3 + 6x = 40 using the cubic formula, 3 81 Indeed, in Ars Magna, Chapter XII Cardano does not know that the terms under the cubic roots are two cubes.…”

Cognitive stories and the image of mathematics

“…The interpretation of an equation can be geometric or arithmetic, theoretical or practical, so the equation serves as a pivot between these contexts (Oaks 2007(Oaks , 2009). Equations subsume their different possible interpretations under their unified language.…”

Mathematical symbols as epistemic actions

“…Even seemingly mundane operations like addition and subtraction are not represented consistently as at least eight synonyms denote subtraction. Remarkably, in his analysis of Arabic algebra, Oaks (2007) found only one instance where shay actually meant an arbitrary quantity. The verbose character of Arabian mathematics, which was intimately tied to mathematical practice, did not allow for the development of formal algebraic proofs or the formulation of general solutions.…”