Abstract
I examine the way a relevant conceptual novelty in mathematics, that is, the notion of group, has been constructed in order to show the kinds of heuristic reasoning that enabled its manufacturing. To this end, I examine salient aspects of the works of Lagrange, Cauchy, Galois and Cayley (Sect. 2). In more detail, I examine the seminal idea resulting from Lagrange’s heuristics and how Cauchy, Galois and Cayley develop it. This analysis shows us how new mathematical entities are generated, and also how what counts as a solution to a problem is shaped and changed. Finally, I argue that this case study shows us that we have to study inferential micro-structures (Sect. 3), that is, the ways similarities and regularities are sought, in order to understand how theoretical novelty is constructed and heuristic reasoning is put forward.
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Notes
That is a polynomial that has been set equal to zero—e.g. x2 + 7x + 5 = 0.
This problem requires, once provided the edge of a cube, finding the edge of a second cube whose volume is double that of the first (by using only the tools of a compass and straightedge), in modern algebraic symbolism\(x=\sqrt[3]{2}\) .
Another interesting issue that characterizes this process (which we do not have room to treat here) is the heuristic role of new symbolisms. Partial solutions discovered in the sixteenth century increasingly made use of symbolism in a way that made possible theoretical questions to be posed and answered by mathematicians. For example, the refinement of the algebraic symbolism also allowed questions about the relation of roots and factors to be formulated and pursued (see Barnett 2010, p. 2 on this point).
To provide a bit more detail, first, negative solutions were accepted, thus extending the domain of integers to the negative axis, so that, e.g., every first-degree equation in the normal form ax + b = 0 (even with positive coefficients a, b) has a solution. Then, by observing that second-degree equations, like e.g. x2 + 1 = 0 have no solutions even in the extended realm of negative numbers, a further extension to complex numbers was introduced.
He retained the term angulus solidus (solid angle) for his point-like vertices.
For instance the methods of Cardano, Tschirnhaus, Euler, Bezout for cubics; and the methods of Cardano, Descartes, Tschirnhaus, Euler, Bezout for quartics.
For instance, for m = 2, by exchanging x1 with x2 we have that A = x2 + x1 and B x2x1. Both are equal to the original expressions, where A = x1 + x2 and B = x1x2.
The invariance is expressed by Lagrange (1770) by saying that the resolvent “does not change” when t is replaced in a specific way.
The roots of unity are all found on the unit circle, as they must have modulus equal to one. Now, every point on the unit circle can be identified by an arrow starting at the origin of coordinates and pointing at it. This arrow creates an angle alpha with the real axis (called argument of the complex number). Each multiplication of a root by itself rotates its arrow counterclockwise by the same angle alpha. So the m-roots of unity are such that after rotating m times by alpha one gets back to the point x = 1, i.e., m alpha = 360°, which has exactly m distinct solutions alphak=(k/m) 360°, with k = 0,…, m−1. Increasing k we obtain solutions that are equivalent, e.g. k = m gives alpha = 360° which is the same arrow as alpha = 0, k = m + 1 gives alpha = 360° + 360°/m, which is the same arrow as alpha = 360°/m, and so on.
The roots are the points on the circle: their values have a real part and an imaginary part, and are measured respectively along the horizontal and vertical coordinates.
By ‘algebraic form’ we mean that the roots of a given equation can be determined from its coefficients by means of a finite number of steps that involve only elementary arithmetic operations (+; −, ×, ÷), and the extraction of roots.
On this point see for instance (Gillies 1995; Ippoliti 2016). A recent interesting account is the one described by Fisch (2017), who deals with George Peacock’s struggles that also ultimately failed but that, again, proved heuristically crucial in helping others to develop a new branch of mathematics.
I won’t discuss here another very interesting, crucial issue emerging from this case study, that is, the heuristic role of notation (e.g. notation allows generality, permits a useful polysemy, makes calculation possible, etc.), and in particular the way it allows problems to be solved and posed.
Permutation here denotes the change of one arrangement (say abc) to another arrangement (e.g. acb)—in modern terms, we would say that it is a function that maps one set of entities (in this case, the letters a, b, c) onto the same set in a one-to-one manner.
See Thomas (2011) on the role of assimilation in mathematics.
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Acknowledgements
I would like to thank the two anonymous referees for the valuable advice on the both the philosophical and the mathematical parts of my paper.
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Ippoliti, E. Manufacturing a Mathematical Group: A Study in Heuristics. Topoi 39, 963–971 (2020). https://doi.org/10.1007/s11245-018-9549-1
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DOI: https://doi.org/10.1007/s11245-018-9549-1