Abstract
The paper describes the introductory approach to some questions of differential calculus (the program of mathematics for the final year of the course of the secondary humanistic studies), focussing on the question on the graph of a function. While the point is to find the graph of the function using the fundamental notions of calculus such as equations, limits, and derivatives, the conceptual framework is enriched by notions from historiography (Carlo Ginzburg), literature (Agatha Christie), and philosophy of mathematics. The aim is to teach students to solve a problem of graphing a function as Hercule Poirot would solve a case of murder.
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Notes
Let me point out that the goal of this article is a discussion about graphing a function by using metaphors borrowed from other areas of knowledge, to make it more rooted in the intellectual path of pupils, not to lessen its mathematical rigour. Let me also add that, since this is aimed at students of humanities-oriented schools, I found it appropriate to address poets in the title, in order to remind that the beauty of mathematics and poetry, of both the world and literature, moves (in all senses) us in the same way in all cases.
See for instance the procedure Poirot describes in [12].
After all, we only see with our mind’s eye [12]; indeed, what we see by just looking does not always correspond to the actual sequence of facts and could sometimes be misleading. More than once, Poirot points out that actual events may not correspond to the reality perceived through our eyes. This is how we have to understand what he says towards the end of his case on the Nile: “we had only the evidence of visual appearance, and though apparently that was perfectly sound, it was no longer certain” [11].
The phrasing by Charles Sanders Peirce has been slightly adapted by me. We have to give a correspondence between the two formulations: T corresponds to the result, A to the rule (which is here guessed at), of course assuming that the situation described in the case occurs. In the words of Ernan McMullin, this is an inference that makes science [28], and in those of Sherlock Holmes, “the curious analytical reasoning from effects to causes” [16].
As James Robert Brown writes: “Two structures are isomorphic when (a) they have the same number of elements or objects, and (b) the relations among the elements of one structure have the same pattern as the relations among the elements of the other” [3, p. 39].
See [14]. Perhaps elsewhere more than here, but it is to be stressed the importance of psychology so may time pointed out by Poirot (see [26, p. 270–271]). Just one example: “‘Les femmes,’ generalized Poirot. ‘They are marvellous! They invent haphazard—and by miracle they are right. Not that it is that, really. Women observe subconsciously a thousand little details, without knowing that they are doing so. Their subconscious mind adds these little things together—and they call the result intuition. Me, I am very skilled in psychology. I know these things’” [6].
The division of facts into important ones and non-important ones means, in my opinion, that some of the facts (traces) are less relevant to the case being studied. For instance, in the case of function (1) (see Fig. 5), analysing the second derivative we would find three points of inflection (that is, the points where the second derivative changes its sign). However, two of them, x = −2 and x = 1, are less important while graphing the function, since at those points two vertical asymptotes exist, found while studying the domain and the limits of the function.
As Poirot himself would say, out of the confusion of clues methodically collected, order emerges (see [26], p. 277).
This is a paraphrase of the remarks on the consistency of a set of postulates on which a system is based, as given in [29, p. 14]: “A given set of postulates serving as foundation of a system” can be considered consistent if “no mutually contradictory theorems can be deduced from the postulates”. In other, Anna Sfard’s words: “Any two mathematicians charged with the task of determining the endorsability of a narrative are expected to arrive at the same conclusion. if they don’t, at least one of them is suspected to have deviated from the rules of mathematical endorsement” [32, p. 225].
See [26]. Such an experiment could concern not just the search for the missing link, but also a kind of falsification, in an almost Popperian sense, of the foundation of the proposed narrative: “When a thing arranges itself so, one realises that it must be so, one only looks for reasons why it should not be so. If one does not find the reasons why it should not be so, then one is strengthened in one’s opinion” [13].
Just like it happens to Poirot in the case described by Agatha Christie in [5].
It would be of course one of the non-technical formulations of Gödel’s incompleteness theorems, open to philosophical and even theological interpretations, such as the one suggested for example by John P. van Gigch: “Gödel’s Theorem states that in any formal, logical, and consistent system there exist true statements, expressible in the language of the system, which cannot be proved within the boundaries of the system itself (i.e., there always exists an undecidable statement which is not provable)” [23, p. 20]. .
Something analogous happens in Hercule Poirot’s career. Late in his life, he meets a person of whose guilt in a murder he is convinced, but without being able to prove it [15]. In a letter to his friend Hastings, Poirot admits: “And I saw that I had come across at last, at the end of my career, the perfect criminal, the criminal who had invented such a technique that he could never he convicted of crime”. This technique is described resorting to a metaphor borrowed from chemistry: the way of acting of X is compared by Poirot to a case of catalysis, that is “a reaction between two substances that takes place only in the presence of a third substance, that third substance apparently taking no part in the reaction and remaining unchanged… It means that where X was present, crimes took place - but X did not actively take part in these crimes” [15]; see also [26].
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Sierotowicz, T. Calculus for poets, or, the graph of a function and Monsieur Poirot: an evidential paradigm in the teaching of mathematics. Lett Mat Int 4, 181–188 (2017). https://doi.org/10.1007/s40329-016-0148-y
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DOI: https://doi.org/10.1007/s40329-016-0148-y