Mathematical Logic in High School: Hints and Proposals

In school, students learn how to reason and argue, and logic is the art of reasoning. Aristotle, who first developed it, held it to be so, i.e., the foundation of all science. But one certainly cannot impose on girls and boys an institutional course in logic as a prerequisite to all other knowledge. Not even in high school, when the matu- ration process of the students allows the teacher some more abstraction. In truth, in the Archive of Public Education - Cultural, Educational and Professional Pro- file of High Schools [129], it is stated that the logical-argumentative area assumes a central role, because it contributes to the formation of a citizen who “supports her/his own convictions, bringing adequate examples and counterexamples and us- ing concatenations of statements; accepts to change her/his opinion by recognising the logical consequences of a correct argumentation”. But education in logic must be done prudently, in the right formative ways. Some would argue that the practice of mathematics, often based on reasoning, in particular the model of Euclidean geometry, is in itself a cue to progressively in- sinuate logical mechanisms. Unfortunately, in recent times, however, Euclidean geometry seems to be a subject in disgrace, often neglected or forgotten. Instead, there are those who emphasise, in mathematics, the importance of intuition, dis- covery, experience and error, contrasting it with the excessive rigour of too many proofs. The purpose of this thesis is to propose various ideas that, within the fun- damental programmes of high school, specifically of Italian Liceo Classico and Liceo Scientifico, attempt to insinuate logic and accustom the students to logic in a way that we hope is light, clear and pleasant. We therefore do not propose a systematic treatment. We prefer to recall basic logic and then to give scattered ideas rather than a structured and definitive theory. But, as mentioned, we are confident that these hints can best prepare students of high school for logic. Indeed we address ourselves primarily to teachers and we believe that their knowl-edge of logic is useful and indeed necessary. But through them we also wish to address students. The thesis is organised as follows. The first chapter introduces and discusses the whole topic and explains why in our opinion logic is important in high schools. We also discuss how and when to propose it to students. The next two chapters introduce basic logic to teachers and students. The second illustrates the simplest logic, the Boolean one, recapitulating its essential points and emphasising in particular the use of connectives. The third deals with first-order logic, which we may consider the most classical of logics. Here we highlight in par- ticular the function of quantifiers. The following chapters propose several topics, belonging to logic or related to logic, that seem very intriguing and could be considered in high school. First, in chapter four, we treat Aristotelian syllogistics, which, even in recent times, frequently appears in various access tests. We will present some amusing introduc- tions to it, such as those of Lewis Carroll [25] or Pagnan-Rosolini [86]. Chapter five is dedicated to proofs without words. Relying on various examples from geometry, number theory and combinatorial calculus, it illustrates how reason- ing can sometimes be successfully expressed and developed through the images and intuitions they suggest. In this chapter, we also discuss the logic of images proposed by Leonardo in the Codex Atlanticus [73] to address and solve geometric questions often linked to the Pythagorean theorem. The sixth chapter is dedicated to what Henri Poincaré called the “mathematical reasoning” par excellence, namely the principle of induction. This law governs nat- ural numbers and is often used as a powerful demonstrative tool in various exercises. However, students seldom learn it and above all understand it properly. Drawing on the history of the principle of induction, from its primitive intuitions to its for- malisation by Peano and Dedekind, we attempt to approach it in what we hope will be a pleasant and appealing way, also o ering a wide range of examples. Logical paradoxes are another logical theme that is impossible to forget: mental games that not only disorientate but also intrigue and amuse, which are the heart of the seventh chapter. To the classical logics considered at the beginning of the thesis, based on two truth values, yes or no, we then contrast multi-valued and fuzzy logics, which are better suited to analysing situations of uncertainty. We link them, in chapter eight, to the Rényi-Ulam game, which searches for truth in contexts in which the information received may be lying and deceptive. The final chapter takes up a basic topic of high school mathematics: equations. Diophantine games show us how they can be an opportunity for challenge and fun, as well as suggesting intriguing insights into fundamental themes of modern math- ematics: not only number theory and algebra, but also game theory and the theory of computability and computational complexity. For a general and in-depth overview of mathematical logic, we refer to [10] and [114]. For the theory of computability and computational complexity we refer the reader to [37] and [83], for number theory to [60].