Mathematical Induction: A Focus on the Conceptual Framework

International Journal of Educational Studies in Mathematics, 2015

In this study, we report on a professor of mathematics who also teaches mathematics in a secondary school. We checked the influence of the professor's presenting the mathematical induction (MI) principle on his students' perceptions of the principle. Extensive observations on the MI content were performed to determine the professor's conception the MI principle. Moreover, seventy four students answered a questionnaire which was built to examine the students' perceptions of the MI principle. Finally, both data analysis were triangulated and crosschecked. It was concluded that the students' main perception of MI was: mathematical induction identifies the natural number set, including formulation; following four representations of the MI that were exposed by the teacher. This means that the learners mainly, perceived MI by reflecting on the properties of the set of natural numbers, in addition to the usual perception which is as "a method of proof". Consequently, it can be argued that this representation has pedagogical advantages.

2003

The paper explores symmetry between the argument forms of induction and hypothesis (or abduction). The symmetry is inspired by an analysis of arguments due to Peirce. In fact, the symmetry is between a more general form of induction, symmetric induction, and generalized hypothesis. Under very generic assumptions, symmetric induction is equivalent to standard enumerative induction. Generalized hypothesis is a considerably more general form of argument than Peirce's hypothesis, which is derived as a special case. The connection between generalized hypothesis and Peirce's hypothesis maps by the symmetry to a connection between induction and another form of inference, qualification of abstraction. Ultimately, both argument forms are shown to be instances of a general inference form about identity of sets based on identity of subsets.

The Australian Association of Mathematics Teachers, 2017

Mathematical induction is a proof technique that can be applied to establish the veracity of mathematical statements. This professional practice paper offers insight into mathematical induction as it pertains to the Australian Curriculum: Mathematics (ACMSM065, ACMSM066) and implications for how secondary teachers might approach this technique to students. In particular, literature on proof – and specifically, mathematical induction – will be presented, and several worked examples will outline the key steps involved in solving problems. After various teaching and learning caveats have been explored, the paper will conclude with some mathematical induction example problems that can be used in the secondary classroom.

The philosophy professor says, -This swan is white. That swan is white. And so is the other,‖ and asks, -May we legitimately infer that all swans are white?‖ In this way the students are introduced to induction or, as we seem always to call it, the -problem‖ of induction.

A MATHEMATICAL REVIEW: Hintikka, Jaakko Aristotelian induction. Rev. Internat. Philos. 34 (1980), no. 3-4, 422--439. MR0630683 (82m:00016) The author considers the question of how Aristotle explains knowledge of axioms, i.e., basic premises of axiomatic science. "Induction" is the conventional translation of Aristotle's term "epagogé" indicating a process through which knowledge of the truth of axioms is drawn from experience. Thus, it has no special relation to mathematical induction, the number-theoretic principle. The fact that Aristotle traced all knowledge of axioms to experience is already clear and well known to scholars despite contrary information persistently appearing in popularizations. It is true, however, as pointed out in this paper, that Aristotle gave apparently conflicting accounts of how induction functions and also that some of Aristotle's remarks about axioms may seem to conflict with his view that knowledge of them is based on experience. This paper proposes to solve these problems through a plausible premise, perhaps original with the author, to the effect that Aristotle's induction involves a combination of empirical examination of examples with conceptual analysis of exemplified notions. Reviewed by J. Corcoran NOTE ADDED 121715: Given the various ways ‘induction’ is used today, it is misleading to use "induction" as the translation of Aristotle's term "epagoge" indicating a process through which knowledge of the truth of axioms is drawn from experience. “Experience-based intuition” might work better. On this subject, one should note that there are many sentences, propositions, rules, processes, and so on that are all called ‘mathematical induction’—an expression totally unsuited for any of them. It the first place, the adjective ‘mathematical’ is far too general, ‘arithmetic’, ‘numerical’, or ‘number-theoretic’ would be better. In the second place, the noun ‘induction’ in ‘mathematical induction’, if justified at all, cannot be justified by reference to the most common uses of the word in normal English, in science, in philosophy, or in methodology of science. In current senses, mathematical induction does not produce knowledge of mathematical axioms. In fact, in some current senses, mathematical induction is a mathematical axiom, knowledge of which is produced by induction in the sense associated with Aristotle. I am glad to point out that Hintikka’s use of the word ‘axiom’ for “fundamental truth” is in full accord with its use by Tarski, Church, and other giants of mathematical logic. Moreover, it reveals the pathetic inadequacy of ill-advised and unjustified attempts to use the word for “arbitrary assumption”.

Among the three predominant types of logic—deductive, inductive, and abductive—inductive logic has long captured the attention of Western philosophy; this is due to its concomitant collection of seemingly incalcitrant challenges. In this essay, I shall discuss both the old and new riddle of induction. In so doing, my hope is to discuss the core issues associated with each, respectively, as well as lay bare the rational motivation for the transition from the old problem of induction to induction's 20th century renewed challenge. As such, I shall also provide critical reflections regarding attempts to resolve both the old and new riddles of induction, along with suggestions for further relevant avenues of research.

Journal of Baltic Science Education

Among the mathematical methods which are taught in the last years of almost every high school, the mathematical induction deserves particular attention. It can be used both to define mathematical entities and to prove theorems. The second use is more common at high school level and is easier. Thus, I will basically focus on it, though analysing in depth two definitions by induction. The aim of this contribution is to offer the basic elements for a didact unit which could be developed in six/seven hours of lesson.

Theoretical Computer Science, 1997

Several concise formulations of mathematical induction are presented and proved equivalent. The formulations are expressed in variable-free relation algebra and thus are in terms of relations only, without mentioning the related objects. It is shown that the induction principle in this form, when combined with the explicit use of Galois connections, lends itself very well for use in calculational proofs. Two non-trivial examples are presented. The first is a proof of Newman's lemma. The second is a calculation of a condition under which the union of two well-founded relations is well-founded.

1997

Gentzen [7] proved the coinsistency of PA (Peano Arithemetic) by using the transfinite induction up to the first epsilon number fo. Here fo is limk Wk, where wo = 0 and Wi:+l = ww•. Later in [8] he proved that the accessibility (i.e., transfinite induction) proof up to any ordinal less than fo, eg., Wk for any natural number k, is provable in P A.