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The many facets of a definition: The case of periodicity
Journal of Mathematical Behavior 22 (2003) 91–106
The many facets of a definition: The case of periodicity Joop Van Dormolen, Orit Zaslavsky∗ Technion – Israel Institute of Technology, Technion, Haifa 32000, Israel
Abstract This paper was triggered by an authentic conversation between two mathematics teacher educators who debated whether a constant function is a periodic function, within the framework of a professional development program for secondary mathematics teachers. Their initial conversation led to deep mathematical and pedagogical musing surrounding mathematical definitions. In this paper, we present various aspects of a mathematical definition, including the role and nature of definitions in school mathematics, critical versus preferable features of a definition, and the arbitrariness underlying the choice of definition. We discuss the interplay between logical and pedagogical considerations with respect to definitions, drawing on the definition of a periodic function as an example. © 2003 Elsevier Science Inc. All rights reserved. Keywords: Mathematical definition; Periodicity; Logic; Pedagogy; Teacher education; Secondary school mathematics
1. Vignette LL: Look here on this definition of a periodic function. It says that a constant function is not a periodic function. Do you agree? JD: Well that depends, of course, on the definition. Some definitions accept a constant function as a periodic function. LL: What do you mean by “some definitions.” There cannot be more than one definition of a concept. JD: Why not? LL: We cannot communicate if we each have a different definition. JD: True, but we can agree to use one particular definition and on that base we could communicate very well. LL: But we must use the definition that is used in mathematics.
∗ Corresponding author. Tel.: +972-4-8262798; fax: +972-4-8241819. E-mail address: [email protected] (O. Zaslavsky).
0732-3123/03/$ – see front matter © 2003 Elsevier Science Inc. All rights reserved. doi:10.1016/S0732-3123(03)00006-3
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JD: I have no idea what you mean by that. A definition is just an agreement between people to give a name to a certain object or a certain idea. By the way, I saw in the definition that you showed me just now that the definition of a period is not correct. They say that every p = 0 that fulfills the requirements of a periodic function is called a period of the function. That cannot be true. Then a periodic function can have many different periods. But the period is the smallest positive p that fulfills the conditions (if there exists a smallest positive). LL: Well, didn’t you say just now that it depends on how you define a concept? You are talking about “the” period. Here they talk about “a period.” Why would it be impossible for a periodic function to have many periods? JD: We are free to give any definition as long as it fulfills certain logical criteria. LL: Show me then where we would get a contradiction if there were many periods. JD: I think you are right. We have to consider all the criteria of a definition. Some are necessary, but not all.
2. Pedagogy before logic? The vignette above gives the essence of a conversation that took place between one of the authors and a colleague. The discussion was the trigger for the current article. As can be expected of mathematics educators with a sound mathematical background, who are fond of mathematics, the above conversation evoked a much deeper discussion of its implications. The discussion moved on from technicalities about definitions to more subtle questions regarding the role and nature of definitions in school mathematics. Also the question was raised whether a definition must be a formal definition or could it be an example-based description, such as: A periodic function is a function that can be constructed in the following way: Divide the x-axis into equal-length segments, such as for example . . . , [−39, −26], [−26, −13], [−13, 0], [0, 13], [13, 26], [26, 39], . . . Take any of these segments, no matter which one, and define a function on it, no matter how (e.g., as in Fig. 1). Then define another function on the whole x-axis, such that on each segment it behaves in the same way as the first function (as in Fig. 2). Then the new function is a periodic function. Its values are repeated regularly.
Fig. 1.
Fig. 2.
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From a formal standpoint this is not a “good” definition. In fact, it is not a definition at all, because it does not fulfill certain criteria, which are discussed in the next section. However, from a pedagogical standpoint this example-based description has many advantages, and can serve as an informal definition. It may be seen as following the spirit of many mathematics educators and learning psychologists who agree that, in general, the learning of a new concept can best be started in an informal way through one or more well-chosen examples. There are differences in opinions regarding several related issues, such as the specific circumstances in which this learning may take place, or the interplay between a given concept-definition and the sequencing of examples and non-examples of the concept. (For example, Ausubel & Robinson, 1969; Brousseau, 1997; DeCecco, 1968; Fischbein, 1987; Freudenthal, 1973, 1978; Hershkowitz, Bruckheimer, & Vinner, 1987; Leikin, Zaslavsky, & Berman, 1998; Mason & Pimm, 1984; Skemp, 1971; Tall & Vinner, 1981; Van Dormolen, 1991; Van Dormolen & Arcavi, 2000; Vinner, 1991; Winicki-Landman & Leikin, 2000). Our paper addresses some of these issues only indirectly. However, its main goal is to provoke thinking about the nature of mathematical definitions from both pedagogical and formal mathematical perspectives and to consider ways to define a concept at a given moment. In order to work well with formal definitions, or at least in order to teach the use of formal definitions, it is helpful to examine some criteria that people consider as significant for a definition to be a “good” definition. A definition is like a cut diamond with many facets. In this paper we attempt to shed light on the formal side of the diamond, hoping that the sparkling lights will catch our readers’ eyes. We realize that many of our readers are familiar with several issues discussed in this paper but, as the vignette shows, many may not be aware of all of them. We hope to inspire some of the readers to re-think and gain insight into some salient issues associated with definitions, similarly to the process that we ourselves have encountered.
2.1. Necessities and elegance of a definition Some of the criteria that we demand of a definition are necessary. They are fundamental parts of a deductive system. They are logical necessities. These criteria include: Criterion of hierarchy Criterion of existence Criterion of equivalence Criterion of axiomatization Other criteria are not necessary from a logical standpoint, but they are part of a general culture. For example, the criterion that demands that no more properties of a concept be mentioned unless it is necessary to establish the concept. Is it often called: Criterion of minimality Another criterion comes to play when, for example, a textbook author has to choose between two definitions, that are equivalent, but one looks nicer, needs fewer words or less symbols, or uses more general basic concepts from which the newly defined concept is derived. We call it Criterion of elegance A criterion that sometimes is a necessity and sometimes a beautification is: Criterion of degenerations We turn to a comprehensive discussion of the various criteria.
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2.1.1. Criterion of hierarchy Aristotle gave criteria for defining a concept. (See for example, Heath, 1956, pp. 116–124.) One of them contains a description of the structure of a definition. According to that criterion any new concept must be described as a special case of a more general concept. One or more properties must be used to describe this special case. For example, in “A right angle is an angle of which the legs are perpendicular to each other” the general concept is “angle” and the property is: “the legs are perpendicular to each other.” This way of defining a new concept has an important logical consequence: One may only use the general concept and the properties if they have been determined previously. This may seem a trivial condition, but in practice sometimes there are problems when we want to explain intuitively what a certain concept is. For example, a periodic function is a function for which a certain phenomenon keeps recurring. This is not a good definition in the Aristotelian sense, because “a certain phenomenon” is not well defined. Nor is “keeps coming back.” As an introduction to the concept of periodicity, however, it is acceptable as long as it is accompanied by examples that students can easily recognize. In this case, it is more than acceptable. It is a good didactical practice because (a) it uses ideas that are well known by the students and (b) it involves the students themselves in answering the question of how to clearly define the concept of a periodic function. 2.1.2. Criterion of existence A definition tells us what a concept is, but usually it does not say whether there exists an instance of such a concept within the current system. For this reason, Aristotle required, as a second criterion, that it must be proven that at least one instance of the newly defined concept exists in the current context. One can take a strictly formal standpoint and leave out this criterion. In that case it might be possible that the set of instances of the newly defined concept is empty. However, it is not advantageous to do so. For one thing, it opens the way to defining absurdities. For example, one could try defining, within the context of Euclidean geometry, a circlesquare as a square for which all points have the same distance to a certain point. Such a definition fulfills the first criterion, but not the second: in Euclidean geometry there is no such thing.1 The circlesquare in the system of Euclidean geometry is an example of an absurdity. One gets entangled in a philosophical web of discussions about the possibility of defining a thing that does not exist and therefore it seems to exist and not-exist at the same time.2 Moreover, it might lead to unforeseen contradictions in the system.3 This existence-criterion was the reason why Euclid wrote his “constructions.” For example, the construction of the bisector of an angle was not presented in order to show how this can be done, but rather in order to prove that it can be done. In other words: there exists such a thing as “bisector of an angle.” 1
The impossibility lies in the definition of Euclidean distance. There are many other possibilities to define distance. For example, taxi-distance in a Cartesian coordinate system: The taxi-distance between two points (a, b) and (p, q) is equal to |a − p| + |b − q|. The set of all points (x, y) that have the distance 1 to the origin is a circle by definition, but looks like a square: |x| + |y| = 1. 2 Quine (1953) gives an example of two philosophers that discuss whether or not Pegasus (the flying horse) exists. One of them remarks that it is impossible to say that he does not exist, because the moment one starts talking about Pegasus, one assumes its existence, otherwise one cannot talk about Pegasus. Also see Davis and Hersch (1981). 3 What to do when we cannot prove that a concept exists? Is it true that a concept either exists or does not exist? See for example Davis and Hersch (1981). In one of his stimulating books Stewart (1997) shows that one can prove that “The largest non-zero whole number is 1”(p. 194) if one does not bother to ask whether such a number exists.
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The first theorem in Euclid’s books was the construction of an isosceles triangle. Even though he had defined this concept before, he had to prove its existence. The consequence of this criterion—both from a logical as well from a pedagogical standpoint—is that, after a well-defined concept one has to give an instance. If a teacher gave an intuitive introduction before the definition was developed, s/he could refer to the introductory examples as a proof that the definition was a good one in this sense. In the case of periodic functions this is not a difficult task at all. In any case, Aristotle’s criterion seems to be a good and sensible one from a pedagogical perspective as well, if only to remind us that it is good practice to clarify a definition with an example. 2.1.3. Criterion of equivalence A third criterion (not explicitly given by Aristotle) is that, when one gives more than one formulation for the same concept, one must prove that they are equivalent. In practice this means that one has to choose one of the formulations as the definition and consider the other formulations as theorems that have to be proved. Consider, for example, the following formulations: A parallelogram is a quadrilateral in which the opposite sides are parallel. A parallelogram is a quadrilateral in which the opposite sides are equal. A parallelogram is a quadrilateral in which the sides of one pair of opposite sides are both equal and parallel. A parallelogram is a quadrilateral that is symmetrical with respect to a point. One can choose each of these formulations as the definition of a parallelogram. Thus, each of the other formulations is then a theorem to be proved.4 Which definition to actually choose amongst a number of equivalent definitions is a matter of taste or convenience. It can also depend on the context. Whatever the choice is, the mere idea that there is freedom to choose in mathematics will probably come as a surprise to many students. The “freedom to choose” is a friendlier way to look at what often is considered “arbitrariness” of a definition. Yet, freedom is never unconditional, and in order to have the freedom to choose a definition of a concept, one must make sure that all “candidates” are equivalent. Moreover, as we discuss below, one often has the freedom to choose between two definitions even if they are not equivalent. However, then the consequences of each choice should be carefully examined. 2.1.4. Criterion of axiomatization This criterion implies that a definition fits in and is part of a deductive system. Here is a short outline of what we mean by a deductive system. All the above relates to explicit definitions of a concept. An explicit definition tells us what the concept is. Like: “A parallelogram is . . . ,” or “A periodic function is . . . .” It does so with the use of a more general concept, such as quadrilateral (for a parallelogram) of function (for periodic function). These more general concepts must also be defined, based on even more general concepts. It is not possible however to continue this endlessly. At a certain point one gets concepts that cannot be defined with more 4 In the same book of Quine, there is a discussion about the influence of context. Some philosophers might say that the parallelograms are not the same, because they have different contexts, while they are the same. (In the book, the discussion is about the question of whether the Morningstar and the Eveningstar are the same. They refer to the same physical object and yet are not the same because they function in different contexts.) Quine solves this problem by stating that the definitions refer to the same object, but that they have different meanings.
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general ones, like point, line, plane in case of the parallelogram. A point cannot be described in a sentence like: “A point is . . . .” Nor can a line or a plane. In short: some concepts cannot be defined according to the Aristotelian criterion of hierarchy. Aristotle knew this and therefore wrote that axioms or postulates implicitly define such concepts. In geometry such concepts are point, line, and plane. Those concepts are implicitly defined in terms of axioms by their relations to each other: “There is one and only one line through any two points,” “On each line lie at least two points,” etcetera. To establish these relations in axioms is not enough. One can only talk about the concepts if they exist, but one cannot prove that they exist. Therefore, there are also axioms that establish their existence, such as: “There exist at least three points that do not lie on the same line.” Note that in these kinds of axioms relationships such as “go through” and “lie on” are defined implicitly. Another example is the definition of natural numbers. There are several possibilities. One can adopt the Peano axioms, or define natural numbers with the help of the cardinality of finite sets. In the first case one of the axioms states the existence of a “first element,” that we may call “one.” In the second case one has to look deeper at the axioms of sets in which there is a statement that the empty set exists. Now one can ask how to verify if a certain (explicit) definition satisfies the criterion of axiomatization. In other words: one must verify if it fits in and is part of a deductive system. In practice this would mean that it has to be verified if all the concepts used in the definition are, in their turn, again defined within the same deductive system, like in the example of the parallelogram above. A student once described a circle as “a figure that everywhere has the same roundness.” In the mind of the student, this was a perfect description of what she understood to be a circle. In the context of Euclidean geometry, seen as a deductive system, it is not (for the time being) a good definition as long as concepts like “everywhere” and “roundness” are not (yet) defined. 2.1.5. Criterion of minimality This criterion demands that no more properties of the concept be mentioned than is required for its existence. According to this criterion the following description would not be a good definition within Euclidean geometry: A rectangle is a quadrilateral with four right angles. In Euclidean geometry we can prove that the sum of the four angles of a quadrilateral is 360 degrees. Therefore, it is enough to define: A rectangle is a quadrilateral with three right angles. Let us refer to the first definition as the “4-angle-description” and to the second one as the “3-angledescription.” On first sight the criterion of minimality seems to be more of an aesthetic or philosophical nature than a logical one. Indeed, describing a rectangle as quadrilateral with four right angles will not result in a contradiction in the system, and may have advantages from a pedagogical perspective. Moreover, often at the time when a concept is defined there may not be sufficient knowledge to determine whether it is minimal. Thus, insisting on this criterion may impede the development of certain concepts or theories. There are, however, good reasons to accept this criterion. According to the criterion of existence it is necessary to establish that the new concept exists. This means that one has to investigate the new definition. When one investigates the 3-angle-description of a
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rectangle, it is inevitable that one gains the insight that a quadrilateral with four right angles is precisely what one had in mind, but defining it with three right angles is enough. So why not use the more economical one?5 There is, however, more to say than just an economical argument. By the very fact that one, by investigating a description that is meant to serve as a definition, discovers that some properties are superfluous, one has implicitly proved a theorem, namely that an object that satisfied the more frugal description has the properties that are mentioned in the more extensive non-minimal description. In the example of the rectangle, by investigating the 3-angle-definition of a rectangle, one can prove the theorem that if a quadrilateral has three right angles, the fourth angle is also a right one. So, by accepting the criterion of minimality one has to take the minimal description as definition and as such reject the non-minimal description. Thus, from a strictly formal point of view, a non-minimal description that has more than the necessary characteristics is not considered a definition: it consists of a definition and at least one theorem. If we want our definitions to tell us only the necessary information about the concept of which we talk, then we must accept the criterion of minimality. On the other hand, working against that criterion may have strong pedagogical advantages. In order to get a sound and rich mental image of the new concept, it can be advantageous to start with a description that does not satisfy the criterion of minimality. For example, we expect young students to better construct the concept of rectangle when they are exposed to the 4-angle-description instead of 3-angle-description. (The fact that there are other ways to describe a rectangle is not relevant in the current discussion.) In the second part of this article we present a description of a periodic function that does not satisfy the criterion of minimality, but seems to be easier to understand than a definition that does satisfy this criterion. Learning about logic and deductive systems in general is a completely different matter. In that case one has to understand the distinction between statements that must be proved and definitions that describe something. In that case, also the necessity of having axioms has to be clarified. What one wants to do with it afterwards is another matter, as a non-minimal definition is not an impediment for the structure of a deductive system. As for teaching mathematics at school, it depends on text book writers and teachers what to choose: a formal way of accepting only “pure” definitions, or starting with more extensive descriptions and postponing the formalities until students understand more about differences between statements and descriptions. 2.1.6. Criterion of elegance Sometimes a textbook author, for example, has to choose between two definitions that are equivalent, but one looks nicer, needs fewer words or less symbols, or uses more general basic concepts from which the newly defined concept is derived. Here is an example: Definition I. The distance between two objects is the minimum length of a segment that has one endpoint on one object and the other endpoint on the other object. 5
Philosophers of mathematics and logicians seem not to be very interested in this issue. We could not find discussions about it. They are much more concerned in deductive systems as a whole and do not agree at all in what is meant by that terminology. See for example Lakatos (1976).
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Fig. 3.
Definition II. Let two objects be given in a Cartesian coordinate system by the equations F(x, y, z) = 0 and G(x, y, z) = 0. Then the distance between the two objects is the minimum of (xF − xG )2 + (yF − yG )2 + (zF − zG )2 where F(xF , yF , zF ) = 0 and G(xG , yG , zG ) = 0. We speculate that most authors would choose Definition I as the more elegant one. Not only is it shorter in text, it is also more general, as it is not restricted to a three-dimensional Cartesian coordinate system and even not restricted to an Euclidean definition of length, and gives the essence of the notion of distance. Yet there is something to say for Definition I: It tells you precisely what you have to do in order to calculate the distance between the two objects. So why not use that definition as long as the context is restricted to a three-dimensional Cartesian coordinate system? The criterion of elegance is the most subjective of all criteria discussed. It heavily relies on personal values. Elegance is an issue not only in the choice of definitions. One can also talk about an elegant proof, or an elegant theorem.6 2.1.7. Criterion of degenerations The consequence of a definition is sometimes that it allows instances that do not conform to our intuitive idea of the concept. Such instances we shall call degenerations. The following three examples illustrate the nature and role of this criterion: • From geometry: ◦ A quadrilateral is a set of four points A, B, C, D of which no three are collinear and four segments AB, BC, CD and DA. This definition allows three kinds of quadrangles, as represented in Fig. 3. One has the feeling that Fig. 3c does not belong to this group. Now one either may accept this as a degeneration of what one intuitively thinks the concept should be, or can extend the definition so that this kind of quadrilateral is excluded. It seems to be a matter of taste. There is, however, a more serious reason to exclude type (c). Consider the theorem that the sum of the angles of a quadrilateral is 360 degrees? What are the angles of type (c)? When one wants to preserve the theorem for type (c) one has to come up with a rather complicated and artificial definition of the angles of type (c). It might be more sensible, at least for school mathematics, to declare that type (c) is not a quadrilateral, or in any case declare that type (c) will not be investigated. By the way: Is it reasonable to demand that no three vertices lay on one straight line?7 6
For a more elaborate discussion on this subject see, for example, Rota’s essay The Phenomenology of Mathematical Beauty (Rota, 1997). 7 Leaving out the condition of collinearity would lead to logical absurdities.
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• From algebra: ◦ A quadratic equation is an equation of the form ax2 + bx + c = 0, with a = 0. It is common practice in school mathematics to demand that a = 0. Apparently, the case in which a = 0 is considered as an unacceptable degeneration. Indeed, some theorems would not be true anymore for a = 0. For example, the theorem that for b2 − 4ac > 0 the equation has two different solutions. By analogy, it is similar for equations of higher degrees. Is it therefore not strange that for linear equations of the form ax + b = 0 the degeneration of the case with a = 0 is accepted? (At least in some school curricula). It is customary to say that 0x + b = 0 is a linear equation with no solutions if b = 0 and with an infinite number of solutions if b = 0. Apparently, there is no problem with theorems about such linear equations, so there is no reason to exclude this kind of degeneration. • From analytic geometry: ◦ A conic section is a curve with the equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 with the condition that A2 + B2 + C2 = 0. The condition is a measure to ensure that at least one of the coefficients A, B, C is non-zero and therefore the equation is really of second degree. Otherwise the line with equation x = 0 could be called a conic section. Even the false statement 1 = 0 could be called a conic section. These kinds of degenerations are avoided by including the condition regarding the coefficients. However, the condition allows for “curves” like x2 + y2 = 0 (a circle with zero radius), xy = 0 (two intersecting lines) and x2 − 1 = 0 (two parallel lines) and even x2 + y2 + 1 = 0. It would not be very difficult to exclude such degenerations, but there is no need for that as it does not disturb the development of a consistent theory of conic sections. On the contrary, it would make the formulation of many theorems more complicated if those degenerations are excluded. Below we shall discuss a certain kind of periodic function that one might consider as a degeneration. Degenerations are instances of a concept that we do not expect to be included when defining the concept. They are a logical outcome of the definition. One might not like the occurrence of such instances and in that case one might change the conditions of the definition so that such instances are not included. Lakatos (1976) called this process monster barring. To call a certain instance a degeneration is, of course, highly subjective and therefore there is no objective criterion for such a decision. Whether or not to exclude a degeneration is often a matter of taste. It makes sense to do so or not to do so in cases where their rejection or their acceptance would disturb the development of a theory. Most often one accepts them because one cannot foresee the consequences of rejection or, as Lakatos argued, because it stimulates mathematical investigation. Anyway it should be shown that, if the definition allows for degenerations, properties that are proved for “normal” instances also apply to degenerations. If this is not the case then either the definition or the properties must be modified and reformulated.
3. The case of periodicity Our general discussion of the notion of definition was provoked by examining several definitions of two concepts related to periodicity — a periodic function and a period of a function. Some of the
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authors who address the issue of definitions in mathematics education (Leikin & Winicki-Landman, 2000; Leikin, Zaslavsky, & Berman, 1998; Tall & Vinner, 1981; Thurston, 1974; Vinner, 1977; Vinner, 1991; Winicki-Landman & Leikin, 2000) raise pedagogical and mathematical considerations with respect to the definitions of other specific concepts (e.g., exponentiation, tangent, limit, derivative, symmetry). The specific concept can serve as examples to make some points regarding definitions in general or as a focal issue, to gain better understanding of the possible definitions of the given concepts themselves. We attempt to do both. Having dealt with the notion of definition in a more general way, we now turn to a discussion on the definitions of the concepts associated with periodicity. Once a certain formulation is chosen as the definition, and it is shown that it fulfills the Aristotelian criteria, all what follows (theorems, definition of other concepts, etc.) must be in accordance with that definition. Here is a discussion that arises from this requirement. Intuitively a periodic function is a phenomenon that keeps recurring with the same values, like the tides of the sea, the voltage of alternative current, the height of the needle of a sowing machine. Logically there is no requirement that a formal definition of periodicity must relate as close as possible to our intuition and experience. For didactical reasons, however, it is desirable that it does so. Here are two definitions of a periodic function that concur with such a desire. (We restrict ourselves to real functions.) Definition A. A function f is called periodic if there exists a non-zero number p, such that for every x that belongs to the domain of f, the following conditions are fulfilled: (a) x ± p belongs to the domain of f,8 (b) f(x ± p) = f(x). Definition G. A function f is called periodic if there exists a non-trivial translation9 of the graph of f along the horizontal axis10 such that the image coincides with the original. The geometrical Definition G is a global one: it tells us something about the function as a whole. The algebraic Definition A is an analytic one: it tells us something about the function taking a point-wise perspective. Both definitions are equivalent. According to the criterion of equivalence one has to choose one of them as a definition and formulate the other as a theorem, for example, by leaving out the word “called” from one of them. We think that teachers and textbook writers who use both the global geometric and the analytic approach should make the equivalence plausible. 8 Sometimes one sees definitions in which condition (a) is absent. The rationale for that seems to be that mentioning f(x + p) in condition (b) implies that x + p belongs to the domain. Such an understanding is a matter of taste. If one declares at the beginning of a context that the use of the expression f(x) implies that x belongs to the domain of x, there is also no formal objection against the omission of condition (a). In that case the word “that belongs to the domain of f ” is also superfluous. We shall not follow this habit here. 9 A trivial translation is a translation along a zero vector. In practice, this is not a translation at all. The term “non-trivial” is necessary, because otherwise every function would be periodic. Its equivalent in Definition A is the condition that p should be non-zero. In the following we shall always mean a non-trivial translation. 10 The word “horizontal” is essential in order that this definition is equivalent to Definition A. If we leave it out then we can find many functions that are periodic according to Definition G, but not according to Definition A. For example, the graph of the function f defined by f(x) = x + sin(x) can be translated along the line y = x such that the image coincides with the origin. This would be a periodic function if we leave out the condition horizontal, but it would not be periodic according to Definition A.
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Most often the definition of a periodic function is immediately followed by a definition of the period of a periodic function. Before we do that, we want to make some remarks about the definitions that are given above. Some people would say that Definition G is more elegant than Definition A, because it is more global. Others can argue that just for that reason Definition A is the more elegant of the two. A close look at the two equivalent definitions and some of their implications may lead to several questions worth pursuing. We discuss a few of them. 3.1. If f(x + p) = f(p), then what about f(x + 12345p)? In a textbook (De Lange & Kindt, 1985) the following definition appears: Definition T. A function f is called periodic if there exists a non-zero number p, such that for every x that belongs to the domain of f the following conditions are fulfilled: (a) x ± p, x ± 2p, x ± 3p, x ± 4p, etcetera, belong to the domain of f, (b) f(x) = f(x ± p) = f(x ± 2p) = f(x ± 3p) = . . . etcetera. Although such a definition indeed comes close to our experience, this is logically not necessary and therefore violates the criterion of minimality. It can easily be shown by complete induction that the above Definition T is in fact a property of periodic functions that are defined by Definition A. This is an example of the fact that, in order to be rigorously logical we cannot always follow our intuition literally. It is also an example of the contrary: The writers of the textbook did not want this rigor at the moment that they introduced the idea of periodicity and therefore used the more intuitive way of describing a periodic function. 3.2. Is the constant function periodic? An intuitive notion of a periodic function is that the values of it change, but keep repeating themselves regularly. Someone with that notion could try and find a definition of periodic functions and could come up with Definition A or G. Later he would discover that according to these definitions the constant function is a periodic function. For example, Let f be defined as: f(x) = 1564.78 for every x. Then there is a number, say 821, such that (a) for every x in the domain of f, also x + 821 belongs to the domain of f, and (b) f(x) = f(x + 821). He would get the feeling that the constant function is a degeneration of the concept. It does not comply with his intuitive idea of periodicity that has to do with change. Some writers (e.g., Shama, 1995), therefore, explicitly exclude constant functions from their definition of a periodic function. This seems more a matter of aesthetics and of personal values. Indeed, as long as one is consistent in what follows there is no logical objection to it. Here, however, is an example of a problem that might arise under certain conditions:11 the sum of two periodic functions is also a periodic function: f(x) = sin2 (x)
and
g(x) = cos2 (x)
11 If two functions that are periodic under Definition A, have the same domain and the ratio between their periods is a rational number, the sum of the two functions is a function that is periodic under Definition A.
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These two functions satisfy the specific conditions necessary for the theorem to apply; thus, their sum should be a periodic function. However, the sum of these two functions is a constant function: (f +g)(x) = 1 for all x. So if one excludes a constant function from the definition of a periodic function, one must reformulate the theorem under the indicated conditions, “. . . the sum of two periodic functions is either a periodic function or a constant function.” We suggest that a more convenient way to treat situations like this is to allow degenerations. 3.3. Do we really need the ± sign? The use of the ± sign in Definition A made us wonder why it is not enough to use the + sign only. In other words: is the use of the ± sign not a violation of the criterion of minimality? Let us examine the following definition that appears in a textbook of trigonometry instead of Definition A (Van Dormolen, 1970). Definition D. A function f is called periodic on R (the set of real numbers) if there exists a non-zero number p, such that for every x that belongs to the domain of f, the following condition holds: f(x + p) = f(x). There are a number of differences between Definition D and Definition A. First, instead of the ± sign in Definition A only the + sign appears in Definition D. Secondly, in Definition D there is a restriction on the domain, namely, this definition requires that f has R as its domain. Finally, the condition that x + p belongs to the domain does not appear in Definition D. This condition need not be included because it is implied by the fact that f has R as its domain. And indeed it is easy to prove that for any number p that fulfills the condition f(x + p) = f(x) for every x in R, also the condition f(x − p) = f(x) holds.12 In fact, in Definition D we need not restrict ourselves to R. We can define a periodic function on the set of rational numbers, or the set of integers, or the set of all integers that are divisible by 13. Example: The function of parity: f is defined on the set of integers and f(x) = 0 if x is even and f(x) = 1 if x is odd. This function f is periodic. Again one wonders what is so special about domains like those mentioned above. Can we omit any reference to the domain and just use the following definition? Definition A+ . A function f is called periodic if there exists a non-zero number p, such that for every x that belongs to the domain of f, the following conditions are fulfilled: (a) x + p belongs to the domain of f, (b) f(x + p) = f(x). We shall show by two counter-examples that Definition A+ is not equivalent to Definition A. Let f be a function defined on the real numbers and let p be a number such that for all x: f(x) = f(x + p). The number x − p is a real number and therefore it belongs to the domain of f. Now, because x − p is an element of the domain of f, we can apply Definition D to that number: f((x − p) + p) = f(x − p). This is the same as f(x) = f(x − p). In summary: For every x in the domain the number x − p also belongs to the domain and f(x) = f(x − p). 12
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Fig. 4.
First counter-example: Let f describe a certain process that starts at a certain time (x = 0), continues for 1 hour and then starts all over again. This happens at every whole hour. Fig. 4 is the graph of such a function f. Here f(x + 1) = f(x) for all x of the domain, which is the set of non-negative real numbers. However, nothing is defined for negative numbers x and therefore it is not true that f(x − 1) = f(x) for all x of the domain.13 This is a case of a bounded domain. Intuitively, we would be inclined to accept a function like this as a periodic function, because in essence it behaves like a periodic function and can represent a real life phenomenon that repeats itself over and over in the same way. However, according to Definition A this is not a periodic function.14 It would not be difficult to develop a consistent theory on the basis of Definition A+ , although (as we shall indicate with the next example) we may not be inclined to accept as periodic functions all the additional functions that satisfy Definition A+ . Following is the second counter-example in which the domain is unbounded. Let D be the set that consists of all negative (whole) multiples of and all non-negative real numbers. Let f be defined on that domain such that f(x) = 0 if x < 0 and f(x) = sin(x) if x ≥ 0 (see Fig. 5). The number or any (whole) multiple of has the property of p in Definition A+ , but not in Definition A: It is not true that f(x − ) = f(x) for every x of the domain. There is no other possibility for p. Thus, according to Definition A+ the function f is periodic, but not according to Definition A. Actually, this is an example that we would not be inclined to accept as a periodic function even before checking whether it satisfies the conditions of Definition A. Going back to the question: What is so special about domains of functions like the set of real numbers, the set of rational numbers, the set of integers, or the set of all integers that are divisible by 13? These domains satisfy the following condition: 13 It is possible to extend a function f that fulfils Definition A+ to the other side of its domain (if its domain is lower bounded) and to obtain by that a periodic function according to Definition A. This can be done by a statement that on every interval (n, n + 1) that starts with an integer n and ends with the consecutive integer n + 1, the function behaves like it behaves on (0, 1). Or more formally: If G is defined on [0,1), for every integer n and for every x in the interval (n, n + 1): f(x) = G(x − n). 14 A function like this that can algebraically be constructed (and be drawn on a graphic calculator) as follows. Let h be defined by: h(x) = 1 for x ≥ 0 and h(x) = −1 for x < 0. Let g be any function that is defined on the segment [0,1]. Let [x] be the entire √ function: [x] is the largest integer that is smaller than or equal to x. Then the function f, defined by f(x) = g(x − [x]) (h(x)) has domain [0,→) and for every x : f(x + 1) = f(x), while f(x) does not exist for x < 0.
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Fig. 5.
For every x in the domain and every non-zero number z: If x + z belongs to the domain, then also x − z belongs to the domain. One way to modify Definition A+ is to include functions that satisfy our intuitive notion of periodicity can be adding to the definition this condition on the domain. We do not know if this condition is necessary to turn Definition A+ into an acceptable definition. It certainly is sufficient.
4. How to define the concept of period of a function? As before, starting with our intuition and experience, we can ask ourselves what we mean by the word period. A periodic function describes a situation that repeats itself after a certain time, or after a certain number, or after a certain distance. This “certain time,” or “certain number,” or “certain distance” we call period. On the basis of this intuition we can define such a number more formally. Writers define period in different ways: Definition B. Let f be a periodic function (according to Definition A). Every number p (=0) that fulfills the conditions of Definition A is called a period of f. If there exists in the set of all periods a smallest positive one, then that smallest positive one is called the minimal period of f or basic period of f. Definition R. Let f be a periodic function (according to Definition A). If there exists a smallest positive number in the set of all numbers p that fulfill the conditions of Definition A, that smallest positive one is called the period of f.15 This time the two definitions of period are not equivalent. If one uses the broader Definition B, a periodic function has an infinite number of periods. There is no such thing as the period. It is, however, possible for a periodic function to have the minimal (or basic) period. If one prefers the restricted Definition R, it is possible to have at the most just one period. The period according to Definition R is the same as the minimal (basic) period in Definition B. Why would one prefer Definition R to Definition B or vice versa? Some people think that the really interesting number is the smallest positive number (if such a number exists) and therefore they prefer 15 Analogue definitions can be given that concur with Definition G. Instead of “numbers p” one must talk then about “distances of translation” or “vectors of translation” or any other term to that effect.
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Definition R. They can then talk about the period p of a periodic function. Others argue that it is sometimes difficult, maybe even impossible, to find the smallest number, even when they know that such a number exists for a certain function. Those prefer Definition B. Whatever choice one makes, one has to be consistent. For example, different statements describe the same property of (a) period: If one uses Definition B, the property mentioned in If f(x+p) = f(p), then what about f(x+12345p)? should be stated as following: If there exists a minimal period, every period is a (positive or negative) multiple of that minimal period. If one uses Definition R, the property should be rephrased to: If there exists a period, every (positive or negative) multiple of it fulfills the conditions of Definition A. 4.1. Is there always a period? We wrote that sometimes a writer excludes constant functions from being a periodic function. We assumed that this was because the values of a constant function do not change. There is another possibility that has to do with the idea of the basis period (or the period in case of Definition R). A constant function does not have a basic period. That is why we had to add the words “if there exists” in both Definitions B and R. One might be inclined to consider this as the reason for excluding constant functions as being counter-intuitive and as a result omit the condition “if there exists” from the definitions of period. This, however, does not work, because it requires that other degenerations of a similar nature be excluded too. Take for example Dirichlet’s function: f(x) = 0 if x is a rational number and f(x) = 1 if it is not. If the constant function is excluded, so should Dirichlet’s function be exluded. In order to be consistent, we want to be sure that if we exclude degenerations like these, we exclude all possible ones. As long as we do not have a good criterion that describes all of these degenerations, we have to include the requirement of existence of a lowest positive p. 5. Closing remarks: logic before pedagogy? The vignette in the beginning of this article is actually a short resume of a much longer and disordered series of discussions. It focused on the definition of period. LL thought that Definition B was the correct one while JD defended Definition R until both realized that each of them was confined to what s/he had learned many years ago and did not refer to any logical criteria. The same happened with the question of whether a constant function was a periodic function. This dialogue triggered a much longer and deeper discussion within a community of colleagues and led first to several workshops within the framework of a professional development program for secondary mathematics teachers, followed by written resource material for these teachers, and finally resulted in the current article. We (re-)learned that in doing mathematics some rules have to be followed for logical reasons, some for conventional reasons and some for pedagogical reasons and that it is dangerous to mix-up the three reasons. We also (re-)learned that it is worthwhile for teachers as well as for teacher educators to investigate in depth the logical aspects of the mathematical activity that we are planning to teach, even if we do not
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intend to overburden our students with formal logic. Teachers and teacher-educators need to engage in such kinds of investigation in order to be able to make educated pedagogical decisions. In that sense, for them, logic comes before pedagogy. And above all, we (re-)learned that mathematics is a human activity in which we have the freedom to investigate and the freedom to make decisions, as long as we are aware of their consequences.16 References Ausubel, D. P., & Robinson, F. G. (1969). School learning: an introduction to educational psychology. New York: Holt, Rinehart and Winston. Brousseau, G. (1997). Theory of didactical situations in mathematics. Edited and translated by N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield. Dordrecht: Kluwer Academic Publishers. Davis, P. J., & Hersch, R. (1981). The mathematical experience. Boston: Houghton Mifflin Company. DeCecco, J. P. (1968). The psychology of learning and instruction: Educational psychology. Englewood Cliffs: Prentice-Hall. De Lange Jzn., J., & Kindt, M. (1985). Periodieke functies. Culemborg: Educaboek. Fischbein, E. (1987). Intuition in science and mathematics: an educational approach. Dordrecht: Kluwer Academic Publishers. Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel. Freudenthal, H. (1978). Weeding and sowing. Dordrecht: Reidel. Heath, T. L. (1956). Euclid. The thirteen books of the elements. New York: Dover Publications. Hershkowitz, R., Bruckheimer, M., & Vinner, S. (1987). Activities with teachers based on cognitive research. In: M. M. Lindquist, & A. P. Shulte (Eds.), Learning and teaching geometry, K-12, 1987 Yearbook. NCTM. Lakatos, I. (1976). Proofs and refutations. Cambridge: CUP. Leikin, R., Berman, A., & Zaslavsky, O. (1998). Definition of symmetry. Symmetry: Culture and Science, 9(2/4), 375–382. Leikin, R., & Winicki-Landman, G. (2000). On equivalent and non-equivalent definitions II. For the Learning of Mathematics, 20(2), 24–29. Mason, J., & Pimm, D. (1984). Generic examples: seeing the general in the particular. Educational Studies in Mathematics, 15, 277–289. Quine, W. V. O. (1953). From a logical point of view. Cambridge, MA: Harvard University Press. Rota, G.-C. (1997). Indiscrete thoughts. Boston: Birkhauser. Shama, G. (1995). The concept of periodicity. Unpublished Doctoral Dissertation, Tecnion, Israel Institute of Technology, Haifa (in Hebrew). Skemp, R. R. (1971). Psychology of learning mathematics. Harmondsworth: Penguin Books. Stewart, I. (1997). The mathematical maze: seeing the world through mathematical eyes. London: Weidenfeld & Nicolson. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169. Thurston, H. (1974). Clarity in the calculus. Mathematics Teaching, 67, 58–60. Van Dormolen, J. (1970). Goniometrische functies. The Hague: Van Goor. Van Dormolen, J. (1991). Metaphors mediating the teaching and understanding of mathematics. In J. A. Bishop, S. Mellin-Q-Sen, & J. Van Dormolen (Eds.), Mathematical knowledge: Its growth through teaching (pp. 89–106). Dordrecht: Kluwer Academic Publishers. Van Dormolen, J., & Arcavi, A. (2000). What is a circle. Mathematics in School, 20(5), 15–19. Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In: Tall, D. (Ed.), Advanced mathematical thinking. Dordrecht: Kluwer Academic Publishers. Winicki-Landman, G., & Leikin, R. (2000). On equivalent and non-equivalent definitions I. For the Learning of Mathematics, 20(1), 17–21. The function x → 3sin x − 4sin3 x has certainly a period 2 (in the terminology of Definition B). Some might find it hard to show that it has the minimal period 2/3. The function x → (sin4 x + cos4 x)/(2 − sin2 (2x)) has certainly a period of 2 (in the terminology of Definition B). Some might find it hard to proof that it has no minimal period. 16
The many facets of a definition: The case of periodicity Joop Van Dormolen, Orit Zaslavsky∗ Technion – Israel Institute of Technology, Technion, Haifa 32000, Israel
Abstract This paper was triggered by an authentic conversation between two mathematics teacher educators who debated whether a constant function is a periodic function, within the framework of a professional development program for secondary mathematics teachers. Their initial conversation led to deep mathematical and pedagogical musing surrounding mathematical definitions. In this paper, we present various aspects of a mathematical definition, including the role and nature of definitions in school mathematics, critical versus preferable features of a definition, and the arbitrariness underlying the choice of definition. We discuss the interplay between logical and pedagogical considerations with respect to definitions, drawing on the definition of a periodic function as an example. © 2003 Elsevier Science Inc. All rights reserved. Keywords: Mathematical definition; Periodicity; Logic; Pedagogy; Teacher education; Secondary school mathematics
1. Vignette LL: Look here on this definition of a periodic function. It says that a constant function is not a periodic function. Do you agree? JD: Well that depends, of course, on the definition. Some definitions accept a constant function as a periodic function. LL: What do you mean by “some definitions.” There cannot be more than one definition of a concept. JD: Why not? LL: We cannot communicate if we each have a different definition. JD: True, but we can agree to use one particular definition and on that base we could communicate very well. LL: But we must use the definition that is used in mathematics.
∗ Corresponding author. Tel.: +972-4-8262798; fax: +972-4-8241819. E-mail address: [email protected] (O. Zaslavsky).
0732-3123/03/$ – see front matter © 2003 Elsevier Science Inc. All rights reserved. doi:10.1016/S0732-3123(03)00006-3
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JD: I have no idea what you mean by that. A definition is just an agreement between people to give a name to a certain object or a certain idea. By the way, I saw in the definition that you showed me just now that the definition of a period is not correct. They say that every p = 0 that fulfills the requirements of a periodic function is called a period of the function. That cannot be true. Then a periodic function can have many different periods. But the period is the smallest positive p that fulfills the conditions (if there exists a smallest positive). LL: Well, didn’t you say just now that it depends on how you define a concept? You are talking about “the” period. Here they talk about “a period.” Why would it be impossible for a periodic function to have many periods? JD: We are free to give any definition as long as it fulfills certain logical criteria. LL: Show me then where we would get a contradiction if there were many periods. JD: I think you are right. We have to consider all the criteria of a definition. Some are necessary, but not all.
2. Pedagogy before logic? The vignette above gives the essence of a conversation that took place between one of the authors and a colleague. The discussion was the trigger for the current article. As can be expected of mathematics educators with a sound mathematical background, who are fond of mathematics, the above conversation evoked a much deeper discussion of its implications. The discussion moved on from technicalities about definitions to more subtle questions regarding the role and nature of definitions in school mathematics. Also the question was raised whether a definition must be a formal definition or could it be an example-based description, such as: A periodic function is a function that can be constructed in the following way: Divide the x-axis into equal-length segments, such as for example . . . , [−39, −26], [−26, −13], [−13, 0], [0, 13], [13, 26], [26, 39], . . . Take any of these segments, no matter which one, and define a function on it, no matter how (e.g., as in Fig. 1). Then define another function on the whole x-axis, such that on each segment it behaves in the same way as the first function (as in Fig. 2). Then the new function is a periodic function. Its values are repeated regularly.
Fig. 1.
Fig. 2.
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From a formal standpoint this is not a “good” definition. In fact, it is not a definition at all, because it does not fulfill certain criteria, which are discussed in the next section. However, from a pedagogical standpoint this example-based description has many advantages, and can serve as an informal definition. It may be seen as following the spirit of many mathematics educators and learning psychologists who agree that, in general, the learning of a new concept can best be started in an informal way through one or more well-chosen examples. There are differences in opinions regarding several related issues, such as the specific circumstances in which this learning may take place, or the interplay between a given concept-definition and the sequencing of examples and non-examples of the concept. (For example, Ausubel & Robinson, 1969; Brousseau, 1997; DeCecco, 1968; Fischbein, 1987; Freudenthal, 1973, 1978; Hershkowitz, Bruckheimer, & Vinner, 1987; Leikin, Zaslavsky, & Berman, 1998; Mason & Pimm, 1984; Skemp, 1971; Tall & Vinner, 1981; Van Dormolen, 1991; Van Dormolen & Arcavi, 2000; Vinner, 1991; Winicki-Landman & Leikin, 2000). Our paper addresses some of these issues only indirectly. However, its main goal is to provoke thinking about the nature of mathematical definitions from both pedagogical and formal mathematical perspectives and to consider ways to define a concept at a given moment. In order to work well with formal definitions, or at least in order to teach the use of formal definitions, it is helpful to examine some criteria that people consider as significant for a definition to be a “good” definition. A definition is like a cut diamond with many facets. In this paper we attempt to shed light on the formal side of the diamond, hoping that the sparkling lights will catch our readers’ eyes. We realize that many of our readers are familiar with several issues discussed in this paper but, as the vignette shows, many may not be aware of all of them. We hope to inspire some of the readers to re-think and gain insight into some salient issues associated with definitions, similarly to the process that we ourselves have encountered.
2.1. Necessities and elegance of a definition Some of the criteria that we demand of a definition are necessary. They are fundamental parts of a deductive system. They are logical necessities. These criteria include: Criterion of hierarchy Criterion of existence Criterion of equivalence Criterion of axiomatization Other criteria are not necessary from a logical standpoint, but they are part of a general culture. For example, the criterion that demands that no more properties of a concept be mentioned unless it is necessary to establish the concept. Is it often called: Criterion of minimality Another criterion comes to play when, for example, a textbook author has to choose between two definitions, that are equivalent, but one looks nicer, needs fewer words or less symbols, or uses more general basic concepts from which the newly defined concept is derived. We call it Criterion of elegance A criterion that sometimes is a necessity and sometimes a beautification is: Criterion of degenerations We turn to a comprehensive discussion of the various criteria.
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2.1.1. Criterion of hierarchy Aristotle gave criteria for defining a concept. (See for example, Heath, 1956, pp. 116–124.) One of them contains a description of the structure of a definition. According to that criterion any new concept must be described as a special case of a more general concept. One or more properties must be used to describe this special case. For example, in “A right angle is an angle of which the legs are perpendicular to each other” the general concept is “angle” and the property is: “the legs are perpendicular to each other.” This way of defining a new concept has an important logical consequence: One may only use the general concept and the properties if they have been determined previously. This may seem a trivial condition, but in practice sometimes there are problems when we want to explain intuitively what a certain concept is. For example, a periodic function is a function for which a certain phenomenon keeps recurring. This is not a good definition in the Aristotelian sense, because “a certain phenomenon” is not well defined. Nor is “keeps coming back.” As an introduction to the concept of periodicity, however, it is acceptable as long as it is accompanied by examples that students can easily recognize. In this case, it is more than acceptable. It is a good didactical practice because (a) it uses ideas that are well known by the students and (b) it involves the students themselves in answering the question of how to clearly define the concept of a periodic function. 2.1.2. Criterion of existence A definition tells us what a concept is, but usually it does not say whether there exists an instance of such a concept within the current system. For this reason, Aristotle required, as a second criterion, that it must be proven that at least one instance of the newly defined concept exists in the current context. One can take a strictly formal standpoint and leave out this criterion. In that case it might be possible that the set of instances of the newly defined concept is empty. However, it is not advantageous to do so. For one thing, it opens the way to defining absurdities. For example, one could try defining, within the context of Euclidean geometry, a circlesquare as a square for which all points have the same distance to a certain point. Such a definition fulfills the first criterion, but not the second: in Euclidean geometry there is no such thing.1 The circlesquare in the system of Euclidean geometry is an example of an absurdity. One gets entangled in a philosophical web of discussions about the possibility of defining a thing that does not exist and therefore it seems to exist and not-exist at the same time.2 Moreover, it might lead to unforeseen contradictions in the system.3 This existence-criterion was the reason why Euclid wrote his “constructions.” For example, the construction of the bisector of an angle was not presented in order to show how this can be done, but rather in order to prove that it can be done. In other words: there exists such a thing as “bisector of an angle.” 1
The impossibility lies in the definition of Euclidean distance. There are many other possibilities to define distance. For example, taxi-distance in a Cartesian coordinate system: The taxi-distance between two points (a, b) and (p, q) is equal to |a − p| + |b − q|. The set of all points (x, y) that have the distance 1 to the origin is a circle by definition, but looks like a square: |x| + |y| = 1. 2 Quine (1953) gives an example of two philosophers that discuss whether or not Pegasus (the flying horse) exists. One of them remarks that it is impossible to say that he does not exist, because the moment one starts talking about Pegasus, one assumes its existence, otherwise one cannot talk about Pegasus. Also see Davis and Hersch (1981). 3 What to do when we cannot prove that a concept exists? Is it true that a concept either exists or does not exist? See for example Davis and Hersch (1981). In one of his stimulating books Stewart (1997) shows that one can prove that “The largest non-zero whole number is 1”(p. 194) if one does not bother to ask whether such a number exists.
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The first theorem in Euclid’s books was the construction of an isosceles triangle. Even though he had defined this concept before, he had to prove its existence. The consequence of this criterion—both from a logical as well from a pedagogical standpoint—is that, after a well-defined concept one has to give an instance. If a teacher gave an intuitive introduction before the definition was developed, s/he could refer to the introductory examples as a proof that the definition was a good one in this sense. In the case of periodic functions this is not a difficult task at all. In any case, Aristotle’s criterion seems to be a good and sensible one from a pedagogical perspective as well, if only to remind us that it is good practice to clarify a definition with an example. 2.1.3. Criterion of equivalence A third criterion (not explicitly given by Aristotle) is that, when one gives more than one formulation for the same concept, one must prove that they are equivalent. In practice this means that one has to choose one of the formulations as the definition and consider the other formulations as theorems that have to be proved. Consider, for example, the following formulations: A parallelogram is a quadrilateral in which the opposite sides are parallel. A parallelogram is a quadrilateral in which the opposite sides are equal. A parallelogram is a quadrilateral in which the sides of one pair of opposite sides are both equal and parallel. A parallelogram is a quadrilateral that is symmetrical with respect to a point. One can choose each of these formulations as the definition of a parallelogram. Thus, each of the other formulations is then a theorem to be proved.4 Which definition to actually choose amongst a number of equivalent definitions is a matter of taste or convenience. It can also depend on the context. Whatever the choice is, the mere idea that there is freedom to choose in mathematics will probably come as a surprise to many students. The “freedom to choose” is a friendlier way to look at what often is considered “arbitrariness” of a definition. Yet, freedom is never unconditional, and in order to have the freedom to choose a definition of a concept, one must make sure that all “candidates” are equivalent. Moreover, as we discuss below, one often has the freedom to choose between two definitions even if they are not equivalent. However, then the consequences of each choice should be carefully examined. 2.1.4. Criterion of axiomatization This criterion implies that a definition fits in and is part of a deductive system. Here is a short outline of what we mean by a deductive system. All the above relates to explicit definitions of a concept. An explicit definition tells us what the concept is. Like: “A parallelogram is . . . ,” or “A periodic function is . . . .” It does so with the use of a more general concept, such as quadrilateral (for a parallelogram) of function (for periodic function). These more general concepts must also be defined, based on even more general concepts. It is not possible however to continue this endlessly. At a certain point one gets concepts that cannot be defined with more 4 In the same book of Quine, there is a discussion about the influence of context. Some philosophers might say that the parallelograms are not the same, because they have different contexts, while they are the same. (In the book, the discussion is about the question of whether the Morningstar and the Eveningstar are the same. They refer to the same physical object and yet are not the same because they function in different contexts.) Quine solves this problem by stating that the definitions refer to the same object, but that they have different meanings.
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general ones, like point, line, plane in case of the parallelogram. A point cannot be described in a sentence like: “A point is . . . .” Nor can a line or a plane. In short: some concepts cannot be defined according to the Aristotelian criterion of hierarchy. Aristotle knew this and therefore wrote that axioms or postulates implicitly define such concepts. In geometry such concepts are point, line, and plane. Those concepts are implicitly defined in terms of axioms by their relations to each other: “There is one and only one line through any two points,” “On each line lie at least two points,” etcetera. To establish these relations in axioms is not enough. One can only talk about the concepts if they exist, but one cannot prove that they exist. Therefore, there are also axioms that establish their existence, such as: “There exist at least three points that do not lie on the same line.” Note that in these kinds of axioms relationships such as “go through” and “lie on” are defined implicitly. Another example is the definition of natural numbers. There are several possibilities. One can adopt the Peano axioms, or define natural numbers with the help of the cardinality of finite sets. In the first case one of the axioms states the existence of a “first element,” that we may call “one.” In the second case one has to look deeper at the axioms of sets in which there is a statement that the empty set exists. Now one can ask how to verify if a certain (explicit) definition satisfies the criterion of axiomatization. In other words: one must verify if it fits in and is part of a deductive system. In practice this would mean that it has to be verified if all the concepts used in the definition are, in their turn, again defined within the same deductive system, like in the example of the parallelogram above. A student once described a circle as “a figure that everywhere has the same roundness.” In the mind of the student, this was a perfect description of what she understood to be a circle. In the context of Euclidean geometry, seen as a deductive system, it is not (for the time being) a good definition as long as concepts like “everywhere” and “roundness” are not (yet) defined. 2.1.5. Criterion of minimality This criterion demands that no more properties of the concept be mentioned than is required for its existence. According to this criterion the following description would not be a good definition within Euclidean geometry: A rectangle is a quadrilateral with four right angles. In Euclidean geometry we can prove that the sum of the four angles of a quadrilateral is 360 degrees. Therefore, it is enough to define: A rectangle is a quadrilateral with three right angles. Let us refer to the first definition as the “4-angle-description” and to the second one as the “3-angledescription.” On first sight the criterion of minimality seems to be more of an aesthetic or philosophical nature than a logical one. Indeed, describing a rectangle as quadrilateral with four right angles will not result in a contradiction in the system, and may have advantages from a pedagogical perspective. Moreover, often at the time when a concept is defined there may not be sufficient knowledge to determine whether it is minimal. Thus, insisting on this criterion may impede the development of certain concepts or theories. There are, however, good reasons to accept this criterion. According to the criterion of existence it is necessary to establish that the new concept exists. This means that one has to investigate the new definition. When one investigates the 3-angle-description of a
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rectangle, it is inevitable that one gains the insight that a quadrilateral with four right angles is precisely what one had in mind, but defining it with three right angles is enough. So why not use the more economical one?5 There is, however, more to say than just an economical argument. By the very fact that one, by investigating a description that is meant to serve as a definition, discovers that some properties are superfluous, one has implicitly proved a theorem, namely that an object that satisfied the more frugal description has the properties that are mentioned in the more extensive non-minimal description. In the example of the rectangle, by investigating the 3-angle-definition of a rectangle, one can prove the theorem that if a quadrilateral has three right angles, the fourth angle is also a right one. So, by accepting the criterion of minimality one has to take the minimal description as definition and as such reject the non-minimal description. Thus, from a strictly formal point of view, a non-minimal description that has more than the necessary characteristics is not considered a definition: it consists of a definition and at least one theorem. If we want our definitions to tell us only the necessary information about the concept of which we talk, then we must accept the criterion of minimality. On the other hand, working against that criterion may have strong pedagogical advantages. In order to get a sound and rich mental image of the new concept, it can be advantageous to start with a description that does not satisfy the criterion of minimality. For example, we expect young students to better construct the concept of rectangle when they are exposed to the 4-angle-description instead of 3-angle-description. (The fact that there are other ways to describe a rectangle is not relevant in the current discussion.) In the second part of this article we present a description of a periodic function that does not satisfy the criterion of minimality, but seems to be easier to understand than a definition that does satisfy this criterion. Learning about logic and deductive systems in general is a completely different matter. In that case one has to understand the distinction between statements that must be proved and definitions that describe something. In that case, also the necessity of having axioms has to be clarified. What one wants to do with it afterwards is another matter, as a non-minimal definition is not an impediment for the structure of a deductive system. As for teaching mathematics at school, it depends on text book writers and teachers what to choose: a formal way of accepting only “pure” definitions, or starting with more extensive descriptions and postponing the formalities until students understand more about differences between statements and descriptions. 2.1.6. Criterion of elegance Sometimes a textbook author, for example, has to choose between two definitions that are equivalent, but one looks nicer, needs fewer words or less symbols, or uses more general basic concepts from which the newly defined concept is derived. Here is an example: Definition I. The distance between two objects is the minimum length of a segment that has one endpoint on one object and the other endpoint on the other object. 5
Philosophers of mathematics and logicians seem not to be very interested in this issue. We could not find discussions about it. They are much more concerned in deductive systems as a whole and do not agree at all in what is meant by that terminology. See for example Lakatos (1976).
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Fig. 3.
Definition II. Let two objects be given in a Cartesian coordinate system by the equations F(x, y, z) = 0 and G(x, y, z) = 0. Then the distance between the two objects is the minimum of (xF − xG )2 + (yF − yG )2 + (zF − zG )2 where F(xF , yF , zF ) = 0 and G(xG , yG , zG ) = 0. We speculate that most authors would choose Definition I as the more elegant one. Not only is it shorter in text, it is also more general, as it is not restricted to a three-dimensional Cartesian coordinate system and even not restricted to an Euclidean definition of length, and gives the essence of the notion of distance. Yet there is something to say for Definition I: It tells you precisely what you have to do in order to calculate the distance between the two objects. So why not use that definition as long as the context is restricted to a three-dimensional Cartesian coordinate system? The criterion of elegance is the most subjective of all criteria discussed. It heavily relies on personal values. Elegance is an issue not only in the choice of definitions. One can also talk about an elegant proof, or an elegant theorem.6 2.1.7. Criterion of degenerations The consequence of a definition is sometimes that it allows instances that do not conform to our intuitive idea of the concept. Such instances we shall call degenerations. The following three examples illustrate the nature and role of this criterion: • From geometry: ◦ A quadrilateral is a set of four points A, B, C, D of which no three are collinear and four segments AB, BC, CD and DA. This definition allows three kinds of quadrangles, as represented in Fig. 3. One has the feeling that Fig. 3c does not belong to this group. Now one either may accept this as a degeneration of what one intuitively thinks the concept should be, or can extend the definition so that this kind of quadrilateral is excluded. It seems to be a matter of taste. There is, however, a more serious reason to exclude type (c). Consider the theorem that the sum of the angles of a quadrilateral is 360 degrees? What are the angles of type (c)? When one wants to preserve the theorem for type (c) one has to come up with a rather complicated and artificial definition of the angles of type (c). It might be more sensible, at least for school mathematics, to declare that type (c) is not a quadrilateral, or in any case declare that type (c) will not be investigated. By the way: Is it reasonable to demand that no three vertices lay on one straight line?7 6
For a more elaborate discussion on this subject see, for example, Rota’s essay The Phenomenology of Mathematical Beauty (Rota, 1997). 7 Leaving out the condition of collinearity would lead to logical absurdities.
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• From algebra: ◦ A quadratic equation is an equation of the form ax2 + bx + c = 0, with a = 0. It is common practice in school mathematics to demand that a = 0. Apparently, the case in which a = 0 is considered as an unacceptable degeneration. Indeed, some theorems would not be true anymore for a = 0. For example, the theorem that for b2 − 4ac > 0 the equation has two different solutions. By analogy, it is similar for equations of higher degrees. Is it therefore not strange that for linear equations of the form ax + b = 0 the degeneration of the case with a = 0 is accepted? (At least in some school curricula). It is customary to say that 0x + b = 0 is a linear equation with no solutions if b = 0 and with an infinite number of solutions if b = 0. Apparently, there is no problem with theorems about such linear equations, so there is no reason to exclude this kind of degeneration. • From analytic geometry: ◦ A conic section is a curve with the equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 with the condition that A2 + B2 + C2 = 0. The condition is a measure to ensure that at least one of the coefficients A, B, C is non-zero and therefore the equation is really of second degree. Otherwise the line with equation x = 0 could be called a conic section. Even the false statement 1 = 0 could be called a conic section. These kinds of degenerations are avoided by including the condition regarding the coefficients. However, the condition allows for “curves” like x2 + y2 = 0 (a circle with zero radius), xy = 0 (two intersecting lines) and x2 − 1 = 0 (two parallel lines) and even x2 + y2 + 1 = 0. It would not be very difficult to exclude such degenerations, but there is no need for that as it does not disturb the development of a consistent theory of conic sections. On the contrary, it would make the formulation of many theorems more complicated if those degenerations are excluded. Below we shall discuss a certain kind of periodic function that one might consider as a degeneration. Degenerations are instances of a concept that we do not expect to be included when defining the concept. They are a logical outcome of the definition. One might not like the occurrence of such instances and in that case one might change the conditions of the definition so that such instances are not included. Lakatos (1976) called this process monster barring. To call a certain instance a degeneration is, of course, highly subjective and therefore there is no objective criterion for such a decision. Whether or not to exclude a degeneration is often a matter of taste. It makes sense to do so or not to do so in cases where their rejection or their acceptance would disturb the development of a theory. Most often one accepts them because one cannot foresee the consequences of rejection or, as Lakatos argued, because it stimulates mathematical investigation. Anyway it should be shown that, if the definition allows for degenerations, properties that are proved for “normal” instances also apply to degenerations. If this is not the case then either the definition or the properties must be modified and reformulated.
3. The case of periodicity Our general discussion of the notion of definition was provoked by examining several definitions of two concepts related to periodicity — a periodic function and a period of a function. Some of the
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authors who address the issue of definitions in mathematics education (Leikin & Winicki-Landman, 2000; Leikin, Zaslavsky, & Berman, 1998; Tall & Vinner, 1981; Thurston, 1974; Vinner, 1977; Vinner, 1991; Winicki-Landman & Leikin, 2000) raise pedagogical and mathematical considerations with respect to the definitions of other specific concepts (e.g., exponentiation, tangent, limit, derivative, symmetry). The specific concept can serve as examples to make some points regarding definitions in general or as a focal issue, to gain better understanding of the possible definitions of the given concepts themselves. We attempt to do both. Having dealt with the notion of definition in a more general way, we now turn to a discussion on the definitions of the concepts associated with periodicity. Once a certain formulation is chosen as the definition, and it is shown that it fulfills the Aristotelian criteria, all what follows (theorems, definition of other concepts, etc.) must be in accordance with that definition. Here is a discussion that arises from this requirement. Intuitively a periodic function is a phenomenon that keeps recurring with the same values, like the tides of the sea, the voltage of alternative current, the height of the needle of a sowing machine. Logically there is no requirement that a formal definition of periodicity must relate as close as possible to our intuition and experience. For didactical reasons, however, it is desirable that it does so. Here are two definitions of a periodic function that concur with such a desire. (We restrict ourselves to real functions.) Definition A. A function f is called periodic if there exists a non-zero number p, such that for every x that belongs to the domain of f, the following conditions are fulfilled: (a) x ± p belongs to the domain of f,8 (b) f(x ± p) = f(x). Definition G. A function f is called periodic if there exists a non-trivial translation9 of the graph of f along the horizontal axis10 such that the image coincides with the original. The geometrical Definition G is a global one: it tells us something about the function as a whole. The algebraic Definition A is an analytic one: it tells us something about the function taking a point-wise perspective. Both definitions are equivalent. According to the criterion of equivalence one has to choose one of them as a definition and formulate the other as a theorem, for example, by leaving out the word “called” from one of them. We think that teachers and textbook writers who use both the global geometric and the analytic approach should make the equivalence plausible. 8 Sometimes one sees definitions in which condition (a) is absent. The rationale for that seems to be that mentioning f(x + p) in condition (b) implies that x + p belongs to the domain. Such an understanding is a matter of taste. If one declares at the beginning of a context that the use of the expression f(x) implies that x belongs to the domain of x, there is also no formal objection against the omission of condition (a). In that case the word “that belongs to the domain of f ” is also superfluous. We shall not follow this habit here. 9 A trivial translation is a translation along a zero vector. In practice, this is not a translation at all. The term “non-trivial” is necessary, because otherwise every function would be periodic. Its equivalent in Definition A is the condition that p should be non-zero. In the following we shall always mean a non-trivial translation. 10 The word “horizontal” is essential in order that this definition is equivalent to Definition A. If we leave it out then we can find many functions that are periodic according to Definition G, but not according to Definition A. For example, the graph of the function f defined by f(x) = x + sin(x) can be translated along the line y = x such that the image coincides with the origin. This would be a periodic function if we leave out the condition horizontal, but it would not be periodic according to Definition A.
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Most often the definition of a periodic function is immediately followed by a definition of the period of a periodic function. Before we do that, we want to make some remarks about the definitions that are given above. Some people would say that Definition G is more elegant than Definition A, because it is more global. Others can argue that just for that reason Definition A is the more elegant of the two. A close look at the two equivalent definitions and some of their implications may lead to several questions worth pursuing. We discuss a few of them. 3.1. If f(x + p) = f(p), then what about f(x + 12345p)? In a textbook (De Lange & Kindt, 1985) the following definition appears: Definition T. A function f is called periodic if there exists a non-zero number p, such that for every x that belongs to the domain of f the following conditions are fulfilled: (a) x ± p, x ± 2p, x ± 3p, x ± 4p, etcetera, belong to the domain of f, (b) f(x) = f(x ± p) = f(x ± 2p) = f(x ± 3p) = . . . etcetera. Although such a definition indeed comes close to our experience, this is logically not necessary and therefore violates the criterion of minimality. It can easily be shown by complete induction that the above Definition T is in fact a property of periodic functions that are defined by Definition A. This is an example of the fact that, in order to be rigorously logical we cannot always follow our intuition literally. It is also an example of the contrary: The writers of the textbook did not want this rigor at the moment that they introduced the idea of periodicity and therefore used the more intuitive way of describing a periodic function. 3.2. Is the constant function periodic? An intuitive notion of a periodic function is that the values of it change, but keep repeating themselves regularly. Someone with that notion could try and find a definition of periodic functions and could come up with Definition A or G. Later he would discover that according to these definitions the constant function is a periodic function. For example, Let f be defined as: f(x) = 1564.78 for every x. Then there is a number, say 821, such that (a) for every x in the domain of f, also x + 821 belongs to the domain of f, and (b) f(x) = f(x + 821). He would get the feeling that the constant function is a degeneration of the concept. It does not comply with his intuitive idea of periodicity that has to do with change. Some writers (e.g., Shama, 1995), therefore, explicitly exclude constant functions from their definition of a periodic function. This seems more a matter of aesthetics and of personal values. Indeed, as long as one is consistent in what follows there is no logical objection to it. Here, however, is an example of a problem that might arise under certain conditions:11 the sum of two periodic functions is also a periodic function: f(x) = sin2 (x)
and
g(x) = cos2 (x)
11 If two functions that are periodic under Definition A, have the same domain and the ratio between their periods is a rational number, the sum of the two functions is a function that is periodic under Definition A.
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These two functions satisfy the specific conditions necessary for the theorem to apply; thus, their sum should be a periodic function. However, the sum of these two functions is a constant function: (f +g)(x) = 1 for all x. So if one excludes a constant function from the definition of a periodic function, one must reformulate the theorem under the indicated conditions, “. . . the sum of two periodic functions is either a periodic function or a constant function.” We suggest that a more convenient way to treat situations like this is to allow degenerations. 3.3. Do we really need the ± sign? The use of the ± sign in Definition A made us wonder why it is not enough to use the + sign only. In other words: is the use of the ± sign not a violation of the criterion of minimality? Let us examine the following definition that appears in a textbook of trigonometry instead of Definition A (Van Dormolen, 1970). Definition D. A function f is called periodic on R (the set of real numbers) if there exists a non-zero number p, such that for every x that belongs to the domain of f, the following condition holds: f(x + p) = f(x). There are a number of differences between Definition D and Definition A. First, instead of the ± sign in Definition A only the + sign appears in Definition D. Secondly, in Definition D there is a restriction on the domain, namely, this definition requires that f has R as its domain. Finally, the condition that x + p belongs to the domain does not appear in Definition D. This condition need not be included because it is implied by the fact that f has R as its domain. And indeed it is easy to prove that for any number p that fulfills the condition f(x + p) = f(x) for every x in R, also the condition f(x − p) = f(x) holds.12 In fact, in Definition D we need not restrict ourselves to R. We can define a periodic function on the set of rational numbers, or the set of integers, or the set of all integers that are divisible by 13. Example: The function of parity: f is defined on the set of integers and f(x) = 0 if x is even and f(x) = 1 if x is odd. This function f is periodic. Again one wonders what is so special about domains like those mentioned above. Can we omit any reference to the domain and just use the following definition? Definition A+ . A function f is called periodic if there exists a non-zero number p, such that for every x that belongs to the domain of f, the following conditions are fulfilled: (a) x + p belongs to the domain of f, (b) f(x + p) = f(x). We shall show by two counter-examples that Definition A+ is not equivalent to Definition A. Let f be a function defined on the real numbers and let p be a number such that for all x: f(x) = f(x + p). The number x − p is a real number and therefore it belongs to the domain of f. Now, because x − p is an element of the domain of f, we can apply Definition D to that number: f((x − p) + p) = f(x − p). This is the same as f(x) = f(x − p). In summary: For every x in the domain the number x − p also belongs to the domain and f(x) = f(x − p). 12
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Fig. 4.
First counter-example: Let f describe a certain process that starts at a certain time (x = 0), continues for 1 hour and then starts all over again. This happens at every whole hour. Fig. 4 is the graph of such a function f. Here f(x + 1) = f(x) for all x of the domain, which is the set of non-negative real numbers. However, nothing is defined for negative numbers x and therefore it is not true that f(x − 1) = f(x) for all x of the domain.13 This is a case of a bounded domain. Intuitively, we would be inclined to accept a function like this as a periodic function, because in essence it behaves like a periodic function and can represent a real life phenomenon that repeats itself over and over in the same way. However, according to Definition A this is not a periodic function.14 It would not be difficult to develop a consistent theory on the basis of Definition A+ , although (as we shall indicate with the next example) we may not be inclined to accept as periodic functions all the additional functions that satisfy Definition A+ . Following is the second counter-example in which the domain is unbounded. Let D be the set that consists of all negative (whole) multiples of and all non-negative real numbers. Let f be defined on that domain such that f(x) = 0 if x < 0 and f(x) = sin(x) if x ≥ 0 (see Fig. 5). The number or any (whole) multiple of has the property of p in Definition A+ , but not in Definition A: It is not true that f(x − ) = f(x) for every x of the domain. There is no other possibility for p. Thus, according to Definition A+ the function f is periodic, but not according to Definition A. Actually, this is an example that we would not be inclined to accept as a periodic function even before checking whether it satisfies the conditions of Definition A. Going back to the question: What is so special about domains of functions like the set of real numbers, the set of rational numbers, the set of integers, or the set of all integers that are divisible by 13? These domains satisfy the following condition: 13 It is possible to extend a function f that fulfils Definition A+ to the other side of its domain (if its domain is lower bounded) and to obtain by that a periodic function according to Definition A. This can be done by a statement that on every interval (n, n + 1) that starts with an integer n and ends with the consecutive integer n + 1, the function behaves like it behaves on (0, 1). Or more formally: If G is defined on [0,1), for every integer n and for every x in the interval (n, n + 1): f(x) = G(x − n). 14 A function like this that can algebraically be constructed (and be drawn on a graphic calculator) as follows. Let h be defined by: h(x) = 1 for x ≥ 0 and h(x) = −1 for x < 0. Let g be any function that is defined on the segment [0,1]. Let [x] be the entire √ function: [x] is the largest integer that is smaller than or equal to x. Then the function f, defined by f(x) = g(x − [x]) (h(x)) has domain [0,→) and for every x : f(x + 1) = f(x), while f(x) does not exist for x < 0.
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Fig. 5.
For every x in the domain and every non-zero number z: If x + z belongs to the domain, then also x − z belongs to the domain. One way to modify Definition A+ is to include functions that satisfy our intuitive notion of periodicity can be adding to the definition this condition on the domain. We do not know if this condition is necessary to turn Definition A+ into an acceptable definition. It certainly is sufficient.
4. How to define the concept of period of a function? As before, starting with our intuition and experience, we can ask ourselves what we mean by the word period. A periodic function describes a situation that repeats itself after a certain time, or after a certain number, or after a certain distance. This “certain time,” or “certain number,” or “certain distance” we call period. On the basis of this intuition we can define such a number more formally. Writers define period in different ways: Definition B. Let f be a periodic function (according to Definition A). Every number p (=0) that fulfills the conditions of Definition A is called a period of f. If there exists in the set of all periods a smallest positive one, then that smallest positive one is called the minimal period of f or basic period of f. Definition R. Let f be a periodic function (according to Definition A). If there exists a smallest positive number in the set of all numbers p that fulfill the conditions of Definition A, that smallest positive one is called the period of f.15 This time the two definitions of period are not equivalent. If one uses the broader Definition B, a periodic function has an infinite number of periods. There is no such thing as the period. It is, however, possible for a periodic function to have the minimal (or basic) period. If one prefers the restricted Definition R, it is possible to have at the most just one period. The period according to Definition R is the same as the minimal (basic) period in Definition B. Why would one prefer Definition R to Definition B or vice versa? Some people think that the really interesting number is the smallest positive number (if such a number exists) and therefore they prefer 15 Analogue definitions can be given that concur with Definition G. Instead of “numbers p” one must talk then about “distances of translation” or “vectors of translation” or any other term to that effect.
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Definition R. They can then talk about the period p of a periodic function. Others argue that it is sometimes difficult, maybe even impossible, to find the smallest number, even when they know that such a number exists for a certain function. Those prefer Definition B. Whatever choice one makes, one has to be consistent. For example, different statements describe the same property of (a) period: If one uses Definition B, the property mentioned in If f(x+p) = f(p), then what about f(x+12345p)? should be stated as following: If there exists a minimal period, every period is a (positive or negative) multiple of that minimal period. If one uses Definition R, the property should be rephrased to: If there exists a period, every (positive or negative) multiple of it fulfills the conditions of Definition A. 4.1. Is there always a period? We wrote that sometimes a writer excludes constant functions from being a periodic function. We assumed that this was because the values of a constant function do not change. There is another possibility that has to do with the idea of the basis period (or the period in case of Definition R). A constant function does not have a basic period. That is why we had to add the words “if there exists” in both Definitions B and R. One might be inclined to consider this as the reason for excluding constant functions as being counter-intuitive and as a result omit the condition “if there exists” from the definitions of period. This, however, does not work, because it requires that other degenerations of a similar nature be excluded too. Take for example Dirichlet’s function: f(x) = 0 if x is a rational number and f(x) = 1 if it is not. If the constant function is excluded, so should Dirichlet’s function be exluded. In order to be consistent, we want to be sure that if we exclude degenerations like these, we exclude all possible ones. As long as we do not have a good criterion that describes all of these degenerations, we have to include the requirement of existence of a lowest positive p. 5. Closing remarks: logic before pedagogy? The vignette in the beginning of this article is actually a short resume of a much longer and disordered series of discussions. It focused on the definition of period. LL thought that Definition B was the correct one while JD defended Definition R until both realized that each of them was confined to what s/he had learned many years ago and did not refer to any logical criteria. The same happened with the question of whether a constant function was a periodic function. This dialogue triggered a much longer and deeper discussion within a community of colleagues and led first to several workshops within the framework of a professional development program for secondary mathematics teachers, followed by written resource material for these teachers, and finally resulted in the current article. We (re-)learned that in doing mathematics some rules have to be followed for logical reasons, some for conventional reasons and some for pedagogical reasons and that it is dangerous to mix-up the three reasons. We also (re-)learned that it is worthwhile for teachers as well as for teacher educators to investigate in depth the logical aspects of the mathematical activity that we are planning to teach, even if we do not
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intend to overburden our students with formal logic. Teachers and teacher-educators need to engage in such kinds of investigation in order to be able to make educated pedagogical decisions. In that sense, for them, logic comes before pedagogy. And above all, we (re-)learned that mathematics is a human activity in which we have the freedom to investigate and the freedom to make decisions, as long as we are aware of their consequences.16 References Ausubel, D. P., & Robinson, F. G. (1969). School learning: an introduction to educational psychology. New York: Holt, Rinehart and Winston. Brousseau, G. (1997). Theory of didactical situations in mathematics. Edited and translated by N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield. Dordrecht: Kluwer Academic Publishers. Davis, P. J., & Hersch, R. (1981). The mathematical experience. Boston: Houghton Mifflin Company. DeCecco, J. P. (1968). The psychology of learning and instruction: Educational psychology. Englewood Cliffs: Prentice-Hall. De Lange Jzn., J., & Kindt, M. (1985). Periodieke functies. Culemborg: Educaboek. Fischbein, E. (1987). Intuition in science and mathematics: an educational approach. Dordrecht: Kluwer Academic Publishers. Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel. Freudenthal, H. (1978). Weeding and sowing. Dordrecht: Reidel. Heath, T. L. (1956). Euclid. The thirteen books of the elements. New York: Dover Publications. Hershkowitz, R., Bruckheimer, M., & Vinner, S. (1987). Activities with teachers based on cognitive research. In: M. M. Lindquist, & A. P. Shulte (Eds.), Learning and teaching geometry, K-12, 1987 Yearbook. NCTM. Lakatos, I. (1976). Proofs and refutations. Cambridge: CUP. Leikin, R., Berman, A., & Zaslavsky, O. (1998). Definition of symmetry. Symmetry: Culture and Science, 9(2/4), 375–382. Leikin, R., & Winicki-Landman, G. (2000). On equivalent and non-equivalent definitions II. For the Learning of Mathematics, 20(2), 24–29. Mason, J., & Pimm, D. (1984). Generic examples: seeing the general in the particular. Educational Studies in Mathematics, 15, 277–289. Quine, W. V. O. (1953). From a logical point of view. Cambridge, MA: Harvard University Press. Rota, G.-C. (1997). Indiscrete thoughts. Boston: Birkhauser. Shama, G. (1995). The concept of periodicity. Unpublished Doctoral Dissertation, Tecnion, Israel Institute of Technology, Haifa (in Hebrew). Skemp, R. R. (1971). Psychology of learning mathematics. Harmondsworth: Penguin Books. Stewart, I. (1997). The mathematical maze: seeing the world through mathematical eyes. London: Weidenfeld & Nicolson. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169. Thurston, H. (1974). Clarity in the calculus. Mathematics Teaching, 67, 58–60. Van Dormolen, J. (1970). Goniometrische functies. The Hague: Van Goor. Van Dormolen, J. (1991). Metaphors mediating the teaching and understanding of mathematics. In J. A. Bishop, S. Mellin-Q-Sen, & J. Van Dormolen (Eds.), Mathematical knowledge: Its growth through teaching (pp. 89–106). Dordrecht: Kluwer Academic Publishers. Van Dormolen, J., & Arcavi, A. (2000). What is a circle. Mathematics in School, 20(5), 15–19. Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In: Tall, D. (Ed.), Advanced mathematical thinking. Dordrecht: Kluwer Academic Publishers. Winicki-Landman, G., & Leikin, R. (2000). On equivalent and non-equivalent definitions I. For the Learning of Mathematics, 20(1), 17–21. The function x → 3sin x − 4sin3 x has certainly a period 2 (in the terminology of Definition B). Some might find it hard to show that it has the minimal period 2/3. The function x → (sin4 x + cos4 x)/(2 − sin2 (2x)) has certainly a period of 2 (in the terminology of Definition B). Some might find it hard to proof that it has no minimal period. 16