1 Introduction: The Interaction Between Mathematics and Physics

According to Wigner (1960) “the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and … there is no rational explanation of it” (Wigner 1960, 117). The reason Wigner considered the usefulness of mathematics a “miracle” (Wigner 1960, 120) has a lot to do with his view of the interplay of mathematics and physics and his view of the nature of mathematics. The interplay between mathematics and physics is often described as an application of the former to the latter. This view suggests that the particular applied mathematical principles are taken from an already developed storehouse of mathematical theories. According to Wigner, “mathematics is the science of skilful operations with concepts and rules invented for this purpose” (Wigner 1960, 117). This quote places Wigner within the prevailing formalist philosophy of mathematics according to which mathematics is a theory of axiomatic systems driven by internal criteria of beauty and logical rigour. And indeed, if mathematics and its application are viewed in this way, the applicability of mathematics in the natural sciences seems to be unreasonable.

However, as even the most superficial historical research will show, and as the father of the formalist philosophy of mathematics Hilbert repeatedly emphasized, mathematics does not develop in this way. Axiomatization of mathematics is philosophically important, but it is rarely the driving force behind the development of mathematics. As Hilbert pointed out, the driving force of mathematics has always been problems in need of solutions. Some of these problems are internal mathematical problems, but many of the problems result from motivations to describe the natural or social world surrounding us. In fact, in many cases where mathematics is “applied” to physics, the appropriate kind of mathematics is not available on the shelf in advance but must be developed during the process of “application”. In such a case, the encounter between mathematics and physics is not appropriately described by the word “application”; it is rather a mutual interaction between mathematics and physics that develops and shapes both disciplines. And even in a case where physics develops by applying an already existing piece of mathematics from the shelf, this piece of mathematics has often been developed previously in connection with the treatment of other (and perhaps similar) physical phenomena, see Lützen (2011a, 2011b, 2013).

As Wigner correctly pointed out, the creation of mathematical concepts is central to the development of mathematics. In this paper, we shall show how these concepts can be shaped in connections between mathematics and physics. When the external and often physical origins of mathematical theories and concepts are taken into account and the interaction between mathematics and physics is understood as a mutual process, it becomes less unreasonable that mathematics can be effective in the description of the physical world.

Introducing an awareness of the interaction of physics and mathematics into our teaching of mathematics also has the potential to benefit students’ learning of mathematics in several respects. In traditional mathematics education, mathematical terms are often presented as timeless entities with no connection to nature or our empirical world in a broader sense. They are presented as ideal objects outside of space and time, and the theories about them as eternal truths. We shall argue that introducing mathematical concepts and theories embedded in a rich historical context with emphasis on the interaction between mathematics and other areas, in particular physics, in an explicit–reflective frameworkFootnote 1 has at least three advantages: (1) it provides a meaningful context for developing students’ conception of the nature of mathematics, their understanding of the field of mathematics and how mathematical knowledge is created. (2) It gives students an opportunity to become aware of the existence of and to reflect upon meta-discursive rules governing mathematics, and to realize that these rules change over time and may be determined by extra-mathematical concerns. (3) It becomes possible to teach inquiry in mathematics in the same sense as in the sciences, as a mathematical practice that aims at obtaining new mathematical knowledge. We shall illustrate such a historical teaching and learning of mathematics within an explicit and reflective framework through two examples of student-directed, problem-oriented project work from Roskilde University, Denmark, in which the connection to physics is explicit and provides a learning space where the nature of mathematics and mathematical practices are linked to natural science.

2 The Main Stages in the Development of the Concept of Function

As an example of how mathematical concepts can be shaped by physics, we shall investigate the development of the concept of function. This notoriously difficult concept has undergone major changes during its 4-century-long history. Here are the main stages in the development:

  1. 1.

    The first widely used concept of function was the Eulerian one: a function y of x is an analytic expression [a formula denoted f(x)] expressing y in terms of x

  2. 2.

    The Dirichlet concept: y is a function of x if to each x there is associated one value of y [called f(x)].

  3. 3.

    The Bourbaki concept: a function from a set A to a set B is a subset C of the Cartesian product A × B with the property that for each x in A there is exactly one y in B such that (x, y) is in C.

  4. 4.

    Distributions or generalized functions

We shall begin by analysing how these successive concepts came into being. In particular, we shall focus on those driving forces that were connected with physics.Footnote 2 After this purely historical analysis, we shall describe and discuss how it can be used for teaching purposes. In particular, we shall present two student projects from Roskilde University dealing with aspects of the history of the concept of function. The first deals with step 1 and 2 above, the second with step 4.

3 Functions Before the Concept of Function

It is easy to trace Dirichlet-type functions at least 4,000 years back in time. For example, the Babylonian tables of squares and cubes of natural numbers can be thought of as tables of the function x 2 and x 3. Ancient astronomy offers many other examples of tables of “functions”. From Seleucid times (about 300 BC), the Babylonians tabulated various astronomical phenomena as “functions” of time. In his influential astronomical main work, the Almagest Ptolemy (c. 150 AD) published the first surviving trigonometric table of the cords in a circle as a “function” of the angle.Footnote 3 The astronomical and the trigonometric tables were computed for the purpose of describing the natural world, and so one might consider these tables as evidence of a physical origin of the function concept. However, it is anachronistic to attribute a function concept to the Babylonians or the classical Greek mathematicians. Indeed, the astronomical or trigonometric tables were not reified or conceptualized. Cords were considered as lines in a circle and the process of going from an angle to a cord was not thought of as a mathematical thing, but rather as a problem that the table helped to solve.

4 Influence from the Calculus

As has also been pointed out by Anna Sfard (1991), it is probably a general law that processes or problem complexes are reified when someone invents an operation which operates on the process or the problem complex as a whole, which was precisely what happened in the case of functions. Functions were conceptualized after the invention of an operation, differentiation, operating on the entire function rather than on each “function value” separately. In that way, the explicit introduction of the concept of function can be intimately linked with the invention of the calculus. And the calculus owed much to physics. To be sure, the calculus was invented primarily as a means to study curves. After the invention of analytic geometry, which attaches a curve to any equation in two variables, it became necessary to develop means for studying their properties, and so infinitesimal techniques were developed as early as the 1630s. But even in this early period of pre-calculus, curves were often connected to physical phenomena. Indeed, since ancient times some curves (like Archimedes’ spiral) had been defined by a continuous motion and after Galileo’s investigations of kinematics this became a standard way to think about curves. And as curves were considered from a kinematic point of view, the idea of a variable quantity became a central concept in the emerging field of analysis.Footnote 4

Pre-calculus methods were often used or developed with physical problems in mind. For example, Fermat used his method of maximum and minimum to derive the law of refraction, and Descartes was interested in normals to curves in connection with the investigation of caustic curves. In Newton’s fluxional calculus, kinematical and mathematical ideas were totally integrated. Curves were generated by continuous motion, making, for example, the x and y coordinates into fluents or flowing quantities whose velocities were what Newton called the fluxions of these fluents. Leibniz’ version of the differential calculus was less imbued with ideas in physics. However, along with his continental followers such as the brothers Bernoulli, he developed many new techniques in connection with problems of a physical nature such as the brachistochrone problem, which asks for the curve along which a heavy point-mass will slide from one point to another in the shortest time.

The first introduction of the word function appeared in the geometric paradigm of curves. In 1673, Leibniz used this word to denote a quantity that varies from point to point on a curve, such as the tangent, the normal, or the ordinate. However, as mathematical analysis was gradually removed further and further from geometry during the eighteenth century, the concept of function followed suit.

5 Euler’s Concept of Function

Though Leibniz’s differential calculus dealt with variables and curves (Bos 1974), its forceful algorithms operated on the analytic (mostly algebraic) expressions describing the curves. Therefore, it is not surprising that Johann Bernoulli in 1718 suggested a non-geometric version of the function concept:

Here a function of a variable quantity denotes a quantity composed in any arbitrary manner from this variable quantity and constants. (Bernoulli 1718, 241)

This definition was reformulated by Euler in his very influential textbook Introductio in Analysin Infinitorum (1748a):

A function of a variable quantity is an analytic expression which is composed in any manner of this variable and of numbers or constants. (Euler 1748a, book 1, chapter 1, §4)

Thus, for Euler, a function was a formula. In addition to the algebraic operations, Euler also allowed transcendental operations such as sin and cos (which he was the first to consider as functions) and infinitely repeated operations such as infinite sums and products. It is also worth remarking that Euler insisted that a variable quantity can in principle attain all (complex) values, so that all his functions were in principle “defined” for all complex values of the independent variable. It is well known how he attained this goal in the case of trigonometric functions by way of the surprising Euler’s formulas.

According to Euler, analysis was a science of functions and not of curves. It could be applied to geometry, as he showed in the second volume of the Introductio, and to mechanics, but in principle, analysis was completely free from geometric considerations.

6 The Vibrating String and “Discontinuous” Functions

However, in connection with a debate with D’Alembert about the motion of a vibrating string, Euler reformulated his function concept. The debate began in 1747 with D’Alembert’s (1747) famous complete solution of the wave equation. This equation is a partial differential equation satisfied by the function f(t,x) that represents the oscillation away from the equilibrium position of a taunt string, where t represents time and x represents the position along the equilibrium position. This equation, which Euler soon wrote on the form

$$ \frac{{\partial^{2} f}}{{\partial t^{2} }} = \frac{{\partial^{2} f}}{{\partial x^{2} }} , $$
(0)

was one of the first partial differential equations to be solved.Footnote 5 D’Alembert showed that its complete solution was of the form:

$$ f(t,x) = g(x + t) + h(x - t) , $$
(1)

where g and h are “arbitrary functions”. If the string is fixed at its end points, as a guitar string, and the end points are situated at x = 0 and x = l and its initial velocity is assumed to be zero, Euler was able to deduce that the solution is of the form:

$$ \frac{1}{2}f\left( {x + t} \right) + \frac{1}{2}f\left( {x - t} \right) , $$
(2)

where f is an odd function that is periodic with period 2l. If one inserts t = 0, one can see that f denotes the initial shape of the string.

Now Euler (1748b) conversely argued that if one gives the string the shape f in the interval (0, l) at time zero and let it loose, the string will at any later time have a shape described by (2) where f has been extended beyond the interval (0, l) such that it is odd and periodic with period 2l. Euler was well aware that this continuation of f would in general be different from the analytic continuation of the function f beyond the interval (0, l). Thus, the odd and periodic continuation was not really a function in his own sense of the word “function”, but he decided to extend the meaning of the word function to these more general objects. He even allowed the initial curve to be described by different analytic expressions in different subintervals of the interval (0, l). In this way, he was able, for example, to describe the motion of a plucked string, where the initial shape is described by one linear expression in the left part of the interval and by another linear expression in the right part.

D’Alembert strongly opposed Euler’s solution of the initial value problem. He insisted that his solution requires that the functions g and h in (1) are analytic expressions. Thus, Euler’s solution (2) is only a solution in the case where the analytic continuation of the function f [initially defined on (0, l)] is odd and has period 2l.Footnote 6 His opposition to Euler’s ideas was not merely a result of conservatism. He correctly pointed out that if the continuation of f is not analytic, its curvature could change abruptly at a point and at this point, the wave equation would not be satisfied (D’Alembert 1761, §7) (we would say that the curve is not twice differentiable).

The disagreement between the two savants continued for several years. Other mathematicians such as Daniel Bernoulli and Lagrange joined the debate, but in the end the question was left unsolved until the more rigorous approach to the calculus was introduced in the nineteenth century. We shall not follow the subsequent discussion, but we emphasize that the disagreement between Euler and D’Alembert is a classical conflict between an internalist mathematical rigorist (in this case D’Alembert)Footnote 7 and an externalist (here Euler), for whom mathematics had to be shaped to fit its use in physics. Indeed, Euler agreed with D’Alembert that “as it has been treated until now, analysis could only be applied to curves whose nature can be included in one analytic equation” (Euler 1765a, §7). However, he argued that the problem of the vibrating string forces mathematicians to generalize the concept of function and thereby “opens for us a new chapter in analysis by allowing us to apply calculus to curves that are not subject to any law of continuity” (Euler 1765, §7).

Here, continuity has a different meaning from the modern one. In a paper, De usu functionum discontinuarum in analysi Euler (1763) pointed out that, while one has usually assumed that to any function there corresponds a curve (the graph), he will now assume that conversely any curve defines a function. If the function is described by the same analytic expression everywhere (i.e. if it is a function in his earlier sense of that word), he called it a continuous function. If it is not described by one analytic expression, he called it discontinuous. Discontinuity can happen in two ways:

  1. 1.

    Functions can correspond to mixed curves, i.e. curves that are described by different analytic expressions in different intervals

  2. 2.

    Functions can correspond to entirely arbitrary hand-drawn curves, where the curve is not even described by an analytic expression in small intervals, but where the laws so to speak change from point to point

So physics forced Euler to extend the concept of function, and it is hard to imagine that such an extension could have been suggested by mathematics itself. In fact, as pointed out by D’Alembert, the formalism of analysis did not quite make sense for the new kind of functions. And in the beginning, the extension did not add beauty to mathematical analysis; on the contrary, it strained its foundation. But it enhanced to the physical applicability.

7 A General Definition Narrowly Interpreted

It is remarkable that Euler as late as 1763 continued to talk about analytic expressions in connection with his new function concept, even in cases where such an expression is only valid at an isolated point. This shows how ingrained his earlier function concept was in the contemporary practice of analysis. It is even more remarkable when we notice that in his book on differential calculus of 1755, he gave a very general definition of a function:

When x denotes a variable quantity then all quantities that depend on x or are determined by it in some way are called functions of x. (Euler 1755, introduction)

This definition sounds very much like the later Dirichlet concept. However, it is not clear what kind of dependences Euler had in mind. He did not explain to his readers that there was a difference between the new definition and the previous one in the Introductio. In fact, in the book of 1755, he only dealt with analytic expressions. Therefore, it is not certain that he thought of his new definition as a proper generalization of his previous definition. In particular, it is unclear whether he linked the reformulation to the new discontinuous functions that he met in connection with the vibrating string around the same time. If he had thought about his new definition in that way, he would probably have used it as a framework for his discussion in 1763 of discontinuous functions, but in the 1763 paper, there is no reference to the definition of 1755.

Euler’s general 1755 definition was widely quoted or rephrased during the next half century by his followers such as Lagrange (1806),Footnote 8 Lacroix and Cauchy. However, its generality was not emphasized during this period, and, for example, Lagrange explicitly stated that any function was given by an analytic expression. Thus, for more than half a century mathematicians formulated a very general definition of the concept of function, but in actual practice interpreted this definition in a rather narrow sense.

8 Heat Conduction and the Dirichlet Concept of Function

It was a new physical phenomenon, heat conduction, that finally led to a clear distinction between analytic expressions and general functions. In his Théorie analytique de la chaleur (1822), Fourier studied the conduction of heat in homogeneous materials. In this connection, he wanted to allow an arbitrary temperature distribution described by an arbitrary function that he defined as follows:

In general, the function f(x) represents a succession of values or ordinates each of which is arbitrary. An infinity of values being given to the abscissa x, there are an equal number of ordinates f(x). All have actual numerical values, either positive or negative or nul. We do not suppose these ordinates to be subject to a common law; they succeed each other in any manner whatever, and each of them is given as it were a single quantity. (Fourier 1822, 430)

where Euler had allowed the law to change from point to point in a function corresponding to a hand-drawn curve, Fourier now discarded the laws altogether and presented the function values f(x) as given independently of each other. As in the case of the discussion of the vibrating string, the new definition of a function raised the problem of the interpretation of the basic analytic operations. In particular, Fourier pointed out that one could no longer think of the integral as an anti-derivative, as had been the tradition since Johann Bernoulli. Instead, he simply defined it as the area under the curve.

The new notion of integral was important for Fourier in connection with his solution of the heat equation and the problem of developing arbitrary functions in trigonometric series (Fourier series). The problem of convergence of Fourier series was taken up by Fourier’s student Dirichlet in two papers of 1829 and 1837. In the latter, he defined a continuous function on the interval (a, b) as follows:

If every x gives a unique y in such a way that when x runs continuously through the interval from a to b then y = f(x) varies little by little, then y is called a continuous function of x in this interval. It is not necessary that y depends on x according to the same law in the entire interval. One does not even need to think of a dependence that can be expressed through mathematical operations. (Dirichlet 1837)

The condition of continuity had been defined more rigorously by Cauchy in his Cours d’analyse (1821). What is important in Dirichlet’s definition is the explicit rejection of the necessity of a common law. The value of f(x) can be determined in any way from x. In his previous paper, he had even given an explicit example

$$ f\left( x \right)= c\quad{\text{for}}\,x\,{\text{rational \quad and}}\quad f\left( x \right)=d\quad{\text{for}}\,x\,{\text{irrational}} $$

as an example of a function which cannot be integrated.

Since Dirichlet had formulated the new function concept in such a clear way, Hermann Hankel (1870) later named it Dirichlet’s concept of function.

9 Critique of Dirichlet’s Concept of Function. The Bourbaki Concept

Dirichlet’s general concept of function was accepted and used by most of his successors starting with Riemann. But it was also criticized. Some mathematicians such as Weierstrass considered it so general, that one could deduce nothing from it. He insisted that the goal is always to find an analytic expression for a function (Laugwitz 1992). It was also gradually revealed that general functions could defy our intuition about their behaviour. In fact, many pathological functions were constructed beginning with Dirichlet’s everywhere discontinuous function, mentioned above, and culminating with Weierstrass’s continuous but nowhere differentiable function (Weierstrass 1872). And finally towards the end of the nineteenth century, a discussion arose about the allowable ways by which one could define a function. The question was this: According to Dirichlet each x should give a unique f(x). But by which means could all f(x)s be given? Lebesgue, Baire, and Borel discussed the problem [see, e.g. Monna (1972)] and came up with various more or less constructive means, but the problem was really only solved within set theory.

Already with Cantor’s naïve set theory and his idea of a mapping between two sets, functions could be considered as special cases of mappings, namely between sets of numbers. But which are the means by which mappings can be defined? This question was answered within the axiomatic set theory of Zermelo and Fraenkel: a mapping or function between A and B is a subset of A × B with the property that for each x in A there is precisely one y in B such that (x, y) is in A × B. In this way, the question of the means of constructing functions was reduced to the means of building sets, and they are described in the axioms of set theory.

This modern set-theoretic definition of the concept of a function was suggested by Bourbaki in 1939. The development leading to it was driven by internal mathematical foundational considerations and shall not be discussed any further here. However, it is necessary for the teacher to understand that the definition resulted from a striving for logical hygiene rather than a pursuit of applicability.

10 Distributions

Physics’ demand on the function concept did not end with the introduction of the Dirichlet concept of function. During the end of the nineteenth century and the beginning of the twentieth century, physicists began to use bold generalized functions that could not exist according to the Dirichlet definition. For example, in his operational calculus, the controversial electrical engineer and physicist Oliver Heaviside represented an electric impulse as the derivative of the non-differentiable function:

$$ F\left( x \right) = 0\quad{\text{ for}}\,x < 0\quad{\text{ and}}\quad F\left( x \right) = 1\quad{\text{ for}}\,x \geq 0 $$

According to Heaviside, this derivative is zero except at zero where it is infinite.

This derivative, which Dirac named the delta function, came to play an essential role in his formulation of quantum mechanics (1926). The delta function has the property that its integral over the real axis (or any interval containing 0) is equal to one. But since the delta function is zero except in one point its Lebesgue integral must be zero. Thus, it is clear that the delta function does not exist as a proper function according to Dirichlet.

Other problems of a similar kind turned up during the beginning of the twentieth century. For example, mathematicians and electrical engineers tried to use the Laplace transform in order to make Heaviside’s operational calculus more meaningful. However, the Laplace transforms of many of the functions used in the description of electrical networks were not proper functions. Moreover, the problem of the vibrating string that had been left undecided by Euler and D’Alembert resurfaced. There are good physical reasons to consider the solution (1) to be a solution of the wave Eq. (0) even if g and h are not differentiable functions. Several concepts of generalized solutions to differential equations were put forward during the early part of the twentieth century. Using these new concepts, (1) could in fact be considered as a generalized solution of (0) even if g and h are not differentiable. However, these theories did not give a meaning to the derivatives of the non-differentiable functions themselves.

All these problems were solved by Laurent Schwartz when he generalized the concept of function. In 1945, he had the idea to define a distribution as a functional on the space of infinitely often differentiable functions with compact support. He developed a theory of these generalized functions during the following 5 years and published his theory in two volumes in 1950–1951. Where problems solved by the theory of distributions were to a large degree suggested by physics, Schwartz’s solution of these problems was suggested by the new theory of functional analysis which was developed primarily for intrinsic mathematical reasons. Still, physics also played a certain part in the development of functional analysis. For example, von Neumann’s introduction of the notion of abstract Hilbert space was published in a fundamental paper in which quantum mechanics was presented in a mathematically rigorous way to circumvent the delta function (Von Neumann 1927).Footnote 9

11 Bringing Awareness of the Interplay Between Mathematics and Physics to University Mathematics Students

In the following, we will present two examples of student-centred, problem-oriented project work at the university level through which students developed informed conceptions about the nature of mathematics as a scholarly (scientific) discipline and about the production of mathematical knowledge. We will see that by focusing on the interplay between mathematics and physics during the gradual formation of a mathematical concept (here the concept of a function) and the later development of mathematical theory (theory of distributions), it is possible to focus on actual mathematical practices in students’ learning of mathematics. In this way, the students can not only experience mathematics as a knowledge-producing endeavour, but also become aware that our conception of mathematics changes over time. Last but not least, they will be alerted to the fact that the driving forces both for producing new mathematics and for forming our conception of mathematics are often influenced by factors and elements in extra-mathematical realms, which in this case is physics.

The two student projects were conducted by groups of four and three students, respectively, at the mathematics master programme at Roskilde University in Denmark. This programme is rather special in the sense that all students participate in student-directed, problem-oriented project work each semester. The projects run for the entire semester and take up half of the students’ working load. The other half is taken up by regular courses. The students work on the projects in groups of three to eight students. In each semester, they form new groups and choose new problems to work on. Each group has a supervisor, which is a faculty member whose research area is closest to the students’ choice of problem for their project work. The curriculum of the project is not specified in advance, but the various semesters have a specific type of project. The two projects presented below were both completed at the first semester of the master’s programme, and hence were supposed to deal with “mathematics as a scholarly discipline”. In this type of project, the students have to work with a representative problem through which they will experience mathematics as a knowledge-producing and/or proof-based enterprise. These projects often, but not always, feature aspects of the history and/or philosophy of mathematics.Footnote 10

11.1 Project 1: Fourier and the Concept of Function: The Transition from Euler’s to Dirichlet’s Concept of a Function

Students explained in their project report that they wanted to “investigate Fourier’s significance for the development of the concept of function” (Godiksen et al. 2003, p. 2). The students focused on the origin on the concept of function within the mathematical practices of Euler, Fourier and Dirichlet, as they emphasized in their report:

The strength of focusing on these three mathematicians is that it has given us the opportunity to study their original works (sometimes in translations) in depth. That has given us a more direct impression of their thoughts than secondary literature could have given us. (Godiksen et al. 2003, pp. 2–3).

The students analysed the primary sources with respect to discussions about the function concept, changes in the perception of the function concept and discussions about the proper way to argue with functions. Interpreted within the framework of Anna Sfard’s (2008) discursive approach to mathematics, the students used the sources as “interlocutors” as explained in Kjeldsen and Blomhøj (2012). The students used the sources as ‘voices’ from the past. They took notice of the central ideas of these interlocutors and the differences between their perceptions of mathematics in general and the function concept in particular. The students explained how they used the sources as “windows” into the practice of the historical interlocutors:

The main elements of Euler’s conception of a function could easily be explained very briefly, but that would not contribute to any deep understanding of the concept. In order to obtain this, one has to look at how Euler worked with functions. (Godiksen et al. 2003, p. 17; italic in the original)

The students emphasized that to get a sense of Euler’s conception of functions, they studied not only how Euler explicitly defined functions, but also how he used them in his mathematical practice. The students gave the following interpretation of Euler’s conception:

The definition of a function [Euler’s definition] does not contain any specific information about its domain and image. This is because in Euler’s theory, variable quantities are ascribed a property that render specifications of such sets superfluous. …

… Euler conceived a variable as an arbitrary element, quite like our conception, but no constraints are allowed. The variable should be able to take all values … (it is universal). (Godiksen et al. 2003, p. 18)

Euler conceived of variables as universal. They were not limited in scope. Moreover, Euler’s analysis was global in the sense that its rules were supposed to hold for all functions and for all values of the variables. These two aspects of Euler’s analysis—the generality of the variable and the general validity of analysis—are meta-rules of Euler’s mathematical discourse. The quote below from the students’ report show that the interplay between physics and mathematics in the origin of the function concept provided a learning landscape in which the students came to reflect upon these meta-level rules of Euler’s mathematical discourse:Footnote 11

This property which […] has been named the criteria of the generality of the variable clearly reflects the earlier mentioned paradigm of the general validity of analysis.

[…]

Even though the use of the methods of analysis often created weird results the methods were used frequently in Euler’s concept of a function. The reason why there weren’t that many contradictions and paradoxes was that almost all Euler functions, which consist of analytical expressions, have all the above mentioned properties [they were nice], except maybe in isolated points. … Hence, there was no natural driving force that led to a clarification of the concepts of continuity, differentiability, and integration, since these properties so to speak were built into the concept of a function.

[…]

Euler … was of the opinion that the analysis had to be developed such that it was able to describe all situations that occur in nature. (Godiksen et al., 2003, pp. 22–23; italic in the original)

This awareness of the significance of physics in the rules that governed the meta-level of Euler’s mathematical discourse also surfaced in the students’ treatment of Fourier’s work, of which they wrote:

Fourier expresses clearly that mathematics is a tool for describing nature and mathematics had to be governed by nature. (Godiksen et al., 2003, p. 53)

The idea that mathematics has to be governed by nature is a meta-level rule of (past) mathematical discourse that became exposed for the students—a meta-level rule that is no longer a rule in modern mathematics in the same sense—but, as we shall see below, perhaps in another sense. Hence, the students learnt that even though mathematics has a certain stability, conceptions of mathematics change over time, and mathematical objects are not timeless entities, that they have not necessarily come into being as ideal elements from inside mathematics itself—but have originated in our attempts to describe and master the empirical world.

11.2 Project 2: The Dissemination of the Theory of Distributions After 1950

This project report opens with a tribute to functions: “The function concept occurs at many places in mathematics and is also used in physics and applied mathematics to model many different objects” (Albrechtsen et al. 2004, p. 3, italic in the original). Then, the students continued by problematizing the limits of the function concept in physics with reference to Dirac’s delta function, which is very useful in physics, but does not satisfy the criteria for being a function. As explained above, these problems were taken care of by Laurent Schwartz when he developed the theory of distributions. The students were interested in the status of distribution theory within the mathematical community, whether it was merely a “tidying up” job—for hygienic reasons, or whether it played, or came to play, a significant role in mathematics, in particular, in the theory of partial differential equations.

This project is rather different from the previous one, primarily due to a different method of approach. Instead of reading primary papers or other expositions of the new theory, the students analysed (1) the writings of Schwartz in which he explained his motivations for developing the theory of distributions; (2) the literature about distribution theory from 1950 to 2004, with focus on speeches, reviews, biographies, texts books in partial differential equations; and 3) interviews with mathematicians including the professors Lars Hörmander from Sweden and Gerd Grubb, Denmark. Hence, the students did not dig deep into the technical side of the development of the theory of distributions, but relied primarily on what could be called “meta-literature” with respect to mathematics.

They also realized that as in the case of the function concept, Schwartz’ development of the theory of distributions had at least part of its origin in physics: “Laurent Scwhartz’s development of distribution theory was motivated by some concrete problems which partly originated in physicists’ practice of using the delta function and partly in some considerations of the wave equation, but perhaps mainly in previous mathematical work on generalized solutions of partial differential equations.” (Albrechtsen et al. 2004, p. 17).

The students discussed the reception and dissemination of the theory of distributions. Did the theory come to play a role in different subjects in and outside of mathematics and in higher education, and, if so, to what extent? They realized that even though Schwartz received the Fields Medal for his work on distributions, the importance of his work was questioned. Some mathematicians thought it was too simple to be of any use, and others that it was too abstract (Albrechtsen et al. 2004, p. 47). However, through their investigations the students found that during the 1980s, the doubts about the importance of distributions in the theory of partial differential equations were no longer present. They also realized that even though the theory belongs to abstract modern mathematics, physics still plays a role for some mathematicians’ perception of mathematics—as a field that can function as a critique of mathematics. In the words of Gerd Grubb quoted from the students’ interview:

There are so many different differential equations you could study, so I use the applications as guiding principle for what it is worth spending time on. I believe there is feedback between applications and math both ways. Of course there is the conception that mathematics provides some tools for applications, but applications also provide some motivation for mathematics; they provide quality. You can write some sick weird equation and dedicate your life to solving it or saying something about the solutions. It has a much lower quality than taking an equation that users of mathematics are passionate about and want to know something about… That’s my opinion on applications that I try both to deliver something that is of value for users of mathematics and use applications as a quality parameter for what I want to spend my time on. (Albrechtsen et al. 2004, p. 38).

12 Interplay Between Mathematics and Physics: Teaching with and about the Nature of Mathematics

Implementing inquiry-based teaching and learning in mathematics and science education has been on the agenda for many reforms in mathematics and science education in many places (Abd-El-Khalick 2013, p. 2087). The hope is to provide students with learning experiences that to some extent resemble authentic scientific practices, thereby enhancing students’ understanding of the nature of mathematics and science as well as forcing them to develop skills and critical insights into how science and mathematics works and generates knowledge. However, research from science education has shown that despite such efforts, precollege students tend to continue to hold naïve conceptions of the nature of science (NOS).Footnote 12

Research in science education has also shown that engaging in inquiry- (or history and philosophy of science)-oriented activities without directly addressing issues of NOS will probably not lead to students’ understanding of NOS. Abd-El-Khalick’s and Lederman’s (2000) work has shown that for this to happen an explicit–reflective framework is needed. Abd-El-Khalick (2013) emphasizes that “explicit” relates to curriculum, while “reflective” relates to students reflecting on their learning experiences from within an epistemological framework (p. 2091). Abd-El-Khalick and Lederman’s research has been directed towards the sciences and cannot necessarily be transferred to mathematics. However, since we are dealing with the interplay between mathematics and physics in connection with the formation of mathematical concepts and development of mathematical theory—that is, nature of mathematics from the perspective of the significance of physics, it is rather plausible, that their research results will also hold for teaching and learning of the nature of mathematics (NOM). And in general, inquiry as well as history and philosophy of mathematics oriented activities that do not directly address issues about the nature of mathematics will probably not lead to students’ understanding of such issues. Hence, it is more likely than not that Abd-El-Khalick’s (2013, p. 2088) quote from Rutherford (1964, p. 84) will also hold true for mathematics:

Science teachers must come to understand just how inquiry is in fact conducted in the sciences. Until science teachers have acquired a rather thorough grounding in the history and philosophy of the sciences they teach, this kind of understanding will elude them, in which event not much progress towards the teaching of science as inquiry can be expected. (Quoted in Abd-El-Khalick (2013, p. 2089).

Because of the very nature of mathematics, it is difficult for students to gain insights into mathematical inquiry by “looking” at how modern research mathematicians work. However, they may be able to gain those insights more readily by studying historical cases where they can examine concept formation and knowledge production in technically less demanding environments.

Abd-El-Khalick (2013, p. 2088) has introduced the distinction between teaching with and about the nature of science (NOS) to provide a framework through which one can couple a development of informed conceptions of NOS with the teaching of science as inquiry in a meaningful way. By these notions, he understands the following:

Teaching about NOS refers to instruction aimed at enabling students to achieve learning objectives focused on informed epistemological understandings about the generation and validation of scientific knowledge and the nature of the resultant knowledge. […] Teaching with NOS entails designing and implementing science learning environments that take into consideration these robust epistemological understandings about the generation and validation of scientific knowledge. (Abd-El-Khalick 2013, p. 2090).

If we consider the two student projects presented above in Abd-El-Khalick’s terminology, we can say that the student-centred and problem-oriented project work (the Roskilde Model) provides an explicit–reflective framework that allows for students to achieve informed understandings of the nature of mathematics. It is explicitly formulated in the study-regulating rules for the “mathematics as a scholarly (scientific) discipline” project work, that students should gain insights into mathematics as a knowledge-producing and validating endeavour. Hence, specific NOM learning objectives are included in the curriculum. On the instructional level, it is required that the students obtain the above learning outcomes through problem-oriented project work, and that they formulate and work with a problem that exemplifies mathematics as a societal and cultural phenomenon. This establishes a reflective framework in the sense expressed by Abd-El-Khalick above. In both projects, the students’ project reports document that the students achieved informed understandings of NOM through their participation in the student-directed, problem-oriented project within the explicit–reflective framework of the theme of “mathematics as a scholarly/scientific discipline” embedded in rich historical case studies exhibiting the interaction between mathematics and physics. To be more specific, the students learnt about how and where mathematicians find inspiration to introduce new mathematical concepts, how meta-discursive rules of mathematics change over time, and how the content/object level rules of mathematics as well as the meta-level rules are influenced by physics. In this way, they obtained an informed and more nuanced understanding of Wigner’s seeming dilemma concerning the effectiveness of mathematics. The rationale behind this part of Roskilde University’s programme is to make the students learn about NOM. In particular, making mathematics university students aware of the physical origin of (some) mathematical concepts and having them learn about NOM through rich history and philosophy of mathematics cases give them a necessary prerequisite to be able to teach with NOM when/if they become high school mathematics teachers. This also equips them with the tools to teach inquiry in mathematics in the same sense as inquiry is thought of and experimented with in science, namely as scientific practices in the development of scientific knowledge. When inquiry is mentioned in relation to mathematics it mostly refers to mathematical modelling, that is, to the application of mathematics already on the shelf, as discussed in the introduction. By teaching students about the interplay between physics and mathematics in the history of mathematics, one can teach inquiry in mathematics in the same sense that it is taught in physics, namely as a mathematical practice that aims at obtaining new mathematical knowledge. Moreover, it will reveal meta-discursive rules of mathematics and make them objects of students’ reflections. It is thereby possible to give students a glimpse into the rationale behind the establishment of mathematical concepts and the processes of and choices involved in developing new mathematical knowledge.

This stands in sharp contrast to how the students are usually introduced to mathematical concepts and theories. For example, in the case of the theory of function that we have discussed in this paper, students are traditionally presented with some variant of the Bourbaki definition which completely disregards the origin of this concept in general and its relationship to physics in particular. One may stipulate that a historical introduction of the concept of function can provide a deeper understanding of the concept, its use in mathematical practice, and in particular how it cannot be used.

13 Concluding Remarks

We have argued that the concept of function and its generalization to distributions gradually developed through a process that was driven by physics at crucial moments. The eighteenth century discussions about the vibrating string and the early nineteenth century study of heat conduction led mathematicians to define and gradually use functions given as a variable that depends in an arbitrary way on the value of another variable. The modern definition of function as a Cartesian product of two sets came about through an internal mathematical process of axiomatization and rigourization, but its generalization to distributions was again partly a result of physical applications. The influence from physics during the formation of mathematical concepts may partly explain the effectiveness of mathematics in the modelling of nature.

We have also shown how students can benefit from studying the historical interaction between mathematics and physics. Such historical studies can deepen the student’s understanding of the nature of mathematics and in particular inform them about concept formation.