exemplarity in mathematics education: from a romanticist viewpoint to a modern hermeneutical one

Exemplarity in Mathematics Education:from a Romanticist Viewpoint to a ModernHermeneutical One

Interpreting the Teaching of Infinitesimals as an Exemplary Theme

Tasos Patronis • Dimitris Spanos

Published online: 30 January 2013� Springer Science+Business Media Dordrecht 2013

Abstract This paper proposes a setting of exemplarity different from the already known

one, which is basically a Romanticist philosophical setting. Our general aim is to describe

and explore the nature of some exemplary themes and interpretive models in advanced

mathematics teaching and learning. In order to do so, we move from Romanticism towards

the viewpoint of Modern Hermeneutics, by applying ideas appearing mainly in Gadamer

and Ricoeur. We use this new setting as a philosophical framework, to interpret some

results from two didactical research studies that have already appeared on infinitesimals.

1 Introductory Remarks and Explanations

Traditional epistemology has opposed mathematics and science to humanities, and this

opposition has prevailed for a long time in education. However, as Martin Eger observes, it

is difficult to think of a more important problem in science (and mathematics) education

than the problem of meaning (Eger 1992, p. 337). Mathematical abstractions and gener-

alizations are considered, within traditional epistemology, as objective knowledge which is

justified, either by its efficiency as a tool in natural sciences, or by its self-consistency and

rationality, while the teaching of mathematics pays almost no attention in questions of

meaning.

Formalism in logic, as well as mathematical Structuralism have offered a pedagogical

argument against studying ‘‘old-fashioned’’ and ‘‘vague’’ themes and questions. Since, at

least, Frege’s Begriffsschrift (Conceptual Notation), mathematical propositions are con-

sidered to be formulated in a precise hypothetical form, instead of the (ancient) ambiguous

logical form of subject-predicate. Although Frege himself did not agree with Hilbert’s

spirit of axiomatization (Frege 1980), he considered that the utility of the modern hypo-

thetical form of propositions lies in the possibility of inferring a mass of particular

knowledge from a general law (Potaga 2002, p. 81). Program reformers and textbook

writers in Mathematics, especially those with the formalist inclination, often transfer this

T. Patronis � D. Spanos (&)Department of Mathematics, University of Patras, 26110 Patras, Greecee-mail: [email protected]

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Sci & Educ (2013) 22:1993–2005DOI 10.1007/s11191-013-9577-6

logical form to educational practice, mostly forgetting that ‘‘precise meaning’’ in mathe-

matics is historically determined (see Khait 2004, for a related discussion). As a result, the

initial questions that led to definitions and theorems are kept apart from the learners. This,

more or less, is the case of some current textbooks of Nonstandard Analysis treating the

‘‘hyperreal numbers’’ axiomatically (see e.g. Davis 2005; Robert 2003), by modifying set

theory or by using filters of real sequences as a formal construction.

On the other hand, according to Gadamer (1972), the first step of any process of

interpretation is to search for the questions to which a given statement is supposed to

‘‘answer’’.1 By considering reading and writing scientific texts as basically an interpretive

activity, the philosophy of science and mathematics, as well as of science and mathematics

education, turn to a hermeneutical endeavor. For example, Imre Lakatos’ fictitious dia-

logue in Proofs and Refutations (Lakatos 1976) can be read as a hermeneutical recon-

struction of the development of a historical theme (polyhedra, Euler’s formula and related

concepts).

In parallel directions in education, Martin Eger and Tony Brown have already focused

on teaching and learning as an interpretive dialogue producing meaning from linguistic

and symbolic forms (Eger 1992, 1993a, b; Brown 1991, 1994, 2001), thus introducing a

hermeneutical approach in science and mathematics education.

Now, when we speak of ‘‘the particular’’ as exemplifying ‘‘the general’’ in mathematics

teaching and learning, we usually mean a particular instance of a class of objects, which in

a way represents (and gives some idea of) a general concept that is to be taught or learned.

This is what Efraim Fischbein has called a (tacit) paradigm or paradigmatic model (Fis-

chbein 1987, pp. 143–144), which he distinguishes from mere examples or other non-

paradigmatic models.

In this paper, besides, we focus on a more radical way of linking ‘‘the particular’’ with

the ‘‘general’’: we speak of ‘‘the particular’’ as critically reflecting the complexity of ‘‘the

general’’. What we mean here is that some particular, more or less familiar but rather

problematic, theme of discussion may uncover a complex totality or field of research. In

this direction teaching starts with a crucial opening question, by which the learners are

interrogated and engaged in action. This is the pedagogical idea (or principle) of exemp-

larity, which is the main idea to be discussed in this paper. As we shall see below (Sect. 2

of this paper), the pedagogical idea of exemplarity, introduced within the Romanticist

German philosophical tradition of education by Martin Wagenschein in the 1950s, was

reformulated by Skovsmose (1994), in the context of Critical Education.

Ole Skovsmose has already discussed exemplarity as a source of inspiration for Critical

Mathematics Education. What we aim to do here is to extend this discussion to the

philosophy of teaching (advanced) mathematics; an attempt which, we think, will help

towards an alternative to the objectivist perspective. In order to do so, we shall move

towards a more comprehensive aspect of exemplarity, based on a framework of charac-

teristics of exemplary themes and interpretive models, under the viewpoint of modern

Hermeneutics (Sect. 3 of this paper). The last part (Sect. 4) of our paper is a re-inter-

pretation of didactical research already published in teaching infinitesimals. Two PhD

studies (Spagnolo 1995; Stergiou 2008) are re-examined in terms of their aims and

1 Hans-Georg Gadamer, a leading thinker in modern (philosophical) Hermeneutics, has considered inter-pretation and understanding of a text as an act of meeting and being confronted with something which isradically different from us but shares with us some common ‘‘ground’’. This point of view questionsobjectification of meaning and seems to be particularly important in epistemology and pedagogy.

1994 T. Patronis, D. Spanos

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research findings; some of these are re-interpreted in the light of the hermeneutical

framework of Sect. 3.

2 The Romanticist Point of View

2.1 Harmonic Wholes and Exemplary Parts or Individuals

Although the idea of exemplarity was introduced in twentieth century education by Martin

Wagenschein, it has probably originated in the context of Medieval thinking and re-

appeared in the Enlightenment and the Romanticist tradition of philosophical and educa-

tional thought. The idea, first, of a harmonic universe (later a ‘‘system of the world’’),

which does not simply consist of isolated elements—and its analogy with living organ-

isms—is already present in the ancient and medieval thought. At least as early as in 1194,

Moses Maimonides wrote that ‘‘the Universe, in its entirety, is nothing else but one

individual being’’ and explained universal motion in analogy with beating of the heart in

human body (Frank 1962, pp. 96–97). Similar ideas appeared more systematically in

Newton’s thought and in the French and German Enlightenment. According to Hanna

Arendt (Arendt 1991), Kant held a similar view, exemplifying general ideas by particular

instances or individual persons of special moral validity.

As a result of this stream of thought, the Humboldtian educational reform in Germany

was part of a broader social drive to produce ‘‘a new man’’ and at the same time ‘‘a

responsible and rational citizen’’ (Jahnke 1989, p. 60). Let us also note here that the ideals

of an integrated human and of a rational citizen were not separated; ideal human persons

were considered as representing the whole of society.

However, a critique of the scientific and logical spirit of Enlightenment appears in the

Romanticist Philosophy and Art of the nineteenth century, which not only admires

‘‘harmonic wholes’’, but also accuses science and logic for destroying them. Thus

Friedrich Schiller (1990), in his Sixth (1795) Letter On the Aesthetic Education of Man,

argues that an ancient Greek, considered as an individual person, can essentially rep-

resent the society and culture of his time, in contrast to a modern individual person who

is unable, by necessity, to represent the whole modern society; this is because, says

Schiller, ancient Greeks were created directly by Nature, while modern people are the

product of Logic, which separates Nature into pieces. Schiller’s argument seems to

address among other things, the problem of loss of unity between ‘‘the particular’’ and

‘‘the essential in the general.’’ This loss was, later, dramatically expressed by (young)

Lukacs in his Theorie des Romans (1915). Lukacs, who was also influenced by Hegel,

mentions Schiller and considers ancient greek art as an exemplary synthesis, in the

‘‘particular’’, of properties that are essential in the ‘‘general’’—something being lossed in

our world (Robinet 1968).

According to Jahnke (1989), a general pedagogical attitude with an enormous impact on

nineteenth century mathematics education was that of ‘‘organic thinking’’. As early as in

1845, the famous mathematician A. L. Crelle, founder of the Journal fur die reine und

angewandte Mathematik, considered that the ‘‘mathematical spirit’’ (or the ‘‘mathematical

way of thinking’’) was to be awakened in students by teaching mathematics as a coherent

system closed upon itself. We need here to distinguish between this purist tendency of

mathematicians of nineteenth century (as Crelle) and the romanticist tendency of the same

period towards organic wholes in mathematics and science, which is expressed, for

example, by Schopenhauer and Goethe.

Exemplarity in Mathematics Education 1995

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2.2 The Deductive Treatment of Geometry, and Wagenschein’s Idea

Wagenschein’s pedagogical idea of exemplarity was historically preceded (at least as

regards the teaching of mathematics) by a philosophical critique of the classical deductive

treatment of geometry.

From Descartes and Spinoza to Husserl and Mach, Western philosophy has treated

geometry as a paradigm of idealization of experience and of systematic thinking. Arthur

Schopenhauer, however, as early as in 1818, in The World as Will and Representation, has

criticized the deductive presentation of Euclid from a philosophical and pedagogical point

of view. Instead of giving a complete conception of the natural world, as Schopenhauer

thinks that a treatise of geometry might do, Euclid’s Elements start with some propositions,

arbitrarily chosen as principles, and proceed to tiresome proofs of other propositions. By

the principle of non-contradiction we are forced to admit, says Schopenhauer, that

everything demonstrated by Euclid is true; but we do not get to know why it is so

(Schopenhauer 1996, vol. II, §15, p. 70). Perhaps things are more complex than the way

this critique makes them to appear: Popper (2002, Essay 9) has conjectured that Euclid’s

Elements were intended to function as a mathematical treatise of cosmology and hence

indeed included an implicit conception of the natural world, contrary to Schopenhauer ‘s

opinion.

We find in Ernst Mach a similar critique of the deductive treatment of geometry, as well

as a description of the reaction and efforts for change. Mach’s critique is focused, once

again, on the need to perceive the ‘‘organic connection’’ between the various geometric

propositions (Mach 1976, p. 113). Mach refers to Schopenhauer (among others) at a

footnote of the same page, addressing the problem of a genetic approach:

…therefore, arose in Germany, among philosophers and educationists, a healthy reaction, whichproceeded mainly from Herbart, Schopenhauer and Trendelenburg. The effort was made to introducegreater perspicuity, more genetic methods and logically more lucid demonstrations into geometry(ibid, p. 113, footnote; our emphasis).

Almost 10 years before the educational reform of the 1960s (known widely as ‘‘New

Mathematics’’), a conference was held at Tubingen, in 1951, on secondary and upper

secondary education. One of the main questions at this conference (as well as at the

beginning of the 1960s) was how to realize aims for education in a situation of ever-

expanding knowledge. As Skovsmose (1994) observes, one possibility to overcome this

problem is to identify the basic structures which underly a given subject, and to hope that

these structures can be grasped by the learners without splitting the logical connections of

the subject into a mass of information. This approach has been transferred, through a

formal axiomatic presentation in the Bourbakist style, in mathematics education. A dif-

ferent possibility, however, has appeared at the conference of Tubingen: fundamental

concepts and educational values can be experienced and understood on the basis of the

concentrated study of some exemplary theme. Actually the expressions ‘‘exemplary theme’’

and ‘‘exemplarity’’ were not used at the conference, but the main idea was already present.

After attending this conference at Tubingen, Martin Wagenschein became active in

developing the idea of exemplarity (Wagenschein 1965, 1968) and introducing it in

mathematics and science education, as well as in the teaching of history. He supposed that

some individual phenomena (in the natural sciences, mathematics or history) may be

viewed as ‘‘mirrors’’ of a totality (Wagenschein 1965; p. 300, Skovsmose 1994, p. 75).

It is interesting to consider here a specific theme, by which Wagenschein illustrated his

pedagogical idea of exemplarity in the teaching of geometry. As such a theme he chose the


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proof of the theorem of Pythagoras (Proposition 47 of Book I of Euclid’s Elements), which,

according to Wagenschein, reflects the logical structure of all Euclid’s system and hence

provides a global understanding of school geometry (Wagenschein 1965). This proof, as a

matter of fact, uses, explicitly or implicitly, four definitions, five postulates, and twenty

seven other propositions, which practically means more than half of Euclid’s Book I of the

Elements (Nikolantonakis 2003). But even if the students of geometry succeed in grasping

this logical complexity, this is a different thing from understanding of the reason why does

the Pythagorean Theorem hold. Thus this example fails to respond to Schopenhauer’s and

Mach’s critique of the deductive treatment of geometry and shows the limitations of

Wagenschein’s idea of exemplarity, even within the romanticist philosophical educational

tradition.

2.3 Skovsmose’s Reformulation of Wagenschein’s Idea of Examplarity

Ole Skovsmose has reformulated Wagenschein’s pedagogical idea of exemplarity into a

different (although closely related) aspect that provides inspiration for Critical Education.

Thus the idea (or principle of exemplarity) that the learning of a particular phenomenon

can engage the learner in understanding a complex totality, was reinterpreted in the same

sense in which a particular socio-political event can critically reflect a political totality

(Skovsmose 1994, p. 77). This ‘‘totality’’, however, as Skovsmose says, need not be

considered as being clear of every sort of cultural conflict or social crisis, but, on the

contrary, it can be interpreted as a controversy.

Setting up a ‘‘scene’’ in the mathematics classroom, with a crucial ‘‘opening question’’

in the beginning, may provide a rich field to initiate a dialogue and give the opportunity for

knowledge conflicts and negotiation of meaning. As Skovsmose (ibid, chapter 4) indicates

by his examples of project work in the classroom, his reformulation of exemplarity may

become a link between educational theory and practice, by planning a thematic approach

in mathematics education. We need, however, to explore further the nature of ‘‘exemplary

themes’’ in mathematics, which we intend to do now, moving towards a theoretical

direction which questions the objectivist trend in mathematics education.

3 Towards a Modern Hermeneutical Aspect

3.1 Structuralism as Antipode of Romanticism and Historicism

The teaching of modern mathematical ideas has been often related to Structuralism, a

philosophical perspective that denies the significance of conflicting interpretations in

mathematics. This perspective holds that the objective meaning of (mathematical) prop-

ositions can be reproduced if one follows logical strict guidelines and general structural

patterns, which would allow one to transcend all subjective or accidental limitations. In this

sense Structuralism can be seen as a biased hermeneutics: the extreme antipode of

Romanticism and Historicism, according to which meaning is hidden in texts and limited

into their own historical context.

Since (modern) philosophical Hermeneutics also opposes Historicism, we have to

explain, first, why Structuralism cannot help a true understanding of mathematical activity.

Abstract formulations in terms of closed systems are considered by structuralists as the

only objective source of explanation in mathematics. The lattice structure, for example, of


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the divisors of the number 60 should explain everything about the importance of this

number as base in representational and metric systems.

Thus in Structuralism there is no need for interpretation of mathematical activity

other than the evolution of dominant mathematical structures themselves.2 In such

closed systems one cannot easily perceive, however, the internal mathematical features

that are responsible for the dialectical change of key notions and of the meaning of

propositions. This point is obscured by the fact that only the final products are

described, separated from the true adventure of conflicting interpretations and active

mathematical thought.

3.2 Characteristics of Exemplary Themes

Comparing ancient Greek philosophy with modern thought, Hans-Georg Gadamer has

chosen two leading themes to show the limits of objectification through the heritage of

ancient Greek thought. The first theme is the natural experience of human body, as

opposed to res extensa of the Cartesian notion of corpus. A conflict is set up, says Gad-

amer, between natural bodily experience and the experience of a patient being treated by

technical medical means and thus placed in the situation of an ‘‘object’’ (Gadamer 2002,

pp. 122–123).

The second theme chosen by Gadamer is the freedom of human beings, which also has a

structure of ‘‘essential non-objectivity’’. However, the idea of freedom, as developed

especially by Kant, was not interpreted as a fact in nature but rather a fact of reason,

‘‘something we must think, because without thinking of ourselves as free we cannot

understand ourselves at all’’ (ibid, p. 123). At the same time, Kant, says Gadamer,

developed the idea that freedom is not graspable and provable by the epistemological

possibilities of modern science. This sounds rather paradoxically as being in conflict with

the previous interpretation of freedom as ‘‘a fact of reason’’, since modern science is a fact

of reason par excellence.

Both themes discussed by Gadamer as above share two fundamental characteristics:

(a) they resist a complete objectification, although they are related to science’s natural

origins;

(b) they are subject to different and possibly conflicting (or controversial) interpretations,

thus holding an open dialogue on their meaning

We propose considering any theme sharing the above characteristics (a) and (b) as an

‘‘exemplary theme’’ in a modern hermeneutical perspective. This proposal can be justified

as denoting an aspect of exemplarity which is more comprehensive than that of

Wagenschein, because now a particular theme can reflect a totality of possibly conflicting

interpretations and not necessarily a ‘‘harmonic whole’’. As regards mathematics and

science, an exemplary theme in the above sense cannot be limited into a unique scientific

paradigm and may reflect various, possibly conflicting, currents of thought. Ratios and

incommensurability in early Greek mathematics, existence of infinitesimals, and

axiomatization of Non-Euclidean geometries, are such exemplary themes in mathematics

(see Kaisari and Patronis 2010, for a similar criterion for selection of themes introducing

Elliptic Geometry to students).

2 This is e.g. the general view in the Bourbaki exposition of mathematics, in which interpretations areconfined to the Notes Historiques at the end of each book of the Elements de Mathematique.


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The second of the above characteristics is in fact more closely related to the work of

another leading figure of modern Hermeneutics, namely Paul Ricoeur.3 A distinctive

feature of Ricoeur’s hermeneutics is his attention to the conflictual nature of human

experience. Thus the title of his first collection of essays (1969) was Le Conflit des

Interpretations. In one of his more recent essays (Ricoeur 1981), Ricoeur considers the

reading and interpretation of texts as a paradigmatic model of ‘‘dialogue’’ and of under-

standing others’ actions. This is emphasized by Ricoeur as a critical opposition to his-

toricist tradition of Hermeneutics (see Jervolino 1996, pp. 219–220). The meaning of a text

is not something lying ‘‘behind’’ it, but something situated ‘‘before’’ it, that possibly has

interrogated the writer and may also engage the reader in action.

We think that Ricoeur’s representation of understanding social activity as interpreting a

‘‘text’’ could also hold for understanding mathematical activity as interpretation of

(explicit or implicit) ‘‘texts’’. This is in accordance with Gadamer’s thesis that the first step

of any process of interpretation is to search for the questions to which a given statement (in

our case, a definition, an axiom, a lemma, or a theorem) is supposed to ‘‘answer’’.

3.3 The Notion of Interpretive Model

Exemplary themes in the above sense can be also related to the historical practice of

interpretation via models. We suggest that in the history of mathematics there are cases in

which some problematic (or controversial) mathematical ideas are newly interpreted and

partly objectified in terms of paradigmatic models; we think of this term in a sense parallel

to that of Fischbein (1987, p. 143). This means, for mathematical research, to represent the

ideas in question on a more or less familiar construction used in a new determinant way.

For example, Beltrami’s Pseudosphere can be thought as a paradigmatic model of

Hyperbolic Geometry, since, soon after its publication, Beltrami proved that Hyperbolic

Geometry is a Riemannian geometry of constant negative curvature (Beltrami 1868–1869).

Moreover as Rodin (2006) observes, although the notion of interpretation got involved in

mathematics since Gauss’ geometrical work (in correspondence with other geometers), as

an explicit term ‘‘interpretation’’ appears first in the title of Eugenio Beltrami’s paper

‘‘Saggio di interpretazione della geometria non-euclidea’’ (1868).

In view of the preceding example, we propose to call a paradigmatic model (or rep-

resentation) of the above kind ‘‘an interpretive model’’. The epistemic function of an

interpretive model in mathematics is not simply an assignment of ‘‘meaning’’ to symbols

and undefined terms of a mathematical theory (formalized or not): by interpreting the terms

in a new, determinant way and then verifying the axioms of the theory, an interpretive

model helps a partial overcoming of the conflicts of mathematical experience and a

transformation of them into a new understanding. Thus, perhaps the concept of interpretive

model offers a missing link between the epistemology of mathematics and (modern)

philosophical Hermeneutics.

In current university courses of mathematics, interpretive models such as Beltrami’s

Pseudosphere and Riemann’s Sphere usually appear only as a method of representing

mathematical structures or proving the relative consistency of sets of axioms; thus their

history and interpretive value is missing from tertiary mathematics learning. It should be

3 This second characteristic was first suggested to us by Ricoeur’s own view of the character of Xerxes, inthe tragedy Perses (Persians) of Aeschylus, as exemplary: Xerxes is accused as having provoked thePersians’ disaster but, at the same time, he is considered to be a victim of the Gods and therefore he is ‘‘anexemplary human person’’ who deserves the sympathy of the Athenian audience (Ricoeur 1988, p. 363).


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stressed here, in particular, that the Pseudosphere is only a ‘‘partial model’’ of plane

Hyperbolic Geometry in the standard model theoretical sense. Thus, although the notion of

interpretive model bears something in common with the standard notion of interpretation

and model in logic, the two notions essentially differ, as they have a different function and

purpose.

4 Infinitesimals as Exemplary Theme(s)

4.1 General Remarks

The idea of infinitesimals shares both characteristics (a) and (b) considered above, in the

preceding section, about exemplary themes in a modern hermeneutical sense. This idea

makes an early appearance in Greek mathematics and philosophy. According to F. Enri-

ques’ interpretation of Presocratic geometrical arguments (Enriques and Mazziotti 1982),

two of Zeno’s arguments (Dichotomy and Achilles) are addressed against Pythagorean

geometric atomism. Enriques argues that Zeno might have an early intuition of Eudoxus–

Archimedes Principle and its violation. Thus, if a Pythagorean ‘‘unit’’ was conceived as a

linear infinitesimal element e[ 0 (in modern notation), then for any finite segment of

length a we would have:

aþ a

2þ a

4þ � � � � a � ðeþ eþ eþ � � �Þ

since no one of the fractions 1=2n (n = 1, 2, … would be less than e; but the infinite sum at

the right part of the above inequality could be thought to be infinitely great.

Infinitesimals were probably banished by Eudoxus, in what was to become ‘‘official

Euclidean mathematics’’ (Bell 2005). Infinitesimals, since then, were for several times

‘‘driven underground’’ (it was not the case, however, for the concept of a horn-shaped

angle, which has been distinctively discussed by Euclid and his commentators). They

reappeared in the method of indivisibles, an instrumental mathematical tool in sixteenth

and seventeenth centuries. Also they appeared as evanescent quantities in Newton, as

differentials in Leibniz and as infinitely small line segments composing curved lines in de

l’Hopital and other mathematicians of early modernity. They finally became a subject of

intense debate in the Paris Academie des Sciences (Mancosu 1989). A contemporary

reconstruction of Cauchy’s notion of the continuum by Lakatos (1978) re-introduces

infinitesimals in view of the recently developed theory of Nonstandard Analysis.

From the viewpoint of modern Hermeneutics, this amazing evolution of ideas about

infinitesimals and its different interpretations through the ages can be considered as a

resistance to complete objectification. Although infinitesimals have being for several times

a subject of controversy among mathematicians, they were ‘‘driven out’’ of scientific

discourse because their idea seemed contradictory and not logically graspable. This con-

troversial character of infinitesimals can be possibly explained by the failure of embracing

all of their conflicting interpretations into a single closed system of ‘‘hyperreal numbers’’.

For example, if we admit that infinitesimals are nilpotent (i.e. that the square of an

infinitesimal or some higher power of it must equal zero), then we cannot consider infi-

nitely great quantities as inverse to infinitesimals (since nilpotent elements in algebra

cannot be inverted, being zero divisors). If, on the other hand, we reject nilpotency, then

our calculations with infinitesimals become much more complicated and we enter into a

labyrinth of ‘‘scales’’ of infinitesimals.


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Could ratios (instead of differences) be used to compare infinitesimals, thus allowing a

linear ordering that does not obey the Archimedean Postulate? Such a question may have

led Paul du Bois-Reymond, a mathematician of the late nineteenth century, to consider real

functions of a real variable as an interpretive model of ‘‘hyperreal numbers’’, by ordering

any two of them according to the limit of their ratio (as the variable x tends to infinity). As

du Bois-Reymond himself has written in 1875, he has been led to this criterion by analogy

with the ordinary domain of numbers:

(…) I have distinguished the different infinities of functions by their different magnitudes, so thatthey form a domain of quantities (the infinitary) with the stipulation that the infinity of u(x) is to beregarded as larger than that of w(x) or equal to it (…) according as the quotient u(x)/w(x) is infiniteor finite. Thus in the infinitary domain of quantities the quotient enters in place of the difference inthe ordinary domain of numbers. (du Bois-Reymond, quoted in Fisher, 1981, p. 106).

The analyst Paul du Bois-Reymond and the geometer Giuseppe Veronese may be, at first,

pictured as ‘‘dissenting voices’’ or ‘‘divergent’’ thinkers of the mathematical continuum in

nineteenth and early twentieth centuries (Bell 2005, chapter. 5). Considered, however,

from a modern hermeneutical viewpoint, these mathematicians are in the heart of the

problems of the continuum as much as their opponents of the ‘‘main stream’’ in the second

half of nineteenth century. Perhaps a critical difference of attitude between these ‘‘diver-

gent’’ thinkers and those of the ‘‘main stream’’ appears in the way that the work of the

former is reminiscent of the past. Thus du Bois-Reymond provides rules of calculation for

infinitesimal segments (Fisher 1981, p. 115), while Veronese cares also about infinitesimal

parts of the continuum and horn-shaped (or curvilinear) angles (Spagnolo 1999). These

subjects, of course, have been at the center of a long debate since the ancient Greeks and

until, at least, the seventeenth and eighteenth centuries.

Nevertheless it can be said that both above two thinkers were in a constant dialogue (in

the form of debate) with the ‘‘main stream’’ tendency of arithmetization of the continuum.

They expressed their opposition and their own philosophical ideas in their work, thus

becoming targets of Cantor’s criticism (Bell, op. cit.).

Contrary to what happened with Non-Euclidean Geometry, the du Bois-Reymond’s

interpretive model of ‘‘hyperreal numbers’’ appeared many years before the formulation of

axiomatic systems. In fact, it was only during the 1960s and the beginning of 1970s, that

Abraham Robinson and his followers developed Nonstandard Analysis as ‘‘a formal lan-

guage in the sense of modern logic’’ (Davis 2005).4

Several researches in Mathematics Education, related to the concept of limit, have

revealed a spontaneous ‘‘infinitesimalist’’ model of limit in students’ cognitive stories (see

e.g. Sierpinska 1987, pp. 389–394). Students’ conceptions of entities that do not obey the

Archimedean Postulate (such as the curvilinear angles, as well as real functions considered

as an interpretive model of hyperreal numbers) have been the subject of two PhD research

studies in Mathematics Education (Spagnolo 1995, 1999; Stergiou 2008; Stergiou and

4 The following extract makes things clearer: ‘‘Leibniz postulated a system of numbers having the sameproperties as ordinary numbers but which included infinitesimals. …Yet Leibniz’ position seems absurd onits face. The ordinary real numbers obviously have at least one property not shared by Leibniz’ desiredextension. Namely, in the real numbers, there are no infinitesimals. This paradox is avoided by specifying aformal language in the sense of modern logic (mercilessly precise in the same way that programminglanguages for computers are). Leibniz’ principle is then reinterpreted: there is an extension of the reals thatincludes infinitesimal elements and has the same properties as the real numbers insofar as those propertiescan be expressed in the specified formal language. One concludes that the property of being infinitesimalcannot be so expressed, or, as we shall learn to say: the set of infinitesimals is an external set.’’ (Davis 2005,p.2. Author’s emphasis).


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Patronis 2002). Filippo Spagnolo, as well as Virginia Stergiou, have decided not to teach

these entities axiomatically. In the rest of this paper we precede to a new interpretation of

some results from these two research studies, in view of our preceding discussion of

exemplarity in a modern hermeneutical perspective.

4.2 Curvilinear Angles as Entities not Satisfying the Archimedean Postulate

Filippo Spagnolo was interested in the teaching of Analysis at the secondary and tertiary level

and, particularly, in ‘‘the promises of progress’’ that was hoped by using Nonstandard

Analysis (Spagnolo 1999). He rejected the idea of axiomatic presentation of this theory in

teaching to young students, because, in this way there is a ‘‘loss of sense’’ for them: they

calculate without understanding, while logicians and (modern) mathematicians justify their

calculations within their theories which are not accessible to students (op. cit., p. 126).

Spagnolo then turned to a ‘‘semantic’’ approach, searching for models that satisfy the axioms

for hyper real numbers, in order to find a ‘‘realization’’ of hyper reals which would appear to

students as more simple and ‘‘natural’’. He could not find a better example than curvilinear

angles which, however, was ‘‘insufficient’’. It was difficult for students to enter the world of

non-Archimedean structures with so little a ‘‘semantic field’’ in their possession.

In view of our preceding discussion, Spagnolo’s attempt is basically hermeneutical, as

he is searching for what we have called an interpretive model. And indeed there exists such

a model for the idea of curvilinear angles, which was first introduced by Veronese in late

nineteenth century and which is mentioned by Spagnolo; but this model is also inaccessible

to young students, since it is a set of entities expressed by formal power series.

Spagnolo’s own interpretation of students’ difficulties follows a view of Didactics of

Mathematics expressed mainly during the 1980s via the notion of epistemological obsta-

cles. Thus, he maintains that the Postulate of Eudoxus–Archimedes becomes an episte-

mological obstacle in understanding of non-Archimedean structures. But even within this

framework (which goes back to the philosopher Gaston Bachelard), the controversial

interpretations of infinitesimal entities such as curvilinear angles are apparent. These

controversial interpretations can justify the selection of these infinitesimal entities as an

exemplary theme in a modern hermeneutical sense.

4.3 Rates of Convergence as Reflecting Infinitesimal Orders

Curvilinear angles may be well compared to each other and ordered, together with usual

(rectilinear) angles, but is not at all easy to think how to multiply or divide a curvilinear

angle by another. There is, however, another theme, encountered in Analysis that reflects

both a totality of different ‘‘orders’’ and algebraic operations between infinitesimals,

without so much difficulty as in the case of curvilinear angles.

‘‘Consider the sequences: 1=2n, 1=n2 and 1=n! (n = 1, 2, …). Which one, do you

believe, tends to zero faster than the others?’’. This was the opening question that was

asked by Virginia Stergiou in her Ph.D., research study (Stergiou and Patronis 2002;

Stergiou 2008) to a class of second-year mathematics students in Greece. A reason for

asking such a particular question was that no attention had been paid to a lack of con-

ceptual coherence between the courses of Calculus and Numerical Analysis. The idea of

rate of convergence penetrates both these mathematical areas and forms an intuitive

background for a modern construction of infinitesimals, as well as of the infinitely great

quantities, and their ‘‘orders’’.


123

Among the case studies of second-year mathematics students analyzed in Stergiou and

Patronis 2002, we wish to reinterpret here the case of Dimitris, a student with many

difficulties in the courses of Real Analysis and Abstract Algebra, who wrote the following

in his essay:

‘‘(…) we realize that when n increases, the difference of the denominators increases too

as follows: limn!1ð2n � n2Þ ¼ 1. We conclude that the sequence 1=2n tends faster than

the sequence 1=n2’’.

By examining this statement rigorously, it is clear that this is not a valid mathematical

statement. Moreover, the limit of the difference of two sequences is not a criterion of

ordering sequences with respect to their rate of convergence. However, if the meaning of

the above statement is not interpreted as an ‘‘objective’’ meaning, but as something that has

possibly interrogated the student, then Dimitris’ intended meaning of the expression

‘‘limn!1ð2n � n2Þ ¼ 1’’ can be read as follows:

‘‘The sequence 2n (n = 1, 2, …) tends to infinity in an infinitely faster way than n2

(n = 1, 2, …)’’.

Indeed, as the limit expression.

limn!1ðan � bnÞ ¼ � � �

is in use, in general, in Calculus and Analysis lessons, Dimitris might had no other

opportunity to express symbolically his own image of the ‘‘orders of infinity’’.

5 Conclusion and Related Perspectives

Seen from a formalist or a structuralist perspective, the problem of teaching infinitesimals

appears as the teaching of Nonstandard Analysis (at an undergraduate or graduate level),

which is reduced to selection and presentation of the suitable axiomatic system. Thus the

knowledge about the infinitely small and the infinitely great is codified into a system of

axioms, which generalizes classical operations of real numbers and permit some properties,

which had been banished from Analysis (as e.g. the property of being infinitesimal).

What this formalist view may fail to perceive is that, in order to understand, the learners

need first to interpret the previously banished properties as ‘‘external properties’’ of the

system.;5 and conversely, to interpret the axioms of the system within an interpretive

model accessible to them (if any). Evidently, such a double interpretive process needs a lot

of effort as a teaching–learning enterprise. For younger students it seems preferable not to

present an axiomatic system at all, but use some exemplary theme(s).

Exemplarity, in its hermeneutical aspect proposed in this paper, is an attempt to explore

and interpret the ‘‘objective world’’ of mathematics in such a way that historical conflicts

and different interpretations are apparent to the learners. It is important to note here that

such an attempt cannot be identified (although it shares some common features) with the

well known ‘‘genetic approach’’ to mathematics teaching (as described and exemplified

e.g. in Toeplitz 1963). In the ‘‘genetic approach’’ (as in Toeplitz) conflicting interpretations

and conceptualizations different from the main stream of thought are put aside and what

finally matters is to explain how mathematical concepts and structures have reached their

final form, as if there existed one only possible path of development.

5 See note 4.


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Our attempt can be considered as parallel to Skovsmose’s critical discussion of the

teaching of applications of mathematics, in everyday life situations reflecting complex

social structures (or ‘‘totalities’’). In both cases the complexity of situations arises not only

from mathematics previously invented, but also from conflicting interpretations of ideas. In

our case, the exemplary themes and interpretive models discussed here were intended to

question the traditional objectivist perspective and help a hermeneutical understanding of

advanced mathematical ideas.

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