1.2.1 What Does ‘Practice’ Mean?

We have noted that PMP introduces new topics in the philosophy of mathematics. Another way to characterise PMP takes as a starting point different interpretations of ‘practice’ and how they shape which questions are asked and what methods are drawn upon. Carter (Reference Carter2019) describes three main understandings of ‘practice’ that give rise to different articulations of PMP: agent-based, historical, and epistemological PMP. It is important to note that these strands are not mutually exclusive. Most often more than one of these perspectives are in play.

In the agent-based perspective of PMP, ‘practice’ refers ‘mainly to the fact that mathematics is a human activity focussing on the agents, real or idealised, doing mathematics’ (Carter Reference Carter2019, p. 11). Practice thus refers to the different activities associated with doing mathematics. Activities may consist of applying, learning or developing mathematics (Giaquinto, Reference Giaquinto, Mancosu, Jørgensen and Pedersen2005). Focussing on agents’ activities might also mean considering mathematics in its cultural, social, and educational contexts as noted by Van Bendegem (Reference Van Bendegem, Soler, Zwart, Lynch and Israel-Jost2014).

An agent-based perspective might consider general human traits or abilities and how they affect our engagement with mathematics. One might note that human beings have limited visual and cognitive capacities and so what we are able to perceive (and know, perhaps) depends on how information is presented to us. Such an assumption lies behind some of the studies of the role of visual representations that are presented in Section 3. Similar considerations are implicit in the studies of, and comparisons between, the motivations for different notational systems that are used in, for example, logic (see Waszek and Schlimm (Reference Waszek and Schlimm2021) and Shin (Reference Shin2002)).

On a broader scale, scholars have objected to the traditional account of mathematical knowledge because it leaves out the role of human agents. Recently, Silvia De Toffoli (Reference De Toffoli2021) has proposed an epistemological framework for mathematics that puts human practitioners at the center of inquiry. On the received view, De Toffoli writes, we acquire knowledge of a mathematical proposition when we have a proof of it, where proof is conceived as a deduction from axioms. That is, knowledge is tied to justification that is obtained via formal proofs.Footnote ¹³ Considered as a description of practice this is too strict. Her first point is that standard proofs in mathematics rarely are in the form of formal proofs. Second, the description leaves out the human dimension, in particular the fact that human beings are fallible. She points to numerous cases where mathematicians thought they had found a proof of a theorem that turned out to contain gaps. One example is Wiles’ first proof of Fermat’s last theorem. De Toffoli formulates instead what she refers to as a ‘fallibilist account of mathematical justification’. This involves the notion of a simil-proof which aspires to model the proofs that mathematicians make, but takes into account that sometimes when formulating arguments and believing they are proofs, they turn out not to be. An argument is a simil-proof ‘when it is shareable, and some agents who have judged all its parts to be correct as a result of checking accept it as a proof. Moreover, the argument broadly satisfies the standards of acceptability of the mathematical community to which it is addressed’ (p. 835). To be justified in the belief of a proposition then means to be able to come up with a simil-proof for it and be able to appropriately defend this simil-proof.

A few studies provide more detailed descriptions of the notion of a mathematical practice understood as the activities of agents. One is an elaboration of how human activities such as collecting, counting and measuring give rise to abstract mathematical concepts and theories. Mac Lane’s Mathematics: Form and Function (1986) presents such a picture from a mathematical point of view whereas Ferreirós (Reference Ferreirós2016) has further historical and pragmatic perspectives. A different perspective is offered by Valeria Giardino (Reference Giardino2023) who responds to the question ‘What is essential to the practice that we call mathematical?’. Her response has two components. One is that mathematics is an activity concerned with conceptual content with an inherent inferential component. The second component emphasises the material tools such as notations and figures used in mathematical reasoning. Giardino combines Brandom’s inferentialism with Hutchins’ theory of distributed cognition to form what she refers to as enhanced material inferentialism.Footnote ¹⁴

José Ferreirós (Reference Ferreirós and Sriraman2024) formulates a slightly different characterisation of a practice that emphasises the formulation of abstract mathematical theories and use of material tools in the form of symbolic frameworks. A practice is ‘recognized as mathematical due to its links with the notions of number and form (i.e., spatial figures), which may be explicit or implicit; generally speaking, mathematics has to do with the study of patterns, numerical and geometric patterns being prominent’ (pp. 2799–2800).

A related question is: when is a practice, understood as a human activity, to be counted as a ‘mathematical practice’ and in which case do such practices give rise to mathematical knowledge? Giardino and Ferreirós argue that elementary practices based on basic perception and ‘core knowledge systems’ that underlie our number sense do not yet constitute mathematical knowledge proper. Ferreirós further discusses whether ethnomathematics is a mathematical practice. According to Ferreirós’ characterisation, the activities of ethnomathematics, for example, weaving baskets with intricate mathematical patterns, do not constitute a mathematical practice since they lack both the symbolic framework and the theoretical dimension. Both forms of activities belong to a different level that he refers to as proto-mathematics. It is one of his (see Ferreirós (Reference Ferreirós2016)) points, however, that the theoretical parts of mathematics depend on these lower levels, referring to ‘the interplay of practices’ or a ‘web-of-knowledge’.

Returning to the different strands of mathematical practice, historical PMP considers mathematics as the outcome of certain human activities and events. More important is the underlying assumption that mathematics has changed across time and therefore cannot be perceived as a static body of truths. From an historical point of view, one could be interested in the various internal and external factors that influence, and have influenced the development of mathematics over time. The philosopher of mathematical practice might wish to ask if there are any general things to say regarding its development. In the historical PMP, the interconnections between history and philosophy can further be employed for various purposes: Philosophy may offer general categories that help uncover new insights in historical studies (Epple, Reference Epple2004). Conversely, philosophical questions may be investigated by analysing historical case studies.

Historical perspectives may enrich a philosophical study by revealing sources of ideas and by suggesting new categories and different interpretations. To illustrate how, we give an example concerning the phenomenon ‘reverse mathematics’. In contemporary mathematical logic ‘reverse mathematics’ refers to the program of Harvey Friedman and Stephen Simpson (see, e.g., Simpson (Reference Simpson2009)). The point is to determine which set existence axioms are needed to prove particular mathematical results. They have formulated a hierarchy of logical subsystems that have been used to determine the strength of major results from all parts of mathematics, including logic, arithmetic, analysis and algebra. The fundamental idea when determining the strength of a proposition is that one does not only prove that the proposition follows from a particular subsystem – one also proves that the subsystem can be derived from the proposition (modulo the base system in which reasoning takes place). In this way, it is certain that the subsystem is the weakest possible system in which the result is provable.

Taking an historical perspective reveals that the idea of asking which axioms are needed to prove a particular proposition is not a recent invention. Victor Pambuccian (Reference Pambuccian2009) traces it back to the ancient Greeks, more precisely to Pappus of Alexandria, and refers to it as ‘reverse geometry’ (see also Arana (Reference Arana2008)). It further appears in Hilbert’s Foundations of Geometry. At the end of this work, Hilbert ties attempts at proving impossibility results to what he refers to as the ‘purity’ of methods in demonstration and writes: ‘the preceding geometrical study attempts … to explain what are the axioms, hypotheses, or means necessary to the demonstration of a truth of elementary geometry’ (Hilbert Reference Hilbert and Townsend1950, p. 82).

Andrew Arana (Reference Arana2008, Reference Arana, Arana and Panza2022) further argues that different concerns seem to be at stake when mathematicians ask for which axioms are necessary to prove a particular result. Friedman and Simpson’s reverse mathematics uses tools from proof theory and computability theory to analyse the computational or combinatorial content of mathematical theorems, which Arana (Reference Arana2008, p. 37) refers to as ‘strong logical purity’. In contrast, one may search for the ‘right, proper, essential, or appropriate’ axioms (2022, p. 403). Arana ties the latter to Aristotle’s concern of finding definitions that reveal the essence of a thing, which further entails that definitions, or axioms, cannot introduce ‘foreign elements’: a definition of a concept in number theory, for example, cannot rely on geometry.

Second, historical studies might throw light on why certain philosophical programs such as the foundational schools were initiated. Such studies do not at the outset belong to philosophy. But awareness of the historical circumstances under which a program or text has been produced might affect how it is interpreted. Tappenden (Reference Tappenden2006) is a good example illustrating how this is the case in some of Frege’s writings. James Tappenden claims that Frege has often been interpreted from a narrow and distorted point of view and, if one reads him according to his contemporary mathematical background, a different and richer picture emerges. Tappenden considers the following passage:

On a formalist reading, one might wish to emphasise the passages in italics and claim that Frege’s main concerns were the validity of propositions and rigorous proofs. If one reads the passage bearing in mind the state of affairs in Frege’s mathematical field, one could instead focus on the other issues mentioned: how to define fundamental concepts and determine the domain of application of central principles. Tappenden tells us that Frege’s field was geometry and complex analysis from a Riemannian perspective. This field was in wild disarray in the second half of the nineteenth century. There was a dispute about Weierstrass’ computational and Riemann’s conceptual approaches (see Stein (Reference Stein, Aspray and Kitcher1988)). One disagreement concerned how to define a complex function. Riemann’s conceptual approach defined a complex function by the today well-known Cauchy–Riemann equations, that is, in terms of a given property. Weierstrass believed that the proper way to treat complex functions was by their power series or as analytic expressions and to calculate. Today we know that these two characterisations are essentially co-extensive. But this fact was not known before 1900. Another concern was the domain of application of various principles. One fundamental result is Dirichlet’s principle, which establishes the existence of certain functions under given conditions. It was used, for example, in Riemann’s famous work on Abelian functions to establish the existence of functions on a Riemann surface of genus $g$ with poles at specified points. It turned out that Dirichlet’s principle was used in cases where it did not hold. A precise formulation and a rigorous proof were given by Hilbert in 1901. It may be some of these concerns that Frege refers to when he writes that ‘[t]he concepts of function, of continuity, of limit, and of infinity have been shown to stand in need of a sharper definition’ and that ‘the same ideals can be seen at work – rigor of proof, precise delimitation of extent of validity, and as a means to this, sharp definition of concepts’.

The final approach, epistemological PMP, considers ‘practice’ in a more abstract or general way, referring to mathematics itself, or the many ways that it is presented or accessible to human beings. In this case, ‘practice’ may refer to mathematical results or theories as presented in articles, books, at talks, and so on, or what mathematicians themselves say or write about mathematics. The requirement in this strand is that whatever philosophical question is asked, we need to consider the relevant part of mathematics in order to respond to it.

As part of epistemological PMP, philosophers are examining how to capture what mathematicians might mean when they say that a result is deep or a proof is explanatory. A special issue has been devoted to different characterisations of depth in mathematics (Philosophia Mathematica, Volume 23 (2), published in 2015). One may also consider qualities from an abstract point of view and discuss the general notion of virtues in mathematics, see Aberdein et al. (Reference Aberdein, Rittberg and Tanswell2021).

The topic of ‘mathematical explanations’ has received much attention. When referring to ‘explanations’ in mathematics it is useful to distinguish between an external and an internal point of view (Mancosu Reference Mancosu2008b). The external perspective discusses whether mathematical facts can serve as explanations in other domains and the implications this has for the ontology of mathematics. Philosophers also consider explanations that are internal to mathematics, in particular how to characterise explanatory proofs. Two main proposals (Steiner (Reference Steiner1978) and Kitcher (Reference Kitcher1984)) have been critically examined.Footnote ¹⁵ Steiner defends the view that an explanatory proof makes evident how a characterising property of an entity mentioned in the proposition leads to the conclusion.Footnote ¹⁶ Kitcher characterises explanations in terms of a notion of unification. Others have suggested modifications to these views (see, for example, Lange (Reference Lange2018)). I refer to Mancosu et al. (Reference Mancosu, Poggiolesi and Pincock2023) for further details as well as an overview on the topic of mathematical explanations. One might also take a broader perspective and consider explanations in mathematics, not only in connection with proofs, but also as related to mathematical understanding (see Carter (Reference Carterforthcoming)).

The last example of epistemological PMP returns to the much discussed concept, the infinite. This example further illustrates that typically many different aspects of practice such as historical case studies or results from contemporary mathematics are involved in PMP studies. The example is based on a forthcoming contribution by Mancosu (forthcoming). Mancosu discusses, among other things, an interpretation of the infinitely large formulated by the medieval scholar Robert Grosseteste (ca 1168–1253) and – using recent results from measure theory – argues that Grosseteste’s idea can be made mathematically rigorous. Today, the standard conception of the size, or cardinality of a set, states that two sets have the same size if their elements can be brought in a one-one correspondence.Footnote ¹⁷ Grosseteste, an Oxford theologian, had a different intuition. According to this conception, if two infinite collections $A,B$ fulfil that , then the size of $A$ is strictly less than the size of $B$. Mancosu refers to some passages of Grosseteste’s work Commentary on the Physics where he proposes certain ideas concerning measuring the number of points of finite line segments. The basic idea is that we choose a specific line segment as our unit and that it will contain a given number of points. (According to Grosseteste, whereas human beings can only measure line segments relative to a unit, God assigns an absolute infinite number to the number of points of the unit segment.) The unit line segment is used as a ‘measure’, that is, any other (finite) line segment can be measured relative to it. The idea includes a number of assumptions: If two line segments have equal lengths, then they contain the same number of points. If the length of a line segment is a rational number times the unit line, then their numbers of points are also in the same proportion.

Grosseteste’s ideas, as Mancosu points out, were challenged by later scholars. One is Fishacre (1205–1248) whose most famous challenge was a demonstration that there is a one-one correspondence between any two finite line segments. Although Fishacre is mostly unknown today, the ensuing discussion influenced other thinkers such as Oresme and Galileo (see Mancosu (Reference Mancosuforthcoming) for details).

Mancosu demonstrates how tools from measure theory and the notion of an elementary numerosity (that counts the number of points of a line segment) can be used to implement Grosseteste’s idea. Grosseteste’s conception of the measurement of finite line segments and their corresponding number of points, assumes that there is a correlation between the measure of the finite segments and the numerosities of points they contain. Mancosu refers to Benci, Bottazzi, and Di Nasso (Reference Benci, Bottazzi and Di Nasso2015) who show that if $L$ is the Lebesgue measure on $R$, then it is possible to construct an elementary numerosity on $R$ that has the desired properties: Finite line segments of equal length have exactly the same (infinite) numerosity of points. This concludes the summary of the relevant sections of Mancosu’s forthcoming book.

Based on the different interpretations of practice, it might appear as if the philosophy of mathematical practice consists of a number of diverse and incompatible approaches. Carter (Reference Carter2019) suggests this need not be the case and presents a simple overall framework to capture the various approaches to PMP. It consists of two components, mathematics and agents.Footnote ¹⁸ The component ‘mathematics’ contains the part of mathematics that is considered, for example, theories, concepts, propositions, axioms, proofs, and so on. The second component refers to the human agents, real or idealised and their mathematical activities. The idea is that philosophical studies considers variations of these two components as well as relations between them. The strands presented earlier emphasise different aspects of mathematical practice and so will take different starting points: whereas the agent-based strand’s main focus is on the human agents and how, for example, mathematical knowledge is shaped through their ‘practices’, the starting point of the epistemological strand is mathematics itself – sometimes leaving out considerations of agents completely. The historical strand primarily focuses on examining the interplay between mathematics, its concepts, theorems, theories, and so on, and the practices (of human agents) that shape its development.

2.2.1 Hilbert on the Axiomatic Method

Bourbaki (Reference Bourbaki1950) stresses the importance of the axiomatic method as a component of structuralism and elsewhere attributes it to David Hilbert (1862–1943). In the philosophy of mathematics, Hilbert is mainly known for his later work on the foundations of mathematics, as the founder of formalism. As part of this programme Hilbert created proof theory with the intent to give a finitist proof of the consistency of mathematics. He was, however, concerned with the foundations of mathematics throughout his career when he conceived and tackled questions regarding foundations in different ways. Sieg (Reference Sieg, Reck and Schiemer2020) explains that Hilbert’s contributions to the foundations of mathematics can be characterised in two main ways: The first involves a structural axiomatic approach, while the second is based on proof theory and formal methods. In the late nineteenth century, thinking about the foundations of arithmetic, Hilbert was influenced by Dedekind’s logicist views. By 1904 Hilbert expressed that the consistency of the axioms of arithmetic would be demonstrated if one could show that no contradictions could be derived from the axioms. Sieg refers to this as a ‘quasi-syntactic’ view, since the idea of a formal system including deduction rules had not yet been developed. It was only after 1917 that Hilbert began to develop his meta-mathematics. This came after studies of Whitehead and Russell’s Principia Mathematica and a realisation that the underlying principles of reasoning had to be spelled out in order to formulate a foundation for mathematics (using mathematical tools as pointed out in the previous section). By principles of reasoning, I refer to both the basic (logical) assumptions as well as the inference rules. In this way reasoning becomes a calculus and the rules and assumptions can be studied by mathematical tools.

Hilbert formulated the axiomatic method during his earlier work on the foundations of mathematics and as part of the structural axiomatic approach. It is explicitly addressed in Hilbert’s ‘Axiomatic thought’ (1918) but he uses the basic ideas much earlier in his Grundlagen der Geometrie which was first published in 1899. In the rest of this section, I refer to the English title, Foundations of Geometry (FoG). The Foundations of Geometry has its roots in the plane geometry of Euclid’s Elements and its axiomatic presentation. Some of the more recent developments of geometry are added and it is presented in a form where all assumptions are spelled out. Concerning assumptions, it was long since known that the Elements drew on certain assumptions that were not included among the axioms. Already the first proposition of book I reads off information from the accompanying diagram, that is, that two circles intersect in a point (see Section 3 for further details).Footnote ²² The missing assumption in this case is an axiom of contnuity. The fourth proposition states that if two triangles have an angle in common and the two line segments that enclose the angles are pairwise equal then the two other angles and remaining line segments will also be pairwise equal. The demonstration superimposes the two triangles and so exploits that properties are not changed when a geometric figure is placed at a different location. Further assumptions were identified and spelled out in the nineteenth century by Pasch (Reference Pasch1882/1926) in his Vorlesungen uber neuere Geometrie. The axiom that is now named after him states that if a line passes one side in a triangle, then, if continued, it will eventually go through one of the other two sides; see Figure 1. The non-Euclidean geometry that was formulated by Gauss, Bolyai and Lobachevsky during the first half of the nineteenth century is included in the FoG, as is the projective geometry developed by Desargues and later Poncelet, among others.

Figure 1 Pasch’s axiom: Let $A$, $B$, $C$ be three points that do not lie on a line and let $a$ be a line in the plane $ABC$ which does not meet any of the points $A$, $B$, $C$. If the line $a$ passes through a point of the segment $AB$, it also passes through a point of the segment $AC$, or through a point of segment $BC$.

(Hilbert 1950, p. 5)

Another major difference between Euclid’s Elements and Hilbert’s geometry concerns the content of the axioms. Book I of the Elements starts out by defining all considered geometrical objects, a point, line segment, circle, and so on, and some of their properties. The postulates (except number IV) tell us which constructions are permissible under given configurations of points and line segments. In this way, further points, lines and geometric figures are constructed as we work our way through the propositions. Each proposition starts out with certain given geometric objects, for example, a line segment (and so also its endpoints). By systematically using the postulates further geometric objects appear (Panza Reference Panza2012). In contrast, there are no definitions of the basic geometric objects in Hilbert’s geometry. He does not say what a point, line or plane is; Hilbert assumes the ‘three distinct systems of things’ exist and gives them different labels according to the category they belong to. The purpose of the listed axioms is then to tell us how the elements of the three systems are related:

The axioms are presented in five different groups. The groups consist of axioms of connection, order (defining the relation ‘between’ on points on line segments), parallels (containing the parallel postulate), congruence, and continuity.Footnote ²³ The first group of axioms characterises the relations between the system of things, that is, the points, lines and planes. The axioms tell us, for example, that two distinct points determine a line and conversely that two lines determine a point.Footnote ²⁴ Other groups include axioms that were drawn upon in Euclid’s Elements but not explicitly mentioned such as the beforementioned axiom of continuity. The axioms of congruence enable a proof of a proposition corresponding to the beforementioned proposition I.4.

The presentation in the Foundations of Geometry has certain structural implications. The first is that we can deal only with objects to the degree that they are ‘fixed’ by the relations defined on the system. A point, for example, has no intrinsic properties, it can only be determined as the point of intersection between two lines. The second is that a group of axioms together characterises a given structure, or theory (in this case a geometry). Third, the axioms do not come with a fixed ontology: the theory applies to any system of objects that fulfils the axioms. This is often illustrated by Hilbert’s famous quote: ‘one must be able to say “tables, chairs, beer-mugs” each time in place of “points, lines, planes” ’ which Hilbert allegedly said to Otto Blumenthal at the Berlin train station in 1891.

Around the same time as Hilbert published the Foundations of Geometry, he wrote the short article ‘Über den Zahlbegriff’ (1900) where he notes that the axiomatic method is also the right method to use when one wishes to study the logical foundation of other areas than geometry – including arithmetic. The article presents the axioms for the real numbers in a similar way as in the Foundations of Geometry, that is, as belonging to different groups: axioms of connection, calculations (commutative associative and distributive laws), order and continuity of the numbers. In brief, together they characterise the real numbers as a complete, ordered field that fulfils the Archimedean principle. Later the axiomatic method is applied to all of mathematics. In ‘Axiomatic thought’, the method and all its advantages are spelled out, mentioning numerous examples from both mathematics and physics, including set theory, thermodynamics and mechanics.

The aim of formulating mathematics on an axiomatic framework is connected with Hilbert’s concerns with the foundations of mathematics. In the introduction to the FoG, certain now familiar foundational concepts are explicitly mentioned:

One purpose is therefore to offer (groups of) axioms that are mutually independent and complete. Corry (Reference Corry, Ferreirós and Gray2006) points out that by ‘complete’ Hilbert at this time intends a pragmatic notion, that is, the aim is that all known results of geometry should be possible to prove. The intention is not that all possible ‘true’ results of geometry are provable (thus avoiding the consequence of Gödel’s first incompleteness theorem that was published later in 1931). These aims are also mentioned in his ‘Über den Zahlbegriff’ as well as in his later ‘Axiomatic thought’. But note also the last sentence in the quote that mentions a different aim, that of ‘exploring the significance and consequences of the different groups of axioms’. This is related to the progressive part of the axiomatic method (Hilbert Reference Hilbert and Ewald1918, paragraph 6)

In addition to simply deriving consequences from the formulated axioms of a theory, Hilbert emphasises the fruitfulness of the method. Several paragraphs of ‘Axiomatic thought’ are devoted to explaining how a deepening of the foundations and exploring the dependence and independence of propositions lead to new results. Hilbert mentions examples from mathematics as well as physics. One obvious example is the independence of the parallel postulate which led to the construction of entirely new geometries.

Besides the fact that axioms define relations and that a group of axioms characterises a structure, a different interpretation of ‘structure’ arises from the axiomatic method. It concerns how one chooses to build – or structure – a particular theory, that is, which principles are chosen as one’s axioms or first principles. In this case, one might refer to the logical structure of a theory. The logical structure of a theory is mentioned in the beginning of ‘Axiomatic thought’ using words like ‘order’ and ‘logical relations’:

In this way mathematical theories are organised in terms of their logical structure. In Bourbaki (Reference Bourbaki1950) we find a different way to organise mathematics that we turn to later in this section.

How to structure – or present – a theory becomes relevant as a consequence of a change in the conception of the status of axioms. One might hold that an axiom is a ‘true’ characterisation of the mathematical subject matter or that the purpose of axioms is to reveal the essence of the characterised concepts. After the publication of Hilbert’s Foundations of Geometry a famous correspondence followed between Hilbert and Frege debating the nature and role of axioms. (Frege further criticised the lack of definitions fixing the meaning of concepts in the FoG.) Frege defended the view that axioms are true and that it therefore was not necessary to prove that groups of axioms are consistent. Hilbert, on the other hand, responded – with the much discussed claim – that

Other contemporary writers tended to agree with Hilbert’s view that axioms should not be considered as true, but merely hypothetical, as suppositions from which to reason. I refer to Feferman (Reference Feferman1999), Schlimm (Reference Schlimm2013) and Cantù (2022) for different roles of axioms in mathematics and to Hellman and Shapiro (Reference Hellman and Shapiro2018) on the Hilbert–Frege correspondence.

2.2.2 Noether and Algebraic Structures

Emmy Noether introduced axioms that characterise abstract algebraic structures such as rings and ideals which become ‘freestanding’ mathematical structures. One might claim that Hilbert’s structures also are freestanding in the sense that they apply to any collection of objects fulfilling the relations formulated by the axioms.Footnote ²⁵ But the structures mentioned are intended to characterise a specific field or domain, for example, geometry or the real numbers. Noether’s algebraic structures are truly general or non-categorical. In Noether’s work we also find the emergence of further structuralist themes that are characteristic for contemporary mathematics. She formulates what is referred to as structure theorems and establishes formal relations between structures. It is also important to notice the influence Noether had through her students and other mathematicians that she talked to. Her student van der Waerden mentions her as a main source of inspiration in his influential textbook Moderne Algebra. She also promoted the use of algebraic invariants which contributed to the development of algebraic topology and had an impact on Mac Lane, who, together with Eilenberg, later developed category theory (McLarty Reference McLarty, Ferreirós and Gray2006).

Noether, born in 1882 in Erlangen, was the daughter of Max Noether, professor in mathematics there. Max Noether was a colleague of Paul Gordan, who would become Noether’s doctoral advisor, and a friend of Klein and Hilbert, both located in Göttingen. In 1915 Hilbert and Klein invited Emmy Noether to come to Göttingen to work on invariant theory. From 1919 she started her work on algebra which is the area most often mentioned in connection with her contributions to a structuralist view of mathematics. She was invited to work at Bryn Mawr College in the US in 1933 and died in Princeton in 1935.

Noether (Reference Noether1921, Reference Noether1927) wrote two influentual papers on algebra, the first, ‘Idealtheorie in Ringbereichen’, the second, ‘Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern’. Both appeared in the Mathematische Annalen. In the first of these articles Noether proves a generalised version of the fundamental theorem of arithmetic, more precisely that an ideal of a module (or ring) can be written uniquely as a product of prime ideals. A major accomplishment is that she identifies precisely which assumptions are required to prove this result. In addition to the axioms of a ring, the proof requires the ascending chain condition which is an original contribution by Noether. A ring fulfilling this condition is today called a Noetherian ring.

Besides ideals and rings, the algebraic structures of a field and group were already known by the time Noether wrote her papers. What is new to her approach is, firstly, that she considers the algebraic structures as abstract entities where, before, they were thought of as generalisations of the various number domains. Because of this, rings or fields were implicitly assumed to have properties that these domains shared. By considering them as abstract entities in their own right, Noether was able to identify all assumptions needed to prove the considered results. Van der Waerden (Reference Van der Waerden1935) later characterised her style saying that

Algebra has its roots in computations and solving equations. Solutions to the cubic and quartic equations were found during the sixteenth century and collected in Cardano’s Ars Magna (published in 1545). These accomplishments were followed by a desire to find similar solutions to the quintic which turned out not to be possible. Following methods set out by Lagrange, Abel and Galois were able to demonstrate the insolubility of polynomial equations of a degree higher than four. Two of the fundamental concepts that came from these studies are permutation groups and polynomials over a field. Problems in number theory also led to advances in algebra. The most famous problem is Fermat’s last theorem; another influential result was that of ‘the law of quadratic reciprocity’ formulated by Euler while working on results of Fermat. In the simplest form, this asks which primes, $p$, can be written as a square modulo another prime $q$. The surprising fact is that if $p,q$ are odd primes, $p\ne q$ and either $p$ or $q$ is equal to 1 modulo 4, then $p$ is a square mod $q$ if and only if $q$ is a square mod $p$. Gauss’s work on higher reciprocity laws led him to introduce the Gaussian integers, $Z\left[\sqrt{-1}\right]$, which are numbers of the form $a+b\cdot \sqrt{-1}$ for $a$ and $b$ integers. A generalisation of such numbers includes numbers of the form $a+b\sqrt{-n}$, for $n$ square free.

While considering these different number domains it became clear that the usual properties of the integers do not always hold. An important property, stated in the fundamental theorem of arithmetic, is that any number can be uniquely decomposed as a product of prime numbers. An example of a ring where this is not true is $Z\left[\sqrt{-5}\right]$. In this ring it is possible to write, for example, 6 as a product of irreducible factors in two ways: $3\cdot 2=(1+\sqrt{-5})(1-\sqrt{-5})$. The numbers, 2, 3 and $1\pm \sqrt{-5}$ can all be shown to be irreducible in $Z\left[\sqrt{-5}\right]$. To restore this property, Kummer, Kronecker and later Dedekind introduced notions that grew into the concepts of an ideal and a prime ideal. Dedekind’s definition of an ideal is essentially the one that is used today and it is Dedekind’s work that was the starting point of Noether’s contributions to algebra.Footnote ²⁶

Noether’s Idealtheorie in Ringbereichen formulates first the conditions that define a commutative ring and ideals in such a ring. Noether only considers finitely generated ideals and proves that the ascending chain condition (a.c.c.) follows.Footnote ²⁷ She also notes that if one assumes the a.c.c., then ideals of a given ring will be finitely generated and consequently that one may assume the a.c.c. condition instead. In other words, these two conditions are equivalent over the axioms of a commutative ring.

Noether defines a commutative ring as a system of things fulfilling six axioms quite similar to how it is defined today. The first axioms assert the commutativity and associativity of addition and multiplication and the distributive law. The last axiom, called the unlimited and uniqueness of subtraction, states that for any two elements $a,b$ in the ring, there exists a unique element $x$ that solves the equation $a+x=b$. The existence and uniqueness of a zero-object follow from the last axiom. The main result, the unique decomposition of ideals, states that any two decompositions will contain the same number of primary ideals, in Noether’s words:

Theorem IX Given two different shortest representations of an ideal as the least common multiple of primary ideals, the number of components will be the same.Footnote ²⁸

The decomposition of ideals into prime ideals is an example of what is referred to as a ‘structure theorem’, that is, a theorem that tells us how a type of structure can be decomposed into simpler objects. The basic idea of such theorems is that a particular object can be written (uniquely) as, for example, a product or direct sum of certain basic building blocks (primes, prime ideals, etc.).

As mentioned, Noether carefully chooses her axioms so that they suffice to prove the unique decomposition of ideals. The approach of searching for just the right, or minimal, axioms required to prove a given result is seen to a greater extent in her 1927 paper where she also explores which conclusions are obtainable from given groups of axioms. It has been suggested that her style and the concern of finding the ‘right’ axioms are inspired by Dedekind (McLarty Reference McLarty2017).Footnote ²⁹ In addition, her approach can be seen as an application of Hilbert’s axiomatic method, an instance of what he refers to as ‘deepening of foundations’.Footnote ³⁰

The final structural theme, namely, the formulation of formal relations between structures emerges, for example, in the shape of the isomorphism theorems found in Noether’s Reference Noether1927 paper. These theorems establish that certain abstract relations exist between structures and certain specified subsets and constructions on them (equivalence classes). Furthermore, these relations are expressed without mentioning any elements of the structures or the operations defined on them.

The theorems are formulated in the context of modules and later in the setting of ideals of a ring, $R$. The section begins with a definition stating that two $R$-modules, $M$ and $\bar{M}$, are homomorphic when there is a function from $M$ to $\bar{M}$ that preserves the operations on elements. If the map is bijective, it is an isomorphism in which case $M$ and $\bar{M}$ are said to be isomorphic. Noether then introduces congruence classes and formulates the homomorphism theorem (which is sometimes today referred to as the first isomorphism theorem). The theorem states that if there is a homomorphism, $h$, between two modules, $M,\bar{M}$ over the same ring and $U$ is the subset of $M$ that is mapped to zero by $h$, namely, the kernel of $h$, then the congruence classes $M|U$ is isomorphic to $\bar{M}$. Given the homomorphism theorem, Noether demonstrates that the two isomorphism theorems follow.

There are versions of both the homomorphism theorem and the isomorphism theorems in Dedekind’s writings. But they are formulated in the context of groups and as arithmetical results. More importantly, they are not presented as being dependent on each other as in Noether’s writings and certainly not in the general form that she gave them:

2.2.3 Bourbaki and Structures as a Tool

Various structural themes emerge in the work of Hilbert and Noether. Neither of them, though, says that mathematics is the study of structures. This changes in the writings of Bourbaki. Bourbaki explicitly claims that structures play a fundamental role in mathematics. In addition to giving a bit of background information, I shall focus on how Bourbaki characterises the axiomatic method and his conception of the fundamental structures of mathematics – the mother structures – and how they form the building blocks of modern mathematics.

Nicolas Bourbaki is the pseudonym of a group of mainly French mathematicians. One of their main accomplishments is the book series, Éléments de Mathématique, in English Elements of Mathematics. Another is their influence on how modern mathematics is conceived and done:

The idea of the Bourbaki group was conceived in 1934 by André Weil and Henri Cartan who at the time were teaching at the university of Strasbourg. They were worried that French students were lagging behind after the war and found that one way to improve the situation would be to write an updated textbook in analysis. They presented the idea at one of their weekly Paris meetings with former student colleagues from l’École Normal and so the group was formed in 1935. Some of the first members included, besides Cartan and Weil, Claude Chevalley, Jean Delsarte, Jean Dieudonné and Charles Ehresman. Later other prominent mathematicians were invited to join, for example, Eilenberg, Grothendieck, Samuel and Serre. When formed, the group realised that they did not just need a book on analysis and so they decided to rewrite all of mathematics. A task that continues to this day.

The name ‘Elements’ refers to Euclid’s Elements. Bourbaki’s intention was to adapt the axiomatic style used in Hilbert’s Foundations of Geometry and van der Waerden’s Moderne Algebra and apply it more generally to all of mathematics.

Cartan (Reference Cartan1980) notes that the axiomatic style as introduced by Hilbert and used by the German school of algebra (referring to Noether and her students) ‘has penetrated the whole of mathematics’ (Cartan, p. 176). Today Bourbaki’s Elements comprise multiple volumes on topics such as algebra, topology, analysis, spectral theory and differential and analytic manifolds.

‘The architecture of mathematics’ (Bourbaki Reference Bourbaki1950) tells us how Bourbaki understands the axiomatic method. The method involves a few steps and is exemplified by the introduction of the structure of a group. In the first step one realises that seemingly differently looking collections have similar properties. Consider, for example, the collection of real numbers under addition, the collection of integers modulo a prime number $p$, and finally the displacements in 3-dimensional space under composition. Viewed as collections on which there is defined an operation, they are alike. In the next step, one investigates the logical structure of their shared properties and asks if a number of them suffice to derive the rest. In the given example, it is found that the axioms characterising a group have as a consequence the remaining shared properties.

As presented here, the method resembles to some degree Hilbert’s description of how to deepen the foundations. But there are some noticeable differences. Bourbaki compares three different collections and abstracts from them to form the general concept of a group. The elements of the before-considered collections become empty positions. The only focus is on properties of the operations defined on them. Second, although Bourbaki mentions further foundational issues that are related to formal studies of mathematics, such as the consistency of the chosen axioms, he does not seem worried about these issues (see the introduction of Bourbaki (Reference Bourbaki1960)). He does, however, share Hilbert’s attitude regarding the fruitfulness of the axiomatic method. One advantage mentioned is that whenever we establish that a given collection fulfils the axioms of a group (or any of the other fundamental structures), we will know – at no further mental cost – that it also has all the properties of a group. As such the axiomatic method is associated with unification.Footnote ³¹ This is not the only mentioned advantage. He refers to the obtained structures as ‘tools’ giving examples of the advances made when realising that a given set or structure can be represented in a different domain. One such example is the geometric representation of the complex numbers which they write eventually enabled Gauss, Abel, Cauchy and Riemann to transform analysis. We return to Riemann’s visual representation of complex functions in Section 3.

In regard to ontology, Cartan stresses that a structuralist view has the consequence that one gets rid of objects – mathematics concerns properties and reasoning – and so mathematics is based on hypothetical reasoning: A mathematician having found a proof of a proposition will go through the proof to see which assumptions are actually required. ‘Instead of declaring which objects are to be investigated, one has only to list the properties of the objects to be used in the investigation. These properties are then brought to the fore expressed by axioms; whereupon it ceases to be important to express what the objects are. Instead, the proof can be constructed in such a way as to hold true for every object that satisfies the axioms’ (Cartan Reference Cartan1980, pp. 176–177).

A group is but one example of the algebraic structures. The algebraic structures constitute one of Bourbaki’s fundamental structures referred to as ‘mother structures’. These are the structures from which modern mathematics is built and constitute a sort of foundation for mathematics. Besides the algebraic structures, there are the order structures (i.e., collections on which there is defined an order relation) and the topological structures. These structures form the basic building blocks. In addition to these there are more specific structures and mixed structures that, together with the mother structures, constitute a hierarchy of structures. Specific structures are obtained by adding further properties, for example, considering abelian groups or finitely generated groups. We further encounter mixed structures: collections that have, or can be given, multiple structures. The set of real numbers is one example: ‘the set of real numbers is provided with three kinds of structures: an algebraic structure, defined by arithmetical operations (addition and multiplication); an order structure, since inequalities are defined between real numbers; and finally a topological structure based on the notion of limit’ (Cartan Reference Cartan1980, p. 177). Further examples abound and include topological groups, analytical fibre spaces and $C^{*}$-algebras.

The hierarchy of structures and the various interrelations between the different types of structure provide the means to organise all of mathematics. Bourbaki uses the metaphor ‘the architecture of mathematics’ for the picture of mathematics as founded on the mother structures. He likens mathematics with all its interrelations to a big city

2.2.4 Components of a Methodological Structuralism

We are now able to describe some of the components of a structuralist view of mathematics. One component consists of various facets of the axiomatic method. From one perspective, the axiomatic method can be seen as a tool to create structures. The individual axioms characterise relations between the considered entities, and together, a collection of axioms defines a structure such as a ring or a group. From another perspective, the method can be used to organise a theory, choosing which propositions should be taken as primitive, that is, as the foundation for the rest. Bourbaki refers to a different metaphor, that of a big city, for how mathematics is organised (and constantly re-organised); the axiomatic method is used to create the fundamental mother structures which are then combined and interrelate in multiple ways to form the complex that is mathematics. Note therefore that, although I refer to ‘the’ axiomatic method as used by Hilbert, Noether and Bourbaki, they do this in quite distinct ways with different outcomes and motivations.

A related theme is the shift of focus concerning the subject matter of mathematics; mathematics becomes the science of structures rather than individual objects. In some cases, it is even claimed that structuralism gets rid of objects and that there is no need to say anything about their nature; see Cartan’s comment that mathematics is only based on reasoning and hypotheses that are formulated in the form of axioms. A structuralist view thus opens up for a hypothetical view of mathematics. I refer to Awodey (Reference Awodey2004), Carter (Reference Carter2014) and Ferreirós (Reference Ferreirós2016) for different positions that consider mathematics as hypothetical.

A second component concerns the different uses of ‘structure’. In addition to the basic structures such as a group, topological space, and so on, we have noted the existence of structure theorems that say how entities of various sorts can be decomposed into simpler ones. Structure theorems come in various forms and constitute a useful tool in mathematics.Footnote ³² That is, mathematics does not only consider structures as ‘units’. Structures may themselves have ‘a structure’ (which can be understood in different ways). Moreover there are relations, or interrelations between structures that we return to shortly.

The final component concerns the claim that the structural point of view (via the axiomatic method) is an effective tool in mathematics. Recall Hilbert’s claim that the axiomatic method, namely, the deepening of foundations, considerations of independence, and so on, is a fruitful endeavour that has led to many new discoveries. Bourbaki further stresses the methodological point of view of structuralism. First is the role of unification, the identification of certain basic structures – the mother structures – that he uses to build mathematics. The second is the use of these basic structures to economise thought. The last subsection of this section considers in more detail the methodological aspect of structuralism. This aspect is further combined with the observation that mathematics rarely considers structures in isolation but exploits many different types of interrelations between them.Footnote ³³

2.3.1 Formal Maps, Analogies and Representations

Relations between structures come in many shapes, ranging from formally specified maps to informal correspondences. They are exploited in various ways to determine properties of objects or structures. A formal map, for example, an isomorphism, between collections with different structure can be exploited to transfer properties from one collection to the other. Less formal correspondences may also be useful when looking for properties of a given structure. Consider as an example a finite field, $(F,+,\cdot )$. We might be interested to know something about its subfields. To do this, we notice that the field under multiplication (with the zero-object removed) has the structure of a group. We therefore consider the subset $F\backslash \left\{0\right\}$ of the field as a group under multiplication. Considered as a group, we may use Lagrange’s theorem to tell us about its possible subgroups. (Lagrange’s theorem states that the order of subgroups divides the order of the group). Equipped with the knowledge about possible subgroups, noting that they become possible subfields of $F$ if one adds the zero-object, in combination with the fact that the order of finite fields are powers of primes, one may be able to say something about the subfields of $F$.

Another type of informal relation is analogy. Analogy is often mentioned as an important part of the mathematician’s toolbox – in contemporary mathematics as well as historically:

To this list one could add John von Neumann, who allegedly claimed that he considered his bicommutant theorem for von Neumann algebras as an analogue to the Galois correspondence, and Michael Atiyah stressing its importance in relation to the unification of mathematics (Atiyah Reference Atiyah1978). More recent is Barry Mazur (Reference Mazur2021) who writes that an analogy ‘connects two disparate concepts by some similarity in their structure’ (p. 12). Mazur considers as an example knots in 3-manifolds as analogous to primes of number fields which makes it possible to unify topological structures with arithmetic structures.Footnote ³⁴ While highly useful, it is difficult to give a precise definition of an analogy. Characterisations range from stating that an analogy consists merely of ‘perceived similarities’ between domains to the actual identification of some ‘shared structure’.

The last interrelation to be considered here is representation. A representation is a sign that stands for some (mathematical) entity to somebody (who is the interpretant).Footnote ³⁵ I consider cases where we represent a particular situation, sometimes referred to as the target, such as an expression, concept, relations between concepts, and so on. A representation, then, is often composed of other, more basic signs such as notational elements. When forming a representation, a specific correspondence, or rule, between the representation and the target is established. The correspondence is not always stated explicitly, but is exploited in practice. A simple example is when we consider the graph of a real function, $y=f\left(x\right)$, given by some expression. In this case, the convention is that corresponding values, $x$ and $f\left(x\right)$, of the function are represented by the point $(x,f(x\left)\right)$.

As an example of a fruitful representation, we consider $C{}^{*}$-algebras that can be represented by directed graphs. $C{}^{*}$-algebras generated by directed graphs are referred to as graph-algebras. A $C{}^{*}$-algebra can be characterised as a set (a vector space over the complex numbers) that has three types of structures (cf. Bourbaki’s mixed structures). Each structure is given by a different type of operation: it has an algebraic structure obtained by the operations of addition and multiplication as well as multiplication by complex scalars. There is a norm defined on its elements which can be used to define a topology, and, finally, there is an involution operation, referred to as the $*$-operation. A $C{}^{*}$-algebra therefore has both an algebraic and a topological structure. The final operation, the *-operation, provides a link between the algebraic and topological structure.Footnote ³⁶

An important research question concerns the classification of $C{}^{*}$-algebras, that is, determining up to isomorphism which algebras are possible to define. A tool to achieve this is $K$-theory. The idea is that two $C{}^{*}$-algebras are isomorphic if their corresponding $K$-groups, $K_0$ and $K_1$, are pairwise isomorphic.Footnote ³⁷ It turns out that it is generally quite complicated to calculate $K$-groups even for simple $C{}^{*}$-algebras. It has recently been discovered that a large class of $C{}^{*}$-algebras can be generated from directed graphs. One advantage of having this alternative form of representation is that it gives a much easier calculation of the $K$-groups. A directed graph is formally defined as a four-tuple: $D=\{V,E,r,s\}$, where $V$ consists of a collection of vertices $v_i$ and $E$ consists of edges $e_i$ that each has a source and a range among the vertices. $r,s:E\to V$ are the range and source maps that inform about the range and source of all the edges. Figure 2 shows an example of a directed graph.

Figure 2 A visual representation of a directed graph with four vertices and five edges.

A particular graph can be read according to a specified rule to define generators and relations between them which together generate a specific $C{}^{*}$-algebra. Read in a different way, one obtains a linear map from which it is possible to calculate the two $K$-groups, $K_1$ and $K_0$ (as the kernel and co-kernel of the linear map, respectively). See Raeburn and Szymanski (Reference Raeburn and Szymanski2004) and Carter (Reference Carter2018) for details. Figure 3 illustrates the connections.

Figure 3 A directed graph generates a $C{}^{*}$-algebra and gives rise to a linear map from which the two $K$-groups can be calculated.

One point in this example is that mathematics frequently represents problems in other domains where they become easier (or even possible) to solve. Another point concerns the fact that for such representations to be useful, one needs first to establish formal relations between the domains. In the case considered, there are theorems that establish the connections, for example, a theorem that mentions the correspondences shown in Figure 3 – roughly:

If $D$ is a directed graph subject to some conditions and ${\Delta }_D$ a particular map, then the co-kernel of ${\Delta }_D$ is isomorphic to $K_0$ of the graph algebra generated from $D$ and the kernel of ${\Delta }_D$ is isomorphic to $K_1$ of the graph algebra generated from $D$. (Raeburn and Szymanski Reference Raeburn and Szymanski2004, theorem 3.2)

Mathematics is also about these connections – not only the individual structures such as a $C{}^{*}$-algebra. Furthermore, we determine properties of $C{}^{*}$-algebras by exploiting this connection (and others like it). I illustrate these claims in more detail with a simpler example on the complex numbers.

2.3.2 Representations of Complex Numbers

Besides serving as an illustration of the usefulness of represention in mathematics and how this depends on established correspondences, the complex numbers have given rise to a much discussed problem. The challenge is that a certain structuralist position, the so-called non-eliminative structuralism, seems to entail that we are not able to distinguish between the two complex numbers $i$ and $-i$ (Keränen Reference Keränen2001). Non-eliminative structuralism holds that structures are in the ontology (they cannot be ‘eliminated’) and mathematical objects are considered as places in structures. Furthermore, places have only structural properties. The problem of indiscernibles occurs when one asks about the identity conditions of places. Considering the complex numbers as an algebraic field, Keränen argues that there is no relation that makes it possible to distinguish between $i$ and $-i$. An obvious candidate is an automorphism, but conjugation (which is an automorphism on $C$) takes $i$ to $-i$ and vice versa. There is a mathematical solution to this problem. McLarty (Reference McLarty2020) notes that our ability to distinguish between the two complex numbers depends on whether we consider the complex numbers as an algebraic structure or as part of complex analysis. In the context of algebra, complex conjugation is an automorphism, whereas in complex analysis, it is not.

There are a number of ways to introduce the complex numbers. A textbook in complex analysis defines them as ordered pairs of real numbers, $C=\left\{\right(x,y)|x,y\in R\}$, equipped with addition and multiplication:

$\begin{array}{c}(x,y)+(a,b)=(x+a,y+b),(x,y)\cdot (a,b)=(xa-yb,xb+ya)\end{array}$

$(C,+,\cdot )$ so defined can be shown to be a field. $i$ is defined as the element $(0,1)$. (It follows that $-i$ is the number $-(0,1)=(0,-1)$).

After introducing different representations of the numbers the authors comment:

It is possible to perform basic calculations such as addition and multiplication on numbers from the given definition. But when it comes to other types of calculations, some of the other ways of treating the complex numbers are more convenient. Let us consider an example. Suppose we wish to calculate the 6th power of the number $z=1-i$ (now presented in the rectangular form). In order to do this we represent the number in polar form. We first consider the corresponding coordinates $(1,-1)$ in a Cartesian coordinate system and notice that the point can also be characterised by the length, $r$, of the line segment between the point and the origin (referred to as the modulus) and the angle, $\theta$, that this line makes with the positive $x$-axis (the argument). That is, it can be determined by the polar coordinates $(r,\theta )$; see Figure 4. The corresponding polar form of representation is .

Figure 4 The complex number $z=1-i$ represented in a Cartesian coordinate system that also shows the polar coordinates $(r,\theta )$.

The first task is to determine the values of $r$ and $\theta$, that is, establish a correspondence between the two forms of representation. Noticing that we have a right angled triangle, we can use a trigonometric relation, for example, $tan\left(\theta \right)=y/x$, and Pythagoras’ theorem to determine the two numbers. We write $z$ in the polar form as $z=\sqrt{2}(cos(-\pi /4)+i\cdot sin(-\pi /4\left)\right)=\sqrt{2}\cdot e^{-\left(\pi /4\right)\cdot i}$.

Before we can perform the calculation, we need to establish what the operations defined on the Cartesian coordinates correspond to in the new form of representation. Using a variety of trigonometric identities, it is possible to show that multiplication of two numbers given in polar coordinates, $(r_1,{\theta }_1)$ and $(r_2,{\theta }_2)$, is given by $(r_1\cdot r_2,{\theta }_1+{\theta }_2)$. That is, the modulus of the product is the product of the moduli of the multiplied terms and the argument is found by adding the given arguments. Using this fact, it is easy to calculate the result:

$\begin{array}{c}{z}^6=(\sqrt{2}\cdot e^{-\left(\pi /4\right)\cdot i}{)}^6=(\sqrt{2}{)}^6\cdot e^{6\cdot (-(\pi /4)\cdot i)}=8\cdot e^{-\left(3\pi /2\right)\cdot i}\end{array}$

The geometric representation of the complex numbers is not only a useful tool when performing simple calculations on them. The geometric representation of complex numbers turned out to be extremely fruitful in the hands of Riemann as previously mentioned. Riemann (Reference Riemann and Weber1851) represented complex functions visually and was thereby able to formulate their characterising property in geometric terms, see Section 3.3.2. Riemann further considered the multivalued Abelian functions and found a way to represent them by Riemann surfaces. This example illustrates well that a representation in a different context sometimes paves the way to a whole new field, a field that does not seem imaginable from the original form of presentation. In the next section, we consider this example as well as other examples of fruitful representations and discuss some of their properties.

3.3.1 What Is a Diagram?

When discussing the role of diagrams it is often useful to contrast them with other types of representation such as sentential representations. Similarly, when investigating the properties of fruitful representations it may beneficial to determine whether they rely on characterising properties specific to diagrams or if they hold more generally. It is therefore appropriate to state what a ‘diagram’ is. When referring to visual representations, it is common to distinguish between external and internal representations. Internal representations refer to mental images produced by our visual imagery whereas external representations are signs that are written or constructed on some physical media such as a piece of paper or a blackboard. In the following we consider only external representations.

Many different types of representations are used in mathematics; among visual representations there are diagrams, figures, graphs, tables and illustrations. See, for example, Bertin (Reference Bertin and Berg2011) for a distinction between some of these. Another category consists of sentential representations, where it may be relevant to distinguish between natural language and mathematical expressions (that are formed using mathematical notation). It may not even be clear to which category mathematical expressions belong.

There are further cases where a characterisation of ‘diagram’ is useful. We mentioned the question whether proofs can be based on diagrams. Responses to this question depend on what type of information can reliably be read off from diagrams and diagram discipline (Manders Reference Manders2008b) or whether diagrams can be considered to be a type of formal system (Shin Reference Shin2002). One might also argue that diagrams (or a subclass of them) constitute a notation (Goodman Reference Goodman1976, pp. 170–171) and that they therefore can be parts of proofs. I refer to the forthcoming Element by Dirk Schlimm for a philosophy of notations.

In the philosophy of mathematics only a few contributions explicitly say what diagrams are, but in information and visualisation theory there are a number of proposals on how to characterise diagrams in contrast to other types of representations such as natural languages.Footnote ⁴⁶ Keith Stenning (Reference Stenning, Anderson, Cheng and Haarslev2000) and Atsushi Shimojima (Reference Shimojima2001) discuss a number of proposed distinctions between diagrams and sentential representations in the context of information theory. Their aims, however, differ. Whereas Stenning seeks to understand how diagrams communicate, Shimojima notices that graphic representations hold the capacity to offer ‘free rides’ and wishes to explain this feature. Some of their distinctions make sense also in the context of mathematics. An often mentioned difference between diagrams and sentential representations is the apparent two-dimensionality of diagrams in contrast to the linearity of sentential representations. Several counter examples have been offered, however, arguing that this distinction is neither sufficient nor necessary. One is the use of two-dimensional notations, for example exponentials, in mathematics. Another type of counter-example consists of linear representations that intuitively seem to be diagrams such as the linear graph, $E$, shown in Figure 11. Whereas this distinction concerns the appearance of the representations, it is also possible to make a distinction based on how they are interpreted, that is, their semantics. Stenning (Reference Stenning, Anderson, Cheng and Haarslev2000) notes that sentential representations are read and interpreted in a fixed direction, often from left to right, reading one character at a time, as when reading this text. In contrast, diagrams need not be interpreted in a fixed direction. Because of this, it is possible to read a particular diagram in multiple ways (as we noticed about Peirce’s existential graphs and graph-algebras in Section 2.3.1). It is therefore of vital importance to supply rules that say how a particular diagram should be read.

Figure 11 Two directed graphs, $E$ and $F$.

Another distinction that sometimes applies considers whether letters, points and other basic components act as types or as tokens. In sentential representations, letters typically are taken as types. The ‘t’ in ‘typically’ and in ‘type’ both refer to the same letter, the type ‘t’, regardless of its location on this page. In diagrams, it is often the case that the reference of a character or sign is determined by its location, that is, the same letter, say, placed at two different locations means that they refer to two different objects. Consider the graphs in Figure 11. They are composed of points and arrows placed at different locations. It is implicitly understood that the different points (and arrows) do not refer to the same, but to different vertices (or edges). The type of reference in this case is denoted token-reference whereas the first is type-reference.

A final distinction concerns how reasoning is displayed or the representation is formed. When making calculations in sentential representations one usually rewrites the transformed expression below or next to the previous one when making changes. This is referred to as discursive reasoning. When operating on diagrams, it is possible to add information successively to the same representation, as we shall see, forming agglomerative representations.

I offer the following characterisation of diagrams that I find useful when discussing the heuristic role of diagrams, in particular the question why they sometimes contribute new insights (see also Carter (Reference Carter2021, Reference Carter2018)). It is not intended as a sharp distinction between different types of representations which I doubt is possible. First, I consider and distinguish between three different levels: the visual appearance of the representation, its semantics and, finally, how it is used. When considering the appearance of a representation, I note that mathematical representations are typically composed of certain basic elements according to syntactic rules. The basic constituents of diagrams consist of geometric elements such as points, line segments and curves and other types of notational elements such as letters (standing for mathematical objects). The syntactic rules of diagrams further allow production of two-dimensional objects. The syntactic rules of directed graphs, for example, make it possible to form two-dimensional displays as shown by the graph, $F$, in Figure 11. Similarly, some of the basic constituents of the Elements are line segments that can be composed into complex two-dimensional figures. The exponential expression referred to earlier may be considered as a notational unit. The syntax, however, makes it possible to compose exponentials into two-dimensional displays. The reason that they are not diagrams, I propose, is because of their semantics. Sentential representations come with a fixed reading direction (which is the case for exponentials; we must not confuse $2^3$ with $3^2$) whereas diagrams typically may be read in multiple directions according to specified rules. Finally, the uses of a particular representation may draw on the beforementioned distinctions between type and token reference and agglomerative versus discursive reasoning, where token reference and agglomerative reasoning are effective ways of using diagrammatic representations.

3.3.2 Free Rides in Mathematics

Suppose you are given the following information about three sets, $A,B$ and $C$, that $A$ and $B$ are disjoint and that $C$ is a subset of $B$. Formally, $A\cap B=\varnothing$ and $C\subseteq B$. The phenomenon of a free ride can be illustrated by representing these two facts by so-called Euler diagrams. Euler diagrams are constructed by using the convention that a set is represented by a closed curve in the plane such as a circle, and that elements of the set are contained inside the closed domain. It follows that a circle that is fully contained in another represents set inclusion and that two non-intersecting circles represent disjoint sets. According to this convention the given information leads to the configuration shown in Figure 12.

Figure 12 A diagrammatic representation of $A\cap B=\varnothing$ and $C\subseteq B$.

Observing the diagram, one notices that the two circles marked $C$ and $A$ do not overlap. This translates into the set theoretic claim that $A\cap C=\varnothing$. Since this is a piece of information that was not put into the representation when we constructed it, it is consequential information that comes for free, that is, what Shimojima calls a free ride. Notice also that Proposition I.1 in the Elements contains a free ride: The consequential piece of information that is possible to read off is the point of intersection between the two drawn circles.

The property that we read off from the Euler diagram follows from topological properties of the plane. The diagram consists of two non-intersecting circles, marked $A$ and $B$ and another circle that is fully contained in circle $B$. It is then impossible for circle $C$ to have any points in common with circle $A$. That is, the free ride depends on geometric or topological properties of the representation itself. Shimojima denotes geometric, topological (and physical) properties of a representation as nomic constraints of the representation and argues that they govern free rides. It is not important for our concerns exactly what this involves; I refer to (Shimojima Reference Shimojima2001) for details. We focus instead on free rides as they occur in mathematical practice and extend the notion accordingly.

My recent article ‘ “Free rides” in mathematics’ (Carter Reference Carter2021) analyses a number of occurrences of mathematical free rides. Some of the examples resemble the ones considered earlier: free rides depend on geometric and topological properties of the representations. But this is not always the case. Free rides, defined as ‘consequential pieces of information that can be read off from a diagram,’ occurring in mathematics are more varied. In some cases they depend on the fact that diagrams allow multiple readings and theoretical results that connect these readings. This is the case in the example of graph-algebras that we presented in Section 2.3.1. A graph-algebra is a certain type of $C{}^{*}$-algebra that can be represented by directed graphs. An advantage is that the directed graphs can be read in two distinct ways. One reading of these graphs gives generators and relations that define a $C{}^{*}$-algebra, referred to as a graph-algebra. When read in a different way, one obtains a linear map from which it is easy to calculate the two $K$-groups of the $C{}^{*}$-algebra. Formal results establish the connection between the linear map and the corresponding $C{}^{*}$-algebra (Raeburn & Szymanski Reference Raeburn and Szymanski2004).

In other cases the essential feature is a combination of the possibility to form two-dimensional displays with what was referred to as agglomerative reasoning. One of the considered examples is a commutative diagram from category theory that is read as a map in order to see that different expressions are identical. That the considered morphisms are successively added to the same diagram is as essential to the use of it as is its two-dimensionality. We illustrate these features by looking at Riemann’s geometric representation of the property that characterises a complex function. We return to this example in the next section. Further details can be found in Carter (Reference Carter2021, pp. 10484–10486).

Riemann (Reference Riemann and Weber1851) defines a function of a complex variable, $z=x+i\cdot y$, stating that $f\left(z\right)=u+i\cdot v$ is a complex function if the differential quotient $\frac{df}{dz}$ is independent of $dz$. (Today we would say that a function so defined is differentiable at $z$.) He further comments that this idea is easier to grasp when using one’s ‘spatial intuition’ and so represents values of the variable and the function in different planes as illustrated in Figure 13. Using this geometric representation, he was able to express geometrically the condition of being a complex function. The condition entails that the two angles $v$ and $w$ will be identical and that the two triangles shown are similar. That is, the free ride that is obtained in this case is the similarity of the triangles. Note that we do not observe this fact. It is calculated based on visual information that it is possible to represent the difference between points (e.g., $dz=z-z_1$) in polar coordinates. (See Section 2.3.3 on different forms of presentations of complex numbers.) Inserting differences expressed in polar coordinates in the expression $\frac{df}{dz}=\frac{df}{d{z}^{\prime }}$, it is possible to derive the result.

Figure 13 A geometric representation of the condition to be a complex function.

Returning to the preceding points, note that it is crucial that the objects considered are added to the same diagram, for example, that the three considered complex numbers, $z,z_1,z_2$ are represented in the same complex plane forming a triangle. That is, we construct an agglomerative representation. In addition, the representation formed clearly depends on the possibility to form two-dimensional displays.

To sum up, we have observed that mathematical free rides depend on a variety of characterising properties of diagrams. One is the possibility of forming two-dimensional displays, that is, a consequence of the appearance and syntax of diagrams. Second, I noted the multiple readability of diagrams which is made possible because diagrams need not come with a fixed semantics. Thirdly, is the possibility to form agglomerative representations that was mentioned among the uses of a diagram. The last mentioned feature, that of token versus type reference, is mixed in the considered cases. The $z$ shown in Figure 13 refers as a type so as to tell us that $f\left(z\right)$ is the value of this particular $z$ whereas the line segments should be taken as tokens.

3.3.3 Observing and Manipulating Iconic Representations

Free rides are obtained by representing certain information and observing the result. This can be considered as a special case of the more general case of constructing and manipulating (or experimenting on) iconic representations. In order to explain what this means, we present a few details about iconic representations as they are used in mathematics. Recall that a representation is a sign that stands for some entity, or object, to somebody, the ‘interpretant’. That is, a representation is a triad. When referring to iconic representations, we are considering the relation between the sign and the object it stands for, that is, in which capacity the sign accomplishes standing for the object(s) it represents. According to Peirce, there are three main possibilities, referred to as icons, indices and symbols. Icons represent because they share some kind of likeness with what they represent. A drawn rectangle could be an iconic representation of the shape of my floor. Indices represent because they are either physically connected with what they represent, as in smoke indicating fire, or because of a ‘purposeful connection’. The latter occurs when we name, for example, a particular set ‘ $A$’ allowing us to refer to and reason about this set. Symbols are signs that represent their objects by conventions. Words are symbols as are the signs like ‘ $\cdot$’ and ‘ $=$’ that we use in mathematics.

The characterisation of iconic signs, that they represent by likeness, has been criticised, most notably by Goodman (Reference Goodman1976). Goodman argues, among other points, that since a similarity relation is an equivalence relation while representation is not, signs cannot represent because of likeness.Footnote ⁴⁷ This objection does not take into account that an icon is also a sign. The latter presupposes an intention: someone is taking the sign as representing an object with a specific purpose. This entails that a sign (no matter what type) is neither reflexive, symmetric, nor transitive.Footnote ⁴⁸ I refer to Stjernfelt (Reference Stjernfelt2007, Section 3) for a detailed discussion.

The icons used in mathematics rarely resemble as images; they typically express characterising properties of objects or relations between them and according to some convention (as noted in the case of Euler diagrams representing relations between sets). But they are still icons in the sense that they convey ideas of what they represent. In fact, this is an important characteristic of an icon:

Note also that iconic representations need not take the form of diagrams (as they are usually conceived) – algebraic, or formal, expressions act equally well as icons. To see how, we consider the following simple example about numbers. A characterising property of an odd number is that it always has remainder one when dividing it by two. This fact can be represented iconically by the expression $2k+1$, for $k\in Z$. Writing odd numbers in this way makes it possible for us to discover further of their properties. I might be interested in knowing something about the square of an odd number. Representing an odd number as shown, squaring it and manipulating the signs according to stipulated rules, I obtain the following:

$\begin{array}{c}(2k+1{)}^2=4k^2+4k+1=2(2k^2+2k)+1.\end{array}$(1)

Observing the last expression, I notice that it also has the form of an odd number. Thus, representing a mathematical situation (such as an object or relations between objects) iconically and manipulating, or experimenting with, the signs may lead to a new insight. Sometimes when representing certain facts, a conclusion may even be read off immediately as in the case of free rides.

Peirce emphasises this feature of iconic representations. In ‘On the algebra of logic. A contribution to the philosophy of notation’ (published in 1885) he describes how observing signs leads to new discoveries in mathematics:Footnote ⁵⁰

We end this section by two examples that illustrate different ways that icons lead to new insight. The first considers a case of representing a situation in a richer context where new properties become visible. In this case it is possible to re-interpret these properties in the represented configuration (what I have referred to as the target). In the second case that is not possible, that is, the representation gives rise to new concepts and properties that do not correspond to anything in the represented situation and give rise to completely new domains or theories. Recall that when constructing representations, a correspondence is set up saying which objects, properties, relations, and so on, of the target situation correspond to which (visual) features of the representation.

The first example shows how representing natural numbers in a richer context allows us to determine, or discover if you prefer, Pythagorean triples. In this case, we represent numbers geometrically by points as seen in Figure 14. The convention is that each point represents a unit. The representation further exploits the two-dimensionality of the plane to arrange points in the shape of geometric figures. The intention is first to display a square corresponding to the number $c^2$; see Figure 14. Considering the geometric representation on the right then reveals that it is possible to cut up the larger square into a square and a gnomon, corresponding to the stippled lines. We therefore see that the square $c^2$ can be written as the sum of a square that contains $(c-1{)}^2$ points and the number $2(c-1)+1$. The last number need not be a square – but we immediately notice that it is an odd number. If we then search for numbers $c$ so that $2(c-1)+1$ is a square (using the fact that odd squares are squares of odd numbers), we have a systematic way to obtain some of the Pythagorean triples.Footnote ⁵¹ Other Pythagorean triples can be determined by experimenting with the geometric representation, finding different ways to cut up the square.

Figure 14 A geometric representation of Pythagoras’ theorem.

The final example returns to Riemann’s representation of the complex numbers in a plane. We noticed that triangles appear, see Figure 13, and remarked that it is possible to derive that the defining property of a complex function implies that the two triangles that appear are similar. In this case, however, there is no immediate interpretation of this property in the target situation. That is, there is nothing in the analytic context of considering complex functions that corresponds to triangles or other two-dimensional geometric figures. As noted in Section 1.2.1 and the end of Section 2, Riemann exploited the geometric intuition even further and introduced Riemann surfaces as a way to handle multi-valued complex functions. By doing this, he was able to ask questions that turned out to be very fruitful, for example, how many functions are possible to define on such surfaces depending on their geometric properties – a question that would make no sense in the context of the analytic treatment of complex numbers.