Who's Afraid of Mathematical Diagrams?

2.1 Topological Diagrams

Let’s start by considering diagrams of topological surfaces (i.e., two-dimensional manifolds). The Möbius band is the simplest non-orientable surface. It is usually represented either as a sketch of an object in space (Figure 2(a)) or as an arrow diagram (Figure 2(b); ignore the red dashed line for now). In Figure 2(b), the two vertical arrows have to be interpreted as gluing instructions: we have to imagine stretching the square and twisting it in space to align the arrows and identify them. Imagining this process (or implementing it in a physical model made of paper), we obtain something like Figure 2(a) – the arrows can then be erased since they are now internal to the surface.

Figure 2.

The Möbius band

Consider the following question: Which surface do we get if we cut the Möbius band along its core (indicated in the two representations by the dashed red lines)? Likely, Figure 2(a) will not help since our ability to imagine transformations in three dimensions is typically very limited and not especially reliable. We could use a paper model of the Möbius band and see what happens when we cut along its core with a pair of scissors. Alternatively, we can imagine cutting and pasting the arrow diagrams, creating the diagrammatic argument⁸ in Figure 3.

Figure 3.

Cutting a Möbius band along its core

PROPOSITION: Cutting the Möbius band along its core gives rise to a cylinder.

PROOF

The first step consists of cutting along the red dashed line. In order to conserve the original gluing instructions, we introduce the double arrow convention. In this way, we indicate which side of the first piece must be attached to which side of the second and in which direction. The second step consists of gluing the double arrows. To align the arrows, we flip the second piece. The third step is not a transition but a simplification. It consists of erasing the previously used gluing instructions and reshaping the rectangle. The final diagram is a rectangle with two sides identified in the same direction. It is a cylinder.

QED

This is a simple proof of a simple result. But there is a caveat. It is a proof only if the diagrams and diagrammatic transitions are interpreted correctly.⁹ Cutting and pasting is not (only) an intuitive terminology to guide us in imagining diagrammatic transformations. These two operations can be cashed out in precise topological terms: gluing corresponds to forming a quotient space, and cutting can be thought of as the inverse operation of gluing.¹⁰ The notation formed by arrow diagrams is a perspicuous diagrammatic notation that lets us carry out mathematical operations in an intuitive way. Moreover, it tracks relevant topological properties. For example, from Figure 2(b) we see that any band with an odd number of half-twists is also a Möbius band (Figure 4).

Figure 4.

Möbius band with three half-twists

The number of twists is not an intrinsic property of the Möbius band, but one that derives from the way it is embedded in space. Abstracting from the particular embedding is useful in certain circumstances, but it hides how embeddings change with the various transformations. For example, the previous argument does not show that cutting the standard Möbius band along its core yields a twisted cylinder.

I called the argument in Figure 3 a “proof.” It could be objected, however, that it is not a genuine proof but merely an exercise in applying transformations on a specific diagram. Fair enough. As you can see in the following theorem, arrow diagrams generalize and can be used to prove substantive results:

THEOREM 1. (Classification of Surfaces.) Any closed connected surface results from gluing the sides of a polygon according to one of the two gluing instructions in Figure 5 – which are represented by a label (and a color).¹¹ The pattern on the left corresponds to orientable surfaces, while the one on the right to non-orientable surfaces.

Figure 5.

Classification of closed connected surfaces

ONE STEP OF THE PROOF

Figure 6 shows one step of a proof that appears in a graduate textbook in algebraic topology (Massey 1991, 23). The zigzagging convention introduces generality. It indicates that any combination of arrows could be positioned in its place.¹² The general strategy of the proof consists in showing that any collection of arrow diagrams representing a closed connected surface can be transformed into one of the two configurations in Figure 5.¹³ The step in Figure 6 is representative of the technique used. It starts with a diagram containing two pairs of alternating arrows. It proves that such a diagram can be transformed (without altering the surface it represents) into one with two adjacent pairs of arrows following the pattern in Figure 6(4) – which is a fragment of the pattern in Figure 5(a). Let’s see how.

Figure 6.

A step in the proof of the classification theorem for surfaces (Massey 1991, 23)

In order to go from (1) to (2), we cut along c and glue along b. The next two diagrams are equivalent. Figure 6(2) shows that b is now internal to the diagram. Figure 6(3) is obtained by erasing b and introducing the internal arrow d as a new cutting instruction. Figure 6(4) results from cutting along d and gluing along a.

QED

2.2 Syntax and Semantics of Arrow Diagrams

Let us pause briefly before tackling the second mathematical example. I want to draw out the main differences between the two representations for surfaces introduced above: arrow diagrams and sketches. The latter are evocative, pictorial representations of three-dimensional objects. Arrow diagrams are instead topological diagrams with precise mathematical conventions. Diagrams have two central characteristics that pictorial sketches do not have: (i) they satisfy well-formedness conditions, and (ii) they are subject to precise rules of manipulation.

Let us first look at the syntax of arrow diagrams. I limit the discussion to closed connected surfaces. I will thus consider arrow diagrams formed by single polygons with all edges paired – the results can be easily generalized to non-connected surfaces. The primitive objects are digons (i.e., polygons with just two sides) with oriented arrows. As illustrated in Figure 7, there are just two possible basic diagrams, B₁ and B₂. As a matter of fact, the arrows’ direction is not relevant per se; what counts is only the relative orientation of the arrows. This means that inverting the directions of both arrows of B₁ or B₂ would leave them unchanged. We can think of the orientation of each edge with respect to an orientation of the boundary circle. B₁ can then be characterized as the digon in which the two arrows have opposite directions (if you imagine tracing the boundary circle from a node, you are going to have to trace one of the arrows in reverse) and B₂ as the digon in which the two arrows have the same direction.

Figure 7.

Digons

All other arrow diagrams can be obtained from these two basic diagrams, B₁ and B₂, by adding couples of oriented edges. Again, when adding two edges we will have just two choices of orientation: they will go either in the same direction or in opposite directions along the boundary circle. The set $D$ of well-formed arrow diagrams is the smallest set such that:

1. B₁ and B₂ are in $D$.

2. If the arrow diagram P is in $D$, and the arrow diagram P’ is obtained from P by adding two matching oriented edges, then P’ is in $D$.

In order to obtain P’ from P, we add two matching oriented edges, either with the same or with opposite orientation. We can position each of them in the place of a vertex. They can, therefore, be positioned one after the other or separated by one or more preexisting edges.

Figure 8 shows how we can add a pair of oriented edges to B₁, the first digon in Figure 7. There are four possibilities, corresponding to the choices of (i) whether the arrows are in the same or different direction and (ii) whether we insert the arrows one after the other or in different nodes – since the nodes of the digon are equivalent, there are just two possibilities. The next step would be to add two additional oriented edges to one of the configurations above.

Figure 8.

Adding two matching oriented edges

In general, a polygonal arrow diagram is a polygon with directed edges with labels to mark the pairing that results from a finite number of applications of the recursive definition above. Two figures represent the same arrow diagram when there is a bijection taking edges to edges that preserves labels, directions, and adjacency relations. Moreover, given a diagram, we can change the direction of both arrows of a couple that has to be identified.

Now to their semantics. A polygonal arrow diagram represents a closed (i.e., without boundary and compact) connected surface. Surfaces are two-dimensional manifolds. That is, they are topological spaces that are locally indistinguishable from the Euclidean plane.¹⁴

Arrow diagrams represent surfaces as quotient spaces of disks in the plane (all polygons are, in fact, topologically equivalent to circles). In order to specify precisely how to obtain such a quotient space, we can give intrinsic coordinates to each directed edge E_i as follows: 0 at the tail endpoint, 1 at the head endpoint; taking the length of E_i as unit, for each point p on E_i between the endpoints assign t to p, where t (between 0 and 1) is the distance of p from 0 (see Figure 9). Notice that the bottom-left vertex is now labeled 1 but will be labeled 0 if conceived as the starting point of the vertical left edge (all edges will be identified in groups). If E₁ and E₂ have matching labels, gluing the edges is, for all r such that 0 ≤ r ≤ 1, identifying the points of E₁ with the points of E₂ with the same coordinate r.

Figure 9.

How to identify to edges

Given an arrow diagram, for each point p not on an edge, we identify p with itself and nothing else; for each point p on an edge, we identify p with itself and with q, where p and q are on paired edges and have the same coordinate. All the vertices are going to be identified in groups. The relation x ~ y (x is to be identified with y) is an equivalence relation. We thus get a partition of the polygon P into a set of ~ equivalence classes – that is, sets of the form {y ∈ P: x ~ y}.

These ~ equivalence classes are the set-theoretic representatives of points of the surface Σ represented by the polygon P. A topological surface Σ is a set S of points together with a set of subsets of S, which are the open sets of the surface. Which subsets of S are the open sets of Σ? Let g be the function that sends each point p of the polygon P to the ~ equivalence class to which it belongs – that is, g(p) = {y ∈ P: p ~ y}. A subset U of S is open in Σ if and only if g^-1(U) is open in P (where a set of points of P is open in P if and only if it is the intersection of P with an open set in the Euclidean plane in which P is embedded). This is the standard definition of quotient spaces.

The two digons B₁ and B₂ in Figure 7 represent the sphere and the projective plane, which is a non-orientable surface (and therefore cannot be embedded in space). The diagrams in the first line of Figure 8 represent the torus (that is, a closed orientable surface of genus 1) and the Klein bottle, which is a non-orientable closed surface. The diagrams in the second line of Figure 8 can be reduced to B₂ and B₁ by gluing the left and top arrows together.

The last thing we have to show to establish the soundness of the proof passage of Theorem 1 is that the cutting and pasting operation leaves the represented surface invariant. To cut along a segment E running between two vertices, say v1 and v2, is (i) to give E a direction, say from v1 to v2, and (ii) to produce two new diagrams from the parts of the cut diagram on either side of E, which (a) preserves the edges of the pre-cut diagram with the new diagrams’ labels and orientations and (b) gives to each new diagram a new edge (created by the cut) which is a copy of E, each directed to its copy of vertex v2 (see Figure 10).

Figure 10.

Cutting along an internal segment

We can then glue the two diagrams along another pair of matching edges, in this case O1 and O2 (see Figure 11). Doing so, we would obtain the same surface as the one obtained by gluing back directly E1 and E2. This is a consequence of the fact that the order in which we glue the matching edges does not matter.

Figure 11.

Gluing along an edge

Let me now turn to another operation. Surfaces admit the operation of connected sum. For example, performing the connected sum of two tori, we obtain a double torus. Intuitively, starting with the two surfaces, we subtract from each of them an open disk and then glue along the newly formed boundary circles (see Figure 12).

Figure 12.

The connected sum of two tori

This operation corresponds to combining two arrow diagrams together. We insert one diagram on a vertex of the other. In Figure 13 we see how to connect two diagrams representing the torus – we imagine opening up the first diagram at the bottom right and the second diagram at the top left and then joining the two together to get the diagram shown in Figure 14.

Figure 13.

Summing two diagrams

Figure 14.

The diagram of a double torus

The system formed by arrow diagrams is both sound and complete with respect to its interpretation. Each well-formed diagram corresponds to a closed connected surface, and each closed connected surface corresponds to a well-formed diagram. That is, a topological space is such a surface if and only if it can be represented as a well-formed diagram. Performing the cutting and pasting manipulation and the manipulation corresponding to the connected sum, we always obtain other well-formed diagrams representing surfaces.

2.3 Diagrammatic Notational Systems

We saw that sequences of diagrams can be parts of proofs. This is because we can mentally manipulate them in ways that correspond to specific mathematical operations. These manipulations are cognitively simple and can be performed reliably by mathematicians. Crucially, arrow diagrams do not pose the risk of overgeneralization that is typical of geometric diagrams. Overgeneralization arises when a specific diagram is used to obtain general results. For instance, when we sketch a triangle to prove a result about all triangles, how can we be sure that we are not relying on features of our particular representations that are idiosyncratic to it? With arrow diagrams we do not have this problem because they form a sound system with respect to their interpretation. No visible variable features of drawings of an arrow diagram, apart from the number of edges and their pairing and directions, are topologically relevant. Additional variations, such as variations of internal angles and of length ratios of diameters, are geometrically relevant – they do not matter for arrow diagrams, but they are relevant to geometric diagrams.¹⁵ Hence, we are more liable to make a false geometrical generalization (than a false topological generalization) by relying unwittingly on a visible feature of a diagram that is not shared by everything in the range of generalization.

Second, arrow diagrams can be easily generalized. We saw that we can represent non-orientable surfaces – that is, surfaces that cannot be embedded in space. For example, the second diagram in Figure 8 represents the famous Klein bottle.¹⁶ Furthermore, arrow diagrams representing surfaces generalize to solids. Henri Poincaré (1900) explains how to obtain three-dimensional spaces by identifying the faces of solid polyhedra. For example, the Poincaré homology sphere (i.e., a three-manifold whose homology groups are the same as the sphere, but it is topologically different from it) can be obtained by gluing the opposite sides of a solid dodecahedron as in Figure 15.

Figure 15.

A diagram representing the Poincaré homology sphere

This is significant. Three-dimensional spaces cannot, in fact, be represented by something analogous to the sketches of surfaces; in order to see a three-dimensional space from the outside we would have to represent it in a four-dimensional space. The three-dimensional analogs of arrow diagrams for surfaces, however, are not so well behaved. When we consider the three-dimensional spaces resulting from gluing the faces of polyhedra, these are not necessarily manifolds because they may have singular points.

Let us return to arrow diagrams for surfaces. As we have seen, the fact that arrow diagrams can enter into the inferential structure of proofs can be explained by two facts: (i) they satisfy well-formedness conditions, and (ii) they are subject to precise rules of manipulation. In this respect, diagrammatic systems are on all fours with other mathematical notational systems.¹⁷ It will be easier to think of physical instantiations of mathematical notations. We should keep in mind, however, that we can also imagine diagrams and other representations without writing them down. But what are mathematical notations? I will not answer this question in all its generality. Let me just remark that they are formed by elements, in our case, inscribed in a material medium that can be combined in different ways. Not all perceptual features of such elements carry mathematical content; as a result, they are not all relevant for their interpretation.¹⁸ It is helpful, then, to distinguish between perceptual features that are constitutive from perceptual features that are merely enabling. For example, generally the color or width of the lines used in arrow diagrams or linear algebraic notations is merely enabling. In those notations, they cannot carry mathematical content. However, in the proof of the classification theorem for surfaces, the zigzagging has a specific mathematical meaning and is thus constitutive. In the statement of the theorem, colors are also constitutive since they track the pairs of sides to be glued together – although the same information is also conveyed by the labels. Crucially, the syntax of a notation cannot be altered by changing only enabling attributes. The partition of perceptual features of a notation into constitutive and enabling is given by the interpretation, which also determines which manipulations are mathematically meaningful.¹⁹

Arrow diagrams are an example of diagrams that are not merely a subjective representation but form a mathematical notational system. But further doubts concerning the legitimacy of diagrammatic systems in proofs could be raised. Problems might arise, for instance, if some of the constitutive features of a diagram were too difficult to identify or reproduce reliably. As I have argued elsewhere (De Toffoli 2022), notations have to satisfy three basic constraints:

1. a notation should be cognitively accessible: its constitutive perceptual features should be clearly identified, persistent, and stable;

2. a notation should be reproducible: it should be possible for an average practitioner to copy its constitutive perceptual features with relative ease and reliability, possibly with the aid of different tools such as a straightedge and/or computer; and

3. a notation should support calculations and/or inferences: it should be possible for an average practitioner to perform reasonably simple manipulations corresponding to mathematical operations.

In most cases of non-diagrammatic notations, such constraints are met as a matter of course. These constraints come in degrees. However, since we, as humans, share cognitive and perceptual limitations, there is a threshold that all notations have to meet in order to count as notations at all. The fact that such constraints have often been left implicit may have contributed to skepticism vis-à-vis the use of diagrams in proofs.

As Kenneth Manders (2008) has discussed at length with respect to Euclidean geometry, if exact metric properties had been constitutive features of Euclidean diagrams, then ancient Greek geometric practice would not have gotten very far. The resulting representations would not have been shareable and reproducible. That is why such exact metric features were only available from the text.²⁰

The fact that arrow diagrams form a genuine notation is reflected in the fact that they present an algebraic interpretation.²¹ We can label the edges of an arrow diagram in Figure 5 and code it with the sequence, ${a_1{b}_{1}a}_1^{-1} b_1^{-1}{a_2{b}_{2}a}_2^{-1} b_2^{-1}\dots$This can be useful, but transforming the diagrams into such algebraic words would obscure their topological meaning. The fact that arrow diagrams can be easily coded might give the impression that, after all, diagrams are mere abbreviations of other forms of display. But this is incorrect. First, the two proofs include diagrams and not codes. Eliminating the diagrams without substituting them with other representations would leave the arguments incomplete. That is, in those proofs, diagrams are not redundant. Second, although it is true that diagrams might be replaced by algebraic codes, it is far from obvious that such replacement would preserve the original proof. Before proceeding to argue for this last claim (Section 3), it will be fruitful to consider another example of how diagrams are used in mathematics – one that will highlight to an even greater degree the similarities between diagrammatic and non-diagrammatic mathematical notations.

2.4 Algebraic Diagrams

In the mid-twentieth century, after a long exile, diagrams began emerging once again in mathematical papers. This is because new types of diagrams were introduced in the field of homological algebra, namely commutative diagrams. I present an instance of a particular technique used extensively in this field: diagram chasing.²² The mathematical technicalities involved in explaining the technique need not scare the reader off. What is important in what follows is not so much to have a firm grip on each mathematical detail but rather to form an idea of how these diagrams are used in proofs.

Roughly, we begin with a commutative diagram made of nodes and arrows, such as the one in Figure 16, in which the nodes are abelian groups, and the arrows are group homomorphisms.²³

Figure 16.

A commutative diagram

The fact that a diagram is commutative means that if we want to connect two nodes, it does not matter which arrows we follow: any two paths leading from the same starting point to the same endpoint are equivalent – in other words, the journey is precisely not the goal. For example, if we wish to connect $M’$ to $N$ in Figure 16, it makes no difference if we go right and down following arrows $f_{M\text{'}}$ and $d$ or down and right following arrows $d’$ and $f_{N\text{'}}$. Our goal will be to prove certain properties of the maps in a diagram (i.e., the arrows). If we focus on a given node, we can choose a specific element, say $m’$ in $M’$. The technique of diagram chasing consists in moving such an element around the diagram by transforming it following the arrows, for example, moving $m’$ along the arrow $f_{M\text{'}}$ and transforming it into the element $m$ of the group $M$ (where $m=f_{M\text{'}}\left(m’\right)$). That is, we transform a given element belonging to one group in a node of the diagram by applying the homomorphisms corresponding to the arrows.

Each “0” in this diagram denotes a group with its identity element as sole member. This is a customary abuse of notation. In the text “0” denotes the identity of a group; additive notation is used.

We say that a row of a commutative diagram is exact if, in that row, the image of one arrow is the kernel of the next. The kernel of an arrow (i.e., a homomorphism) is the inverse image of 0. That is, it is the set of all the elements that are sent to 0. Saying that the first row of Figure 16 is exact means two things: (1) $\mathrm{I}\mathrm{m} f_{M\text{'}}=\mathrm{K}\mathrm{e}\mathrm{r} g_M$ and (2) $\mathrm{I}\mathrm{m} g_M=\mathrm{M}\mathrm{\text{'}}\mathrm{\text{'}}$, this is because, by definition, the kernel of a homomorphism that sends all elements to 0 is the whole domain of that homomorphism. One last definition is needed. The cokernel of an arrow is the quotient of its codomain by its image. For example, looking again at Figure 16, we have $\mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r} d^{\text{'}}=N^{\text{'}}/ \mathrm{I}\mathrm{m} d^{\text{'}}$.

I now present a passage of the famous snake lemma (named after the shape of the diagram it involves). In an introductory book on homological algebra, Charles A. Weibel writes, “We will not print the proof of the Snake lemma in these notes, because it is best done visually. In fact, a clear proof is given by Jill Clayburgh at the beginning of the movie ‘It’s My Turn’” (1994, 11).²⁴ Although I certainly agree that it is much easier to convey how the diagram chasing technique works with a live performance, the best I can do here is report a short passage of the proof.

SNAKE LEMMA. Given the commutative diagram of abelian groups in Figure 17 in which the rows are exact, we can define the map:

$\delta :\text{ Ker }{d}^{″}\to \text{Coker }d\prime$

with $\delta \left(z^{″}\right)=h\circ f_{N\prime }^{-1}\circ d\circ g_M^{-1}$, where $h:N^{\text{'}}\to \mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r} d\text{'}$ maps $n\text{'}$ to $n\text{'}+\mathrm{I}\mathrm{m}\mathrm{ }d\text{'}$. And we get the following exact sequence:

$\mathrm{K}\mathrm{e}\mathrm{r}\left(d^{\text{'}}\right)\to \mathrm{K}\mathrm{e}\mathrm{r}\left(d\right)\to \mathrm{K}\mathrm{e}\mathrm{r}\left(d^{\text{'}\text{'}}\right)\to \mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}\left(d^{\text{'}}\right)\to \mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}\left(d\right)\to \mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}\left(d^{\text{'}\text{'}}\right)$

Figure 17.

The snake lemma

The lemma is proved by diagram chasing. Each row is exact: the image of one arrow is the kernel of the next – e.g., $\mathrm{I}\mathrm{m} f_{M\text{'}}=\mathrm{K}\mathrm{e}\mathrm{r} g_M$. All rectangles commute. Figure 18 is an extended diagram in which the kernels and the cokernels of the vertical maps are displayed. In this extended diagram, $g_{*}:\mathrm{K}\mathrm{e}\mathrm{r} d\to \mathrm{K}\mathrm{e}\mathrm{r} d^{\text{'}\text{'}}$ is just $g_M$ restricted to $\mathrm{K}\mathrm{e}\mathrm{r} d$ (that is, it is the map induced by $g_M\mathrm{ }$on the kernels). The winding dashed arrow from $\mathrm{K}\mathrm{e}\mathrm{r} d^{\text{'}\text{'}}$ to $\mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r} d^{\text{'}}$ represents the intended function $\delta$. The downward arrows from the kernels are inclusions ($\mathrm{K}\mathrm{e}\mathrm{r} d^{\text{'}}$ is the set of $m\in M’$ that $d^{\text{'}}$ maps to $0_N$; similarly for $d$and $d^{\text{'}\text{'}}$). $\mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r} d^{\text{'}}=N^{\text{'}}/ \mathrm{I}\mathrm{m} d^{\text{'}}$is the group of cosets of $\mathrm{I}\mathrm{m} d^{\text{'}}$ in $N^{\text{'}}$($\mathrm{I}\mathrm{m} d\text{'} is d \text{'}\left[M\text{'}\right])$. The downward arrow to the cokernel of $d\text{'}$ maps $n\text{'}$ to the coset $n^{\text{'}}+\mathrm{I}\mathrm{m} d\text{'}$; similarly for $d$ and $d\text{'}\text{'}$. The middle two rows are exact, and all rectangles commute.

Figure 18.

Extended diagram of the snake lemma

Following Serge Lang (2002, 158), I report only one step of the proof. However, I will prove this step in a much more detailed way, so as to make the argument intelligible to anyone with knowledge of basic group theory and to give some indication (in square brackets) of how it involves a chase around the diagram. Such information would normally be conveyed easily in a live lecture.

We define the homomorphism $\delta : \mathrm{K}\mathrm{e}\mathrm{r} d^{\text{'}\text{'}}\to \mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r} d\text{'}$ in a roundabout way and show that

This has to be the case because of the exactness of the long exact sequence in the statement of the lemma (in fact, the strongest relation$\mathrm{ }\mathrm{K}\mathrm{e}\mathrm{r} \delta =\mathrm{I}\mathrm{m} g_{*}$ must hold). Facts from basic group theory are assumed without mention. Diagram chasing moves are indicated by comments in square brackets.

ONE STEP OF THE PROOF

Let $z\text{'}\text{'}$ be any element of $\mathrm{K}\mathrm{e}\mathrm{r} d\text{'}\text{'}$. By exactness $g_M:M\to M\text{'}\text{'}$ is surjective (as the final arrow $M^{\text{'}\text{'}}\to 0$ has kernel the whole of $M^{\text{'}\text{'}}$). So, $\mathrm{I}\mathrm{m} g_M= M\text{'}\text{'}.$ Now, $z^{\text{'}\text{'}}\in \mathrm{K}\mathrm{e}\mathrm{r} d^{\text{'}\text{'}}\subseteq M^{\text{'}\text{'}}$ (given). So, $z^{\text{'}\text{'}}=g_M\left(z\right)$ for some $z\in M.$

Now [A] we choose one such $z\in M$. [We move backward along $g_M$ transforming $z\text{'}\text{'}$ into $z$; see Figure 19.] As $z^{\text{'}\text{'}}\in \mathrm{K}\mathrm{e}\mathrm{r} d^{\text{'}\text{'}}, 0=d\text{'}\text{'}\left(z\text{'}\text{'}\right)=d\text{'}\text{'}\left(g_M \right(z\left)\right)=g_N \left(d\right(z\left)\right)$ by commutativity (right middle rectangle), so $d\left(z\right)\in \mathrm{K}\mathrm{e}\mathrm{r} g_N=\mathrm{I}\mathrm{m}\mathrm{ }{f}_{N\text{'}}$ by exactness. So, [B] for a unique $z^{\text{'}}\in N^{\text{'}}, d\left(z\right)=f_{N\text{'}}\left(z\text{'}\right)$. It is unique because $f_{N\text{'}}$ is injective (as the initial arrow $0\to N\text{'}$ has image $\left\{0\right\}=\mathrm{K}\mathrm{e}\mathrm{r} f_{N\text{'}}$ by exactness). [We move down along $d$ turning $z$ into $d\left(z\right)$ and then left.]

Figure 19.

Diagram chasing

Now “let” $\delta \left(z\text{'}\text{'}\right)=z\text{'}+\mathrm{I}\mathrm{m} d\text{'}$. Why the scare quotes? Because $z\text{'}$ is defined in terms of $z$, where $z$ is just one choice from $g_M^{-1} \left(z\text{'}\text{'}\right)$ [see A]. So, we need to show that a different choice from $g_M^{-1} \left(z\text{'}\text{'}\right)$, say $z_0$, would make no difference – that is, that $z\text{'}+\mathrm{I}\mathrm{m}\mathrm{ }d\text{'} = z_0^{\text{'}}+\mathrm{I}\mathrm{m}\mathrm{ }d\text{'}$. This holds and we can remove the scare quotes. For brevity we skip the argument.²⁵ We also skip the proof that $\delta$ is a homomorphism.

Suppose now $z^{\text{'}\text{'}}\in \mathrm{K}\mathrm{e}\mathrm{r} \delta$, then $\delta \left(z\text{'}\text{'}\right)= 0$ in $\mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r} d\text{'}=N^{\text{'}}/ \mathrm{I}\mathrm{m}\mathrm{ }d\text{'}$. So $z\text{'}+ \mathrm{I}\mathrm{m} d\text{'}=\delta \left(z\text{'}\text{'}\right)=\mathrm{I}\mathrm{m} d\text{'}.$ Hence $z^{\text{'}}\in \mathrm{I}\mathrm{m}\mathrm{ }d\text{'}$; that is, [C] $z\text{'}=d\text{'}\left(u\text{'}\right)$, for some $u\text{'}$ in $M\text{'}$. Choose one such $u\text{'}$ – it does not matter which.

By commutativity of the central left-hand square, we get $d\left(f_{M^{\mathrm{\text{'}}}}\left(u^{\mathrm{\text{'}}}\right)\right)=f_{N^{\mathrm{\text{'}}}}\left(d^{\mathrm{\text{'}}}\left(u^{\mathrm{\text{'}}}\right)\right).\mathrm{ }$ By [B] and [C], we also get $f_{N^{\mathrm{\text{'}}}}\left(d^{\mathrm{\text{'}}}\left(u^{\mathrm{\text{'}}}\right)\right)=\mathrm{ }{f}_{N\mathrm{\text{'}}}\mathrm{ }\left(z^{\mathrm{\text{'}}}\right)=d\left(z\right)$. Therefore, $d\left(f_{M^{\mathrm{\text{'}}}}\left(u^{\mathrm{\text{'}}}\right)\right)=d\left(z\right).$ It follows that $0=d\left(z\right)-d\left(f_{M^{\mathrm{\text{'}}}}\right(u\mathrm{\text{'}}\left)\right)=d(z-f_{M^{\mathrm{\text{'}}}}(u\mathrm{\text{'}}\left)\right)$. So, $z-f_{M^{\mathrm{\text{'}}}}\left(u\mathrm{\text{'}}\right)\in \mathrm{K}\mathrm{e}\mathrm{r}\mathrm{ }d$. [We can now move right with $g_{\mathrm{*}}.$]

Recall that $g_{\mathrm{*}}$ is $g_M\mathrm{ }$restricted to$\mathrm{ }\mathrm{K}\mathrm{e}\mathrm{r}\mathrm{ }d.\mathrm{ }\mathrm{S}\mathrm{o}\mathrm{ }{g}_{\mathrm{*}}(z-f_{M^{\mathrm{\text{'}}}}(u\mathrm{\text{'}}\left)\right)=g_M(z-f_{M^{\mathrm{\text{'}}}}(u\mathrm{\text{'}}\left)\right)=g_M\left(z\right)-g_M\left(f_{M^{\mathrm{\text{'}}}}\right(u\mathrm{\text{'}}\left)\right).$ Moreover, $f_{M\mathrm{\text{'}}}\left(u^{\mathrm{\text{'}}}\right)\in \mathrm{I}\mathrm{m}\mathrm{ }{f}_{M^{\mathrm{\text{'}}}}=\mathrm{K}\mathrm{e}\mathrm{r}\mathrm{ }{g}_M$ by exactness; so $g_M\left(f_{M^{\mathrm{\text{'}}}}\right(u\mathrm{\text{'}}\left)\right)=0$. Therefore, $g_{\mathrm{*}}(z-f_{M^{\mathrm{\text{'}}}}(u\mathrm{\text{'}}\left)\right)=g_M\left(z\right)$ [previous two lines], $=\mathrm{ }z\mathrm{\text{'}}\mathrm{\text{'}}$ [see A]. So $z^{\mathrm{\text{'}}\mathrm{\text{'}}}\in \mathrm{I}\mathrm{m}\mathrm{ }{g}_{\mathrm{*}}$. As $z\mathrm{\text{'}}\mathrm{\text{'}}$ is an arbitrary member of $\mathrm{K}\mathrm{e}\mathrm{r}\mathrm{ }\delta$, $\mathrm{K}\mathrm{e}\mathrm{r}\mathrm{ }\delta \subseteq \mathrm{I}\mathrm{m}\mathrm{ }{g}_{\mathrm{*}}$.²⁶

QED

As mentioned before, the presentation above is much longer than one we would find in a mathematics textbook or journal article. Many of the moves presented here as symbol manipulations are recognized immediately by practitioners from looking at the diagram, such as the identities established by commutativity of rectangles and exactness of rows.

Any presentation of this proof will make an appeal to exactness and to commutativity, and when such appeals are made, one needs to attend to the diagram to check that the identities claimed are in fact justified by those appeals. This point reveals that the visual thinking supported by the diagram involved in following the argument plays a much larger part than might appear, especially in relation to the expansive text. Moreover, visual thinking, where this includes both visually perceiving a diagram and visually imagining moves around the diagram, is essential to follow this argument.

What is not revealed by following a diagram chasing argument on its own is its cognitive efficiency relative to the mountain of pure symbol manipulation involved in following a diagram-free counterpart of the argument. Of course, one can follow a diagram-free counterpart stepwise, given sufficient indication of the prior information that is drawn on when inferring a new line, but one may still be in the dark about how individual steps contribute to larger sections of the argument and how these sections fit together, because the torchlight of attention tends to be focused narrowly on the individual steps. It is much easier to keep track of what is going on when following the diagram chasing argument, because attention to the diagram itself induces us to zoom out from individual steps (as well as to check identities based on exactness or commutativity). This relative cognitive efficiency applies not only to following an argument but also to discovering an argument. This becomes clearer as one becomes more practiced. It is true that different mathematicians have different preferences and modes of thinking; nevertheless, eliminating the diagrams from diagram chasing arguments would inevitably introduce some cognitive challenges.²⁷

3.1 Criteria of Identity for Proofs

Proofs are abstract objects. They can be considered equivalence classes of proof presentations. However, it is not always clear which equivalence relation is appropriate. At one extreme, we could consider all proofs of the same result to be equivalent. But this coarse-grained way of individuating proofs would be inappropriate to study mathematical practice. For instance, the common practice of proving the same result in multiple ways would be unintelligible. Therefore, we should look for a more fine-grained criterion. As Marcus Giaquinto suggests, proof presentations containing the same “central idea” are generally considered to be equivalent, but

if one’s main concern is with what is involved in thinking through a proof, its central idea is not enough to individuate it: the overall structure, the sequence of steps and perhaps other factors affecting the cognitive processes involved will be relevant.

(2008, 24, emphasis added)

There might be no fact of the matter about whether two presentations present the same proof or different proofs. Timothy Gowers (2007), a Fields medalist, admits in his popular blog that it was not only difficult to sharply characterize what it means for a proof to satisfy informal characterizations such as being explanatory, but “even the seemingly more basic question, ‘When are two proofs the same?’ was pretty hard to answer satisfactorily.” It is plausible to think that there are no context-independent identity conditions for proofs.³⁰

Criteria of identity for proofs are, in fact, determined only in a specific context of investigation.³¹ More precisely, what has to be established to determine the criteria of individuation of a proof is the main concern or purpose for which individuation is required.³² For example, in one context, rigor may be the major concern; in that context, a presentation that makes totally explicit the assumptions, definitions, and logical steps on which the argument depends presents a distinct proof from an informal exposition of the first.³³ But if the major concern is not about rigor but about purity,³⁴ hence about what mathematical concepts are involved, the degree of informality may count as a mere matter of presentation, and so a fully explicit presentation and an informal exposition could present the same proof.³⁵

In the present context of investigation, the main concern is with the epistemic and cognitive benefits and drawbacks for someone working through a proof. In particular, understanding the argument, in the sense of having a grasp of what motivates the major lines of argument, so that the large-scale structure of the argument comes into view, is a major epistemic benefit. And cognitive efficiency, in the sense of the ease of working through the argument, is another significant benefit. For example, it works against efficiency if, in order to see why steps are valid, one has to hold in mind a heavy load of formulas on which later steps may depend or to search back frequently to find the immediate premises on which steps depend. Therefore, in this context, if one proof presentation is significantly better than another with regard to cognitive-epistemic pros and cons for someone working through them, the proofs presented are distinct proofs. This holds even if one of the presentations is obtained from the other by replacing all diagrams with non-diagrammatic encodings.

Let us consider the cases of arrow diagrams for surfaces discussed above. Imagine substituting all diagrams with algebraic codes in the proof of the classification theorem for surfaces presented in Massey (1991), a step of which is reported in Section 2.1. We can perform such a substitution to represent the diagrammatic moves shown in Figure 6 – which I reinserted here as Figure 20 for ease of comparison.

Figure 20.

A step of a proof involving arrow diagrams for surfaces

I will code the diagrams starting with their left edge and using Z as a convention for the zigzag notation that, as we saw above, is used to introduce generality into the diagrammatic representations. Start with the surface aZbZa^-1Zb^-1Z; now cut it so as to separate the surface ZbZ with c, a new edge (this is the first diagram), and glue the two new surfaces along b. We obtain Zac^-1a^-1Zc (second diagram). Now, cut as to separate Za (third diagram) introducing the new edge d. We get dc^-1d^-1Zc, which is equivalent to cdc^-1d^-1Z – which is part of the general pattern for orientable surfaces.

This brief presentation with algebraic words does not involve any diagram, but, as a matter of fact, in order to go through it and recognize it as valid, a mathematician would generally recreate (in the paper or mentally) arrow diagrams similar to the ones in Figure 20. There is, however, an alternative. We can spell out rules of manipulation of the algebraic codes themselves that correspond to the rules of manipulation of the diagrams. After listing such rules, we could check the correctness of the diagram-free proof without having to invoke any diagram whatsoever. But in this case, the diagram-free proof presentation does not present the same proof as the diagrammatic proof presentation. This is because the two presentations differ with regard to epistemic and cognitive benefits and drawbacks. The diagrammatic proof presentation requires the reader to visualize specific spatial manipulations and recognize their validity. The diagram-free proof does no such thing. It is not only much easier to go through the diagrammatic argument, but it also allows us to grasp the proof more vividly. Moreover, the very subject matter of the two presentations differs: one is about surfaces, represented with arrow diagrams in an intuitive way, and the other is about algebraic words, which represent surfaces only indirectly. Arrow diagrams are needed to transition from algebraic words to surfaces since the words are codes for the diagrams and not for the surfaces themselves.

One could concede that this is true but downplay the importance of such a result by pointing at the fact that, after all, if we prove things about geometric or topological objects, it is not surprising that proofs involving geometric or topological representations would stand out compared to ones that do not involve them. There is a twofold answer to this complaint. First, the matter is not whether a result is or is not surprising, but rather whether it is true. Our question was: Can diagrams be essential to proofs? The example of the classification theorem for mathematical diagrams shows that the affirmative response is correct. Second, it is also possible to use the case of commutative diagrams, which are rarely about geometric or topological objects, to make the same point. Let us do that now.

The step of the proof of the snake lemma I presented invokes commutative diagrams. Once again, if we convert the proof into a diagram-free counterpart, a mathematician would generally reconstruct for herself the diagram in order to go through it.³⁶ Without the diagrams it is much, much harder to grasp the large-scale structure of the proof and what motivated the various steps. The diagrams represent in a compact and surveyable way the algebraic situation. They allow mathematicians to identify viable proof strategies. Moreover, also for small-scale grasping, diagrams play a key role. They allow the reader to have a place to return to in order to check whether the algebraic manipulations are correct. Are those the right indices? Is it n or n’? What is the image of x under f? That the particular reasoning at play would change without the diagrams is also supported by the fact that, in jargon, it is common to refer to diagram chasing as a mathematical technique and as the hallmark of certain proofs. We need a diagram to perform the technique of diagram chasing!