## 1. Introduction


Visual thinking is a feature of mathematical practice across many
subject areas and at many levels. It is so pervasive that the question
naturally arises: does visual thinking in mathematics have any
epistemically significant roles? A positive answer begets further
questions. Can we rationally arrive at a belief with the generality
and necessity characteristic of mathematical theorems by attending to
specific diagrams or images? If visual thinking contributes to warrant
for believing a mathematical conclusion, must the outcome be an
empirical belief? How, if at all can visual thinking contribute to
understanding abstract mathematical subject matter?


Visual thinking includes thinking with external visual
representations (e.g., diagrams, symbol arrays, kinematic computer
images) *and* thinking with internal visual imagery; often the
two are used in combination, as when we are required to visually
imagine a certain spatial transformation of an object represented by a
diagram on paper or on screen. Almost always (and perhaps always)
visual thinking in mathematics is used in conjunction with non-visual
thinking. Possible epistemic roles include contributions to evidence,
proof, discovery, understanding and grasp of concepts. The kinds and
the uses of visual thinking in mathematics are numerous and diverse.
This entry will deal with some of the topics in this area that have
received attention and omit others. Among the omissions is the
possible explanatory role of visual representations in
mathematics. The topic of explanation within pure mathematics is
tricky and best dealt with separately; for this an excellent starting
place is the entry on
 explanation in mathematics
 (Mancosu 2011). Two other omissions are the
development of logic diagrams (Euler, Venn, Pierce and Shin) and the
nature and use of geometric diagrams in Euclid's
*Elements*, both of which are well treated in the entry
 diagrams (Shin et al. 2013). The
focus here is on visual thinking generally, which includes thinking
with symbol arrays as well as with diagrams; there will be no attempt
here to formulate a criterion for distinguishing between symbolic and
diagrammatic thinking. However, the use of visual thinking in proving
and in various kinds of discovery will be covered in what
follows. Discussions of some related questions and some studies of
historical cases not considered here are to be found in the collection
*Diagrams in Mathematics: History and Philosophy* (Mumma and
Panza 2012).


## 2. Historical Background


"Mathematics can achieve nothing by concepts alone but
hastens at once to intuition" wrote Kant (1781/9: A715/B743),
before describing the geometrical construction in Euclid's proof
of the angle sum theorem (Euclid, Book 1, proposition 32). In a review
of 1816 Gauss echoes Kant:




> 
> anybody who is acquainted with the essence of geometry knows that [the
> logical principles of identity and contradiction] are able to
> accomplish nothing by themselves, and that they put forth sterile
> blossoms unless the fertile living intuition of the object itself
> prevails everywhere. (Ewald 1996 [Vol. 1]: 300)
> 



The word "intuition" here translates the German
"*Anschauung*", a word which applies to visual
imagination and perception, though it also has more general uses.


By the late 19th century a different view had emerged,
at least in foundational areas. In a celebrated text giving the first
rigorous axiomatization of projective geometry, Pasch wrote:
"the theorem is only truly demonstrated if the proof is
completely independent of the figure" (Pasch 1882), a view
expressed also by Hilbert in writing on the foundations of geometry
(Hilbert 1894). A negative attitude to visual thinking was not
confined to geometry. Dedekind, for example, wrote of an overpowering
feeling of dissatisfaction with appeal to geometric intuitions in
basic infinitesimal analysis (Dedekind 1872, Introduction). The
grounds were felt to be uncertain, the concepts employed vague and
unclear. When such concepts were replaced by precisely defined
alternatives without allusions to space, time or motion, our intuitive
expectations turned out to be unreliable (Hahn 1933).


In some quarters this view turned into a general disdain for visual
thinking in mathematics: "In the best books" Russell
pronounced "there are no figures at all" (Russell 1901).
Although this attitude was opposed by a few mathematicians, notably
Klein (1893), others took it to heart. Landau, for example, wrote a
calculus textbook without a single diagram (Landau 1934). But the
predominant view was not so extreme: thinking in terms of figures was
valued as a means of facilitating grasp of formulae and linguistic
text, but only reasoning expressed by means of formulae and text could
bear any epistemological weight.


By the late 20th century the mood had swung back in
favour of visualization: Mancosu (2005) provides an excellent survey.
Some books advertise their defiance of anti-visual puritanism in their
titles, for example *Visual Geometry and Topology* (Fomenko
1994) and *Visual Complex Analysis* (Needham 1997); mathematics
educators turn their attention to pedagogical uses of visualization
(Zimmerman and Cunningham 1991); the use of computer-generated imagery
begins to bear fruit at research level (Hoffman 1987; Palais 1999),
and diagrams find their way into research papers in abstract fields:
see for example the papers on higher dimensional category theory by
Joyal et al. (1996), Leinster (2004) and Lauda (2005, Other Internet
Resources). But attitudes
to the epistemology of visual thinking remain mixed. The discussion is
mostly concerned with the role of diagrams in proofs.


## 3. Visual thinking and proof


In some cases, it is claimed, a picture alone is a proof (Brown
1999: ch. 3). But that view is rare. Even the editor of *Proofs
without Words: Exercises in Visual Thinking*, writes "Of
course, 'proofs without words' are not really
proofs" (Nelsen 1993: vi). Expressions of the other extreme are
rare but can be found: 



>  [the diagram] has no proper place in the proof as
> such. For the proof is a syntactic object consisting only of sentences
> arranged in a finite and inspectable array. (Tennant 1986)
> 
> 
> 


Between the extremes we find the view that, even if no picture
alone is a proof, visual representations can have a non-superfluous
role in reasoning that constitutes a proof. (This is not to deny that
there may be another proof of the same conclusion which does not involve
any visual representation.) Geometric diagrams, graphs and maps, all
carry information. Taking valid deductive reasoning to be the reliable
extraction of information from information already obtained, Barwise
and Etchemendy (1996:4) pose the following question: Why cannot the
representations composing a proof be visual as well as linguistic? The
sole reason for denying this role to visual representations is the
thought that, with the possible exception of very restricted cases,
visual thinking is unreliable, hence cannot contribute to proof. Is
that right?


Our concern here is thinking through the steps in a proof, either
for the first time (a first successful attempt to construct a proof)
or following a given proof. Clearly we want to distinguish between
visual thinking which merely accompanies the process of thinking
through the steps in a proof and visual thinking which is essential to
the process. This is not always straightforward as a proof can be
presented in different ways. How different can distinct presentations
be and yet be presentations of the same proof? There is no
context-invariant answer to this. Often mathematicians are happy to
regard two presentations as presenting the same proof if the central
idea is the same in both cases. 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 some other factors affecting the cognitive processes
involved will be relevant.


Once individuation of proofs has been settled, we can distinguish
between replaceable thinking and superfluous thinking, where these
attributions are understood as relative to a given argument or proof.
In the process of thinking through a proof, a given part of the
thinking is *replaceable* if thinking of some other kind could
stand in place of the given part in a process that would count as
thinking through the same proof. A given part of the thinking is
*superfluous* if its excision without replacement would be a
process of thinking through the same proof. Superfluous thinking may
be extremely valuable in facilitating grasp of the proof text and in
enabling one to understand the idea underlying the proof steps; but it
is not necessary for thinking through the proof. 


It is uncontentious that the visual thinking involved in symbol
manipulations, for example in following the "algebraic"
steps of proofs of basic lemmas about groups, can be essential, that
is neither superfluous nor replaceable. The worry is about thinking
visually with diagrams, where "diagram" is used widely to
include all non-symbolic visual representations. Let us agree that
there can be superfluous diagrammatic thinking in thinking through a
proof. This leaves several possibilities.


* (a) All diagrammatic thinking in a process of thinking through a proof is superfluous.
* (b)Not all diagrammatic thinking in a process of thinking through a proof is superfluous; but if not superfluous it will be replaceable by non-diagrammatic thinking.
* (c)Some diagrammatic thinking in a process of thinking through a proof is neither superfluous nor replaceable by non-diagrammatic thinking.


### 3.1 The reliability question


The negative view stated earlier that diagrams can have no role in
proof entails claim (a). The idea behind (a) is that, because
diagrammatic reasoning is unreliable, if a process of thinking through
an argument contains some non-superfluous diagrammatic thinking, that
process lacks the epistemic security to be a case of thinking through
a proof.


This view, claim (a) in particular, is threatened by cases in which
the reliability of the diagrammatic thinking is demonstrated
non-visually. The clearest kind of example would be provided by a
formal system which has diagrams in place of formulas among its
syntactic objects, and types of inter-diagram transition for inference
rules. Suppose you take in such a formal system and an interpretation
of it, and then think through a proof of the system's soundness
with respect to that interpretation; suppose you then inspect a
sequence of diagrams, checking along the way that it constitutes a
derivation in the system; suppose finally that you recover the
interpretation to reach a conclusion. (The order is unimportant: one
can go through the derivation first and then follow the soundness
proof.) That entire process would constitute thinking through a proof
of the conclusion; and the diagrammatic thinking involved would not be
superfluous.


Shin et al. (2013) report that formal diagrammatic systems of logic
and geometry have been proven to be sound. People have indeed followed
proofs in these systems. That is enough to refute claim (a), the claim
that all diagrammatic thinking in thinking through a proof is
superfluous. For a concrete example, Figure 1
presents a derivation of Euclid's first theorem, that on any
straight line segment an equilateral triangle is constructible, in a
formal diagrammatic system of a part of Euclidean geometry (Miller
2001).



![[a three by three  array of rectangles each containing a diagram. Going left to right then top to bottom, the first has a line segment with each end having a dot.  The second is a circle with a radius drawn and dots on each end of the radius line segment. The third is the same the second except another overlapping circle is drawn using the same radius line segment but with the first circle's center dot now on the perimeter and the first circle's perimeter dot now the center of the second circle, dots are added at the two points the circles intersect.  The fourth diagram is identical to the third except a line segment is drawn from the top intersection dot to the first circle's center dot.  The fifth diagram is like the fourth except a line segment is drawn from the top intersection dot to the center  dot of the second circle.  ...]](fig1.png)
Figure 1



What about Tennant's claim that a proof is "a syntactic
object consisting only of sentences" as opposed to diagrams? A
proof is *never* a syntactic object. A formal derivation on its
own is a syntactic object but not a proof. Without an interpretation
of the language of the formal system the end-formula of the derivation
says nothing; and so nothing is proved. Without a demonstration of the
system's soundness with respect to the interpretation, one may
lack sufficient reason to believe that all derivable conclusions are
true. A formal derivation *plus* an interpretation and
soundness proof can be a proof of the derived conclusion, but that
whole package is not a syntactic object. Moreover, the part of the
proof which really is a syntactic object, the formal derivation, need
not consist solely of sentences; it can consist of diagrams.


With claim (a) disposed of, consider again claim (b) that, while
not all diagrammatic thinking in a process of thinking through a proof
is superfluous, all non-superfluous diagrammatic thinking will be
replaceable by non-diagrammatic thinking in a process of thinking
through that same proof. The visual thinking in following the proof of
Euclid's first theorem using Miller's formal system
consists in going through a sequence of diagrams and at each step
seeing that the next diagram results from a permitted alteration of
the previous diagram. It is clear that in a process that counts as
thinking through *this* proof, the diagrammatic thinking is
neither superfluous nor replaceable by non-diagrammatic thinking. That
knocks out (b), leaving only (c): some thinking that involves a
diagram in thinking through a proof is neither superfluous nor
replaceable by non-diagrammatic thinking (without changing the
proof).


### 3.2 Visual means in non-formal proving


Mathematical practice almost never proceeds by way of formal
systems. Outside the context of formal diagrammatic systems, the use
of diagrams is widely felt to be unreliable. A diagram can be
unfaithful to the described construction: it may represent something
with a property that is ruled out by the description, or without a
property that is demanded by the description. This is exemplified by
diagrams in the famous argument for the proposition that all triangles
are isosceles: the meeting point of an angle bisector and the
perpendicular bisector of the opposite side is represented as falling
inside the triangle, when it has to be outside (Rouse Ball 1939;
Maxwell 1959). Errors of this sort are comparatively rare, usually
avoidable with a modicum of care, and not inherent in the nature of
diagrams; so they do not warrant a general charge of
unreliability.


 The major sort of error is unwarranted generalisation. Typically
diagrams (and other non-verbal visual representations) do not
represent their objects as having a property that is actually ruled
out by the intention or specification of the object to be
represented. But diagrams very frequently do represent their objects
as having properties that, though not ruled out by the specification,
are not demanded by it. Verbal descriptions can be discrete, in that
they supply no more information than is needed. But visual
representations are typically indiscrete, in that they supply too much
detail. This is often unavoidable, because for many properties or
kinds \(F\), a visual representation cannot represent something as
being \(F\) without representing it as being \(F\) *in a particular
way*. Any diagram of a triangle, for instance, must represent it
as having three acute angles or as having just two acute angles, even
if neither property is required by the specification, as would be the
case if the specification were "Let ABC be a triangle". As
a result there is a danger that in using a diagram to reason about an
arbitrary instance of class \(K\), we will unwittingly rely on a
feature represented in the diagram that is not common to all instances
of the class \(K\). Thus the risk of unwarranted generalisation is a
danger inherent in the use of many diagrams.


Indiscretion of diagrams is not confined to geometrical
figures. The dot or pebble diagrams of ancient mathematics used to
convince one of elementary truths of number theory necessarily display
particular numbers of dots, though the truths are general. Here is an
example, used to justify the formula for the \(n\)th
triangular number, i.e., the sum of the first \(n\) positive
integers.



![[a grid of blue dots 5 wide and 7 deep, on the right side is a brace embracing the right column labeled n+1 and on the bottom a brace embracing the bottom row labeled n]](fig2.png)
Figure 2



The conclusion drawn is that the sum of integers from 1 to \(n\) is
\((n \times n+1)/2\) for any positive integer \(n\), but the diagram
presents the case for \(n = 6\). We can perhaps avoid representing a
particular number of dots when we merely imagine a display of the
relevant kind; or if a particular number is represented, our
experience may not make us aware of the number--just as, when one
imagines the sky on a starry night, for no particular number \(k\) are
we aware that exactly \(k\) stars are represented. Even so, there is
likely to be some extra specificity. For example, in imagining an
array of dots of the form just illustrated, one is unlikely to imagine
just two columns of three dots, the rectangular array for \(n =
2\). Typically the subject will be aware of imagining an array with
more than two columns. This entails that an image is likely to have
unintended exclusions. In this case it would exclude the three-by-two
array. An image of a triangle representing all angles as acute would
exclude triangles with an obtuse angle or a right angle. The danger is
that the visual reasoning will not be valid for the cases that are
unintentionally excluded by the visual representation, with the result
that the step to the conclusion is an unwarranted generalisation.


What should we make of this? First, let us note that in a few cases
the image or diagram will not be over-specific. When in geometry all
instances of the relevant class are congruent to one another, for
instance all circles or all squares, the image or diagram will not be
over-specific for a generalisation about that class; so there will be
no unintended exclusions and no danger of unwarranted generalisation.
Here then are possibilities for reliable visual thinking in
proving.


To get clear about the other cases, where there *is* a
danger of over generalizing, it helps to look at generalisation in
ordinary non-visual reasoning. Schematically put, in reasoning about
things of kind \(K\), once we have shown that from certain premisses
it follows that such-and-such a condition is true of arbitrary
instance \(c\), we can validly infer from those same premisses that
that condition is true of all \(K\)s, with the proviso that neither
the condition nor any premiss mentions \(c\). The proviso is required,
because if a premiss or the condition does mention \(c\), the
reasoning may depend on a property of \(c\) that is not shared by all
other \(K\)s and so the generalisation would be unsafe. For a trivial
example consider a step from "\(x = c\)" to
"\(\forall x [x = c]\)".


A question we face is whether, in order to come to *know*
the truth of a conclusion by following an argument involving
generalisation on an arbitrary instance (a.k.a. universal
generalisation, or universal quantifier introduction), the thinking
must include a conscious, explicit check that the proviso is met. It
is clearly not enough that the proviso is in fact met. For in that
case it might just be the thinker's good luck that the proviso
is met; hence the thinker would not know that the generalisation is
valid and so would not have genuinely thought through the proof at
that step.


This leaves two options. The strict option is that without a
conscious, explicit check one has not really thought through the
proof. The relaxed option is that one *can* properly think
through the proof without checking that the proviso is met, but only
if one is sensitive to the potential error and would detect it in
otherwise similar arguments. For then one is not just lucky that the
proviso is met. Being sensitive in this context consists in being
alert to dependence on features of the arbitrary instance not shared
by all members of the class of generalisation, a state produced by a
combination of past experience and current vigilance. Without
compelling reason to prefer one of these options, decisions on what is
to count as proving or following a proof must be conditional. 


How does all this apply to generalizing from visual thinking about
an arbitrary instance? Take the example of the visual route to the
formula for triangular numbers using the diagram of Figure 2. The diagram reveals that the formula holds
for the 6th triangular number. The generalisation to all
triangular numbers is justified only if the visuo-spatial method used
is applicable to the \(n\)th triangular number for all
positive integers \(n\), that is, provided that the method used does
not depend on a property not shared by all positive integers. A
conscious, explicit check that this proviso is met requires making
explicit the method exemplified for 6 and proving that the method is
applicable for all positive integers in place of 6. (For a similar
idea in the context of automating visual arguments, see Jamnik
2001). This is not done in practice when thinking visually, and so if
we accept the strict option for thinking through a proof involving
generalisation, we would have to accept that the visual route to the
formula for triangular numbers does not amount to thinking through a
proof of it; and the same would apply to the familiar visual routes to
other general positive integer formulas, such as that \(n^2 =\) the
sum of the first \(n\) odd numbers.


But what if the strict option for proving by generalisation on an
arbitrary instance is too strict, and the relaxed option is right?
When arriving at the formula in the visual way indicated, one does not
pay attention to the fact that the visual display represents the
situation for the 6th triangular number; it is as if the
mind had somehow extracted a general schema of visual reasoning from
exposure to the particular case, and had then proceeded to reason
schematically, converting a schematic result into a universal
proposition. What is required, on the relaxed option, is sensitivity
to the possibility that the schema is not applicable to all positive
integers; one must be so alert to ways a schema of the given kind can
fall short of universal applicability that if one had been presented
with a schema that did fall short, one would have detected the
failure. 


In the example at hand, the schema of visual reasoning involves at
the start taking a number \(k\) to be represented by a column of \(k\)
dots, thence taking the triangular array of \(n\) columns to represent
the sum of the first \(n\) positive integers, thence taking that array
combined with an inverted copy to make a rectangular array of \(n\)
columns of \(n+1\) dots. For a schema starting this way to be
universally applicable, it must be possible, given any positive
integer \(n\), for the sum of the first \(n\) positive integers to be
represented in the form of a triangular array, so that combined with
an inverted copy one gets a rectangular array. This actually fails at
the extreme case: \(n = 1\). The formula \((n.(n + 1))/2\) holds for
this case; but that is something we know by substituting
"1" for the variable in the formula, not by the visual
method indicated. That method cannot be applied to \(n = 1\), because
a single dot does not form a triangular array, and combined with a
copy it does not form a rectangular array. But we can check that the
method works for all positive integers after the first, using visual
reasoning to assure ourselves that it works for 2 and that if the
method works for \(k\) it works for \(k+1\). Together with this
reflective thinking, the visual thinking sketched earlier constitutes
following a proof of the formula for the \(n\)th triangular
number for all integers \(n > 1\), at least if the relaxed view of
thinking through a proof is correct. Similar conclusions hold in the
case of other "dot" arguments (Giaquinto 1993, 2007:
ch. 8). So in some cases when the visual representation carries
unwanted detail, the danger of over-generalisation in visual reasoning
can be overcome.


But the fact that this is frequently missed by commentators
suggests that the required sensitivity is often absent. Missing an
untypical case is a common hazard in attempts at visual proving. A
well-known example is the proof of Euler's formula \(V - E + F =
2\) for polyhedra by "removing triangles" of a
triangulated planar projection of a polyhedron. One is easily
convinced by the thinking, but only because the polyhedra we normally
think of are convex, while the exceptions are not convex. But it is
also easy to miss a case which is not untypical or extreme when
thinking visually. An example is Cauchy's attempted proof
(Cauchy 1813) of the claim that if a convex polygon is transformed
into another polygon keeping all but one of the sides constant, then
if some or all of the internal angles at the vertices increase, the
remaining side increases, while if some or all of the internal angles
at the vertices decrease, the remaining side decreases. The argument
proceeds by considering what happens when one transforms a polygon by
increasing (or decreasing) angles, angle by angle. But in a trapezoid,
changing a single angle can turn a convex polygon into a concave
polygon, and this invalidates the argument (Lyusternik 1963).


The frequency of such mistakes indicates that visual arguments
(other than symbol manipulations) often lack the transparency required
for proof. Even when a visual argument is in fact sound, its soundness
may not be clear, in which case the argument is not a way of
*proving* the truth of the conclusion, though it may be a way
of discovering it. But this is consistent with the claim that visual
non-symbolic thinking can be (and often is) part of a way of proving
something.


An example from knot theory will substantiate the modal part of
this claim. To present the example, we need some background
information, which will be given with a minimum of technical
detail.


A *knot* is a tame closed non-self-intersecting curve in
Euclidean 3-space.


In other words, knots are just the tame curves in Euclidean 3-space
which are homeomorphic to a circle. The word "tame" here
stands for a property intended to rule out certain pathological cases,
such as curves with infinitely nested knotting. There is more than one
way of making this mathematically precise, but we have no need for
these details. A knot has a specific geometric shape, size and
axis-relative position. Now imagine it to be made of flexible yet
unbreakable yarn that is stretchable and shrinkable, so that it can be
smoothly transformed into other knots without cutting or gluing. Since
our interest in a knot is the nature of its knottedness regardless of
shape, size or axis-relative position, the real focus of interest is
not just the knot but all its possible transforms. A way to think of
this is to imagine a knot transforming continuously, so that every
possible transform is realized at some time. Then the thing of central
interest would be the object that persists over time in varying forms,
with knots strictly so called being the things captured in each
particular freeze frame. Mathematically, we represent the relevant
entity as an equivalence class of knots.


Two knots are *equivalent* iff one can be smoothly deformed into the other by stretching, shrinking, twisting, flipping, repositioning or in any other way that does not involve cutting, gluing or passing one strand through another.


The relevant kind of deformation forbids eliminating a knotted part
by shrinking it down to a point. Again there are mathematically
precise definitions of knot-equivalence. Figure 3 gives diagrams of
equivalent knots, instances of a trefoil.






![[a closed line which goes under, over, under, over, under, over itself forming a shape with three nodes]](fig3a.png)


![[a closed line which goes under, over, under, over, under, over itself but forming a shop closer to a figure 8 inside an oval]](fig3b.png)




(a)


(b)



Figure 3



Diagrams like these are not merely illustrations; they also have an
operational role in knot theory. But not any picture of a knot will do
for this purpose. We need to specify:


 A *knot diagram* is a regular projection of a knot onto a plane which, when there is a crossing, tells us which strand passes over the other.


Regularity here is a combination of conditions. In particular,
regularity entails that not more than two points of the strict knot
project to the same point on the plane, and that two points of the
strict knot project to the same point on the plane only where there is
a crossing. For more on diagrams in knot theory see (De Toffoli and
Giardino 2014).


A major task of knot theory is to find ways of telling whether two
knot diagrams are diagrams of equivalent knots. In particular we will
want to know if a given knot diagram represents a knot equivalent to
an *unknot*, that is, a knot representable by a knot diagram
without crossings.


One way of showing that a knot diagram represents a knot equivalent
to an unknot is to show that the diagram can be transformed into one
without crossings by a sequence of atomic moves, known as Reidemeister
moves. The relevant background fact is Reidemeister's theorem,
which links the visualizable diagrammatic changes to the
mathematically precise definition of knot equivalence: Two knots are
equivalent if and only if there is a finite sequence of Reidemeister
moves taking a knot diagram of one to a knot diagram of the
other. Figure 4 illustrates. Each knot diagram is changed into the
adjacent knot diagram by a Reidemeister move; hence the knot
represented by the leftmost diagram is equivalent to the unknot.






![[a closed line that goes under, under,  under, over, over, over but forming otherwise a shape much like figure 3a]](fig4a.png)


![[a closed line that goes over, under forming a shape much like a loop within a loop]]](fig4b.png)


![[a closed line that forms a distorted loop with no intersections]](fig4c.png)




(a)


(b) 


(c) 



Figure 4



In contrast to these, the knot presented by the left knot diagram
of Figure 3, a trefoil, may seem impossible to
deform into an unknot. And in fact it is. To prove it, we can use a
knot invariant known as colourability. An arc in a knot diagram is a
maximal part between crossings (or the whole thing if there are no
crossings). Colourability is this:


 A knot diagram is *colourable* if and only if each of its arcs can be coloured one of three different colours so that (a) at least two colours are used and (b) at each crossing the three arcs are all coloured the same or all coloured differently.


The reference to colours here is inessential. Colourability is in
fact a specific case of a kind of combinatorial property known as mod
\(p\) labelling (for \(p\) an odd prime). Colourability is a knot
invariant in the sense that if one diagram of a knot is colourable
every diagram of that knot and of any equivalent knot is
colourable. (By Reidemeister's theorem this can be proved by
showing that each Reidemeister move preserves colourability.) A
standard diagram of an unknot, a diagram without crossings, is clearly
not colourable because it has only one arc (the whole thing) and so
two colours cannot be used. So in order to complete proving that the
trefoil is not equivalent to an unknot, we only need prove that our
trefoil diagram is colourable. This can be done visually. Colour each
arc of the knot diagram one of the three colours red, green or blue so
that no two arcs have the same colour (or visualize this). Then do a
visual check of each crossing, to see that at each crossing the three
meeting arcs are all coloured differently. That visual part of the
proof is clearly non-superfluous and non-replaceable (without changing
the proof). Moreover, the soundness of the argument is quite
transparent. So here is a case of a non-formal, non-symbolic visual
way of proving a mathematical truth.


### 3.3 A dispute: diagrams in proofs in analysis.


Where notions involving the infinite are in play, such as many
involving limits, the use of diagrams is famously risky. For this
reason it has been widely thought that, beyond some very simple cases,
arguments in real and complex analysis in which diagrams have a
non-superfluous role are not genuine proofs. Bolzano [1817] expressed
this attitude with regard to the intermediate value theorem for the
real numbers (IVT) before giving a purely analytic proof, arguing that
spatial thinking could not be used to help justify the IVT. James
Robert Brown (1999) takes issue with Bolzano on this point. The IVT
is this:


If \(f\) is a real-valued function of a real variable continuous on the closed interval \([a, b]\) and \(f(a) < c < f(b)\), then for some \(x\) in \((a, b), f(x) = c\).


Brown focuses on the special case when \(c = 0\). As the IVT can be
deduced easily from this special case using the theorem that the
difference of two continuous functions is continuous, there is no loss
of generality here. Alluding to a diagram like Figure 5, Brown (1999)
writes




>  We have a continuous line running from below to above
> the \(x\)-axis. Clearly, it *must* cross that axis in doing
> so. (1999: 26) 
> 
> 


 Later he claims: 



> 
>  Using the picture alone, we can be certain of this result--if
> we can be certain of anything. (1999: 28) 
> 
> 



![[a first quadrant graph, the x-axis labeled near the left with 'a' and near the right with 'b'; the y-axis labeled at the top with 'f(b)', in the middle with 'c' and near the bottom with 'f(a)'.  A dotted horizontal line lines up with the 'c'.  A solid curve starts the intersection of 'f(b)' and 'a', rambles horizontally for a short while before rising above the 'c' dotted line, dips below then rises again and ending at the intersection of 'f(b)' and 'b'. ]](fig5.png)
Figure 5



Bolzano's diagram-free proof of the IVT is an argument from
what later became known as the Dedekind completeness of the real
numbers: every non-empty set of reals bounded above (below) has a
least upper bound (greatest lower bound). The value of Bolzano's
deduction of the IVT from the Dedekind completeness of the reals,
according to Brown, is not that it proves the IVT but that it gives us
confirmation of Dedekind completeness, just as an empirical hypothesis
in empirical science gets confirmed by deducing some consequence of
the hypothesis and observing those consequence to be true. This view
assumes that we already know the IVT to be true by observing a diagram
relevantly like Figure 5.


That assumption is challenged by Giaquinto (2011). Once we
distinguish graphical concepts from associated analytic concepts, the
underlying argument from the diagram is essentially this.


* 1. Any function \(f\) which is \(\varepsilon\textrm{-}\delta\) continuous on \([a, b]\) with \(f (a) < 0 < f (b)\) has a visually continuous graphical curve from below the horizontal line representing the \(x\)-axis to above.
* 2.  Any visually continuous graphical curve from below a horizontal line to above it meets the line at a crossing point.
* 3.  Any function whose graphical curve meets the line representing the \(x\)-axis at a crossing point has a zero value.
* 4.  So, any \(\varepsilon\textrm{-}\delta\) continuous function \(f\) on \([a, b]\) with \(f (a) < 0< f (b)\) has a zero value.


What is inferred from the diagram is premiss 2. Premisses 1 and 3
are assumptions linking analytical with graphical conditions. These
linking assumptions are disputed. With regard to premiss 1 Giaquinto
(2011) argues that there are functions on the reals which meet the
antecedent condition but do not have graphical curves, such as
continuous but nowhere differentiable functions and functions which
oscillate with unbounded frequency e.g., \(f(x) = x \cdot\sin(1/x)\)
for non-zero \(x\) in \([-1, 1]\) and \(f(0) = 0\).


With regard to premiss 3 it is argued that, under the standard
conventions of graphical representation of functions in a Cartesian
co-ordinate frame, the graphical curve for \(x^2 - 2\) in the
rationals is the same as the graphical curve for \(x^2- 2\) in the
reals. This is because every real is a limit point of rationals; so
for every point \(P\) with one or both co-ordinates irrational, there
are points arbitrarily close to \(P\) with both co-ordinates rational;
so no gaps would appear if irrational points were removed from the
curve for \(x^2- 2\) in the reals. But for \(x\) in the rational
interval [0, 2] the function \(x^2- 2\) has no zero value, even though
it has a graphical curve which visually crosses the line representing
the \(x\)-axis. So one cannot read off the existence of a zero of
\(x^2- 2\) on the reals from the diagram; one needs to appeal to some
property of the reals which the rationals lack, such as Dedekind
completeness.


This raises some obvious questions. Do *any* theorems of
analysis have proofs in which diagrams have a non-superfluous role?
Littlewood (1953: 54-5) thought so and gives an example which is
examined in Giaquinto (1994). If so, can we demarcate this class of
theorems by some mathematical feature of their content? Another
question is whether there is a significantly broad class of functions
on the reals for which we could prove an intermediate value theorem
(i.e., restricted to that class).


If there are theorems of analysis provable with diagrams we do not
yet have a mathematical demarcation criterion for them. A natural
place to look would be O-minimal structures on the reals--this
was brought to the author's attention by Ethan Galebach. This is
because of some remarkable theorems about such structures which
exclude all the pathological (hence vision-defying) functions on the
reals (Van den Dries 1998), such as continuous nowhere differentiable
functions and "space-filling" curves i.e., continuous
surjections \(f:(0, 1)\rightarrow(0, 1)^2\). Is the IVT for functions
in an O-minimal structure on the reals provable by visual means?
Certainly one objection to the visual argument for the unrestricted
IVT does not apply when the restriction is in place. This is the
objection that continuous nowhere differentiable functions, having no
graphical curve, provide counterexamples to the premiss that any
\(\varepsilon\textrm{-}\delta\) continuous function \(f\) on \([a,
b]\) with \(f (a) < c < f (b)\) has a visually continuous
graphical curve from below the horizontal line representing \(y = c\)
to above. But the existence of continuous functions with no graphical
curve is not the only objection to the visual argument, contrary to a
claim of Azzouni (2013: 327). There are also counterexamples to the
premiss that any function that *does* have a graphical curve
which visibly crosses the line representing \(y = c\) takes \(c\) as a
value, e.g., the function \(x^2 - 2\) on the rationals with \(c =
0\). So the question of a visual proof of the IVT restricted to
functions in an O-minimal structure on the reals is still open at the
time of writing.


## 4. Visual thinking and discovery


Though philosophical discussion of visual thinking in mathematics
has concentrated on its role in proof, visual thinking may be more
valuable for discovery than proof. Three kinds of discovery important
in mathematical practice are these:


* (1)propositional discovery (discovering, of a proposition, that it is true),
* (2)discovering a proof strategy (or more loosely, getting the idea for a proof of a proposition), and
* (3)discovering a property or kind of mathematical entity.


In the following subsections visual discovery of these kinds will
be discussed and illustrated.


### 4.1 Propositional discovery


To *discover* a truth, as that expression is being used
here, is to come to believe it by one's own lights (as opposed
to reading it or being told) in a way that is reliable and involves no
violation of epistemic rationality (given one's epistemic
state). One can discover a truth without being the first to discover
it (in this context); it is enough that one comes to believe it in an
independent, reliable and rational way. The difference between merely
discovering a truth and proving it is a matter of transparency: for
proving or following a proof the subject must be aware of the way in
which the conclusion is reached and the soundness of that way; this is
not required for discovery.


Sometimes one discovers something by means of visual thinking using
background knowledge, resulting in a cogent argument from which one
could construct a proof. A nice example is a visual argument that any
knot diagram with a finite number of crossings can be turned into a
diagram of an unknot by interchanging the over-strand and under-strand
of some of its crossings (Adams 2001: 58-90). That argument is a
bit too long to present accessibly here. For a short example, here is
a way of discovering that the geometric mean of two positive numbers
is less than or equal to their arithmetic mean (Eddy 1985) using
Figure 6.



![[two circles of differing sizes next to each other and touching at one point, the larger left circle has a vertical diameter line drawn and adjacent, parallel on the left is a double arrow headed line labelled 'a'.  The smaller circle has a similar vertical diameter line with a double arrow headed line labelled 'b' to the right.  The bottom of the diameter lines are connected by a double headed arrow line labeled 'square root of (ab)'. Another line connects the centers of both circles and has a parallel double arrow headed line labeled '(a+b)/2'.  A dashed horizontal line goes horizontally from the center of the smaller circle until it hits the diameter line of the larger circle.  Between this intersection  and the center of the larger circle is a double arrow headed line labeled '(a-b)/2'.]](fig6.png)
Figure 6



Two circles (with diameters \(a\) and \(b\)) meet at a single
point. A line is drawn between their centres through their common
point; its length is \((a + b)/2\), the sum of the two radii. This
line is the hypotenuse of a right angled triangle with one other side
of length \((a - b)/2\), the difference of the radii.
Pythagoras's theorem is used to infer that the remaining side of
the right-angled triangle has length \(\sqrt{(ab)}\).Then visualizing
what happens to the triangle when the diameter of the smaller circle
varies between 0 and the diameter of the larger circle, one infers
that \(0 < \sqrt{(ab)} < (a + b)/2\); then verifying
symbolically that \(\sqrt{(ab)} = (a + b)/2\) when \(a = b\), one
concludes that for positive \(a\) and \(b\), \(\sqrt{(ab)} \le (a +
b)/2\).


This thinking does not constitute a case of proving or following a
proof of the conclusion, because it involves a step which we cannot
*clearly tell* is valid. This is the step of attempting to
visually imagine what would happen when the smaller circle varies in
diameter between 0 and the diameter of the larger circle and inferring
from the resulting experience that the line joining the centres of the
circles will always be longer than the horizontal line from the centre
of the smaller circle to the vertical diameter of the larger circle.
This step *seems* sound (does not lead us into error) and may
*be* sound; but its soundness is opaque. If in fact it is
sound, the whole thinking process is a reliable way of reaching the
conclusion; so in the absence of factors that would make it irrational
to trust the thinking, it would be a way of discovering the conclusion
to be true.


### 4.2 Discovering a proof strategy


In some cases visual thinking inclines one to believe something on
the basis of assumptions suggested by the visual representation that
remain to be justified given the subject's current knowledge. In
such cases there is always the danger that the subject takes the
visual representation to show the correctness of the assumptions and
ends up with an unwarranted belief. In such a case, even if the belief
is true, the subject has not made a discovery, as the means of
belief-acquisition is unreliable. Here is an example using Figure 7
(Montuchi and Page 1988).



![[A first quadrant graph, on the x-axis are marked (2 squareroot(k), 0) and further to the right (j,0).  On the y-axis is marked (0,2(squareroot(k)) and further up, (0,j).  Solid lines connect (0,2(squareroot(k)) to (2(squareroot(k),0)  and (0,j) to (j,0).  A dotted line goes from the origin in a roughly 45 degree angle the point where it intersects the (0,2(squareroot(k)) to (2(squareroot(k),0) line is labeled (squareroot(k),squareroot(k)).  A curve tangent to that point with one end heading up and the other right is labeled 'xy=k'.]](fig7.png)
Figure 7



Using this diagram one can come to think the following about the
real numbers. When for a constant \(k\) the positive values of \(x\)
and \(y\) are constrained to satisfy the equation \(x \cdot y = k\),
the positive values of \(x\) and \(y\) for which \(x + y\) is minimal
are \(x = \sqrt{k} = y\). (Let "#" denote this claim.)


Suppose that one knows the conventions for representing functions
by graphs in a Cartesian co-ordinate system, knows also that the
diagonal represents the function \(y = x\), and that a line segment
with gradient -1 from \((0, b)\) to \((b, 0)\) represents the
function \(x + y = b\). Then looking at the diagram may incline one to
think that for no positive value of \(x\) does the value of \(y\) in
the function \(x\cdot y = k\) fall below the value of \(y\) in \(x + y
= 2\sqrt{k}\), and that these functions coincide just at the
diagonal. From these beliefs the subject may (correctly) infer the
conclusion #. But mere attention to the diagram cannot warrant
believing that, for a given positive \(x\)-value, the \(y\)-value of
\(x\cdot y = k\) never falls below the \(y\)-value of \(x + y =
2\sqrt{k}\) and that the functions coincide just at the diagonal; for
the conventions of representation do not rule out that the curve of
\(x\cdot y = k\) meets the curve of \(x + y = 2\sqrt{k}\) at two
points extremely close to the diagonal, and that the former curve
falls under the latter in between those two points. So the visual
thinking is not in this case a means of discovering proposition #.


But it is useful because it provides the idea for a proof of the
conclusion--one of the major benefits of visual thinking in
mathematics. In brief: for each equation \((x\cdot y = k\); \(x + y =
2\sqrt{k})\) if \(x = y\), their common value is \(\sqrt{k}\). So the
functions expressed by those equations meet at the diagonal. To show
that, for a fixed positive \(x\)-value, the \(y\)-values of \(x\cdot y
= k\) never fall below the \(y\)-values of \(x + y = 2\sqrt{k}\), it
suffices to show that \(2\sqrt{k} - x \le k/x\). As a geometric mean
is less than or equal to the corresponding arithmetic mean, \(\sqrt{[x
\cdot (k/x)]} \le [x + (k/x)]/2\). So \(2\sqrt{k} \le x + (k/x)\).
So \(2\sqrt{k} - x \le k/x\).


In this example, visual attention to, and reasoning about, the
diagram is not part of a way of discovering the conclusion. But if it
gave one the idea for the argument just given, it would be part of
what led to a way of discovering the conclusion, and that is
important.


Can visual thinking lead to discovery of an idea for a proof in
more advanced contexts? Yes. Carter (2010) gives an example from free
probability theory. The case is about certain permutations (those
denoted by "\(p\)" with a circumflex in Carter 2010) on a
finite set of natural numbers. Using specific kinds of diagram, easily
seen properties of the diagrams lead one naturally to certain
properties of the permutations (crossing and non-crossing, having
neighbouring pairs), and to a certain operation (cancellation of
neighbouring pairs). All of these have algebraic definitions, but the
ideas defined were noticed by thinking in terms of the diagrams. For
the relevant permutations \(\sigma\), \(\sigma(\sigma(n)) = n\); so a
permutation can be represented by a set of lines joining dots. The
permutations represented on the left and right in Figure 8 are
non-crossing and crossing respectively, the former with neighbouring
pairs \(\{2, 3\}\) and \(\{6, 7\}\).






![[a circle with 8 points on the circumference, a point at about 45 degrees is labeled '1', at 15 degrees, '2', at -15 degrees '3', at -45 degrees '4', at -135 degrees '5', at -165 degrees '6', at 165 degrees '7', and at 135 degrees '8'.  Smooth curves in the interior of the circle connect point 1 to 4, 2 to 3, 5 to 8, and 6 to 7.]](fig8a.png)


![[a circle with 8 points on the circumference, a point at about 45 degrees is labeled '1', at 15 degrees, '2', at -15 degrees '3', at -45 degrees '4', at -135 degrees '5', at -165 degrees '6', at 165 degrees '7', and at 135 degrees '8'. Straight lines connect 1 to 6, 2 to 5, 3 to 8, and 4 to 7.]](fig8b.png)



 (a)
 (b)


Figure 8



A permutation \(\sigma\) of \(\{1, 2, \ldots, 2p\}\) is defined to
have a crossing just when there are \(a\), \(b\), \(c\), \(d\) in
\(\{1, 2, \ldots, 2p\}\) such that \(a < b < c < d\) and
\(\sigma(a) = c\) and \(\sigma(b) = d\). The focus is on the proof of
a theorem which employs this notion. (The theorem is that when a
permutation of \(\{1, 2, \ldots, 2p\}\) of the relevant kind is
non-crossing, there will be exactly \(p+1\) R-equivalence classes,
where \(R\) is a certain equivalence relation on \(\{1, 2, \ldots,
2p\}\) defined in terms of the permutation.) Carter says that the
proofs of some lemmas "rely on a visualization of the
setup", in that to grasp the correctness of one or more of the
steps one needs to visualize the situation. There is also a nice
example of some reasoning in terms of a diagram which gives the idea
for a proof ("suggests a proof strategy") for the lemma
that every non-crossing permutation has a neighbouring
pair. Reflection on a diagram such as Figure 9 does the work.



![[A circle, a dashed interior curve connects an unmarked point at about 40 degrees to an unmarked point at -10 degrees (the second point is labeled 'j+1').  Another dashed interior curve connects this point to an unmarked point at about -100 degrees.  A solid interior curve connects and unmarked point at about 10 degrees (labeled 'j') to another unmarked point at about -60 degrees (labeled 'j+a').  Between the labels 'j+1' and 'j+a' is another label 'j+2' and then a dotted line between 'j+2' and 'j+a'.]](fig9.png)
Figure 9



The reasoning is this. Suppose that \(\pi\) has no neighbouring
pair. Choose \(j\) such that \(\pi(j) - j = a\) is minimal, that is,
for all \(k, \pi(j) - j \le \pi(k) - k\). As \(\pi\) has no
neighbouring pair, \(\pi(j+1) \ne j\). So either \(\pi(j+1)\) is less
than \(j\) and we have a crossing, or by minimality of \(\pi(j) - j\),
\(\pi(j+1)\) is greater than \(j+a\) and again we have a
crossing. Carter reports that this disjunction was initially believed
by thinking in term of the diagram, and the proof of the lemma given
in the published paper is a non-diagrammatic "version" of
that reasoning. In this case study, visual thinking is shown to
contribute to discovery in several ways; in particular, by leading the
mathematicians to notice crucial properties--the
"definitions are based on the diagrams"--and in
giving them the ideas for parts of the overall proof.


### 4.3 Discovering properties and kinds


In this section I will illustrate and then discuss the use of
visual thinking in discovering kinds of mathematical entity, by going
through a few of the main steps leading to geometric group theory, a
subject which really took off in the 1980s through the work of Mikhail
Gromov. The material is set out nicely in greater depth in Starikova
(2012).


Sometimes it can be fruitful to think of non-spatial entities, such
as algebraic structures, in terms of a spatial representation. An
example is the representation of a finitely generated group by a
Cayley graph. Let \((G, \cdot)\) be a group and \(S\) a finite subset
of \(G\). Let \(S^{-1}\) be the set of inverses of members of \(S\).
Then \((G, \cdot)\) is *generated by* \(S\) if and only if
every member of \(G\) is the product (with respect to \(\cdot\)) of
members of \(S\cup S^{-1}\). In that case \((G, \cdot, S)\) is said to
be a finitely generated group. Here are a couple of examples.


First consider the group \(S\_{3}\) of permutations of 3 elements
under composition. Letting \(\{a, b, c\}\) be the elements, all six
permutations can be generated by \(\rf\) and \(\rr\) where



\(\rf\) (for "flip") fixes a and swaps \(b\) with \(c\),
i.e., it takes to \(\langle a, b, c\rangle\) to \(\langle a, c,
b\rangle\), and


\(\rr\) (for "rotate") takes \(\langle a, b, c\rangle\) to
\(\langle c, a, b\rangle\).



The Cayley graph for \((S\_{3}, \cdot, \{\rf, \rr\})\) is a graph
whose vertices represent the members of \(S\_{3}\) and two
"colours" of directed edges, representing composition with
\(\rf\) and composition with \(\rr\). Figure 10 illustrates: red
directed edges represent composition with \(\rr\) and black edges
represent composition with \(\rf\). So a red edge from a vertex
\(\rv\) representing \(\rs\) in \(S\_{3}\) ends at a vertex
representing \(\rs\rr\) and a black edge from \(\rv\) ends at a vertex
representing \(\rs\rf\). (Notation: "\(\rs\rr\)"
abbreviates "\(\rs \cdot \rr\)" which here denotes
"\(\rs\) followed by \(\rr\)"; same for
"\(\rf\)" in place of "\(\rr\)".) A black edge
has arrowheads both ways because \(\rf\) is its own inverse, that is,
flipping and flipping again takes you back to where you
started. (Sometimes a pair of edges with arrows in opposite directions
is used instead.) The symbol "\(\re\)" denotes the identity.



![[Two red equilateral triangles, one inside the other.  The smaller triangle has arrows on each side pointing in a clockwise direction; the larger has arrows on each side in a counterclockwise direction.  Black double arrow lines connect the respective vertices of each triangle.  The top vertice of the outside triangle is labeled 'e', of the inside triangle 'f'; the bottom left vertice of the outside triangle is labeled 'r', of the inside triangle 'r'; the bottom right vertix of the outside triangle is labeled with 'rr',of the inside triangle with 'fr'.]](fig10.png)
Figure 10



An example of a finitely generated group of infinite order is
\((\mathbb{Z}, +, \{1\})\). We can get any integer by successively
adding 1 or its additive inverse \(-1\). Since 3 added to the inverse
of 2 is 1, and 2 added to the inverse of 3 is \(-1\), we can get any
integer by adding members of \(\{2, 3\}\) and their inverses. Thus
both \(\{1\}\) and \(\{2, 3\}\) are generating sets for \((\mathbb{Z},
+)\). Figure 11 illustrates part of the Cayley graph for
\((\mathbb{Z}, +, \{2, 3\})\). The horizontal directed edges represent
+2. The directed edges ascending or descending obliquely represent
\(+3\).
 



![[Two horizontal parallel black lines with directional arrows pointing to the right. The top line has equidistant points marked '-2', '0', '2', '4' and the bottom line equidistant points marked '-1' (about half way between the upper line's '-2' and '0'), '1', '3', '5'.  A  red arrow goes from '-2' to '1', from somewhere to the left up to '0', from '0' to '3',  from '-1' to '2', from '1' to '4, from '2' to '5', and from '3' to somewhere to the right up.]](fig11.png)
Figure 11



Another example of a generated group of infinite order is \(F\_2\),
the free group generated by a pair of members. The first few
iterations of its Cayley graph are shown in Figure 12, where \(\{a,
b\}\) is the set of generators and a right horizontal move between
adjacent vertices represents composition with \(a\), an upward
vertical move represents composition with \(b\), and leftward and
downward moves represent composition with the inverse of \(a\) and the
inverse of \(b\) respectively. The central vertex represents the
identity.



![[A blue vertical line pointing up labeled 'b' crossed by a red horizontal line pointing right labeled 'b'. Each line is crossed by two smaller copies of the other line on either side of the main intersection. And, in turn, each of those smaller copies of the line are crossed by two smaller copies of the other line, again on either side of their main intersection.]](fig12.png)
Figure 12



Thinking of generated groups in terms of their Cayley graphs makes
it very natural to view them as metric spaces. A *path* is a
sequence of consecutively adjacent edges, regardless of direction. For
example in the Cayley graph for \((\mathbb{Z}, +, \{2, 3\})\) the
edges from \(-2\) to 1, from 1 to \(-1\), from \(-1\) to 2 (in that
order) constitute a path, representing the action, starting from
\(-2\), of adding 3, then adding \(-2\), then adding 3. Taking each
edge to have unit length, the metric \(d\_S\) for a group \(G\)
generated by a finite subset \(S\) of \(G\) is defined: for any \(g\),
\(h \in G\), \(d\_{S}(g, h) =\) the length of a shortest path from
\(g\) to \(h\) in the Caley graph of \((G, \cdot, S)\). This is the
*word metric* for this generated group.


Viewing a finitely generated group as a metric space allows us to
consider its growth function \(\gamma(n)\) which is the cardinality of
the "ball" of radius \(\le n\) centred on the identity
(the number of members of the group whose distance from the identity
is not greater than \(n\)). A growth function for a given group
depends on the set of generators chosen, but when the group is
infinite the asymptotic behaviour as \(n \rightarrow \infty\) of the
growth functions is independent of the set of generators.


Noticing the possibility of defining a metric on generated groups
did not require first viewing diagrams of their Cayley graphs. This is
because a word in the generators is just a finite sequence of symbols
for the generators or their inverses (we omit the symbol for the group
operation), and so has an obvious length visually suggested by the
written form of the word, namely the number of symbols in the
sequence; and then it is natural to define the distance between group
members \(g\) and \(h\) to be the length of a shortest word that gets
one from \(g\) to \(h\) by right multiplication, that is,
\(\textrm{min}\{\textrm{length}(w): w = g^{-1}h\}\). 


However, viewing generated groups by means of their Cayley graphs
was the necessary starting point for geometric group theory, which
enables us to view finitely generated groups of infinite order not
merely as graphs or metric spaces but as geometric entities. The main
steps on this route will be sketched briefly here; for more detail see
Starikova (2012) and the references therein. The visual key is to
start thinking in terms of the "coarse geometry" of the
Cayley graph of the generated group, by zooming out in visual
imagination so far that the discrete nature of the graph is
transformed into a traditional geometrical object. For example, the
Cayley graph of a generated group of finite order such as \((S\_{3},
\cdot, \{f, r\})\) illustrated in Figure 11
becomes a dot; the Cayley graph for \((\mathbb{Z}, +, \{2, 3\})\)
illustrated in Figure 12 becomes an uninterrupted
line infinite in both directions. 


The word metric of a generated group is discrete: the values are
always in \(N\). How is this visuo-spatial association of a discrete
metric space with a continuous geometrical object achieved
mathematically? By quasi-isometry. While an isometry from one metric
space to another is a distance preserving map, a quasi-isometry is a
map which preserves distances to within fixed linear bounds. Precisely
put, a map \(f\) from \((S, d)\) to \((S', d')\) is a
*quasi-isometry* iff for some real constants \(L > 0\) and
\(K \ge 0\) and all \(x\), \(y\) in \(S\) \[ d(x, y)/L - K \le
d'(f(x), f(y)) \le L \cdot d(x, y) + K. \]


The spaces \((S, d)\) and \((S', d')\) are
*quasi*-*isometric* spaces iff the quasi-isometry \(f\)
is also quasi-surjective, in the sense that there is a real constant
\(M \ge 0\) such that every point of \(S'\) is no further than \(M\)
away from some point in the image of \(f\).


For example, \((\mathbb{Z}, d)\) is quasi-isometric to
\((\mathbb{R}, d)\) where \(d(x, y) = |y - x|\), because the inclusion
map \(\iota\) from \(\mathbb{Z}\) to \(\mathbb{R}\), \(\iota(n) = n\),
is an isometry hence a quasi-isometry with \(L = 1\) and \(K = 0\),
and each point in \(\mathbb{R}\) is no further than \(1/2\) away from
an integer (in \(\mathbb{R}\)). Also, it is easy to see that for any
real number \(x\), if \(g(x) =\) the nearest integer to \(x\) (or the
greatest integer less than \(x\) if it is midway between integers)
then \(g\) is a quasi-isometry from \(\mathbb{R}\) to \(\mathbb{Z}\)
with \(L = 1\) and \(K =\frac{1}{2}\);.


The relation between metric spaces of being quasi-isometric is an
equivalence relation. Also, if \(S\) and \(T\) are generating sets of
a group \((G, \cdot)\), the Cayley graphs of \((G, \cdot, S)\) and
\((G, \cdot, T)\) with their word metrics are quasi-isometric
spaces. This means that properties of a generated group which are
quasi-isometric invariants will be independent of the choice of
generating set, and therefore informative about the group itself.


Moreover, it is easy to show that the Cayley graph of a generated
group with word metric is quasi-isometric to a geodesic 
space.[1] 
 A *triangle*
with vertices \(x\), \(y\), \(z\) in this space is the union of three
geodesic segments, between \(x\) and \(y\), between \(y\) and \(z\),
and between \(z\) and \(x\). This is the gateway for the application
of Gromov's insights, some of which can be grasped with the help
of visual geometric thinking.


Here are some indications. Recall the Poincare open disc
model of hyperbolic geometry: geodesics are diameters or arcs of
circles orthogonal to the boundary, with unit distance represented by
ever shorter Euclidean distances as one moves from the centre towards
the boundary. (The boundary is not part of the model). All triangles
have angle sum \(< \pi\) (Figure 13, left), and
there is a global constant d such that all triangles are
d-thin in the following sense:


A triangle \(T\) is d-*thin* if and
only if any point on one side of \(T\) lies within d of some
point on one of the other two sides.


This condition is equivalent to the condition that each side of
\(T\) lies within the union of the d-neighbourhoods of the other
two sides, as illustrated in Figure 13,
right. There is no constant d such that all triangles in a
Euclidean plane are d-thin, because for any d there are
triangles large enough that the midpoint of a longest side lies
further than d from all points on the other two sides.





![[a circle.  In the interior are three arcs colored  green, blue, and red. For all three smooth curves where each meets the circumference of the circle is marked as at a 90 degree angle.  The green curve may actually be a straight line and goes from about 160 degrees to about -20 degrees.  The blue curve goes from about 170 degrees to about 80 degrees.  The red curve goes from about 90 degrees to about -25 degrees.  Where the green and blue curves intersect is marked as an angle and labelled with the Greek letter alpha; where the blue and the red curves intersect is also marked as an angle and labelled with gamma; and with where the red and the green curves intersect and this labelled with beta.]](fig13a.png)
![[not sure how to describe this]](fig13b.png)


(a)
(b)


Figure 13[2]




The definition of thin triangles is sufficiently general to apply
to any geodesic space and allows a generalisation of the concept of
hyperbolicity beyond its original context:


* A geodesic space is *hyperbolic* iff for some d all
its triangles are d-thin.
* A group is *hyperbolic* iff it has a Cayley graph
quasi-isometric to a hyperbolic geodesic space.



The class of hyperbolic groups is large and includes important
subkinds, such as finite groups, free groups and the fundamental
groups of surfaces of genus \(\ge 2\). Some striking theorems have
been proved for them. For example, for every hyperbolic group the word
problem is solvable, and every hyperbolic group has a finite
presentation. So we can reasonably conclude that the discovery of this
mathematical kind, the hyperbolic groups, has been fruitful.


How important was visual thinking to the discoveries leading to
geometric group theory? Visual thinking was needed to discover Cayley
graphs as a means of representing finitely generated groups. This is
not the triviality it might seem: Cayley graphs must be distinguished
from the diagrams we use to present them visually. A Cayley graph is a
*mathematical* representation of a generated group, not a
*visual* representation. It consists of the following
components: a set \(V\) ("vertices"), a set \(E\) of
ordered pairs of members of \(V\) ("directed edges") and a
partition of \(E\) into distinguished subsets, ("colours",
each one for representing right multiplication by a particular
generator). The Cayley graph of a generated group of infinite order
cannot be fully represented by a diagram given the usual conventions
of representation for diagrams of graphs, and distinct diagrams may
visually represent the same Cayley graph: both diagrams in Figure 14
can be labelled so that under the usual conventions they represent the
Cayley graph of \((S\_{3}, \cdot, \{f, r\})\), already illustrated by
Figure 10. So the Cayley graph cannot be a
diagram.





![[two identical red triangles, one above the other and inverted.   Both have arrows going clockwise around. Black lines with arrows pointing both ways link the respective vertices.]](fig14a.png)
![[two identical red triangles, one above the other.   The bottom one has arrows going clockwise around and the top counterclockwise. Black lines with arrows pointing both ways link the respective vertices.]](fig14b.png)


(a)
(b)


Figure 14



Diagrams of Cayley graphs were important in prompting
mathematicians to think in terms of the coarse-grained geometry of the
graphs, in that this idea arises just when one thinks in terms of
"zooming out" visually. Gromov (1993) makes the point in a
passage quoted in Starikova (2012:138)



>   This space [a Cayley graph with the word metric] may
> appear boring and uneventful to a geometer's eye since it is
> discrete and the traditional (e.g., topological and infinitesimal)
> machinery does not run in [the group] G. To regain the geometric
> perspective one has to change one's position and move the
> observation point far away from G. Then the metric in G
> seen from the distance \(d\) becomes the original distance divided by
> \(d\) and for \(d \rightarrow \infty\) the points in G coalesce
> into a connected continuous solid unity which occupies the visual
> horizon without any gaps and holes and fills our geometer's
> heart with joy.
> 
> 


In saying that one has to move the observation point far away from
G so that the points coalesce into a unity which occupies the
visual horizon, he makes clear that visual imagination is involved in
a crucial step on the road to geometric group theory. Visual thinking
is again involved in discovering hyperbolicity as a property of
general geodesic spaces from thinking about the Poincare disk
model of hyperbolic geometry. It is hard to see how this property
would have been discovered without the use of visual resources. 


## 5. Visual thinking and mental arithmetic



While there is no reason to think that mental arithmetic (mental
calculation in the integers and rational numbers) typically involves
much visual thinking, there is strong evidence of substantial visual
processing in the mental arithmetic of highly trained abacus users.



In earlier times an abacus would be a rectangular board or table
surface marked with lines or grooves along which pebbles or counters
could be moved. The oldest surviving abacus, the Salamis abacus, dated
around 300 BCE, is a white marble slab, with markings designed for
monetary calculation (Fernandes 2015, Other Internet Resources). These were
superseded by rectangular frames within which wires or rods parallel to
the short sides are fixed, with moveable holed beads on them. There are
several kinds of modern abacus -- the Chinese suanpan, the Russian
schoty and the Japanese soroban for example -- each kind with
variations. Evidence for visual processing in mental arithmetic comes
from studies with well trained users of the soroban, an example of
which is shown in Figure 15.



![[Picture of a soroban with 17 columns of beads, each column has 1 bead above the horizontal bar used to represent 5 and 4 beads below the bar each of which represents 1. Together the beads in each column can represent any digit from 0 to 9.]](fig15.png)
Figure 15





Each column of beads represents a power of 10, increasing to the
left. The horizontal bar, sometimes called *the* *reckoning
bar*, separates the beads on each column into one bead of value 5
above and four beads of value 1 below. The number represented in a
column is determined by the beads which are not separated from the
reckoning bar. A column on which all beads are separated by a gap from
the bar represents zero. For example, the number 6059 is represented on
a portion of a schematic soroban in Figure 16.



![[A schematic soroban representing 6059. There are 8 places and the first four from the left are set to 0, then 6, then 0, then 5, then 9 ]](fig16.png)
Figure 16




On some sorobans there is a mark on the reckoning bar at every third
column; if a user chooses one of these as a unit column, the marks will
help the user keep track of which columns represent which powers of
ten. Calculations are made by using forefinger and thumb to move beads
according to procedures for the standard four numerical operations and
for extraction of square and cube roots (Bernazzani 2005,Other Internet Resources). Despite the
fact that the soroban has a decimal place representation of numbers,
the soroban procedures are not 'translations' of the
procedures normally taught for the standard operations using arabic
numerals. For example, multidigit addition on a soroban starts by
adding highest powers of ten and proceeds rightwards to lower powers,
instead of starting with units thence proceeding leftwards to tens,
hundreds and so on.



People trained to use a soroban often learn to do mental arithmetic
by visualizing an abacus and imagining moving beads on it in accordance
with the procedures learned for arithmetical calculations (Frank and
Barner 2012). Mental abacus (MA), as this kind of mental arithmetic is
known, compares favourably with other kinds of mental calculation for
speed and accuracy (Kojima 1954) and MA users are often found among the
medallists in the Mental Calculation World Cup.



Although visual and manual motor imagery is likely to occur,
cognitive scientists have probed the question whether the actual
processes of MA calculation consist in or involve imagining performing
operations on a physical abacus. Brain imaging studies provide one
source of evidence bearing on this question. Comparing well-trained
abacus calculators with matched controls, evidence has been found that
MA involves neural resources of visuospatial working memory with a form
of abacus which does not depend on the modality (visual or auditory) of
the numerical inputs (Chen et al. 2006). Another imaging study found
that, compared to controls without abacus training, subjects with long
term MA training from a young age had enhanced brain white matter
related to motor and visuospatial processes (Hu et al. 2011).



Behavioural studies provide more evidence. Tests on expert and
intermediate level abacus users strongly suggest that MA calculators
mentally manipulate an abacus representation so that it passes through
the same states that an actual abacus would pass through in solving an
addition problem. Without using an actual abacus MA calculators were
able to answer correctly questions about intermediates states unique to
the abacus-based solution of a problem; moreover, their response times
were a monotonic function of the position of the probed state in the
sequence of states of the abacus process for solving the problem
(Stigler 1984). On top of the 'intermediate states'
evidence, there is 'error type' evidence. Mental addition
tests comparing abacus users with American subjects revealed that
abacus users made errors of a kind which the Americans did not make,
but which were predictable from the distribution of errors in physical
abacus addition (Stigler 1984).



Another study found evidence that when a sequence of numbers
is presented auditorily (as a verbal whole "three thousand five
hundred and forty seven" or as a digit sequence "Three,
five, four, seven") abacus experts encode it into an imaged
abacus display, while non-experts encode it verbally (Hishitani
1990).



Further evidence comes from behavioural interference studies. In
these studies subjects have to perform mental calculations, with and
without a task of some other kind to be performed during the
calculation, with the aim of seeing which kinds of task interfere with
calculation as measured by differences of reaction time and error rate.
An early study found that a linguistic task interfered weakly with MA
performance (unless the linguistic task was to answer a mathematical
question), while motor and visual tasks interfered relatively
strongly. These findings suggested to the paper's authors
that MA representations are not linguistic in nature but rely on visual
mechanisms and, for intermediate practitioners, on motor mechanisms as
well (Hatano et al. 1977).



These studies provide impressive evidence that MA does involve mental
manipulation of a visualized abacus. However, limits of the known
capacities for perceiving or representing pluralities of objects seem
to pose a problem. We have a *parallel individuation* system
for keeping track of up to four objects simultaneously and
an *approximate number system* (ANS) which allows us to gauge
roughly the cardinality of a set of things, with an error which
increases with the size of the set. The parallel individuation system
has a limit of three or four objects and the ANS represents
cardinalities greater than four only approximately. Yet mental abacus
users would need to hold in mind with precision abacus representations
involving a much larger number of beads than four (and the way in
which those beads are distributed on the abacus). For example, the
number 439 requires a precise distribution of twelve beads. Frank and
Barner (2012) address this problem. In some circumstances we can
perceive a plurality of objects as a single entity, a set, and
simultaneously perceive those objects as individuals. There is
evidence that we can keep track of up to three such sets in parallel
and simultaneously make reliable estimates of the cardinalities of the
sets (if not more than four). If the sets themselves can be easily
perceived as (a) divided into disjoint subsets, e.g. columns of beads
on an abacus, and (b) structured in a familiar way, e.g. as a
distribution of four beads below a reckoning bar and one above, we
have the resources for recognising a three-digit number from its
abacus representation. The findings of (Frank and Barner 2012) suggest
that this is what happens in MA: a mental abacus is represented in
visuospatial working memory by splitting it into a series of columns
each of which is stored as a unit with its own detailed
substructure.



These cognitive investigations confirm the self-reports of mental
abacus users that they calculate mentally by visualizing operating on
an abacus as they would operate on a physical abacus. (See the 20-second
movie
Brief interview with mental abacus user, at the Stanford Language and Cognition 
Lab, for one such self-report.) There is good evidence that MA often involves
processes linked to motor cognition in addition to active visual
imagination. Intermediate abacus users often make hand movements,
without necessarily attending to those movements during MA calculation,
as shown in the second of the three short movies just mentioned.
Experiments to test the possible role of motor processes in MA resulted
in findings which led the authors to conclude that premotor processes
involved in the planning of hand movements were involved in MA (Brooks
et al. 2018).


## 6. *A priori* and *a posteriori* roles of visual experience


In coming to know a mathematical truth visual experience can play a
merely "enabling" role. For example, visual experience may
have been a factor in a person's getting certain concepts
involved in a mathematical proposition, thus enabling her to
understand the proposition, without giving her reason to believe
it. Or the visual experience of reading an argument in a text book may
enable one to find out just what the argument is, without helping her
tell that the argument is sound. In earlier sections visual experience
has been presented as having roles in proof and propositional
discovery that are not merely enabling. On the face of it this raises
a puzzle: mathematics, as opposed to its application to natural
phenomena, has traditionally been thought to be an *a priori*
science; but if visual experience plays a role in acquiring
mathematical knowledge which is not merely enabling, the result would
surely be *a posteriori* knowledge, not *a priori*
knowledge. Setting aside knowledge acquired by testimony (reading or
hearing that such-&-such is the case), there remain plenty of
cases where sensory experience seems to play an evidential role in
coming to know some mathematical fact.


### 6.1 Evidential uses of visual experience


A plausible example of the evidential use of sensory experience is
the case of a child coming to know that \(5 + 3 = 8\) by counting on
her fingers. While there may be an important \(a\) *priori*
element in the child's appreciation that she can reliably
generalise from the result of her counting experiment, getting that
result by counting is an *a posteriori* route to it. For
another example, consider the question: how many vertices does a cube
have? With the background knowledge that cubes do not vary in shape
and that material cubes do not differ from geometrical cubes in number
of vertices (where a "vertex" of a material cube is a
corner), one can find the answer by visually inspecting a material
cube. Or if one does not have a material cube to hand, one can
visually imagine a cube, and by attending to its top and bottom faces
extract the information that the vertices of the cube are exactly the
vertices of these two quadrangular faces. When one gets the answer by
inspecting a material cube, the visual experience contributes to
one's grounds for believing the answer and that contribution is
part of what makes the belief state knowledge. So the role of the
visual experience is evidential; hence the resulting knowledge is not
*a priori*. When one gets the answer by visually imagining a
cube, one is drawing on the accumulated cognitive effects of past
experiences of seeing material cubes to bring to mind what a cube
looks like; so the experience of visual imagining has an indirectly
evidential role in this case.


Do such examples show that mathematics is not an *a priori*
science? Yes, if an *a priori* science is understood to be one
whose knowable truths are all knowable *only* in an *a
priori* way, without use of sense experience as evidence. No, if
an *a priori* science is one whose knowable truths are all
knowable in an *a priori* way, allowing that some may be
knowable also in an *a posteriori* way.


### 6.2 An evidential use of visual experience in proving


Many cases of proving something (or following a proof of it)
involve making, or imagining making, changes in a symbol array. A
standard presentation of the proof of left-cancellation in group
theory provides an example. "Left-cancellation" is the
claim that for any members \(a\), \(b\), \(c\) of a group with
operation \(\cdot\) and identity element \(\mathbf{e}\), if \(a \cdot
b = a \cdot c\), then \(b = c\). Here is (the core of) a proof of
it:

 \begin{align\*} a \cdot b &= a \cdot c\\ a^{-1} \cdot (a
\cdot b) &= a^{-1} \cdot (a \cdot c)\\ (a^{-1} \cdot a) \cdot b
&= (a^{-1} \cdot a) \cdot c\\ \mathbf{e}\cdot b &= \mathbf{e}
\cdot c\\ b &= c. \end{align\*}

Suppose that one comes to know left-cancellation by following this
sequence of steps. Is this an *a priori* way of getting this
knowledge? Although following a mathematical proof is thought to be a
paradigmatically *a priori* way of getting knowledge, attention
to the role of visual experience here throws this into doubt. The case
for claiming that the visual experience has an evidential role is as
follows.


The visual experience reveals not only what the steps of the
argument are but also that they are valid, thereby contributing to our
grounds for accepting the argument and believing its conclusion.
Consider, for example, the step from the second equation to the third.
The relevant background knowledge, apart from the logic of identity,
is that a group operation is associative. This fact is usually
represented in the form of an equation that simply relocates brackets
in an obvious way:

 \[ x \cdot (y \cdot z) = (x \cdot y) \cdot z \]

We see that relocating the brackets in accord with this format, the
left-hand term of the second equation is transformed into the
left-hand term of the third equation, and the same for the right-hand
terms. So the visual experience plays an evidential role in our
recognising as valid the step from the second equation to the
third. Hence this quite standard route to knowledge of
left-cancellation turns out to be *a posteriori*, even though
it is a clear case of following a proof.


Against this, one may argue that the description just given of what
is going on in following the proof is not strictly correct, as
follows. Exactly the same proof can be expressed in natural language,
using "the composition of \(x\) with \(y\)" for "\(x
\cdot y\)", but the result would be hard to take in. Or the
proof can be presented using a different notational convention, one
which forces a quite different expression of associativity. For
example, we can use the Polish convention of putting the operation
symbol before the operands: instead of "\(x \cdot y\)" we
put "\(\cdot x y\)". In that case associativity would be
expressed in the following way, without brackets:

 \[ \cdot x \cdot
y z = \cdot \cdot x y z. \]

The equations of the proof would then need to be re-symbolised; but
what is expressed by each equation after re-symbolisation and the
steps from one to the next would be exactly as before. So we would be
following the very same proof, step by step. But we would not be using
visual experiences involved to notice the relocation of brackets this
time. This suggests that the role of the different visual experiences
involved in following the argument in its different guises is merely
to give us access to the common reasoning: the role of the experience
is merely enabling. On this account the visual experience does not
strictly and literally enable us to see that any of the steps are
valid; rather, recognition of (or sensitivity to) the validity of the
steps results from cognitive processing at a more abstract level.


Which of these rival views is correct? Does our visual experience
in following the argument presented with brackets (1) reveal to us the
validity of some of the steps, given the relevant background knowledge
? Or (2) merely give us access to the argument? The core of the
argument against view (1) is this:



Seeing the relocation of brackets is not essential to following the
argument.


So seeing merely gives access to the argument; it does not reveal
any step to be valid.



The step to this conclusion is faulty. How one follows a proof may,
and in this case does, depend on how it is presented, and different
ways of following a proof may be different ways of coming to know its
conclusion. While seeing the relocation of brackets is not essential
to all ways of following this argument, it is essential to the normal
way of following the argument when it is symbolically presented with
brackets in the way given above.


Associativity, expressed without symbols, is this: When the binary
group operation is applied twice in succession on an ordered triple of
operands \(\langle a, b, c\rangle\), it makes no difference whether
the first application is to the initial two operands or the final two
operands. While this is the content of associativity, for ease of
processing associativity is almost always expressed as a
symbol-manipulation rule. Visual perception is used to tell in
particular cases whether the rule thus expressed is correctly
implemented, in the context of prior knowledge that the rule is
correct. What is going on here is a familiar division of labour in
mathematical thinking. We first establish the soundness of a rule of
symbol-manipulation (in terms of the governing semantic
conventions--in this case the matter is trivial); then we check
visually that the rule is correctly implemented. Processing at a more
abstract, semantic level is often harder than processing at a purely
syntactic level; it is for this reason that we often resort to
symbol-manipulation techniques as proxy for reasoning directly with
meanings to solve a problem. (What is six eighths *divided by*
three fifths, without using any symbolic technique?) When we do use
symbol-manipulation in proving or following a proof, visual experience
is required to discern that the moves conform to permitted patterns
and thus contributes to our grounds for accepting the argument. Then
the way of coming to know the conclusion has an *a posteriori*
element.


### 6.3 A non-evidential use of visual experience


Must a use of visual experience in knowledge acquisition be
*evidential*, if the visual experience is not merely enabling?
Here is an example which supports a negative answer. Imagine a square
or look at a drawing of one. Each of its four sides has a
midpoint. Now visualize the "inner" square whose sides run
between the midpoints of adjacent sides of the original square (Figure
17, left). By visualizing this figure, it should be clear that the
original square is composed precisely of the inner square plus four
corner triangles, each side of the inner square being the base of a
corner triangle. One can now visualize the corner triangles folding
over, with creases along the sides of the inner square. The starting
and end states of the imagery transformation can be represented by the
left and right diagrams of Figure 17.






![[The first of identical squares in size.  The first has lines connecting the midpoints of each adjacent pair of sides to form another square.  The second has in addition lines connecting the midpoints of opposite pairs of sides.  In addition the outer square of the second has dashed lines instead of solid.]](fig17a.png)


![[The second of identical squares in size.  The first has lines connecting the midpoints of each adjacent pair of sides to form another square.  The second has in addition lines connecting the midpoints of opposite pairs of sides.  In addition the outer square of the second has dashed lines instead of solid.]](fig17b.png)




(a)


(b)



Figure 17



Visualizing the folding-over within the remembered frame of the
original square results in an image of the original square divided
into square quarters, its quadrants, and the sides of the inner square
seem to be diagonals of the quadrants. Many people conclude that the
corner triangles can be arranged to cover the inner square exactly,
without any gap or overlap. Thence they infer that the area of the
original square is twice the size of the inner square. Let us assume
that the propositions concerned are about Euclidean figures. Our
concern is with the visual route to the following:



> The parts of a square beyond its inner square (formed by
> joining midpoints of adjacent sides of the original square) can be
> arranged to fit the inner square exactly, without overlap or gap,
> without change of size or shape.


The experience of visualizing the corner triangles folding over can
lead one to this belief. But it cannot provide good evidence for it.
This is because visual experience (of sight or imagination) has
limited acuity and so does not enable us to discriminate between a
situation in which the outer triangles fit the inner square
*exactly* and a situation in which they fit inexactly but well
enough for the mismatch to escape visual detection. (This contrasts
with the case of discovering the number of vertices of a cube by
seeing or visualizing one.) Even though visualizing the square, the
inner square and then visualizing the corner triangles folding over is
constrained by the results of earlier perceptual experience of scenes
with relevant similarities, we cannot draw from it reliable
information about exact equality of areas, because perception itself
is not reliable about exact equalities (or exact proportions) of
continuous magnitudes.


Though the visual experience could not provide good evidence for
the belief, it is possible that we erroneously *use* the
experience evidentially in reaching the belief. But it is also
possible, when reaching the belief in the way described, that we do
*not* take the experience to provide evidence. A non-evidential
use is more likely, if when one arrives at the belief in this way one
feels fairly certain of it, while aware that visual perception and
imagination have limited acuity and so cannot provide evidence for a
claim of exact fit.


But what could the role of the visualizing experience possibly be,
if it were neither merely enabling nor evidential? One suggestion is
that we already have relevant beliefs and belief-forming dispositions,
and the visualizing experience could serve to bring to mind the
beliefs and to activate the belief-forming dispositions (Giaquinto
2007). These beliefs and dispositions will have resulted from prior
possession of cognitive resources, some subject-specific such as
concepts of geometrical figures, some subject-general such as symmetry
perception about perceptually salient vertical and horizontal axes. A
relevant prior belief in this case might be that a square is symmetric
about a diagonal. A relevant disposition might be the disposition to
believe that the quadrants of a square are congruent squares upon
seeing or visualizing a square with a horizontal base plus the
vertical and horizontal line segments joining midpoints of its
opposite sides. (These dispositions differ from ordinary perceptual
dispositions to believe what we see in that they are not cancelled
when we mistrust the accuracy of the visual experience.) 


The question whether the resulting belief would be knowledge
depends on whether the belief-forming dispositions are reliable
(truth-conducive) and the pre-existing belief states are states of
knowledge. As these conditions can be met without any violation of
epistemic rationality, the visualizing route described incompletely
here can be a route to knowledge. In that case we would have an
example of a use of visual experience which is integral to a way of
knowing a truth, which is not merely enabling and yet not
evidential. A fuller account and discussion is given in chapters 3 and
4 of Giaquinto (2007).


## 7. Further uses of visual representations


There are other significant uses of visual representations in
mathematics. This final section briefly presents a couple of them.


Although the use of diagrams in arguments in analysis faces special
dangers (as noted in 3.3), the use of
diagrams to illustrate symbolically presented operations can be very
helpful. Consider, for example, this pair of operations \(\{ f(x) + k,
f(x + k) \}\). Grasping them and the difference between them can be
aided by a visual illustration; similarly for the sets \(\{ f(x + k),
f(x - k) \}\), \(\{ |f(x)|, f(|x|) \}\), \(\{ f(x)^{-1}, f^{-1}(x),
f(x^{-1}) \}\). While generalization on the basis of a visual
illustration is unreliable, we can use them as checks against
calculation errors and overgeneralization. The same holds for
properties. Consider for example, functions for which \(f(-x) =
f(x)\), known as *even* functions, and functions for which
\(f(-x) = -f(x)\), the *odd* functions: it can be helpful to
have in mind the images of graphs of \(y = x^2\) and \(y = x^{3}\) as
instances of evenness and oddness, to remind one that even functions
are symmetrical about the \(y\)-axis and odd functions have rotation
symmetry by \(\pi\) about the origin. They can serve as a reminder and
check against over-generalisation: any general claim true of all odd
functions, for example, must be true of \(y = x^{3}\) in
particular.


The utility of visual representations in real and complex analysis
is not confined to such simple cases. Visual representations can help
us grasp what motivates certain definitions and arguments, and thereby
deepen our understanding. Abundant confirmation of this claim can be
gathered from working through the text *Visual Complex
Analysis* (Needham 1997). Some mathematical subjects have natural
visual representations, which then give rise to a domain of
mathematical entities in their own right. This is true of geometry but
is also true of subjects which become algebraic in nature very
quickly, such as graph theory, knot theory and braid
theory. Techniques of computer graphics now enable us to use moving
images. For an example of the power of kinematic visual
representations to provide and increase understanding of a subject,
see the first two "chapters" of the online introduction to
braid theory by Ester Dalvit (2012, Other Internet Resources). 


With regard to proofs, a minimal kind of understanding consists in
understanding each line (proposition or formula) and grasping the
validity of each step to a new line from earlier lines. But we can
have that stepwise grasp of proof without any idea of why it proceeds
by those steps. One has a more advanced (or deeper) kind of
understanding when one has the minimal understanding *and* a
grasp of the motivating idea(s) and strategy of the proof. The point
is sharply expressed by Weyl (1995 [1932]: 453), quoted in (Tappenden
2005:150)



>   We are not very pleased when we are forced to accept
> a mathematical truth by virtue of a complicated chain of formal
> conclusions and computations, which we traverse blindly, link by link,
> feeling our way by touch. We want first an overview of the aim and the
> road; we want to understand the *idea* of the proof, the deeper
> context.
> 
> 


Occasionally the author of a proof gives readers the desired
understanding by adding commentary. But this is not always needed, as
the idea of a proof is sometimes revealed in the presentation of the
proof itself. Often this is done by using visual representations. An
example is Fisk's proof of Chvatal's "art
gallery" theorem. This theorem is the answer to a combinatorial
problem in geometry. Put concretely, the problem is this. Let the
\(n\) walls of a single-floored gallery make a polygon. What is the
smallest number of stationary guards needed to ensure that every point
of the gallery wall can be seen by a guard? If the polygon is convex
(all interior angles < 180deg), one guard will suffice, as guards
may rotate. But if the polygon is not convex, as in Figure 18, one
guard may not be enough.



![[An irregular 9 sided polygon.]](fig18.png)
Figure 18



Chvatal's theorem gives the answer: for a gallery with
\(n\) walls, \(\llcorner n/3\lrcorner\) guards suffice, where
\(\llcorner n/3\lrcorner\) is the greatest integer \(\le n/3\). (If
this does not sound to you sufficiently like a mathematical theorem,
it can be restated as follows: Let \(S\) be a subset of the Euclidean
plane. For a subset \(B\) of \(S\) let us say that \(B\)
*supervises* \(S\) iff for each \(x \in S\) there is a \(y \in
B\) such that the segment \(xy\) lies within \(S\). Then the smallest
number \(f(n)\) such that every set bounded by a simple \(n\)-gon is
supervised by a set of \(f(n)\) points is at most \(\llcorner
n/3.\lrcorner\)


Here is Steve Fisk's proof. A short induction shows that
every polygon can be triangulated, i.e., non-crossing edges between
non-adjacent vertices ("diagonals") can be added so that
the polygon is entirely composed of non-overlapping triangles. So take
any \(n\)-sided polygon with a fixed triangulation. Think of it as a
graph, a set of vertices and connected edges, as in Figure 19.



![[10 irregularly placed black dots with a solid black line connecting them to form an irregular 10 sided polygon.  One black dot has dashed lines going to four other dots that are not adjacent to it and one of its adjacent dots has dashed lines going to three other non-adjacent dots (including one dot that was the endpoint for one of the first dots dashed lines), the dashed lines do not intersect.]](fig19.png)
Figure 19



The first part of the proof shows that the graph is 3-colourable,
i.e., every vertex can be coloured with one of just three colours
(red, white and blue, say) so that no edge connects vertices of the
same colour.


The argument proceeds by induction on \(n \ge 3\), the number of
vertices.


For \(n = 3\) it is trivial. Assume it holds for all \(k\), where
\(3 \le k < n\).


Let triangulated polygon \(G\) have \(n\) vertices. Let \(u\) and
\(v\) be any two vertices connected by diagonal edge \(uv\). The
diagonal \(uv\) splits \(G\) into two smaller graphs, both containing
\(uv\). Give \(u\) and \(v\) different colours, say red and white, as
in Figure 20.



![[Same figure as before with one of the black dots split into two red dots side-by-side and another black dot split into two white dots side-by-side. This splits the previously joined figure into two smaller graphs.]](fig20.png)
Figure 20



By the inductive assumption, we may colour each of the smaller
graphs with the three colours so that no edge joins vertices of the
same colour, keeping fixed the colours of \(u\) and \(v\). Pasting
together the two smaller graphs as coloured gives us a 3-colouring of
the whole graph.


What remains is to show that \(\llcorner n/3\lrcorner\) or fewer
guards can be placed on vertices so that every triangle is in the view
of a guard. Let \(b\), \(r\) and \(w\) be the number of vertices
coloured blue, red and white respectively. Let \(b\) be minimal in
\(\{b, r, w\}\). Then \(b \le r\) and \(b \le w\). Then \(2b \le r +
w\). So \(3b \le b + r + w = n\). So \(b \le n/3\) and so \(b \le
\llcorner n/3\lrcorner\). Place a guard on each blue vertex. Done.


 The central idea of this proof, or the proof strategy, is
clear. While the actual diagrams produced here are superfluous to the
proof, some visualizing enables us to grasp the central idea.


## 8. Conclusion


Thinking which involves the use of seen or visualized images, which
may be static or moving, is widespread in mathematical practice. Such
visual thinking may constitute a non-superfluous and non-replaceable
part of thinking through a specific proof. But there is a real danger
of over-generalisation when using images, which we need to guard
against, and in some contexts, such as real and complex analysis, the
apparent soundness of a diagrammatic inference is liable to be
illusory.


Even when visual thinking does not contribute to *proving* a
mathematical truth, it may enable one to *discover* a truth,
where to discover a truth is to come to believe it in an independent,
reliable and rational way. Visual thinking can also play a large role
in discovering a central idea for a proof or a proof-strategy; and in
discovering a kind of mathematical entity or a mathematical
property.


The (non-superfluous) use of visual thinking in coming to know a
mathematical truth does in some cases introduce an *a
posteriori* element into the way one comes to know it, resulting
in *a posteriori* mathematical knowledge. This is not as
revolutionary as it may sound as a truth knowable *a
posteriori* may also be knowable *a priori*. More
interesting is the possibility that one can acquire some mathematical
knowledge in a way in which visual thinking is essential but does not
contribute evidence; in this case the role of the visual thinking may
be to activate one's prior cognitive resources. This opens the
possibility that non-superfluous visual thinking may result in *a
priori* knowledge of a mathematical truth.


Visual thinking may contribute to understanding in more than one
way. Visual illustrations may be extremely useful in providing
examples and non-examples of analytic concepts, thus helping to
sharpen our grasp of those concepts. Also, visual thinking
accompanying a proof may deepen our understanding of the proof, giving
us an awareness of the direction of the proof so that, as Hermann Weyl
put it, we are not forced to traverse the steps blindly, link by link,
feeling our way by touch.