# Reflecting on beauty: the aesthetics of mathematical discovery

Filip D. Jevtić
Mathematical Institute
SASA, Belgrade Jovana Kostić
Faculty of Philosophy
University of Belgrade Katarina Maksimović
Faculty of Philosophy
University of Belgrade

###### Abstract

Mathematical research is often motivated by the desire to reach a beautiful
result or to prove it in an elegant way. Mathematician’s work is thus strongly
influenced by his aesthetic judgments. However, the criteria these judgments
are based on remain unclear. In this article, we focus on the concept of
mathematical beauty, as one of the central aesthetic concepts in mathematics.
We argue that beauty in mathematics reveals connections between apparently
non-related problems or areas and allows a better and wider insight into
mathematical reality as a whole. We also explain the close relationship
between beauty and other important notions such as depth, elegance,
simplicity, fruitfulness, and others.

> In memory of Kosta Došen on the occasion of his seventieth birthday.

##  1 Introduction

There are very few things in the world as widely used and as indispensable in
both science and everyday life as mathematics. However, the motivation for
mathematical research generally does not lie in its utility. Mathematicians do
not tend to choose the subject of their study because of its projected
applications. Rather, they get intrigued by the subject itself, their research
often being motivated by the desire to reach a beautiful result or to prove it
in an elegant way. As Rota observed, “The lack of beauty in a piece of
mathematics is of frequent occurrence, and it is a strong motivation for
further mathematical research” ([1], p. 178). Mathematician’s work is thus
strongly influenced by his aesthetic judgments.111About the significance of
aesthetic judgments for science see [2]. On the other hand, the main objective
of mathematical research is truth. Above all, we want our theories to be
consistent and our results to be correct. It is not immediately clear how
these two objectives are connected and why aesthetic properties such as beauty
serve as the driving force for something that is ultimately unrelated to
aesthetics.

In this article, we analyze the concept of mathematical beauty, as one of the
central aesthetic concepts in mathematics and examine its relationship with
non-aesthetic properties of mathematical results, such as truth, simplicity,
fruitfulness, etc., as well as other aesthetic concepts such as elegance and
depth. We don’t strive to offer a complete characterization of all aesthetic
aspects of mathematics (for a discussion on some of those aspects not
mentioned here see [3]), nor do we aim to give a psychological or a
phenomenological account of aesthetic experience. But we do hope that our
discussion will shed some light on the nature of mathematics itself and what
we believe to be some of the key problems to understanding the subject of
beauty and the aesthetics of mathematical discovery.

Some authors denied the possibility of making any proper aesthetic judgments
involving beauty in mathematics. They assumed beauty must have a perceptual
component which mathematics lacks [4]. It is easy to see that this argument is
unfounded, as non-perceptual beauty is present not only in mathematics but
also in other domains, such as literary art ([5], section 7). So, the first
step in our analysis of mathematical beauty would be to acknowledge its
conceptual nature and the fact that this specific type of beauty requires
prior knowledge to be recognized. Mathematical beauty is “dependent upon the
mathematician’s background knowledge, so it is usually recognizable only to
the well trained” ([5] p. 346). Yet, there is still a lot to be done if we are
to fully understand this concept.

In the literature addressing the question of beauty in mathematics, two main
strands can be distinguished. The first one suggests that the aesthetic
criteria in mathematics are subjective, reflecting the preferences and
expertise of the individual making the aesthetic judgment, as well as the
context in which the judgment occurs. According to this view, asserting the
beauty of a mathematical object or a mathematical result does not derive its
justification from an objective fact, but from the preferences of the subject
making the assertion (e.g. [1]). The second strand takes the criteria for
mathematical beauty as objective and grounded in the content judged as
beautiful. According to this stance, anyone with sufficient knowledge of the
content should be capable of recognizing its beauty. Theories aligned with
this perspective often link mathematical beauty to epistemic qualities such as
truth, clarity, explanatory power, depth, and fruitfulness. Some of them have
even attempted to reduce mathematical beauty to these non-aesthetic qualities,
claiming that aesthetic judgments in mathematics are not really aesthetic at
all ([6], also see [7]).

The main problem with the first perspective is its inability to explain the
connection between beauty and truth (and other epistemic qualities). “Real
beauty is for mathematicians inseparable from truth, but, more than that, it
is a sign of truth, and moreover of important truth” ([8], p. 34). If beauty
is relative to the subjective preferences of mathematicians, how can it serve
not only as the intrinsic motivational force behind mathematical progress but
also as the standard by which we judge the extent of this progress?222Here, we
make a reasonable assumption that both mathematical progress and progress in
science generally are based on objective facts rather than preferences. On the
other hand, the second view faces the problem of how to define the properties
that constitute mathematical beauty while at the same time accounting for
their specific aesthetic qualities. Claiming that mathematical beauty is
reducible to epistemic qualities might suggest that mathematicians, when
making aesthetic judgments of their subject, misconceive its epistemic
properties as aesthetic ones. In our view, this shows there is something
deeply problematic with this claim as it seems to presuppose that
mathematicians aren’t competent enough to differentiate between their own
aesthetic and epistemic judgments, while we possess better insight into the
intentions of their statements.

The account presented in this paper belongs to the second strand. We argue
that mathematical beauty is objective, forming a distinct aesthetic category
irreducible to non-aesthetic or other aesthetic categories from outside
mathematics. However, we maintain that mathematical beauty is closely related
to truth and other epistemic qualities, because it serves as a guiding light,
directing our attention to what is fruitful and deep and guiding us towards
important discoveries. Just like the beautiful color, shape, and overall
aesthetically pleasing appearance of fruit helps us instantly spot pieces of
fruit that are fresher, nutritionally richer, and free from harmful pathogens,
beauty in mathematics helps us recognize those structures and properties of
mathematical objects that are in the mathematical sense more fundamental. An
advantage of our account lies in its ability to explain the enduring focus on
symmetry, simplicity, and harmony within discussions of mathematical beauty,
while also illustrating how these attributes are intertwined with the notion
of truth in mathematics.

The structure of this paper is as follows. In the second section, we present
the proposed account in more detail. As the perspective presented in this
paper draws inspiration from Plato and his idea about the role of beauty in
acquiring knowledge of Forms, in section 3 we briefly discuss Plato’s views on
beauty and their connection to mathematics. We then move on to discuss the
three levels of mathematical entities: objects, theorems, and proofs, and the
three corresponding notions of beauty in section 4. In section 5 we propose a
unified account of mathematical beauty and discuss some possible problems and
solutions.

##  2 A purpose-driven beauty

The account we are proposing includes three components. As we already
mentioned in the introduction, we claim that:

  * $\cdot$

Beauty in mathematics is objective;

  * $\cdot$

It is not reducible to non-aesthetic epistemic properties;

  * $\cdot$

It is connected to truth and other epistemic properties by pointing out to and
guiding us toward them.

Each of these claims requires some explanation. It might be best to begin with
the following example given by the physicist Steven Weinberg ([9], p. 133):

“A physicist who says that a theory is beautiful does not mean quite the same
thing that would be meant in saying that a particular painting or a piece of
music or poetry is beautiful. It is not merely a personal expression of
aesthetic pleasure; it is much closer to what a horse trainer means when he
looks at a racehorse and says that it is a beautiful horse. The horse trainer
is of course expressing a personal opinion, but it is an opinion about an
objective fact: that, on the basis of judgments that the trainer could not
easily put into words, this is the kind of horse that wins races. […] The
physicist’s sense of beauty is also supposed to serve a purpose – it is
supposed to help the physicist select ideas that help us to explain nature.”

This example shows that we are indeed capable of contemplating and assessing
beauty in at least two different ways. On one hand, the judgments about the
beauty of nature are devoid of any purpose or context. We assess the beauty of
an object solely based on how it affects us. For instance, we might call a
running horse beautiful because witnessing a horse running free in the wild is
simply a beautiful sight to behold and an aesthetically pleasing experience.

On the other hand, there is a different kind of aesthetic judgment where
beauty is assessed with a specific purpose in view. In this case, there is an
activity or a predetermined goal, and objects are judged as beautiful to the
extent to which they seem likely to fulfill that purpose or help us attain
that goal. We do not admire the object itself but rather what it represents
for that particular activity. In Weinberg’s example, such would be the
trainer’s assertion that the horse is beautiful. The fact that we can clearly
distinguish between these two contexts and acknowledge that a horse may be
beautiful in the first sense but not in the second one (for example, having a
long mane or a delicate neck may contribute to an aesthetically pleasing
appearance, but a horse with these features might not be beautiful in the eyes
of the horse trainer) shows that there are indeed two distinct types of
aesthetic judgments. The distinction between the two types of aesthetics
judgments is somewhat similar to Harré’s distinction between first order and
second order aesthetic appraisals (see [7]).

But unlike Harré, we believe both types of judgments are aesthetic because
they possess a normative component and they are connected to a specific type
of aesthetic experience, such as satisfaction or admiration for the properties
that evoke the judgment. This enables one to distinguish an aesthetic judgment
from a mere description of an object.333The inability to differentiate between
normative and purely descriptive judgments is in philosophy called the
naturalistic fallacy, which is a concept famously introduced by George Edward
Moore in [10].

While Weinberg addresses the question of beauty in physics rather than
mathematics, his viewpoint can be extended to encompass mathematics as well.
In the subsequent paragraphs, we aim to demonstrate that the beauty
encountered in mathematics is contextual, akin to the beauty of the racehorse
in the earlier example. For instance, a mathematical proof is not beautiful in
the same way a flower is; its beauty lies in its ability to serve a specific
purpose — to offer insight into the realm of mathematics in a way that evokes
fascination or admiration.

Weinberg’s example also serves to illustrate the view that our judgments of
beauty in mathematics are grounded in facts rather than subjective
preferences. Although the aesthetic experience and the associated feelings of
fascination or admiration are inherently subjective, their evocation in
mathematics is always tied to an objective property or relation between
objects under consideration (compare this with how we subjectively experience
warmth, this experience being caused by the objective fact of sitting next to
a fireplace). The horse trainer’s judgment about the horse’s beauty is based
on the properties of the horse that reflect its ability to win races, rather
than on the trainer’s subjective preferences (such as a preference for a
certain color or the length of the tail). In the same way, we argue that the
property or set of properties evoking this particular type of aesthetic
experience in mathematics is not relative to a mathematician’s subjective
preferences but is rather justifiable on objective grounds.

But how exactly to define beauty and explain its connection to truth in
mathematics? According to Weinberg’s proposal, beauty is supposed to help a
mathematician select ideas that can better explain the world of mathematics,
and thus be closer to truth. Along these lines, we offer an account of beauty
according to which:

Beautiful mathematics is one in which a great deal of the entire mathematical
world is reflected.

Our definition admits of varying degrees of beauty depending on the intensity
and extent of this reflection. In other words, the more an object reflects the
totality of the mathematical world, the more beautiful it is. This criterion
should apply not only to mathematical objects but to mathematical results as
well. A beautiful result allows better and wider insight into the mathematical
reality as a whole. It reveals connections between apparently non-related
areas or problems, it is closely related to many other results and thus it
helps us expand our view of mathematical reality. This explains a close
connection between epistemic and aesthetic values in mathematics. If a result
provides better and wider insight connecting non-related areas, it is likely
to be more fruitful and explanatory potent.

The focus of mathematics is not on understanding specific objects and facts,
but rather on using objects and facts as lenses through which we can observe
more general laws and the intricate structures of the mathematical universe as
a whole. A piece of mathematics is considered beautiful inasmuch as it helps
us attain that goal and drives our attention toward results and properties
that offer a more general and deeper understanding of the subject. Some other
views also connect mathematical beauty to some form of comprehension (see [5]
and [1]). Rota centers his account of mathematical beauty on the concept of
enlightenment, and describes beautiful mathematics as one that is
enlightening, while Cellucci prefers the term understanding, as enlightenment
might misleadingly suggest that appreciating mathematical beauty is
instantaneous and doesn’t presuppose prior knowledge of the subject.

Our perspective ultimately draws inspiration from Plato and his idea about the
role of beauty in acquiring knowledge of Forms. In the following chapter, we
briefly discuss Plato’s views on beauty and their connection to mathematics.

##  3 Plato on revealing the unity beneath diversity

In Timaeus, Plato narrates how god created the universe using the most
beautiful mathematical objects as a model. By a beautiful mathematical object,
Plato means “a straight line or a circle and resultant planes and solids […]
these things are not, as other things are, beautiful in a relative way, but
are always beautiful in themselves” ([11], p. 51). With shapes and numbers,
god brought order into the chaos that used to rule. He gave the universe a
perfectly spherical shape “because there is no shape more perfect and none
more similar to itself – similarity being, in his opinion, incomparably
superior to dissimilarity” ([12], 33b). This refers to the fact that a perfect
sphere is completely symmetrical around its center (it has infinitely many
symmetrical transformations). Using other superiorly beautiful figures, god
created fire, air, water, and earth. He made them “as beautiful and as perfect
as they could possibly be” ([12], 53b). These figures are so-called Platonic
solids: tetrahedron, octahedron, cube, and icosahedron. They are composed of
two basic triangles. Both of them have a right angle, but one is isosceles,
and the other is, according to Plato, the most beautiful of scalene triangles,
that is, the half of an equilateral triangle. Besides these four beautiful
solids, Plato mentions dodecahedron as the shape of the elements of ether.

According to Plato, the perfect sphere is aesthetically superior to the other
geometric bodies because it is the most similar to itself, that is, the most
symmetrical and proportionate. The beauty of the five solids can be explained
similarly. The symmetry of these solids is the consequence of the fact that
their faces are identical regular polygons, with the same number of them
meeting in each of the vertices. These objects are nowadays known as regular
polyhedra.

In Plato’s philosophy, the experience of beauty, including that of
mathematical objects, plays a significant role in the pursuit of ultimate
knowledge which is for Plato the knowledge of Forms. Forms, according to
Plato, are universals or concepts that manifest themselves in individual
objects but exist independently of them. Plato believed that only Forms have
real existence, being eternal and changeless, while all other concrete objects
are merely their reflections. Hence, he considered the knowledge of Forms to
be the ultimate and the only true knowledge. The soul, being immortal, has
already come to know the Forms, and the process of learning is the process by
which the soul recalls this knowledge.

In the ascent of the soul toward the perception of the Forms, an important
role is ascribed to mathematics, the science that deals with measure,
symmetry, and shape, in which beauty is contained.444As we use it here, the
term ‘science’ refers to any systematic and methodological study of reality.
Beautiful mathematical objects, such as those mentioned above, are especially
likely to revive the recollection of the Forms. Namely, beautiful things
remind the soul of beauty itself, which belongs to the world of Forms, and
facilitate contemplation or recollection of this world. Through beauty “we are
brought to recognize and value the ideal properties of the Form of beauty and,
indeed, of all the Forms” ([13], p. 205). Beauty has such an important role in
prompting the recollection of the Forms because the Form of beauty can “shine
through its instances” and show in them its ideal properties ([13], p. 214).
It helps the soul find the unity in diversity, that is, recognize the Form of
beauty in its different manifestations.

The motivation for our view on the beauty in mathematics can be found in
different aspects of Plato’s philosophy. Plato took beauty as something
objective and closely related to the possibility of gaining real knowledge,
which is a view we aim to develop here applying it to the beauty in
mathematics. In this, we follow Plato’s idea about the nature of real
knowledge. Plato believed that this knowledge is not related to a particular
object and its specific properties. Rather, it can be reached only when it is
seen past particularities and when the aspects of objects that represent
universal laws or ideas, i.e. Plato’s Forms, are brought to light. These Forms
unite objects that on the surface might seem very diverse.

In the next sections, we present our account of beauty in mathematics inspired
by Plato’s views on the role of beauty in revealing “the unity beneath
diversity”. We explain what this means in the context of mathematics and how
particular features of beautiful mathematical objects, such as symmetry,
follow from this account. Before that, we distinguish different types of
mathematical beauty and present their examples.

##  4 Three notions of beauty in mathematics

We distinguish three types of mathematical beauty, each corresponding to a
different category of entities that can be characterized as beautiful:

  * $\cdot$

the first one is the beauty of mathematical objects and the structures they
form;

  * $\cdot$

the second one is the beauty of theorems that describe these objects;

  * $\cdot$

the third type encompasses the beauty found in the proofs of these theorems
and the tools they employ, including axiomatizations, definitions, methods and
techniques, etc.

The following metaphor describes how one can understand the relationship
between the three categories of mathematical entities. The mathematical world,
consisting of mathematical objects and the structures they form, resembles a
landscape. Mathematical theorems are like photographs of the landscape, which
can be captured from various angles and focused on distinct elements. On the
other hand, proofs and other techniques used in reaching these theorems can be
compared to camera lenses, which allow us to view the mathematical landscape
from specific vantage points. The camera enables us to capture pictures of a
landscape with varying information content and clarity. This depends on the
specific part of the landscape we choose to focus on, our camera setup, and
the lenses used.

According to our perspective, the beauty of any mathematical entity is
connected to its place within the corresponding level of mathematics. Its
beauty arises from its relationship with other entities at the same or
different levels, rather than from any specific properties of the object
itself.

In the following chapters, we provide examples of the three types of beauty
which we hope will further illuminate our perspective. We also try to
determine the common characteristics that unify these notions into a general
underlying concept of mathematical beauty. Although mathematical objects,
theorems, and proofs have different types of aesthetic qualities and manifest
different aspects of beauty, we believe that one of these notions is
fundamental, as it serves as the underlying basis for the other two.

###  4.1 Beauty of mathematical objects

The following examples show the difference between the aesthetic appeal of
mathematical objects and other kinds of beauty we find in art and nature.

(a) Vase

(b) Gongshi

(c) Cube

Figure 1:

Let us consider three objects - (A) a painted Greek vase, (B)
Gongshi555Gongshi, also known as scholar’s rocks, are naturally occurring
rocks which are traditionally appreciated by Chinese scholars., and (C) a cube
- and ask ourselves about their aesthetic value. If an ordinary person, not
particularly interested in mathematics, were asked which of them is (most)
beautiful, the answer would probably be (A) or (B).

However, if a mathematician were asked which of the three has the greatest
aesthetic value of the mathematical type, the answer would be (C). To see more
clearly why, let us examine what aesthetic values these three objects
exemplify, and how they relate to mathematics.

(A) Painted Greek vase. This is clearly an art object, whose aesthetic value
is assessed according to the criteria we use to assess the beauty of any
artwork. However, the fact that the object is a vase, possessing that
particular shape which is characterized by its distinct conic form, is an
important aspect as well. If we were to remove the painting and abstract the
shape by eliminating the handles, we would observe that the shape we are left
with is very mathematical. As a mathematical object, it reflects the concepts
of circle and exhibits perfect rotational symmetry. Therefore, the beauty of
the vase as an artwork is intertwined with the beauty of the mathematical
objects it takes its shape from.

(B) Gongshi. Here we have an object that occurs in nature – a stone that is
awkwardly asymmetrical and textured but also beautiful in its own right. Being
an object of nature, the formation of this rock can be explained through laws
of physics which are in turn expressed through mathematics. For example, the
eroded surface of the rock can be modeled with certain dynamical systems; the
structure of the rock can be explained by crystallography whose laws are
governed by Euclidean geometry, etc. So much mathematics can be connected to
the object, but none of that mathematics is specific to it nor is it inherent
in it in a way the conic shape is inherent in (A). In other words, the beauty
of this object has nothing to do with the mathematics used to describe it.
Since no mathematical object or concept fundamentally underpins the object, we
can say that it doesn’t reflect any mathematics and hence it does not possess
the beauty of mathematical type.

(C) Cube. There are very few things occurring in nature that resemble a cube
and we might safely say that a cube is not an object of nature at all. And
while cubic shapes can occur in artwork (for example, we could have had a
painted cube instead of a painted vase), cube qua cube is no work of art. It
is a mathematical object and it is considered a very beautiful one. But the
aesthetic appeal of a cube is different from the aesthetic appeal of a vase or
a gongshi. The conic shape of a vase, we saw, reflects the concept of
rotational symmetry. The cube, similarly, reflects the notion of cubic
symmetry. However, unlike the conic shape of the vase, the cube reflects so
much more than its particular symmetry. Indeed, it sits at the confluence of
what we would classify as many distinct branches of mathematics. In no
particular order, one sees the cube as (1) convex polyhedron, (2) regular
polytope, (3) zonohedron, (4) Wythoff, i.e. kaleidoscopic construct, (5) unit
ball of a metric, (6) topological sphere, etc. All these “faces” of the cube
open a window to whole fields of mathematics. Moreover, by looking at all the
“faces” together we glimpse at the unity of all those fields as well. This
makes a cube an object of mathematical beauty par excellence. As already
mentioned, our account allows for varying degrees of beauty in mathematics. So
one can say that a cone, although a beautiful mathematical object, is however
less beautiful than a cube since it reflects a smaller part of mathematical
reality.

In some of our examples, and across literature on mathematical beauty, a
prominent place is given to the concept of symmetry. Our account explains why
this is the case. The symmetry is, naturally, often thought of as “geometrical
symmetry”, which has an important place in visual arts as well. However,
symmetry is a much deeper concept, which we’ll briefly clarify. An invariant
is a property of a mathematical object that remains unchanged when
transformations of a certain type are applied to it. For example, the degree
of a polynomial is invariant under the linear change of variables; the area of
a figure in the plane is invariant under isometries of the plane, etc.
Especially important are the transformations under which an entire object is
left unchanged, i.e. under which the object itself is invariant. These
transformations are called symmetries. The set of transformations that fix
invariant forms a group. In particular, when the transformations are
symmetries, we speak of the group of symmetries. Some mathematical objects
share their group of symmetries. For example, the group of symmetries of the
cube is $O_{h}$ ($S_{4}\times S_{2}$). The same is the group of symmetries of
the octahedron (indeed the standard name for the group in question is
“octahedral group”). Therefore, cube, octahedron, and group $O_{h}$ (a bona
fide mathematical object in its own right) are directly and inseparably
connected. In our somewhat poetic language, they are all reflected in any of
them. The more objects that have a particular group of symmetries in common,
the larger segment of mathematics each of them reflects, and the more
mathematically beautiful they are.

According to our account, the concept of symmetry occupies such a significant
role in discussions of mathematical beauty because it makes the connections
between different mathematical objects particularly evident. Nevertheless, it
is only one of many ways in which these connections occur. As we’ve seen in
the case of a cube (and we’ll see in other examples to follow), mathematical
objects and areas are connected in other important ways. Each connection
allows an object to reflect related objects and makes it more beautiful.

The kind of beauty that consists in the ability to reflect different objects
and areas is specific to mathematics and it isn’t necessarily valued in art or
nature. In these areas, what sets an object apart as particularly beautiful
often has nothing to do with its connection to other things. Rather, what is
valued is usually the object’s particularity and uniqueness. The appeal of
expressionistic paintings, for example, lies in distortions of reality which
serve to express a unique and subjective vision of the artist. In this regard,
mathematical beauty is very different from the beauty of artwork. It comprises
a separate aesthetic category, which is bound and driven by a specific purpose
of describing and understanding the abstract domain of mathematics.

###  4.2 Beauty of theorems

In Hardy’s words, mathematical theorems are “notes of our observation” of
mathematical reality ([14], p. 124). Although their purpose is to provide us
with insight into the world of mathematics, they can also be seen as something
that possesses aesthetic value. But what makes a theorem aesthetically
appealing and what is the relationship between the beauty of a mathematical
theorem and the beauty of the object it depicts?

In line with our metaphor (p. 7), a theorem allows us to paint a picture of
the mathematical world. A picture can be beautiful because it depicts a
beautiful part of reality, but also because it is made in a beautiful way.
Similarly, theorems can be beautiful because they describe a beautiful object
or a relation between beautiful objects. On the other hand, they can also be
appreciated because of the light they throw on these objects and their
relations. Beautiful theorems might allow us to observe them from a
particularly suitable angle, or to recognize their particularly important
features, thereby illuminating an aspect of mathematical reality that
significantly improves our understanding of it.

According to this account, the beauty that characterizes a theorem doesn’t
represent a new kind of beauty. Rather, it depends on the beauty of its
content. A beautiful theorem allows us to recognize beautiful aspects of
mathematical reality and it brings our attention to general and recurring
features of mathematical objects that make them beautiful. The theorem,
therefore, owes its beauty to the beauty of the object it describes.

This view explains why mathematicians often say that there can be no beautiful
theorems about trivial facts or uninteresting objects. It can also explain how
the beauty of mathematical theorems, as we see it, can be related to
scientifically valuable results. According to Poincare, those are the results
that show similarities hidden under apparent discrepancies and “unite elements
long since known, but till then scattered and seemingly foreign to each other”
([15], p. 30). They introduce order into apparent disorder and thus help us
navigate physical or mathematical reality.

###  4.3 Beauty of proofs

A mathematical proof is not a mere verification. Proofs are the means by which
we come to comprehend mathematics and advance our knowledge. They too can be
subjected to aesthetic assessment and are often judged as more or less clear,
concise, simple, motivated, elegant, or beautiful. Unlike theorems, proofs
appear to possess a twofold nature. While essentially dependent on the
theorem’s content (as suggested by Rota, who posited that there are likely no
beautiful proofs of ugly theorems in [1], p. 172), proofs also reflect our
approach to understanding and manipulating this content. How do these two
facets of proof interplay in endowing it with aesthetic properties?

On one hand, the aesthetic appeal of a proof can stem from the method used.
Techniques such as visualization or diagonalization, for instance, can make
the proof remarkably simple and effective. However, such proofs are better
qualified as elegant than strictly speaking beautiful. According to Rota,
“mathematical elegance has to do with the presentation of mathematics, and
only tangentially does it relate to content” ([1], p. 178). In other words,
elegance is an aesthetic property that has nothing to do with what is proved
but only with how.666For an interesting analysis of elegance of proofs see
[7].

On the other hand, some proofs are called beautiful because they reveal the
essence of the theorem they prove. The beauty of such proofs hinges on the
theorem’s content as well as the proof’s capacity to elucidate it and to
reveal why the theorem holds (cf. [16]). The proofs that can help us explain
the why of the theorems, so to speak, are also called deep. They are often
general, in the sense that they contain ideas that can be applied to a family
of similar problems. Such ideas, “constituent in many mathematical constructs”
and “used in the proof of theorems of many different kinds”, Hardy calls
significant ([14], pp. 103-109). Deep proofs are based on some fundamental
properties of the mathematical world, not confined to a particular object or
objects of a particular kind.

The beauty of mathematical proof thus rests on two pillars – first, its
effectiveness in verifying the proposition in question (its elegance); and
second, its connection to general mathematical concepts (its depth). We will
try to illuminate the distinction by presenting some examples of proofs. We
engage in their comparative analysis, aiming to determine which one exhibits a
greater degree of beauty and why. Throughout this examination, we delineate
the mathematical facets contributing to their aesthetic appeal. This involves
assessing whether the beauty we find in these proofs falls into our category
of elegance or depth.

###### (A) Proof of Pythagoras Theorem.

Figure 2: (A) Proof of Pythagoras Theorem.

To see that the two pictures contain proof of Pythagoras theorem, one only
needs to realize that the yellow square in the first picture can be decomposed
into two squares from the second one. This is accomplished by rearranging the
four triangles. Since the yellow square represents the square over the
hypothenuse $c$, the red square is a square over the longer of the remaining
sides $a$, and the blue square over the shorter one $b$, this shows that
$c^{2}=a^{2}+b^{2}$. ∎

###### (A’) Another proof of Pythagoras Theorem.

Here, the idea is to show that the areas of the same color are equal, from
which Pythagoras theorem easily follows. That pink-colored areas are equal is
shown by constructing the two triangles, $ABD$ and $FBC$. It is first shown
that the triangle $ABD$ is half of the rectangle $BDLK$ using the lemma
according to which the area of a triangle is half the area of any
parallelogram on the same base ($BD$) and having the same altitude ($AK$). For
the same reason, the triangle $BFC$ is half of the square $ABFG$. Next, it is
shown that the two triangles are congruent because they share two sides and
the angle between them. The equality of the blue areas is shown similarly. ∎

Most mathematicians would agree that (A) is a beautiful proof. It is witty and
a bit unexpected, mostly because it can be expressed entirely in two neat
pictures. What is also pleasing is the ease with which we come to realize that
the theorem holds just by looking at the pictures. However, the aesthetic
quality of the proof derives in great part precisely from those pictures that
are in fact perceived as art (they resemble an abstract art piece). Setting
aside its artistic appeal, one is left with an effective and elegant argument
that, in the course of verifying the theorem in question, presents the
important ideas of congruence and equidecomposability. These ideas make the
proof in question deep.

The proof (A’) of the same theorem is surely less beautiful. It uses an
auxiliary construction that helps to show that the theorem holds but it does
not reveal why. It is made purposely to arrive at the proof in question and
lacks further motivation. The truths it relies on concern the equality of two
figures and the means of establishing it. They cannot be used for proving
similar theorems (such as the one saying that the square over the side
opposite the obtuse angle of the triangle is strictly greater than the sum of
the squares over the two other sides). The proof doesn’t contain deep ideas
that reappear in different places and that can be used in proving other, even
similar, theorems. That is why it is not particularly deep. It is also much
less elegant than the proof (A), i.e. it is longer and more difficult to
follow (for a similar analysis of the aesthetic worth of this proof and its
comparison to another proof of Pythagoras theorem, see [5], pp. 347-348).

###### (B) Proof of the infinity of primes.

Let $\mathbb{P}\subset\mathbb{N}$ be the set of all primes. Suppose that
$\mathbb{P}$ is finite, i.e., $\mathbb{P}=\\{p_{1},\ldots,p_{n}\\}$. Let
$N=p_{1}\cdots p_{n}+1$. Clearly, $N>1$ and is indivisible by any
$p\in\mathbb{P}$. Therefore, there must exist a prime $P$ (possibly $N$
itself) that is not in $\mathbb{P}$, which contradicts the assumption that
$\mathbb{P}$ is the set of all primes. Hence, $\mathbb{P}$ is infinite. ∎

In comparison to the proof in (A), the proof in (B) has no artistic elements
that need to be set aside – its content is purely mathematical. It too gives
an effective and elegant argument (perhaps slightly less elegant than (A)s due
to its slightly lengthier form) and presents an important idea of “proof by
contradiction” and an archetype of a constructive method of refuting the
assumed finiteness of a set. Given how widespread this idea is (much more so
than the strictly geometrical ideas of congruence and equidecomposability
shown in (A)) we can say with confidence that (B) is more beautiful than (A).

###### (C) Proof of the logical truth $((p\rightarrow q)\land p)\rightarrow
q$.

Consider first its proof in a Hilbert-style, axiomatic formal system:

$(((p\rightarrow q)\land p)\rightarrow(p\rightarrow
q))\rightarrow((((p\rightarrow q)\land p)\rightarrow
p)\rightarrow(((p\rightarrow q)\land p)\rightarrow q))$ $((p\rightarrow
q)\land p)\rightarrow(p\rightarrow q)$ $(((p\rightarrow q)\land p)\rightarrow
p)\rightarrow(((p\rightarrow q)\land p)\rightarrow q)$ $((p\rightarrow q)\land
p)\rightarrow p$ $((p\rightarrow q)\land p)\rightarrow q$

∎

###### (C’) Another proof of $((p\rightarrow q)\land p)\rightarrow q$.

This proof is made inside a natural deduction formal system:

$[(p\rightarrow q)\land p]$ $p\rightarrow q$ $[(p\rightarrow q)\land p$] $p$
$q$ $((p\rightarrow q)\land p)\rightarrow q$

∎

Even though here we deal with formal proofs of logical truth, we can still
discern properties similar to those found in mathematical proofs, that make
some of them more beautiful than others.

The proof in (C) is a result of a clever use of available axioms. All we need
to do is find a way to instantiate those axioms that allow us to infer the
formula to be proven by available rules of inference. It shows how
$((p\rightarrow q)\land p)\rightarrow q$ follows from the distributivity of
implication and the axiom standing for the elimination of conjunction, by
repeated application of modus ponens. Since these axioms are logical truths,
and modus ponens is a sound rule of inference, we can be sure that
$((p\rightarrow q)\land p)\rightarrow q$ is a logical truth as well. However,
the idea of the proof remains unclear. How did we know which axioms to use in
the proof, and what is the role of distributivity in making $((p\rightarrow
q)\land p)\rightarrow q$ true? Are the logical truths we used in the proof
more fundamental than the truth we proved? The proof lacks insight into the
meaning of the proven formula and fails to introduce any significant ideas
that could be used in dealing with other formulas of the same kind.

Axiomatic logical systems abound with proofs that seem to be unmotivated by
the formula to be proven and lack a clear, potentially generalizable, idea.
This raises doubts about the suitability of these systems for rigorous study
of deduction.

Alternative formal frameworks, such as natural-deduction systems, offer a more
intuitive and coherent approach to exploring logical truths and deductions.
The proof in (C’) uses general ideas concerning deduction that are closely
related to the meaning of implication. It is based on a simple and clear idea
of how an implication is to be proven: by assuming its antecedent and deducing
the consequent from it. If we manage to do this, we have proven the
implication and our proof no longer depends on the assumption that its
antecedent holds. This assumption can thus be canceled. The proof in (C’) thus
unveils a crucial aspect of implication: an implicative statement can be
interpreted as asserting the potential deduction of its consequent from its
antecedent. Not only are the ideas used in the proof simple and natural, but
they are also very general in the sense that they can be used whenever we want
to prove an implication. They encode a very natural way of reasoning about
implicative statements that is used all over logic and mathematics. This makes
the proof in (C’) deeper than the one in (C). Moreover, its brevity,
directness, and ease of comprehension make it elegant as well.

The ideas contained in the natural-deduction proof are, in a sense, explicated
by its following version in sequent calculus:

$p\ \vdash\ p$ $q\ \vdash\ q$ $p\rightarrow q,p\ \vdash\ q$ $(p\rightarrow
q)\land p\ \vdash\ q$ $\ \vdash\ ((p\rightarrow q)\land p)\rightarrow q$

The turnstile can be read as ‘the formula on the right-hand side is deducible
from the formulas on the left-hand side’. The inference steps then show what
can be inferred about the deducibility relation between some other formulas.
Read in this way, the second row describes the rule of modus ponens. The
inference to the third row corresponds to the conjunction elimination rule as
it is used in natural-deduction proofs: it shows that if a formula is
deducible from the two formulas, then it is also deducible from their
conjunction. The last step describes the main idea of the proof of implication
which is that we prove an implication if we prove that its consequent is
deducible from its antecedent.

The sequent calculus makes the concept of deducibility central. It allows us
to explore the connections between deducibility of different formulas from the
same or different sets of formulas. It thus represents an important part of
the so-called general proof theory that deals with proofs in their own right,
and attempts to understand their nature and mutual relations. This is one of
the central goals of logic that could hardly be approached using axiomatic
formal systems. So, the proofs from the natural-deduction or sequent calculus
are not only more elegant, but they also allow us to approach the subject of
logic in a more direct and revealing way (cf. [8], pp. 32-36). They reflect
other proofs using similar ideas. Moreover, considered in this way, deductions
can be seen to build structures that reflect those found in algebra, topology,
set theory, etc. This is what makes these proofs and their study beautiful
according to our account.

##  5 The unity of mathematical beauty

At the beginning of the paper, we distinguished three kinds of beauty found in
mathematics, each pertaining to different kinds of mathematical entities –
mathematical objects, theorems, or proofs. However, the subsequent discussion
revealed the characteristics the three notions of beauty have in common.

According to our account, each notion of beauty in mathematics has to do with
the possibility of obtaining a wider and more informative insight into the
mathematical world. Mathematical objects are beautiful if they reflect general
features of the mathematical world that are found across different areas of
mathematics. They are the more beautiful the greater share of the mathematical
world they reflect. Theorems have a role to improve and widen our
understanding of the mathematical world and their aesthetic appeal depends on
how well they accomplish this task. It is best accomplished by the theorems
that direct our attention to the very general aspects of mathematical objects
shared with many others. But these are exactly those aspects that make the
objects beautiful. So, the beauty of theorems comes from the beauty of the
objects they describe. Finally, a mathematical proof is the more beautiful
(i.e. deep) the more and the deeper concepts it uses. The possibility of using
the same concepts and ideas in proofs regarding mathematical objects from
different areas depends on connections between the areas that are manifested
in the beauty of these objects.

The suggested perspective entails that the beauty of mathematical objects is
the most fundamental type of beauty in mathematics. Put metaphorically, the
beauty in mathematics is most akin to the beauty of a landscape. Its pictures,
in the form of theorems, are beautiful to the extent to which they manage to
represent faithfully its beauty, and the methods of making the pictures, i.e.
of proving theorems, are beautiful to the extent to which they make this
possible.

Since the beauty of mathematical objects depends on properties that do not
characterize beautiful objects from different areas, mathematical beauty, in
general, is to be considered a sui generis aesthetic category.

> Acknowledgements: We would like to thank Miloš Adžić for his help on
> improving the text with his comments and suggestions.

## References

  * [1] Rota, G.C. (1997) “The Phenomenology of Mathematical Beauty”, Synthese vol. 111, pp. 171-182.
  * [2] McAllister, J. (1996) Beauty and revolution in science, Ithaca: Cornell University Press.
  * [3] Thomas, R.S.D. (2016) “Beauty is not all there is to Aesthetics in Mathematics”, Philosophia Mathematica vol. 25, pp. 116-127.
  * [4] van Gerwen, R. (2011) “Mathematical beauty and perceptual presence”, Philosophical Investigations vol. 34, pp. 249-267.
  * [5] Cellucci, C. (2015) “Mathematical Beauty, Understanding, and Discovery”, Foundations of Science vol. 20, pp. 339-355.
  * [6] Todd, C. (2008) “Unmasking the Truth Beneath the Beauty: Why the Supposed Aesthetic Judgements Made in Science May Not Be Aesthetic at All”, International Studies in the Philosophy of Science vol. 22, pp. 61-79.
  * [7] Harré, R. (1958) “Quasi-aesthetic appraisals”, Philosophy vol. 125, pp. 132-137.
  * [8] Došen, K., and Adžić, M. (2017) “Gödel on Deduction”, Studia Logica vol. 107, pp. 31-51.
  * [9] Weinberg, S. (1994) Dreams of a final theory, New York: Vintage Books.
  * [10] Moore, G. E. (1922) Principia Ethica, Cambridge: Cambridge University Press.
  * [11] Plato (1975) Philebus, Oxford: Clarendon Press.
  * [12] Plato (2008) Timaeus and Critias, Oxford: Oxford University Press.
  * [13] Obdrzalek, S. (2022) “Why Erōs”, in The Cambridge Companion to Plato, second edition, Cambridge: Cambridge University Press, pp. 202-232.
  * [14] Hardy, G.H. (2012) A Mathematician’s Apology, Cambridge: Cambridge University Press.
  * [15] Poincaré, H. (1914) Science and Method, London: Thomas Nelson and Sons.
  * [16] Mordell, L.J. (1959) Reflections of a Mathematician. Canadian Mathematical Congress.