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MATHEMATICS AND VALUES
Paul Ernest
University of Exeter
ABSTRACT
There is a widespread perception that mathematics is objective and value-free and that
values only enter into mathematics subjectively in the personal preferences of
mathematicians. In this paper I argue that, on the contrary, mathematics itself is value-
laden; imbued with epistemological, ontological, aesthetic and ethical values. I identify
a range of values including truth, universalism, objectivism, rationalism, objectism and
ethics, including utility, and purity, that permeate mathematics. Although some of these
are accepted there is controversy both as to whether these are indeed values of
mathematics, or whether, such as in the case of truth, universalism and objectivism,
these are simply necessary concerns or properties of mathematics. I argue that values
are indicated by preferences expressed and choices made in terms of beliefs and
underlying frameworks. The initial choice of the framework of absolutism and its
underlying assumptions is not forced upon us by necessity. Once this choice is made a
consequence is that mathematics is ethics-free and largely values-free. However, other
choices of more humanistic philosophies of mathematics lead to the conclusion that
mathematics is imbued with ethical values, including openness, fairness, democracy and
that even pure mathematics is implicated in human well-being and is therefore imbued
with ethical values.
Introduction
Because of its objectivity it is widely perceived that values only enter into mathematics as a
subjective dimension, such as in the personal preferences or choices of mathematicians.
However, there is a tradition that sees values as objective. Plato (1941) argued for the objectivity
of values such as truth, good, and beauty. In effect, with these choices he proposes that values lie
at the centre of epistemology, ethics, and aesthetics. Furthermore, by positing the objective
existence of truth, good, and beauty as ideal entities Plato makes ontological claims, thus
bringing ontology into the universe of values discourse. If one adopts this perspective the values
at the heart of these four philosophical domains are therefore potentially central to mathematics
too. This leads to a controversial conclusion, namely that epistemology, ethics, aesthetics and
ontology are all central to mathematics and provide the categories within which its values fall.
The fact that each of these domains is not empty with respect to the values of mathematics and
contains specific identifiable mathematical values is something that I argue for below.
Epistemology is concerned with knowledge and truth and so is undoubtedly important for
mathematics, although some would deny that this is a matter of values. The superiority of truth
over falsehood and knowledge over mistaken belief are presumed to be self-evident. As Milton .
(1868: 24) wrote, "Let Truth and Falsehood grapple; who ever knew Truth put to the worse, in a
free and open encounter?" My claim is that some of the assumed goods of philosophy and
mathematics, like truth, rest in fact on the choice or valuation of one property over another, in
other words, on values.
Many agree that aesthetics is central to mathematics, at least in the activities and choices of
mathematicians who discover or invent mathematics. Certainly the claim that some aspects of
mathematics are beautiful is often heard (Hardy 1941, Rota 1997). But the claim that ethics and
its values are central to mathematics too, is perhaps the most controversial of these claims. For it
is often claimed that mathematics is objective, pure, and context and value-free, and that ethics
only enters in when it is applied to human affairs, and not before.
Lastly, the ontology of mathematics is also assumed to be objective and given solely by the
necessary existence of mathematical entities. However, controversies over ontological
frameworks for mathematics go back to the medieval Schoolmen and to the time of the ancient
Greeks, so ascribing the choices of competing ontological frameworks to the domain of values is
not so far-fetched.
In using the term ‘mathematics’ my reference is to the discipline of mathematics or more
properly to the body of mathematical knowledge. This includes the theorems and theories of
mathematics, and it also includes the problems, concepts, signs, proofs and methods embodied in
and related to this knowledge. This knowledge is extended and applied through human activity
but is not the identical with the activities of mathematicians. The issue as to whether
mathematical knowledge is humanly made, that is, is invented rather than discovered, remains
controversial. Although I personally adhere to a social constructivist philosophy of mathematics,
my intention is that the argument that mathematics is value-laden should not rely on the
assumption of this particular philosophy of mathematics.
In this paper I consider the values of mathematics as opposed to the reasons why mathematics is
valued, although in some cases these overlap. There is much to be said about the reasons why
mathematics is valued. Numerous publications have treated the images and perceptions of
mathematics (Ernest 1991 & 1995, Davis and Hersh 1980), attitudes and beliefs about
mathematics (McLeod 1992, Leder et al. 2003), the value accorded to mathematics in society
(Cockcroft 1982, Ernest 2014b), the value and values of mathematics in education (Bishop
1988), and further relevant societal values. However, in this paper I am not concerned with why
mathematics is valued, but with the question: what are the values of mathematics? My
investigation is primarily intended to be philosophical, a contribution to a little investigated
question in the philosophy of mathematics. I believe the results have implications for a number
of adjacent fields, including the history of mathematics, sociology of mathematics, psychology
of mathematics, and mathematics education. All of these fields deal with mathematics and
various aspects of its values. However, it is the philosophy of mathematics that is most resistant
to acknowledging that mathematics is deeply imbued with values and it is towards this domain
that I direct my argument.
Values versus Beliefs
My central claim is that values are manifested in both the prizing of certain characteristics and in
the making of fundamental choices. This fits with the work of Raths et al. (1966), for example,
who argue that values, as expressed through the action of valuing, are composed of three
dimensions: prizing, acting on one's beliefs, and choosing one's beliefs and behaviours. Thus
where there are alternative sets of beliefs expressing values, the preference of one set over
another can be said to demonstrate the values of the person making the choice.
In philosophy, ab initio, there are significant differences between beliefs and values. Within
epistemology beliefs are treated primarily as propositions which may or may not be warranted as
knowledge. Values, as I have argued above, are not the subject of any one branch of philosophy.
The theory of values, termed axiology in philosophy, primarily concerns ethics, morality and
aesthetics (Collins English Dictionary 2009). “'value theory' is roughly synonymous with
'axiology'. Axiology can be thought of as primarily concerned with classifying what things are
good, and how good they are” (Schroeder 2012). However, I use the term values with a broader
meaning here, to include, in addition, epistemological and ontological values.
Despite the initial appearances, the distinction between values and beliefs is less clear-cut that it
first appears in philosophy, and it is not always easy to apply in practice. For example,
universalism is listed below as one of the values of mathematics. Epistemological universalism is
the position that asserts that there are classes of mathematical statements which hold in all
contexts, at all times and for all persons and all possible life forms. This, then, is the belief that
mathematical knowledge is universally valid, at all times and for all rational beings. But this is
not a straightforward belief that can be settled by mathematical warrant or philosophical
argument. It is a meta-level belief about the system of knowledge and its overall constituents,
structure and foundation, so it could also be described as an ideological position or a world-view.
Since there is a range of competing meta-level beliefs in epistemology and elsewhere, the
cleaving to one perspective such as universalism represents the exercise of a preference or
choice, which is the sign of a value. So I include meta-level beliefs, such as universalism,
objectivism and rationalism among the values of mathematics.1
Some holders of absolutist beliefs including universalism, objectivism, foundationalism and
rationalism (Harré and Krausz 1996) argue that these are forced upon them by logical necessity,
and therefore these are not values but logical attributes of knowledge in general and
mathematical knowledge in particular. However, the fact that there are scholars that reject some
or all of these values in mathematics (e.g., Ernest 1998, Hersh 1997) without obvious logical
self-contradiction means that the acceptance of these beliefs cannot rest solely on logical
necessity irrespective of any choices, since others consistently exercise different choices,
preferences and indicators of valuation. This reasoning may well not satisfy absolutists who can
argue that universalism, objectivism, foundationalism and rationalism are an essential part of
mathematics, part of its very nature. It is true that from an absolutist position these ‘values’ are
not the result of any choices but necessary attributes of mathematics itself. Thus what constitute
values of mathematics and what are its necessary attributes depends in part on the
conceptualization of mathematics adopted. Evidently the characterisation of some of the values
described here as values of mathematics is controversial.
1More correctly, universalism, beauty and truth are values and the corresponding belief statements are Mathematics
is universal, beautiful and true (respectively). More generally, if X (in noun form) is a value of mathematics, the
corresponding belief statement is Mathematics is X (in adjectival form). Thus provided that all mathematical values
can be expressed in noun and adjectival form, then they can all be expressed as beliefs in this way. However, the
converse does not hold.
Such considerations notwithstanding, all values can be expressed as beliefs, although not all
beliefs can be expressed as values. Thus to subscribe to the values of truth, beauty and the good
is to believe that these are desirable goals and aspirations. Likewise, to believe that mathematical
objects exist independently of human beings in an ontological realm of their own is to uphold
and subscribe to the values of Platonism or mathematical realism. Thus although a distinction
can be drawn between beliefs and values there is an area of intersection in which some beliefs
are expressions of values and values are expressed as beliefs.
Objective versus Subjective Values
In opening I referred to the widespread view that values only enter into mathematics through
mathematicians’ subjectivity, and contrasted this with Plato’s view that there are a range of
values that can be termed objective. This perspective has certain consequences, for if the values
attributed to mathematics were purely subjective then they could be dismissed as personal
responses to mathematics rather than as objective features of the subject. However, if it can be
shown that mathematics does have objective values then they must be taken more seriously as
attributes of mathematics itself and not merely a feature of individuals’ relationship with
mathematics.
However, there is an ambiguity in the word objective and I wish to contrast two different
meanings. On the one hand, objectivity can refer to having an existence in the physical world as
a brute fact verifiable by the senses, or in the domain of knowledge, by dint of logical necessity.
I term this traditional or absolute objectivity. On the other hand, objectivity can refer to the
opposite of subjectivity in the sense of having an existence that goes beyond any individual
knower’s beliefs. I term this cultural objectivity. Laws, bank account balances, and indeed
language are objective in this cultural sense because their existence is independent of any
particular person or small groups of persons, but not of humankind as a whole. These are
evidently not the same meanings, because mathematical objects could exist in the social and
cultural realm beyond any individual beliefs (cultural objectivity) without having independent
physical existence or existence due to logical necessity (absolute objectivity).2
In the second, cultural sense objectivity is in effect redefined as social, as I argue in Ernest
(1998) drawing on the social theory of objectivity proposed by Bloor (1984), Harding (1986),
Fuller (1988) and others. This is how social constructivism views mathematical objects. Such a
perspective has a strong bearing on the discussion of values in mathematics because it posits that
at least some of mathematics is contingent on human history and culture and thus it allows that
mathematics itself can be imbued with the values of the culture of its human makers. Overall, a
consequence of redefining objectivity as cultural is that mathematics can be understood as being
laden with objective values from human culture, but that these are values that are not reducible to
either subjective beliefs about mathematics or logically necessary properties of mathematics.
Having treated some preliminaries I embark on the exposition of mathematical values themselves
ordered according to the branches of philosophy within which they are located.
2 To posit that mathematical objects exist in the social and cultural realm beyond any individual beliefs means
that they have a complex existence through the interaction of three realms 1. Individual knowledge, 2. symbolic
representations, and 3. The cultural realm, such as the institutions for teaching, researching and warranting
mathematical knowledge, including the actions of their members.
EPISTEMOLOGICAL VALUES - TRUTH
It might seem surprising that the relevance of truth to mathematics is a matter of values, for the
quest for truth and knowledge is intrinsic to the mathematical enterprise. It seems obvious that
truth is to be preferred to falsehood in mathematics. There is, however, mathematical work on
inconsistent and paraconsistent theories and logics in mathematics (Priest et al. 1989), and the
adoption of the falsehood sign ‘f’ as a primitive in mathematical logic (from which truth is
derived) is also valued. Likewise proof by contradiction, and proof by tableaux systems
(resulting in confutation, Bell and Machover 1977) in mathematical logic indicate that falsehood
is sometimes a short-term goal in mathematical quests, admittedly en route to establishing truth.
The same holds of proof by contradiction, a widely used and accepted form of argument used
since the time of the Ancient Greeks 2,500 years ago. Such proofs deliberately set out to derive
logical falsehoods or statements contradictory to the assumptions of the proof.3 Outside of
mathematics falsehoods in the form of statements or writings not expressing literal truths about
the world are widespread and important. These range from models and hypotheses in the
sciences, via hypotheticals and counterfactuals in the humanities and social sciences, to
metaphors, parables and fictions in the realms of literature. Thus falsehood in the sense of being
untrue has important roles across many domains of knowledge and culture.
Putting falsehood to one side, it is clear that truth is central in mathematics. However, the quest
for truth as a widely adopted, if not universally accepted value, is not entirely straightforward.
Establishing the truth of a mathematical statement must be done either directly or indirectly.
Direct access to truth would be by intuition, that is, the ‘mind’s eye’ sees or otherwise perceives
the truth of mathematical knowledge statements directly. This may be enough to engender belief
in the truth of mathematical knowledge for the person experiencing it, but it is not an adequate
basis to properly warrant mathematical truths. For we would need to persuade others that the
mathematical knowledge claim in question can be accepted without any further warranting.
Direct intuitions might be more persuasive if all had identical intuitions, but evidently not all do
share the same intuitions. The philosophical school of Intuitionism promoted the view that the
basis of mathematics is given by a shared pure intuition (Brouwer 1913). But the majority of
mathematicians and philosophers reject some of the so-called truths of mathematics put forward
by Intuitionists, not to mention holding on to much of the classical mathematical knowledge
rejected by Intuitionism. This shows that there is no consensus on the knowledge provided by
mathematical intuition. Furthermore, even if all mathematicians did agree, at any one time, on a
shared mathematical intuition, this would not guarantee that such agreement would last forever.
A shared belief needs an ironclad warrant to turn it into knowledge. Intuition, although valuable
and necessary in mathematics, cannot prove a fully reliable warrant for mathematical knowledge
in every case.
If there is no direct access to mathematical truth then access to mathematical truth must be
indirect, via reason or proof. In order to establish the truth of mathematical knowledge by these
means the following conditions are a minimum requirement. We must have:
1. A starting set of true axioms or postulates as the foundation for reasoning;
3 It should be noted that the Intuitionists including Brouwer (1913) and Heyting (1956) reject of proof by
contradiction as it rests on the principle of the excluded middle which they refuse to accept.
2. An agreed set of procedures and rules of proof that preserve truth, with which to derive truths
from the axioms;
3. A guarantee that the procedures and rules of proof are adequate to establish all the truths of
mathematics or at least of the theory in question (completeness); and
4. A guarantee that the procedures and rules of proof are safe in warranting only truths of
mathematics (consistency).
Elsewhere (Ernest 2014b) I have indicated how each one of these conditions raises technical
problems, the last two running afoul of the limitations imposed by Gödel’s (1931)
incompleteness theorems. But I will focus on values here, pausing only to note that prized as it is
the attainment of the value of truth in mathematics is not without difficulty. What this shows is
both the centrality of the value of truth and that attaining the value of truth in mathematics is
problematic. The attainment of proof for mathematical claims is difficult enough, absorbing
much of mathematicians’ ingenuity, even though it remains a poor surrogate for truth.
Nevertheless, truth and provability remain the central epistemological values within
mathematics.
It might be argued that this claim is trivial, and that all knowledge quests must be concerned with
truth and degrees of proof. My reason for bringing this forward as the first of my overt values of
mathematics is to use this case to breach the widely made claim that mathematics is value-free.
The very heart of the mathematical enterprise is value-driven, and I vanguard my claim with the
most central of these values, namely truth.
FURTHER EPISTEMOLOGICAL VALUES
Epistemology is the philosophical study of knowledge, knowing and the conditions for their
legitimacy. Truth is, of course, the central epistemological value in mathematics. In addition,
absolutism is a widely held position that enters into epistemology and offers further potential
values. Harré and Krausz (1996) identify a number of dimensions of absolutism that can be
regarded as epistemological values, including universalism and objectivism.
1. Epistemological universalism is the position that asserts that there are classes of
mathematical statements which hold in all contexts, at all times and for all persons and
for all possible life forms. They hold universally.
2. Epistemological objectivism proposes that there are classes of mathematical statements
which hold independently of the perspective, beliefs, or conceptual scheme of any
particular person or society or any possible life form. They hold independently of any
knower.
Superficially these are similar positions except that the former stresses the statements hold
everywhere including in all contexts and times, whereas the latter asserts the statements are
independent of all knowers. Both of the assumptions of epistemological universalism and
objectivism are assumed by traditional objectivity but not by cultural objectivity. These are
epistemological (or epistemic) beliefs and they reflect values (absolutist values) because they
represent a choice of ideological perspective or worldview, a preference, whether or not their
subscribers acknowledge that choice is in play.
I discuss below how values such as epistemological universalism and objectivism and the ethical
neutrality of mathematics follow necessarily from the assumptions of absolutism. Thus its
adherents, by choosing the unforced and free adoption of the philosophy of absolutism, are
subsequently committed to the above mentioned values and ethically neutral position
concerning mathematics. While these implications do follow by logical reasoning, the
underlying assumptions leading to these necessary entailments are freely chosen, and hence
embody an implicit values position.
Another epistemological value of mathematics is that of rationalism (Bishop 1988).
3. Rationalism holds that logical thought and abstract reasoning are to be valued in
mathematics above all else, and this includes the valorisation of logical analysis, proof,
argument, critique, testing and counterexamples.
As a value of mathematics rationalism sits close to truth since it concerns the means to uncover
and justify the truth of knowledge claims. However, rationalism extends beyond truth because it
also concerns the preferred means of analysing situations and solving problems. Sometimes this
includes seeking optimal solutions for non-determinate pure problems or for practical and
applied problems. This is opposed to focusing exclusively on uniquely true answers in
mathematics.
Rationalism is a value because it is a prized feature of mathematics. It is difficult to imagine how
mathematics would be possible without this characteristic or feature. Perhaps some of the
number-theoretic conjectures produced by Ramanujan might be said to be derived irrationally,
coming from pure intuition rather that overt processes of reason (Mahalanobis, n.d.).
Nevertheless, one of the reasons mathematics is attractive to many and valued by them is its high
degree of rationality. Thus rationality is an important value of mathematics deserving inclusion
in the discussion.
ONTOLOGICAL VALUES
Ontology is that branch of metaphysics concerning the study of pure existence. Thus its role is to
provide answers to the questions what is there? What exists? There are a number of ontological
assumptions or positions in the philosophy of mathematics. Traditional positions concerning the
existence of universals are nominalism, conceptualism and realism. These go back to the
medieval Schoolmen, or earlier, in some form, to the Ancient Greeks. Modern parallels in the
philosophy of mathematics are formalism, intuitionism and Platonism, respectively. However,
the positions adopted by mathematicians reflecting ontological or ontic value preferences, are
sometimes ambiguous. For example, Cohen (1971), and indeed Bernays (1935) before him, have
suggested that working mathematicians are content to think and act as realists or Platonists, but
when asked to justify their positions may claim to be formalists. I will not explore these
underlying ontological assumptions further except to note that the existence of alternatives
means that adhering to one or other position means that a choice is being exercised. Hence, by
the arguments employed above, ontological values are involved.
Adhering to absolutism in the philosophy of mathematics involves making a number of
ontological commitments. Harré and Krausz (1996) specify universalism and objectivism as
ontological assumptions for absolutism, and I include these as values of mathematics.
4. Ontological universalism: mathematical objects and classes of entities exist for all
persons and for all possible life forms. Mathematical objects are universal so they are the
same for any knower anywhere.
5. Ontological objectivism: there are mathematical objects and classes of entities which
exist independently of the perspective, beliefs, or conceptual scheme of any particular
person or society or any possible life form. Mathematical objects are objective in the
sense of existing independently of all knowers.
Analogous to the epistemological positions of universalism and objectivism these assumptions
stress that mathematical objects exist everywhere and independently of conscious beings. They
are ontological (or ontic) beliefs reflecting absolutist values. As in the epistemological cases,
ontological universalism and objectivism are assumed by traditional objectivity but not by
cultural objectivity.
These beliefs and values, both epistemological and ontological, are widespread in the
communities of mathematicians, philosophers, and further afield. They are long-standing,
legitimate, self-consistent and defensible beliefs, which go at least as far back as Plato. However,
in my view they are not forced on us by logical necessity and are by no means the only possible
positions in the epistemology and ontology of mathematics.
OBJECTISM
Objectism is an ontological or ontic value. It is the assumption of “a world view dominated by
images of material objects” (Bishop 1988: 65). Thus it is a view of the nature or constitution of
the entities that populate the world of our experience, be it phenomenological, conceptual,
physical, objective or some combination of these. I claim that this view has been underpinning
and infusing mathematics since its inception. Mathematics as a science originates in the
systematic accounting conducted in the ancient world (Mesopotamia, Egypt) based on the need
for records for trade, taxation and the like, and the training of scribes for this purpose (Høyrup
1994).4 Thus the basis of mathematics lies in counting, measuring and calculating and in using
the symbolic technology of numeration systems to record and cognitively underpin these
activities.
However, the idea of counting material objects is not a ‘naturally’ given one, as simple and
obvious as it looks to the trained modern eye. “It is really an ad hoc assumption to suppose that
we have before us the universe of things divided into subjects and predicates, ready-made for
theoretical treatment.” (Bernays 1935: 16). According to my analysis counting is based on a set
of prior conceptualizations of the world that include the following assumptions, which make up
objectism. These need to be taken for granted before counting and calculation can take place.
The five assumptions needed are as follows.
1. The world, or at least that part relevant for numeration, is understood to be made up of objects
that are permanent or semi-permanent entities which can be individuated and distinguished.
4Unsystematic oral protomathematics including number words far precedes the ancient civilizations and stretches
back into uncharted prehistory (Lambek 1996).
These entities are not naturally given but a product of the way that we conceptualize and describe
the world (Ingold, 2012). Such a conceptualization results from years of training, from early
childhood on. However, the person counting and calculating may well not be aware that this is a
conceptualization imposed on the world rather than a brute fact of nature. In Kantian terms, what
are perceived as countables are phenomena rather than noumena, and furthermore are culturally
shaped phenomena.
2. Objects as we perceive them fall into kinds, and for the purposes of accounting objects are
understood to be interchangeable within kinds, treatable as equivalent units.
It is our conceptualization of the world that makes what we see and experience as events and
entities individual and distinct and also makes some of them identifiable with each other and
interchangeable. Indeed the objects we individuate and work with are conceptual objects that
represent objects or events in the physical world and are not the physical entities themselves.
3. Objects of a single kind can be physically grouped into unified collections. However we
abstract our ideas of collections to purely abstract ones. Thus the idea of collecting objects
together extends both to multiple kinds within a single collection and to purely abstract
collections which are conceptually based and not designated by physically grouping objects
together.
All collections, physical or conceptual are themselves conceptual objects that represent
designated arrays of objects or events in the physical world and are not the physical entities
themselves.
4. Processes of counting or accounting can be applied to any collection, but during this process
the collection is viewed as a constant conceptual entity free from change, or else the process
of counting is invalidated.
All of the physical world is in the process of flow and change, but for counting and accounting
purposes the objects of our interest both individually and as a collection are timeless. This is
achieved by dealing exclusively with conceptual objects and our conceptions remain static and
timeless throughout all of our imaginary operations on them.
5. Any collection of objects can be counted resulting in a constant and invariant number, that is a
fixed cardinal.
We require this property for counting to serve its social purposes and it emerges from the
specific conceptualizations and operations that we have constructed through the elaboration of
objectism. Continuous as opposed to discrete objects, such as a body of water, piece of land,
heap of grain, etc., are also viewed by objectism as permanent or semi-permanent entities which
are decomposable into multiple units, that is as collections of permanent or semi-permanent
equivalent objects. These can then be conceptualized and treated as discrete objects for counting.
On the basis of these assumptions of objectism the basic laws of arithmetic follow for addition
understood as the operation of combining collections, that is, set union5 (Ernest 2014b).
Furthermore, these seemingly innocuous assumptions provide a foundation for an arithmetic that
5 From the ‘natural’ properties of set union it is easy to derive the properties of symmetry and associativity for the
addition of numbers (Ernest 2013). Of course this depends on our assumptions of objectism and its conceptual
properties which mean that we apply counting to our conceptualization of the world, which formats it
appropriately, not to the raw world itself.
necessarily conserves number under the usual operations of counting and calculating. Without
them arithmetic as we understand it could not exist.
Thus at the very heart of systems of numeration and measurement is the human requirement that
processes of accounting should conserve the material resources being recorded. Otherwise
accounts, trade agreements, taxes, and so on, would not be recorded in stable and fair ways to
perform the required social function, and be trusted and relied upon by all parties. Indeed,
according to Høyrup (1994), in ancient societies the reliability of calculation, measures and
numerical records was understood as part of the idea of justice, taking on an ethical value.
Objectism is not a necessary way of seeing the world. Its necessity arises in part because of the
social needs for numeration systems that are invariant with respect to the processes of counting
and calculating for the purposes of taxation and trade. Mathematics might conceivably have
developed without the assumptions of objectism, for example, if ritual and mystical functions
had been the dominant uses of number, although I do not explore this speculation further here.
The point is that objectism represents a value-dictated choice entered into in the development of
mathematics. This value-laden choice although primarily ontological or ontic has implications in
both the epistemological and ethical domains, and fundamentally shapes all of mathematics. For
number systems are, or can at least be formulated as, the foundation for most of mathematics.
Thus my claim is that even the simple and basic mathematical domains of number and counting
are imbued with values, including those of objectism.
AESTHETIC VALUES
The main aesthetic value prominent in discussions of mathematics is that of beauty. It remains an
open question as to whether beauty is an objective or subjective mathematical value. Rota (1997)
claims it is an objective value, presumably in the traditional sense of objectivity. From a
traditional perspective any determination is at least partly dependent on whether mathematics is
perceived to be superhuman or humanly created. In the former case beauty can be seen as a
human response to something independent of humanity. In the latter case, judgements and values
including beauty are an intrinsic part of the construction of mathematics. For present purposes
this controversy can be sidestepped, for beauty undoubtedly enters into mathematicians’
judgements and activities in appreciating, formulating and creating mathematics, whether it is
discovered or invented. Furthermore, if there were widespread agreement that some
mathematical knowledge and objects were beautiful, this would make such judgements objective
from the perspective of the social theory of objectivity.
The beauty of mathematics is not something appreciated directly through the sense organs, as is
the beauty of paintings, music or landscapes. Naturally, in the appreciation of beauty in paintings
and music the cognitive discernment of, and reflection on, structure plays a part, and this is not
immediately given by the senses. But in mathematics nothing but the symbols, figures or other
representations can be sensed and mathematical beauty is regarded as something deeper in the
domain of meaning and not that of signifiers. Some mathematicians have claimed that there are
beautiful equations, such as
e
iπ
+1 = 0, but in my view it is not the sign, i.e., the string of
signifiers itself that is judged as beautiful, but rather the surprising relationship signified by the
sign string. “It contains five of the most important numbers in maths: 0, 1, e, i, and π, along with
the fundamental concepts of addition, multiplication, and exponentiation - if that's not beautiful,
what is?” (IMA n.d.) This statement relates to the meanings and relations of the concepts
involved, not just the signs that denote them. Some of the features that make this equation
beautiful are its universality, economy, surprise, and the connections exhibited.
Since mathematical beauty cannot be appreciated directly through the sense organs, it must be
appreciated cognitively, through reason and the intellect, together with intuition and affect or
feelings. Mathematical beauty is an attribute that could be ascribed to a variety of mathematical
objects, including mathematical propositions, theorems, concepts, methods, proofs, theories,
applications, models and even, despite the above remarks, the symbolism. What criteria
determine if something is beautiful in mathematics? The term ‘pleasing to the eye’ cannot be
applied in the same sense as it can to paintings, scenery, etc., and ‘pleasing to the mind’s eye’ is
a metaphor that does not take us far towards an understanding of mathematical beauty.
So what makes something mathematical beautiful? The most obvious source of beauty in
mathematics is pattern, structure, and symmetry, as in art. Mathematics is abstract and so the
pattern and structure exhibited that make it beautiful must be appreciated in some abstract sense.
In addition, some of the features of abstraction itself also add to the beauty. These include the
expression of abstraction and generality, and the simplicity and economy of expression used.
Other pleasing aspects of mathematics are surprise and ingenuity in reasoning, as well as
interconnections between ideas in mathematics, as illustrated in the equation shown above. The
use of mathematical modelling to capture aspects of the world can be breathtaking, and
demonstrates its power. Finally the rigour of reasoning in proofs is noted, for example by
Bertrand Russell (1919: 60), as a thing of beauty, albeit “cold and austere”.
Developing these ideas more fully leads to what I propose as seven dimensions of mathematical
beauty. These are as follows.
1. Economy, simplicity, brevity, succinctness, elegance
The compression of a formula or a theorem of wide generality or an argument (proof) into a few
short signs in mathematics is valued and admired.
2. Generality, abstraction, power
The breadth and scope of a generality or a proof also evokes appreciation. (These first two criteria overlap
somewhat.)
3. Surprise, ingenuity, cleverness
Unexpectedness, like wit, is appreciated and valued when it reveals a new knowledge
connection, method or short cut in solving a problem.
4. Pattern, structure, symmetry, regularity, visual design
The discernment of pattern in its various and abstracted forms is the closest the values of
mathematics come to those of art and general aesthetics in the visual field, although in
mathematics these properties are largely abstract. Nevertheless, mathematics is the science par
excellence for elucidating the meaning of structure and pattern.
5. Logicality, rigour, tight reasoning and deduction, pure thought
The development of logical reasoning to its ultimate forms of rigour and purity of thought is a
valued part of mathematics and the steps in a well constructed mathematical proof evoke
admiration like a gold necklace with well forged links.
6. Inter-connectedness, links, unification
The evidence of connections between different concepts and theories within mathematics is
intellectually exciting and attractive. It combines economy, generality, ingenuity and structure
and so it could be argued that it is reducible to these first four dimensions of beauty. Or it can be
seen as sufficiently valuable in its own right as deserving independent listing, as I have done
here.
7. Applicability, modelling power, empirical generality
Like metaphors in poetry the capture of empirical situations in mathematical models and more
generally in applied theories and concepts is something appreciated both within and outside
mathematics as a demonstration of its power and ‘unreasonable effectiveness’ in the physical
world (Wigner 1960), as opposed to the world of pure mathematics.
Elegance is often given as a dimension of mathematical beauty, but I think it is reducible to
several other simpler descriptors in the above list such as economy, generality and power. This
fits with the views of Montano (2014: 182) who writes “elegance is sometimes defined as the
quality of being pleasingly simple yet effective”. It is included above with economy and its
synonyms as this seems to be its primary meaning.
Aesthetic appreciation is of course ultimately irrational in that it cannot be reduced to or replaced
by rational analysis and logical reasoning. These latter can, however, provide a partial
illumination of its components. It the final analysis aesthetic appreciation depends on positive
responses and feelings of humans which give rise to preferred choices and actions. These need
not, however, be purely subjective, as that which is regarded as beautiful is shared within and
possibly across cultures. Thus my claim is that such preferences, and indeed beauty in
mathematics, is objective, in the second, social sense defined above.
In order to make this discussion more concrete I provide a simple example which I claim exhibits
mathematical beauty, drawing on the proof that the sum of the sequence 1, 2, 3, ..., n is n(n+1)/2.
The standard elementary proof involves the following key step (shown in Table 1), the summing
of n pairs of algebraic terms, each totalling n+1:
Table 1: The key step in the elementary derivation of the formula n(n+1)/2
1 2 3 … n-2 n-1 n
n n-1 n-2 … 3 2 1 +
n+1 n+1 n+1 … n+1 n+1 n+1
Figure 1 shows a small relief sketch by the artist John Ernest illustrating this proof (Ernest 2009).
Figure 1: The sum of the first n natural numbers
There is a correspondence between the compound algebraic sum in Table 1 and Figure 1. In the
latter, small black squares represent units, black ‘columns’ (the solid area under the horizontal
dividing line on left hand side and above it on the right hand side) implicitly represent n black
squares, and small white squares represent negative numbers. The figure illustrates the beautiful
symmetry between the matching first three and the last three terms in the general series being
summed, exhibiting a rotational symmetry of order 2. But there is also a near reflective
symmetry about the horizontal and vertical axes, if the complementary colours and some other
minor details are discounted. The figure brings out these pleasing symmetric and structural
features of the proof step.
There are further aesthetic aspects of the proof beyond these structural features, in particular the
ingenuity and cleverness of the proof. By taking the sum 1+ ...+n and reversing it, and combining
the two rows, the n actual column additions involved are sidestepped, since there is a constant
sum, introducing brevity. This features in the well known story of the mathematician Gauss in
elementary school. He is claimed to have summed the numbers 1 to 100 in a few seconds using
this logic. Irrespective of its authenticity, this story is widely told to stress the teacher’s surprise
at Gauss’s ingenuity and cleverness in discovering a short and elegant solution method despite
his youth (Boyer 1989).
Another pleasing aspect of the proof and the formula are their generality and power, applying to
the first n numbers for any n. Thus the elementary derivation of the formula for the first n
numbers discussed here illustrates four of the proposed dimensions of beauty: pattern and
symmetry, generality, brevity and ingenuity. In addition the formula itself exhibits economy and
simplicity.
Of course the elementary derivation of the formula n(n+1)/2 shown is one used only at school
level. A more rigorous derivation employs mathematical induction, such as in the following:
∑
1
1
i
= 1, and if
∑
1
n
i
=
n
(
n+1
)
2
, then
∑
1
n+1
i
=
∑
1
n
i+
(
n+1
)
=
n
(
n+1
)
2
+
(
n+1
)
=
(
n+1
) (
n+2
)
2
.
However this proof loses the arguably more beautiful step shown in Table 2 which both explains
as well as validates the formula (Hersh 1993), albeit at a more elementary level.
In claiming that there is wide agreement that some mathematical knowledge and objects are
beautiful, I am not proposing that this appreciation is intrinsic or necessary. We acquire many of
our values, like our knowledge, from our participation and immersion in social groups and
cultures. Indeed the contribution to this volume of Inglis & Aberdein (2015), Diversity in Proof
Appraisal, suggests that there is significant diversity in mathematicians’ views of beauty and
mathematical aesthetics. So it may be wrong to say that mathematicans’ agreement are shared
and hence objective, even from the perspective of the social theory of objectivity.
ETHICAL VALUES
A further area of values is that of ethics, concerned with the good and right treatment of other
humans, as well as all living things and the environment. A widely held belief concerning the
relationship of mathematics with ethics, another assumption of an absolutist conception of
mathematics, is as follows.
6. Mathematics itself is ethics-free. The independently existing realm of mathematical
objects and knowledge is disconnected from human ethics or any form of social
responsibility.
According to this perspective, only in the application of mathematics do ethics and human
interests enter in. For in the act of interpreting mathematical knowledge, theories or concepts
ethical considerations from within the contexts of application can imbue the applied
mathematics. However, any ethical considerations associated with the application can never
cross the watertight pure/applied divide to attach to the original pure mathematics being used.
Holders of these beliefs, in asserting that mathematics is ethics-free, are denying the relevance of
ethics to mathematics. This claim that mathematics is ethics-free, if not an ethical position itself,
is a meta-ethical values position. It is to prioritise, value and prefer an account of mathematics as
superhuman, neutral, universal, objective, context-independent and absolutist (Harré and Krausz
1996). In support of the ethics-free position it is argued that mathematical objects and knowledge
exist in an independent extra-human realm that is disconnected from humans and their ethics or
any form of social responsibility. The ethics-free and more generally value-free nature of
mathematics, it is claimed, like its universality and objectivity, is simply a property of
mathematics, not a matter of preference or values. From an absolutist perspective of mathematics
this view is sustainable. By an absolutist perspective I mean that mathematical knowledge is
understood to be unequivocally and absolutely true, with mathematical knowledge and objects
understood to be objective in the strong traditional sense, existing universally and independently
of humankind (assumptions 1 to 4 above). From these assumptions it follows that mathematics is
ethics-free, and indeed free from any of the values attributed to it that are not logical implications
of this position.
However, the position of absolutism, with its concomitant assumptions, is not itself forced on us
by logical necessity, for there are legitimate alternatives that reject absolutism and its
assumptions. For example, a social constructivist view of mathematics (Ernest 1998), alongside
several other philosophies of mathematics, rejects the idea that the divide between pure and
applied mathematics is watertight and impermeable to ethics and other values. This echoes the
widespread critique of the fact-value dichotomy in the philosophy of science (Kincaid et al.
2007). Thus in adopting absolutism and its assumptions as the basis for a philosophy of
mathematics a preference is being exercised, that is, a value-laden choice is being made.
In contrast, different philosophies of mathematics such as quasi-empiricism or fallibilism
(Lakatos 1976), humanism (Hersh 1993), naturalism (Kitcher 1984), and social constructivism
(Ernest 1998), hold the view that the divide between pure mathematics and applied mathematics
and other worldly activities is not watertight but is values-permeable. Since applied mathematics
has the purpose of solving theoretical and practical problems outside of mathematics, like
technology it can be said to include the betterment of the human condition among its intrinsic
goals. Thus applied mathematics, at the very least can be said to be underpinned by ethical
values. This is uncontroversial.
However, I also wish to claim that ethical values imbue pure mathematics. It is uncontroversial
to say that at the heart of pure mathematics lies mathematical knowledge. That is, justified
mathematical propositions and their proofs. Proof and justification are arguments and reasoning
applied to persuade, indeed to convince other persons about the truth of mathematical claims,
that is, that mathematical theorems are adequately warranted. However, the very use of proof and
justification can be said to embody the values of openness, fairness and democracy. Proof itself
embodies democracy because it opens up the basis for knowledge to all for verification. Whether
it is the shop keeper presenting a bill for purchases or a mathematician publishing her latest
theorem, the written account allows scrutiny of the correctness of the claims and reasoning.
Indeed the terms justification and justice have the same roots. From the 14th century on
justification has meant the action of justifying and the administration of justice, and justice is the
quality of being fair and just – the exercise of authority in vindication of what is right (Harper
n.d.). True justice depends on the open justification of decisions which is the basis of both
mathematics and democracy. Mathematics, like democracy, is fair because of this openness and
potentially equal treatment of all with respect to knowledge claims, their warranting and
decisions as to their status as knowledge.
Mathematics has long been associated with ideas of justice and fairness. In ancient societies
including those in Mesopotamia the reliability of calculation, measures and numerical records
was understood as part of the idea of justice (Høyrup 1994). Later on, in ancient Greece,
mathematical proof emerged out of a background of philosophical argument and reason that
developed with the first, albeit limited, democracy with its justification of human claims and
rights. It has been argued that some mathematical concepts and methods embody ideas of
fairness. Johnson (2012) argues that fairness underpins probabilistic concepts and probabilistic
methods of reasoning, and that this has implications for the history and present day practices of
market trading.
From an absolutist perspective, the role of proof is purely epistemological in establishing the
truth of a theorem. Its role in the persuasion of persons about the correctness of reasoning is
incidental. From perspective of humanism (Hersh 1993), social constructivism (Ernest 1998), or
compatible philosophies of mathematics, convincing persons about the correctness of reasoning
in a proof lies at the heart of the epistemological function of proof. A demonstration of
correctness of reasoning is always addressed to another.
Elsewhere I propose conversation as the underlying epistemological unit for a social
constructivist philosophy of mathematics (Ernest 1991, 1998). My claim is that conversation,
consisting of symbolically mediated exchanges between persons, underpins mathematics, and
that it does so in four distinct ways.
1. Mathematics is primarily a symbolic activity, using written inscription and language and
inevitably addressing a reader, so mathematical knowledge representations are
conversational.
2. A substantial class of mathematical concepts have a conversational structures (e.g.,
epsilon-delta definitions of limit in analysis, hypothesis testing in statistics, as well as
other concepts, Ernest 1994a).
3. The ancient origins and various modern systems of proof are conversational, through
dialectic or dialogical reasoning, involving the persuasion of others.
4. The epistemological and methodological foundations and acceptance of mathematical
knowledge, including the nature and mechanisms of mathematical knowledge genesis
and warranting are accounted for by social constructivism through the deployment of
conversation in an explicitly and constitutively dialectical way. This account of the
conversational basis of mathematics is based on primarily on the work of Wittgenstein
(1953) and Lakatos' (1976) Logic of Mathematical Discovery (Ernest 1998).
Thus conversation in a number of ways lies at the heart of mathematics, providing it with a
human foundation. It is intrinsic to the fabric of mathematics, underpinning its concepts and
objects, representations, genesis, proof and warranting. But conversation as an interpersonal
activity is inescapably ethical, it is not just about exchanging information (Ernest 1994b,
Johannesen 1996, Gadamer 1986, Rorty 1979). For it entails engaging with a speaker or listener
as another human being with mutual respect and trust, attending to another’s proposals and
responding relevantly, and being aware of reactions to one’s own contributions. In mathematics,
putting one’s proposals in an appropriate and accessible format following received norms of
acceptability is part of one’s ethical responsibility throughout pure, applied and educational
mathematics.
My argument is that the very content of mathematical knowledge – its concepts, methods, proofs
– are conversational, so conversation cannot be dismissed as merely part of the context of
discovery (Popper 1959). These contents as well as the conversational warranting mechanisms
described in Lakatos’ (1976) Logic of Mathematical Discovery and in Ernest’s (1998)
Generalised Logic of Mathematical Discovery are also part of the context of justification (Popper
1959). So mathematics in all of its manifestations is riven through and through by conversation,
throughout its origins, practices, and throughout abstracted mathematical knowledge itself.
It might be argued that as conversation is subsumed into mathematics it becomes vestigial and its
ethical dimensions become attenuated and discountable.6 My rejoinder is twofold. First of all,
conversation does not become vestigial because of its continuing roles in the warranting of
mathematical knowledge. Furthermore, the warranting of mathematical knowledge never ceases,
as every new formulation or publication in mathematics requires warranting. Secondly, as I have
6 A final proof appears monological because all of the anticipated criticisms and responses have been overcome
and incorporated in the final polished result. But as Lakatos (1976) shows the hidden dialogic of the proof
leaves its mark in the refined definitions and lemmas that make up the final proof.
argued above, from humanistic and social constructivist perspectives the distinction between the
contexts of discovery and justification can no longer be claimed to be watertight or absolute.
Some values from the context of discovery cannot be prevented from imbuing the context of
justification. Thus mathematical knowledge and the processes and products of the context of
justification are laden with the values of conversation, and more generally with human values, as
argued above.
Overall, my claim is that in a number of ways mathematics is imbued with ethical values. Its
basis in verifiable truth claims means that it is shot through with the values of openness and
democracy. Its nature as a symbolic activity, a specialized and supplemented form of written
language means that the ethics of human communications are presupposed. If mathematics is
conversational then like all forms of inter-human activities and relationships, it is inescapably
ethical.
UTILITY
A further value of mathematics is that of utility. Utility concerns the usability or applicability of
artefacts, practices or knowledge developed in one domain in the same or in a second domain,
where benefits consequently accrue to the applier. In general, utility and usefulness are desirable
attributes of artefacts, practices and knowledge since human activity should ultimately serve to
improve human flourishing and well-being. According to Bentham (1879: 194) utility is “that
property in an object, whereby it tends to produce benefit, advantage, pleasure, good, or
happiness”. Thus because it concerns the good, utility is an ethical value.
The utility and applicability of mathematics are widely valued, and one of the values that drives
the development of new mathematics is the quest for new useful and applicable theories and
results. The utility of mathematics arises from its use in making mathematical models of the
world in which formal theories are interpreted as models of natural or technological processes
and phenomena, and their underlying mechanisms. These models, that is theories interpreted in
terms of real-world relationships, allow measurements of observable variables to be made,
leading to predicted numerical values and outcomes, according to the logic of the model.
Mathematical theories and models thus act like metaphors in poetry, being analogies or
morphisms that map one domain into another. The mathematical model exists in the pure realm
of theory, while the empirical situation it tries to represent, capture or emulate is a part of human
understanding of the physical world.
The bridging of the gap between the world of pure mathematics and its targeted zone of
application is what allows talk of the ‘unreasonable effectiveness’ of mathematics in the physical
world (Wigner 1960). It is predicated on the assumption of their ontological discreteness.
Positions that reject such ontologies, like social constructivism, can value both the purity of
mathematics invented and extended for its own sake as well as the values of utility and
usefulness of applied mathematics. However, from this perspective the effectiveness of
mathematics in the physical world is no longer seen as quite so surprising or unreasonable. For
the material and physical world all around us is viewed as our only world and is thus both the
source of mathematical concepts and theories as well as being the location of mathematical
work. This is conducted by human beings using the conceptual tools they have invented, as often
as not originally inspired by physical and worldly needs and applications. Irrespective of its
worldly origins and applications, like many crafts and practices, mathematics takes on a life of
its own. It is developed and perfected for its own sake by skilled practitioners, simultaneously
expressing the values of purity and utility. This has been going on ever since the Sumerian and
Egyptian scribes developed their techniques and organised their mathematical knowledge for its
own sake driven by a desire to hone and perfect their professional skills (Høyrup 1994).
Pure mathematics is that field of knowledge in which mathematical concepts, methods, proofs,
problem solutions and theories are refined, developed and extended, often but not always without
any thought of applications outside of mathematics. Thus Newton’s developments in the calculus
are regarded as a triumph of pure mathematics. However, they were at least partly motivated by
the need to solve problems for Newton’s gravitational theory. But even when pure mathematics
is developed with no such applications in mind, not even on the horizon, it represents virtuosity
both in the working mathematician and in the refinement and generalisation of the discipline.
Mathematics is a symbolic technology, whose development was prompted historically by social
problems and needs, and whose subsequent main justification has been its powerful applications.
Between its origins and its applications, the intermediate domain of pure mathematics with its
inwardly directed development is an expression of virtuosity. But the characteristics of this
virtuosity, such as precision, abstractness, generalisability, are what makes the technology so
powerful in its applications. The aim of extending pure knowledge can never escape the
underlying interest of predicting and controlling the world (Habermas 1972). Thus even the
purest of pure mathematics, as an expansion and improvement of knowledge, represents the
desire to improve human understanding, technology, and hence the place of humans in the world.
Because of its focus on the good, all mathematics has an intrinsic ethical dimension. My claim is
that even pure mathematics with seemingly no applications in the world, indirectly helps our
control and mastery of the physical as well as human worlds. All mathematical knowledge is
applied to the expansion of knowledge and the betterment of human understanding: some
directly, called applied mathematics, and some indirectly, called pure mathematics.
Pure mathematics, like other pure activities in the arts and humanities (e.g., painting, dance,
literature, philosophy), can thus be claimed to improve human flourishing and well-being
indirectly, not through improving the immediate material circumstances of life but in providing
thinking tools that may be applied in the future in unanticipated practical and utilitarian ways. Of
course such pure activities also contribute directly through enhancing the spiritual and emotional
life of humankind, thus giving increased life-satisfaction to participants and users, in common
with all of the arts. This chimes with the characterisation of mathematics both as a pure art and
as the queen and servant of science (Bell 1952). Purity is manifested in the role of mathematics
as queen of science, whereas utility is reflected in its role as servant of science.
Since utility is an ethical value, concerned with human good, and mathematics is universally
acknowledged to be useful, utility is an ethical value of mathematics. However, from an
absolutist perspective, it is claimed that utility and hence ethical values only apply to
mathematical knowledge when it is applied, that is interpreted in the physical world. Thus the
attribution of these ethical values to pure mathematics remains contested and controversial.
PURITY: A MIXED VALUE
The last value of mathematics to be discussed here, namely purity, does not fall neatly under a
single branch of philosophy, that is, under epistemology, ontology, aesthetics or ethics. Purity is
a property of an entity or a practice. A pure thing is one that is made of an undiluted or
unpolluted substance. A pure practice is conducted solely for its own sake, for its intrinsic merit,
without ulterior motives or extrinsic goals. However, two senses of purity need to be
distinguished: its descriptive and evaluative meanings. Descriptively, purity applies to something
that is unmixed, undiluted or is in its basic or primal state, be it a substance or practice. The
prescriptive or evaluative sense of purity is something that is unadulterated, unpolluted or
unbesmirched. In this sense impurity applies to something that has been degraded either in terms
of beauty (made ugly) or of being less good (made bad). It is important to distinguish these two
meanings, especially because even when purity is used in the descriptive sense, undertones of
prescription can smuggle in gratuitous negative value-judgements.
Purity is an important value within mathematics. Ancient Greek mathematicians were among the
first to distinguish pure and applied mathematics, with Plato (1941) distinguishing ‘arithmetic’,
which we now term number theory, from ‘logistic’, now called arithmetic. Plato regarded logistic
as appropriate for tradesmen and warriors who “must learn the art of numbers” for practical
purposes, and number theory as appropriate for philosophers because they have to “lay hold of
true being”. Other mathematicians of the era including Euclid of Alexandria and Apollonius of
Perga held similar views. Their purist values were expressed positively in the elevation of
geometry and number theory to the realm of pure thought concerning ideal objects and
relationships. They were expressed negatively in their philosophical dismissal and derogation of
practical numeration and calculation (logistic) and measures, regarded as lowly activities
performed by lesser beings for practical purposes.
In the modern era, calculation and practical mathematics have also been viewed as
mathematically trivial and philosophically uninteresting. Philosophers have been concerned with
the nature of mathematical objects and viewed them as belonging to an ontological category
distinct from the things, tools and beings of the material world. The origins of mathematical
concepts and methods in human historical practices located in the material world are either
dismissed as irrelevant or denied altogether. Such a view is typified by Platonism, concerned
with mathematical objects and truths that are understood as abstract, existing in an unearthly and
idealized (pure) world, like Popper’s (1979) objective World 3, beyond the physical world that
we inhabit as fleshy, embodied and social human beings. Although Platonism is independent of
purism in its prescriptive sense, it facilitates purist values by locating mathematical objects in a
pure and ideal realm disconnected from the material world we inhabit. Given a prescriptive slant,
Platonism elevates mathematical objects to a pure ideal realm unsullied by earthly taint.
From the nineteenth century onwards purity is a prominent value for many mathematicians
including Boole, Hamilton and others. Hardy (1941) extolled pure mathematics as expressing
truths independent of the physical world. He compared pure mathematics to painting and poetry,
claiming it to be the only “real” mathematics that has permanent aesthetic value He contrasted it
with “the dull and elementary parts of mathematics” that have practical uses. Such views
represent purity as a virtue, a quality superior to applicability (and impurity).
Purist values are reflected in the valuation of proof as a higher form of reasoning, and calculation
as a lower form, from Plato onwards. Elsewhere (Ernest 2007) I have argued that the very strong
analogy and structural similarities between proof and calculation, including their
interconvertibility, challenge the view that proof is intellectually superior to calculation in
mathematics. Nevertheless, purism values pure proof-based mathematics as being significant
epistemologically, and pertaining to truth, wisdom, high-mindedness and the transcendent
dimensions of being. Equally this ideology denigrates applied mathematics and calculation as
technical and mechanical, pertaining to the utilitarian, practical, applied, and mundane;
understood as the lowly dimensions of existence.
Restivo (1994) goes further and argues that Purism is an intellectual strategy serving social goals
including the demarcation of knowledge and defending the pursuit of knowledge for its own sake
from outside interests. Although this can be useful in protecting the interests of a discipline, he
claims, it can also focus researchers on concerns interior to their discipline, obscuring the social
contexts and applications of their work. Such a focus may keep researchers from criticizing or
resisting the actions of powerful interests at work outside of their discipline, including
administration and the state. Douglas (1966) argues that purity is a strategy for protecting social
boundaries and that it also strongly resonates with the protection of bodily boundaries from the
threat of pollution, stemming from deep psychological roots. Thus Purism in both its forms,
descriptive and prescriptive, emphasizes boundaries that strongly demarcate disciplines and
social groups, pure mathematics and the community of pure mathematicians, respectively, in the
present case. Overall, purity is a deeply entrenched value applied to a good deal of mathematical
knowledge, and held in high regard by a large section of community of mathematicians.
However, two caveats are worth noting. First, purity has only been a dominant value for a
fraction of the whole history of mathematics. It was powerful in the ancient times of the Greeks
(e.g., Plato and Euclid) and in modern times, beginning with the professionalisation of university
teaching and research in the early- to mid-nineteenth century (Restivo 1994). Second, even in
modern times many professional mathematicians do not fully subscribe to the purist ideology.
Grigutsch and Törner (1998) investigated the views of mathematics of 119 university
mathematicians in Germany. They found that more mathematicians viewed mathematics as
process-based problem solving and applied, concerned with practical use and relevance to
society, than viewed it as purely formalist or Platonic concerned with “aesthetic divine games”.
This suggests that purity was not the dominant value present.
With these caveats purity remains an important value within mathematics, especially within pure
mathematics. It is a value that permeates not only the epistemology of mathematics (pure
knowledge) and metaphysics (existence in a pure, extra-mundane realm), but also its aesthetics
(the pure line of beauty) and ethics (mathematics as pure and free from values and ethics).
Within these domains it is my contention that purity in epistemology and ontology is primarily
descriptive, indicating how mathematical knowledge is demarcated from empirical knowledge in
the first case, and how mathematical objects exist in a separate realm of their own in the second.
This contrasts with the primarily prescriptive and evaluative uses of purity in aesthetics and
ethics. In these domains purity is associated with beauty and the good, their overarching central
values. To fall short of beauty or the good is to be ugly or bad to some degree, and clearly these
are negative evaluations. However, in the history of mathematics and its philosophy these two
orthogonal meanings of purity have been confused so that impurity as a descriptor of applied
mathematics has become entangled with impurity as tainted, dirty or flawed. While purity is
undoubtedly an important value within mathematics there remains an ambiguity in what it means
and how the term is used and understood.
On the face of it purity might seem inconsistent with another mathematical value: utility.
However, it is possible to see both the purity and applicability of mathematics as mathematical
values without any inconsistency. The purity of mathematics is seen to be intrinsic to its
concepts, theorems, methods and theories. The utility of mathematics concerns the applications
of mathematical knowledge in external domains. However, I have argued that even pure
mathematics, although not directly applicable, through being the science of mathematical tools
and technology, shares in the ethical values of applied mathematics. Namely, it is concerned,
however indirectly, with the good in human flourishing.
CONCLUSION
In this paper I argue that mathematics has embedded values. I have identified truth, rationalism,
universalism, objectivism, objectism, beauty, ethics and purity as values of mathematics,
although I do not claim that this list is exhaustive. Some of the values I have identified are
expressed as beliefs, but all of the values proposed are shared, cultural values attributed to
mathematical knowledge. They are objective, in the cultural sense, and not merely the subjective
values of individuals. After truth, perhaps the most widely recognised of these values is that of
beauty, and I offer a novel analysis that individuates seven dimensions of mathematical beauty,
namely economy, abstraction, ingenuity, symmetry, rigour, connectedness, applicability.
Among the more controversial values I identify are objectism and ethics. These are not in general
acknowledged and indeed many mathematicians and philosophers might deny that these are true
values of mathematics. Perhaps the more controversial of these is ethics, and I claim that
mathematics is imbued with ethical values, assumptions and obligations. I claim that
mathematics embodies the values of openness, fairness and democracy, all of which I locate in
the domain of ethics, alongside utility. Identifying conversation as an underpinning
epistemological unit for mathematics, I argue that this brings ethics into the context of
justification of mathematical knowledge as well as being ever-present in the context of
discovery. I also argue that as a symbolic technology, even pure mathematics is concerned with
and a part of human flourishing and hence is underpinned by ethics and the good.
I acknowledge that certain philosophies of mathematics, absolutist philosophies in particular,
reject the attribution of values to mathematics, especially ethical values. Based on the
assumptions of absolutism, mathematics can legitimately be claimed to be ethics-free, and also
partially value-free and at least with regard to some of the values discussed here. However, my
argument is that the choice of an absolutist philosophy of mathematics is itself a values-based
choice, and thus the subsequent implication that mathematics is ethics-free and partly value-free
is a consequence of this initial unforced choice. Thus it cannot be claimed that mathematics is
ethics-free purely on the grounds of logical necessity. It follows on from the choice of a
particular philosophy of mathematics.
The values of mathematics straddle several areas of philosophy. Truth, rationality and purity, at
least the last in its descriptive sense, may be called epistemological or epistemic values, as are
the relevant variants of universalism and objectivism. Beauty naturally pertains to aesthetics, as
partly does purity, in its descriptive sense. Objectism is an ontological or ontic value, as are also
the relevant dimensions of universalism and objectivism. Purity in its descriptive sense of pure
being also belongs to ontology. Lastly democracy, openness and utility, as well as purity in its
prescriptive sense concerning the good, are all ethical values. Furthermore, I have argued that the
conversational foundation of mathematics, as well as the virtuosity exhibited by pure
mathematics bring ethical values into mathematics.
Overall my claim is that far from being value-free, mathematics is imbued with a broad range of
different types of values drawn from epistemology, ontology, aesthetics and ethics. However, I
also acknowledge that, at least in part, an absolutist philosophy of mathematics can legitimately
deny this claim on the basis of its values-based assumptions.
REFERENCES
Bell, E. T. (1952) Mathematics Queen and Servant of Science, London: G. Bell and Sons.
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