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Philosophia Mathematica (III) 22 (2014), 12–34.

Advance Access publication June 12, 2013

Should Philosophers of Mathematics Make Use of Sociology?†
Donald Gillies∗

1. Introduction. Lakatos’s Criticisms of Kuhn
The aim of this paper is to consider whether the philosophy of mathematics
would benefit from the introduction of some sociology. A related question
was discussed by Lakatos in the early 1970s, though Lakatos focussed
on philosophy of science rather than philosophy of mathematics. At that
time, Lakatos was occupied with Kuhn’s views on philosophy of science.
Now Kuhn was obviously in favour of introducing some sociology into the
philosophy of science. This is shown, for example, by the passage in which
he attempts to characterise his concept of normal science. He writes:
‘normal science’ means research firmly based upon one or more past
scientific achievements, achievements that some particular scientific
community acknowledges for a time as supplying the foundation for
its further practice. [1962, p. 10]
† An earlier draft of this paper was read at the conference: ‘What does it mean to do the
philosophy, history and sociology of mathematics in the 21st century?’ held in University
College London on 30 July 2010. I am grateful for the many helpful comments I received on
that occasion, and also for discussions with the conference organiser Josipa Petrunic, during
the year 2009–2010 which she spent as a post-doctoral fellow in UCL’s Department of
Science and Technology Studies. Drafts of the paper were kindly read by Emily Grosholz,
Gianluigi Oliveri, and Paul Ernest. They gave me useful comments which helped with the
revision of the paper. Hauke Riesch sent me the text of his talk ‘Integrating sociology and
philosophy of science’, which was given in Cambridge in October 2010. This too was very
useful to me. Comments from an anonymous referee led to an extensive revision of the
∗ Department of Science and Technology Studies, University College London.
C The Author [2013]. Published by Oxford University Press.
Philosophia Mathematica (III) Vol. 22 No. 1
All rights reserved. For permissions, please e-mail:

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This paper considers whether philosophy of mathematics could benefit
by the introduction of some sociology. It begins by considering Lakatos’s
arguments that philosophy of science should be kept free of any sociology. An attempt is made to criticize these arguments, and then a positive argument is given for introducing a sociological dimension into the
philosophy of mathematics. This argument is illustrated by considering
Brouwer’s account of numbers as mental constructions. The paper concludes with a critical discussion of Azzouni’s view that mathematics differs fundamentally from other social practices.



For Kuhn scientific change — from one ‘paradigm’ to another — is
a mystical conversion which is not and cannot be governed by rules
of reason . . . Scientific change is a kind of religious change. (p. 9)
If even in science there is no other way of judging a theory but by
assessing the number, faith and vocal energy of its supporters, . . .
truth lies in power. (pp. 9–10)
. . . in Kuhn’s view scientific revolution is irrational, a matter for mob
psychology. (p. 91, emphasis original)

Of course in these passages one must allow for Lakatos’s characteristic
tendency to exaggerate. Suppose the merits of two competing theories
T1 and T2 are being debated by some scientific community. After several
years of sober discussion, the collection of new relevant observations, and
the performance of several experiments, the overwhelming majority of the
community decide in favour of T1 . This is certainly a sociological process,
but is it correct to describe it as an example of ‘mob psychology’ at work?
However, Lakatos’s exaggerations are part of the charm of his style and

Emily Grosholz pointed out to me that ‘introducing sociology’ might suggest ‘introducing some ideas from general sociological theories such as those of Durkheim or Weber’.
This Kuhn does not do. He introduces sociology only in the weaker sense of using, in
addition to concepts such as: theory, logical consequence, observation, . . . , sociological
concepts such as: scientific community. He also uses these sociological concepts in his historical case studies, thereby making his history of science, at least partly a social history.
In this paper, I will use ‘introducing sociology’ in this weaker sense.

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What I want to draw attention to here is Kuhn’s reference to ‘some
particular scientific community’. This brings in a scientific community,
and ipso facto sociology.1
Lakatos organised a confrontation between Kuhn and the Popperian
school, the results of which appeared in the famous collection Criticism
and the Growth of Knowledge, published in 1970. Lakatos’s own contribution to this collection: ‘Falsification and the methodology of scientific
research programmes’ actually follows Kuhn in many respects. Lakatos,
like Kuhn, is in favour of basing philosophy of science firmly on history
of science. Moreover, the ‘hard cores’ of Lakatos’s scientific research programmes do seem to bear some resemblance to Kuhn’s paradigms. However, there is one point on which Lakatos diverges very sharply from Kuhn.
Lakatos is not prepared to tolerate the introduction of any sociology into
philosophy of science.
Here are a few quotations from [Lakatos, 1970] where Lakatos criticizes Kuhn.



I do not want to criticize them too much. It is just a matter of taking what
he says with a pinch of salt.
Lakatos’s general position is clear enough. He regards the outcome of
sociological processes as often determined by irrational religious beliefs
or by political power relations. Science, however, should progress in a
rational fashion, and the analysis of rationality belongs to philosophy not
sociology. Lakatos indeed states that he is trying to develop a

Lakatos died in 1974, but in the last years of his life, he continued
to develop his ideas about philosophy and sociology. His work on this
subject took the form of writing an extended review of a book which Toulmin had published in 1972. Toulmin had contributed a paper to Criticism
and the Growth of Knowledge in 1970. He was a follower of Wittgenstein
and, although his position was very different from Kuhn’s, it was still unacceptably sociological as far as Lakatos was concerned. Lakatos’s review
of Toulmin was unfinished at the time of his death. An extract of the
manuscript he left was published as a review of [Toulmin, 1972] in 1976.
In writing the review, Lakatos had started to develop some general ideas
about the appraisal of scientific theories, and a separate paper on this subject was also extracted from the manuscript and published for the first time
in Lakatos’s Philosophical Papers Volume II in 1978 with the title: ‘The
problem of appraising scientific theories: Three approaches’.
The review of Toulmin as published in 1976 contains a harsh attack
not only on Toulmin but also on his mentor Wittgenstein. Lakatos
describes [1976, p. 228] ‘the philosophy of the later Wittgenstein’ as
‘one of the most obscurantist traditions in contemporary philosophy’;
and labels Wittgenstein’s followers (including Toulmin) as ‘the Wittgensteinian “thought-police” ’. I will not, however, discuss Lakatos’s criticisms of Toulmin and Wittgenstein, but rather consider the way Lakatos
develops his attack on sociology in the more general parts of the review of
1976 and the paper of 1978 (both written c. 1973).
Lakatos claims that the most important problem in the philosophy of
science is that of the normative appraisal of allegedly scientific theories.
As he puts it:
. . . the (normative) appraisal of those theories which lay claim to
‘scientific’ status . . ., it seems to me, is the primary problem of the
philosophy of science. To neglect it, or even to assign it a merely secondary role, implies philosophical surrender to a strictly descriptive
sociology and history of science. [1976, pp. 224–225]

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. . . position which, I think, may escape Kuhn’s strictures and present
scientific revolutions not as constituting religious conversions but
rather as rational progress. [1970, p. 10]



Lakatos describes those philosophers of science who try to produce
positive solutions to this problem of normative appraisal as demarcationists. He next argues that demarcationists all believe in the Platonic world
of objective spirit:

This passage very much clarifies Lakatos’s position, and explains why he
wants to exclude sociology from philosophy of science. Philosophy of science should analyse the contents of the Platonic world of ideas, such as
theories, problems, etc. Human society, however, exists outside this realm
of ideas and so its study is not a proper part of philosophy. It could be
remarked that Lakatos should perhaps here speak of a Hegelian rather than
a Platonic world, since his world of ideas is dynamic and changing.
Let us now look at some of the arguments which Lakatos gives for
his thesis that sociology should be excluded from philosophy of science.
He says:
. . . psychologism and sociologism both seem to me to be open to
the following fundamental objection. Everyone . . . is bound to use
normative third-world criteria, whether explicit or hidden, in establishing criteria for a scientific community. [1978a, p. 114]

The argument here is that a sociologist of science who wants to study the
scientific community must first have some criterion for what constitutes
science. So the demarcation problem must first be tackled by philosophers
independently of sociology before the sociology can get off the ground.
Lakatos illustrates this argument by considering the sociologist of science Merton. Merton, he says, ‘must have already decided that Darwinian
biology was scientific, while Catholic theology was not’ [Lakatos, 1978a,
p. 114] before he started his sociological analysis of scientific institutions.
It is questionable, however, whether the sociologist of science is in
any worse position in this respect than the demarcationist philosopher of
science. Let us consider demarcationist philosophers of science trying to
distinguish scientific from non-scientific theories. Surely they will begin
by taking a number of theories which are generally recognised as scientific, such as Newtonian mechanics, quantum mechanics, theories of the
coding of genetic material by DNA, etc. These will then be compared

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Demarcationists . . . all believe in the third of Frege’s and Popper’s
three worlds. . . . [T]he ‘third world’ is the Platonic world of
objective spirit, the world of ideas. Demarcationists appraise the
products of knowledge: propositions, theories, problems, research
programmes, all of which live and grow in the ‘third world’ (whereas
the producers of knowledge live in the first and second worlds).
[1976, p. 226]



. . . one cannot replace philosophy of science by sociology of science
as the supreme watchdog. If both history and sociology of science are
norm-impregnated, rational appraisal of scientific progress must precede, not follow, full scale empirical history . . . [1978a, pp. 115–116]

This passage as applied to history of science seems to contradict Lakatos’s
famous saying : ‘Philosophy of science without history of science is
empty; history of science without philosophy of science is blind’ [1971,
p. 102]. This surely implies a continuous interaction between philosophy
of science and history of science. We begin, say, with some rather crude
philosophy of science. This is used as a guide in the study of history of
science. However, these historical studies suggest some improvements in
our philosophy of science. This improved philosophy of science is used as
a guide to further study of the history of science, and so on. What seems
to be ruled out by Lakatos’s saying is the possibility of first working out a
philosophy of science and then applying this philosophy of science to the
history of science, for a philosophy of science developed before studying
any history of science would be empty. How then could rational appraisal
of scientific progress precede full-scale empirical history? The two have
to be developed together through continual interaction.
Exactly the same position could be adopted as regards philosophy
of science and sociology of science. Surely it makes more sense to say

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with a number of theories generally recognised as non-scientific, such as
astrology, the doctrine of the Trinity, etc. From the study of such cases,
they will try to formulate some criterion to distinguish one type of theory
from the other. As the investigation develops, however, it might happen
that a theory considered initially to be scientific might be re-classified as
non-scientific, or vice versa. The criterion might also give a pronouncement on some doubtful cases.
This procedure could be exactly paralleled by sociologists of science.
Without paying any attention to what demarcationist philosophers of science are saying, they begin by taking a number of institutions which
are generally recognised as scientific, such as the Royal Society. From
a study of such cases, they might formulate a purely sociological characterisation of scientific institutions, and this characterisation might be
used to re-classify some allegedly scientific institutions as non-scientific
or vice versa. It might also give a pronouncement on some doubtful cases.
As far as I can see, sociology of science and demarcationist philosophy
of science are in exactly the same position, and Lakatos’s argument fails.
Yet Lakatos in another passage insists on the primacy of demarcationist
philosophy of science, which he insists must precede both sociology of
science and history of science:



2. Defence of the Introduction of Sociology into Philosophy of
My argument in favour of introducing sociology into philosophy of mathematics is based on Marx’s famous VIIIth Thesis on Feuerbach, which runs
as follows:
Social life is essentially practical. All mysteries which mislead theory to mysticism find their rational solution in human practice and in
the comprehension of this practice. [Marx, 1845, p. 30]

Now both mathematics and its philosophy have a very strong tendency to
lead the mind into mysticism. One can see this very clearly in the case
of Plato’s theory of ideas. As expounded in The Republic, this has all the
characteristics of a typical mystical theory. The World of Ideas according
to Plato is much more real than the World of Everyday Experience. Access
to the World of Ideas is obtained by a select few who, after a long training,

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that these two activities should be developed together through continual
interaction than to claim that one of them, philosophy of science, should
be developed before the other, sociology of science.
Lakatos moreover seems to assume that either philosophy of science
must dominate sociology of science or vice versa, for he writes: ‘. . .
one cannot replace philosophy of science by sociology of science as the
supreme watchdog’ [1978a, p. 115]. But why should there be a supreme
watchdog? Why not two or more watchdogs of equal status?
For these reasons, Lakatos’s arguments against the introduction of
sociology into the philosophy of science do not seem to me to be very
So far I have been considering philosophy of science. Let us now turn
to our proper subject namely philosophy of mathematics. There is no doubt
in my mind that Lakatos would have applied exactly the same arguments
against introducing sociology into philosophy of mathematics. He probably did not mention philosophy of mathematics in this connection because
his main target was Kuhn who never wrote on the philosophy of mathematics. However, some years later in 1992, I edited a collection of papers
on Revolutions in Mathematics. The authors of these papers express a
wide variety of opinions, but some at least are Kuhnian, and others, while
not Kuhnian, make use of sociological concepts. I fear Lakatos would
not have approved of this enterprise, but, as I have suggested, the arguments he might have given against it can be answered. However, to answer
Lakatos’s arguments against a thesis is not the same thing as to provide
positive support for that thesis. In the next three sections of this paper,
I will try to provide such support.



The foremost value of the concept of scientific community is that it
does away with the impression often got from discussions on mathematics and its history, namely that mathematics is a package of
eternal, spiritual truths gradually unwrapping itself in the course of
history, visible only to the inner eye of singular geniuses who make
them accessible to diligent research students. In fact, mathematics is
man-made; its vital basis is the social interactions of mathematicians
in their scientific community. [1976, p. 30]

It is interesting to note that Lakatos in his arguments against the introduction of sociology into philosophy claims that all who think like him
believe in: ‘the Platonic world of objective spirit, the world of ideas’
[1976, p. 226].
Now, to illustrate the de-mystifying powers of empirical sociology,
I could consider the case of Plato’s world of ideas. However, I prefer to
give another, in some ways simpler, example. This is Brouwer’s account
of natural numbers as mental constructions.

3. Brouwer’s Account of Numbers as Mental Constructions
L.E.J. Brouwer (1881–1966) was a Dutch mathematician and philosopher
of mathematics. Indeed he is generally recognised as one of the leading
mathematicians and philosophers of mathematics of the twentieth century. He became famous for having founded the school of intuitionism.
In view of the nature of the criticism I am going to present of Brouwer,
it should be said that Brouwer himself was strongly influenced by mystical views. In 1905, before his academic career began, he wrote a lengthy
manuscript defending a mystical view of the world. Heyting, the editor of
Brouwer’s Collected Works, reprints part of this manuscript with the title:
‘Life, art and mysticism’. He justifies his decision to include this material

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have an insight denied to most of mankind. Moreover, Plato is led to his
theory by philosophising about mathematics.
The claim I would like to make is that in the face of a mystical theory of
this sort, it is very useful to do some empirical sociology: to examine how
mathematical ideas have actually been created by human societies, and
how they are used in those societies. My claim is that such sociological
investigations make the mystical theory less plausible and so constitute an
argument against it. In this way, sociological investigations can make a
useful critical contribution to the philosophy of mathematics. They have
the effect of de-mystifying and producing a more realistic outlook.
A similar line of thought is to be found in Mehrtens, who writes in
defence of Kuhn’s concept of scientific community as follows:



in a footnote in which he says:
. . . according to Brouwer himself there was a narrow connection
between his general philosophical ideas and his philosophy of mathematics and science. For this reason we must give some attention to
the pamphlet . . . [1975, p. 565]

Those imprisoned in life call this mysticism, they think it obscure,
but truly, it is the light that is only darkness to those who are in darkness themselves.

Brouwer’s 1905 pamphlet begins: ‘Originally man lived in isolation
. . .’ (p. 1). Similarly Brouwer takes the isolated mathematician as the starting point for his philosophy of mathematics. Brouwer held that numbers
are the subjective mental constructions of an individual mathematician.
Indeed, for Brouwer, mathematics is essentially the activity of a solitary
human soul (the creative subject) who carries out languageless mental constructions in his or her unique consciousness. In a significant passage from
his 1933 paper, ‘Volition, knowledge, language’, Brouwer writes,
. . . for a human mind equipped with an unlimited memory, pure
mathematics, practised in solitude and without using linguistic signs,
would be exact . . . [1933, p. 443]

Brouwer does admit, of course, that the languageless constructions of
exact mathematics are translated into language so that the creative subject (A, say) can communicate them to another mathematician (B, say).
However, such linguistic exchanges represent a falling off from mathematics as it should be, and exist only to enable B to recreate in his or her
own mind subjective experiences similar to those of A. Thus the passage
quoted above continues
. . . the exactness would be lost in mathematical communication
between human beings with an unlimited memory, because they
would still be thrown upon language as their means of understanding.
[Brouwer, 1933, p. 443]

Let us next examine the process by which, according to Brouwer,
the individual mathematician creates the natural numbers. Here Brouwer

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Heyting goes on to say (p. 565): ‘. . . quotations from Meister Eckhart,
Jakob B¨ohme and the Bhagavad Gita . . . abound in the original’.
In this work [1905, p. 9] Brouwer attacks critics of mysticism as follows:



develops some ideas of Kant’s and argues that the construction is based on
the creative subject’s apprehension of the passage of time. It is indeed hard
to imagine any conscious experience which does not unfold in time, and
it is this temporal development which, according to Brouwer, forms the
basis of the construction of numbers. Here is a passage from Brouwer’s
inaugural address of 1912, in which the process is described:

I do not want to comment in detail on ‘the intuition of the bare twooneness’ which remains for me a somewhat mysterious Brouwerian
concept. One thing is clear however. Brouwer does not imagine the construction of the natural numbers in time in the way that Kant did, as a kind
of counting process 1, 2, 3, . . . . He rather sees the procedure as one of
dividing the temporal continuum.
That concludes my account of Brouwer’s philosophical view that numbers are the subjective mental constructions of an individual mathematician. I now want to cast doubt on this view by giving, in the next section,
an empirically based, sociological and historical account of how numbers
really were constructed by human beings.

4. A Sociological and Historical Account of How Numbers Were
Actually Constructed
The natural numbers {0, 1, 2, 3, . . . , 10, 11, 12, . . . 100, 101, 102, . . .} are
very familiar to most of us these days, and they seem very natural (hence
their name). However, if we look at number systems historically and
anthropologically, we find that our familiar decimalised natural numbers
only came into existence through a long and complicated series of transformations. Different, and generally more primitive, systems existed for
centuries in many parts of the world. Often there was a crucial innovation in some relatively small group, and this change gradually spread elsewhere. The historical picture which emerges is that of the system of natural
numbers being constructed slowly and often painfully over long periods of

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This neo-intuitionism considers the falling apart of moments of life
into qualitatively different parts, to be reunited only while remaining separated by time as the fundamental phenomenon of the human
intellect, passing by abstracting from its emotional content into the
fundamental phenomenon of mathematical thinking, the intuition of
the bare two-oneness. This intuition of two-oneness, the basal intuition of mathematics, creates not only the numbers one and two, but
also all finite ordinal numbers, inasmuch as one of the elements of the
two-oneness may be thought of as a new two-oneness, which process
may be repeated indefinitely; . . . [1912, pp. 127–128]



. . . some Americans I have spoken with3 (who were otherwise of
quick and rational parts enough) could not, as we do, by any means
count to 1000; nor had any distinct idea of that number, though they
could reckon very well to 20. Because their language, being scanty
and accommodated only to the few necessaries of a needy, simple
life, unacquainted either with trade or mathematics, had no words in
it to stand for 1000; so that when they were discoursed with of those
greater numbers, they would show the hairs of their head, to express
a great multitude, which they could not number; which inability,
I suppose, proceeded from their want of names. The Tououpinambos had no names for numbers above 5;4 any number beyond that,
they made out by showing their fingers, and the fingers of others who
were present.


This passage [1690, p. 169] is discussed in [Butterworth, 1999, p. 52].
Presumably native Americans rather than Harvard graduates.
4 The Tououpinambos were native Brazilians. It sounds as if they had the arithmetical
system just mentioned which was found quite recently among the Gumulgal, Baikiri, and

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time by human beings. I will now give a brief sketch of some of the stages
in this human construction of our present-day natural number system.
Of course the beginnings of our present-day system of natural numbers
are lost in the distant past. However, we can get an idea of what these
beginnings might have been like from the very simple number systems
employed even today by some tribes. The number system of some such
tribes consists only of ‘one’, ‘two’, and ‘many’. Assuming that number
systems begin with ‘one’, ‘two’, ‘many’, the next step seems to have been
to have introduced ‘three’ as ‘two and one’, ‘four’ as ‘two and two’, five
as ‘two and two and one’. At any rate a system of this sort has been found
among many diverse peoples, for example the Gumulgal in Australia, the
Bakairi in South America, and the Bushmen in Africa [Barrow, 1992,
p. 53]. It is somewhat ironic that this binary system, which appears at the
beginning of human concern with numbers, has been revived for the very
latest computer technology. However, as far as humans are concerned, the
binary system was usually only employed among tribes who used numbers not greater than five or six. The introduction of larger numbers was
normally associated with a change of base. There are obvious reasons for
this since numbers written to the base 2 soon get unmanageably long for
humans, though of course they are no problem for computers.
Primitive number systems attract the attention of contemporary anthropologists, but have also been discussed by some quite famous earlier
thinkers. Book II, Chapter XVI of John Locke’s An Essay Concerning
Human Understanding is entitled ‘Of Number’, and here Locke writes:2



In practice, whatever they may possess in their language, they certainly use no numeral greater than three. When they wish to express
four, they take to their fingers . . . . They puzzle very much after five,
because no spare hand remains to grasp and secure the fingers that
are required for ‘units’.

The Damaras also experienced some difficulties in trading with Galton.
Here is Galton’s account of a typical transaction:
When bartering is going on, each sheep must be paid for separately.
Thus, suppose two sticks of tobacco to be the rate of exchange for
one sheep, it would sorely puzzle a Damara to take two sheep and
give him four sticks. I have done so, and seen a man first put two
of the sticks apart and take a sight over them at one of the sheep he
was about to sell. Having satisfied himself that that one was honestly
paid for, and finding to his surprise that exactly two sticks remained
in hand to settle the account for the other sheep, he would be afflicted
with doubts; the transaction seemed to come out too ‘pat’ to be correct, and he would refer back to the first couple of sticks, and then
his mind got hazy and confused, and wandered from one sheep to the
other, and he broke off the transaction until two sticks were put into
his hand and one sheep driven away, and then the other two sticks
given him and the second sheep driven away. [1853, p. 133]6

Francis Galton was quite puzzled to explain how the Damaras, who performed quite complicated operations of cattle herding, could manage with
such a simple system of arithmetic. In his account he gives the following
Yet they seldom lose oxen; the way in which they discover the loss
of one, is not by the number of herd being diminished but by the
absence of a face they know. [1853, p. 133]

Galton’s account of arithmetic in tropical South Africa is discussed by Barrow
[1992, p. 36].
6 In fairness to the Damaras, they may have been quite rational to be suspicious about
any exchange conducted with an English explorer such as Galton.

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Francis Galton, who later became famous for his work on statistics,
went in 1850 and 1851, as a young man in his late twenties, on
an expedition to tropical South Africa. In 1853, he published a book
describing his experiences, which included some interesting observations
about the arithmetical abilities of various tribes he met in his travels.5
One of the tribes which he encountered were the Damaras. According
to Galton [1853, p. 133], their arithmetical abilities were rather poor. He
says of them:



In fact Galton goes on to observe that the Damaras compensate for their
lack of arithmetic by having remarkable powers of recognising oxen,
which he was quite unable to acquire:

This shows that in certain circumstances, non-mathematical skills may be
more effective than mathematical ones.
Galton also met a tribe, the Ovampo, who could count. As he says
‘They . . . counted my oxen as quickly as I could have done it myself’
[1853, p. 184]. Interestingly these numerate Ovampo were traders who
‘went about the country bartering’ [1853, p. 184]. This lends support to
Locke’s suggestion that it is trade which stimulates the development of
Let us now return to our study of how number systems developed historically and socially. Our own number system uses the base 10, and this
system seems a very natural choice since humans have 10 fingers. However, history and anthropology show that this choice was by no means
universally adopted. We find systems to the base 5 (corresponding to one
hand), systems to the base 20 (corresponding to the use of toes as well as
fingers), and even in relatively rare cases systems to the base 8. A system
to the base 8 corresponds to counting using the spaces between the fingers
rather than the fingers themselves. Perhaps systems to the base 20 were
destined to disappear when mankind took to wearing shoes, but we still
find remnants of this system in some modern languages. For example the
French word for eighty is quatre-vingts = four-twenties.
Our present-day system of natural numbers uses not only a base (ten)
but also the place-value system and zero. In the place-value system (or
rule of position), the same digit, e.g. 1, has a different meaning depending
on where it appears. For example ‘1’ in 12 means ten, while ‘1’ in 167
means a hundred. This system is not used in Roman numerals where ten
is written X, a hundred C, and so on. Let us now give a sketch of how the
place-value system and zero came to be introduced.

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The Damaras have a wonderful faculty of recollecting any ox that
they have once seen. . . . Hans and John Allen were very quick at
recollecting oxen: I never could succeed in doing so myself: but it is
perfectly essential to a traveller here that some trustworthy persons
of his party should be able to pick out his own oxen from any drove
in which they have become mixed; . . . When fresh oxen are bought
. . . the new oxen are always trying to run off and get home again.
Now the tribe from whom they were bought may be lying at eight
or ten scattered kraals, to any one of which the ox that had been
bought but a few hours before and seen for a few minutes only, may
have made his escape. He has to be picked out from 500 or 600 head
of cattle and this the Damaras can do with perfect certainty. [1853,
pp. 145–146]



Georges Ifrah sums up the history as follows:

Obviously, no civilisation outside of these four ever felt the need to
invent zero; but as soon as the rule of position became the basis for
a numbering system, a zero was needed. All the same, only three of
the four (the Babylonians, the Mayans and the Indians) managed to
develop this final abstraction of number: the Chinese only acquired
it through Indian influences. However, the Babylonian and Mayan
zeros were not conceived of as numbers, and only the Indian zero
had roughly the same potential as the one we use nowadays. [1994a,
p. xxiii]

One curious thing historically about the place-value system was that it
was, as Ifrah points out, introduced quite early on (c. 2000 BCE) by the
Babylonians, but the innovation was not adopted by the Greeks despite the
enormous advances of ancient Greek mathematics in other directions. Perhaps part of the problem was that the Babylonians used a system with base
60. This was a curious choice, which proved too complicated for nearly all
other peoples who preferred the base 10. However, the base 60 still survives in a few areas such as minutes and seconds. Even the Babylonians
themselves used the sexagesimal system only for learned purposes, particularly astronomy, and a decimal, but not positional, system for everyday
reckoning. (See [Ifrah, 1994a, pp. 275, 284].)
In our present-day system, the place-value system is combined with the
use of zero, so that one, ten, hundred, . . . are written 1, 10, 100, . . . . However, the Babylonians did not succeed in fully introducing the very difficult
concept of zero. At first they indicated an empty place by simply leaving a
blank so that 101 would have been written 1 1. This notation meant that the
Babylonians could not distinguish, as we do, between 167, 1670, 16700,
. . . . The order of magnitude of the number had to be inferred from the
context. However, ‘in some instances scribes used special signs to mark
the separation of the orders of magnitude’ [Ifrah, 1994a, p. 291]. Not until
about 200–300 BCE, i.e., 1700 to 1800 years after the first introduction of
the place-value system, did the Babylonians introduce a symbol to denote
the absence of a unit. This symbol was a variant of the old separator sign.7

For an illustration of the symbol, see [Ifrah, 1994a, p. 297].

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Most peoples throughout history failed to discover the rule of position, which was discovered in fact only four times in the history
of the world. . . . The first discovery of this essential tool of mathematics was made in Bablyon in the second millennium BCE. It was
then rediscovered by Chinese arithmeticians at around the start of the
Common Era. In the third to fifth centuries CE, Mayan astronomers
reinvented it, and in the fifth century CE it was rediscovered for the
last time, in India.



Proper to man is a faculty which accompanies all his interactions
with nature, namely the faculty of taking a mathematical view of . . .
life. [1907, p. 53]

For an illustration of this sign, see [Ifrah, 1994a, p. 606].

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However, the Babylonians never understood their zero symbol as standing
for a number on a par with 1, 2 or 3 [Ifrah, 1994a, p. 671]. In this sense
they did not reach the full concept of zero.
The next civilisation to develop a positional system which used zero
was the Mayan civilisation of central America. This civilisation flourished
between the third and tenth centuries CE. However, it had disappeared by
1200 CE, and the newly arrived Spanish were amazed to discover huge
buildings fallen into ruin in the middle of the jungle. These had in fact
been constructed by the Mayans.
The Mayans were remarkably skilled at astronomy as well as architecture. One example is that they calculated the synodic revolution of
Venus as 584 days, a figure which compares well with the modern 583.92
days. As with the Baylonians it was in the learned (and religious) context
of astronomy that the Mayans developed their positional notation with a
zero. Zero was represented by a sign resembling a sea-shell or snail-shell.8
Despite the achievements of the Mayans, their number-system had a flaw.
They used 20 as a base, and so the second level should have been 20 × 20,
i.e., 400. However, the Mayans used 360 rather than 400 for the second
level, so that their system was not a strictly positional one. Why they did
so is not clear. Perhaps it had something to do with the fact that 360 is
closer to the number of days in the year.
Our modern decimal number-system was developed in India, probably before the seventh century CE, and was transmitted from the Indians,
via the Arabs, to Western Europe. The Indians introduced zero in a full
sense. This was certainly a very difficult conceptual step as is shown by
the facts that it took the Babylonians over 1700 years to take it only partially, and by the fact that the step was never taken at all by the ancient
Greeks. Perhaps the difficulty arises because we are used to thinking of
signs or names as denoting something, and so the concept of a symbol,
which denotes nothing, seems at first sight contradictory.
What I have given is of course only a rough sketch of how our familiar
natural numbers were introduced, but it suffices to show that the process
was a long and complicated one, and that it involved many steps which
were difficult to take. In the light of this, our present-day system of natural numbers does appear to be a human construction and one of a rather
elaborate kind. Moreover this construction is very different from the one
described by Brouwer.
Brouwer writes:



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Yet if Brouwer had met the Damaras in Southern Africa, he would have
been surprised to find that their interactions with nature were not accompanied by a mathematical view of life. On the contrary, they recognised that
one of their herd was missing not by the mathematical method of counting,
but by noticing that a familiar face was absent. Indeed during the many
thousands of years before the development of settled agriculture, hardly
any mathematics was known and very few, if any, of the interactions with
nature by humans were accompanied by a mathematical view of life.
Of course the mathematical view of life as it has developed since 1500
in combination with changes in technology and social organisation, has
proved a powerful instrument in dominating nature. But this instrument
is no more natural to man than the petrol engine. Once discovered, the
mathematical view of life has to be transmitted by a long, complicated
(and often painful!), process of instruction in schools and colleges.
In his 1905 pamphlet on ‘Life, art and mysticism’, Brouwer begins with
the claim (p. 1) that ‘Originally man lived in isolation . . .’. However, this is
completely false. It is almost always true that wherever human beings have
existed, they have lived in social groups. There are indeed a few exceptions
to this rule. For example, there are cases of individuals who were stranded
on desert islands or in uninhabited lands where they managed to survive
on their own for a number of years. Another exceptional case is that of
hermits who lived a solitary and isolated existence for religious reasons.
However, these are indeed exceptional cases, and we can safely say that
social groups are the usual environment for human beings.
In accordance with his ideas of man living in isolation, Brouwer imagines an individual mathematician constructing the numbers in solitude
from the a priori intuition of time. Perhaps some individual mathematicians might have the illusion they could do this, but this would only
be because they had learnt all about the numbers through long years of
instruction. If we examine how our current system of natural numbers in
the decimal notation was actually constructed historically as has just been
done, it seems almost impossible that an individual could do it on his or her
own without prior instruction. Think only of the difficulties of introducing
the concept of zero. It took the Babylonians between 1700 and 1800 years
to introduce a sign for zero, and even then they did not reach a full concept of zero. Some mathematical concepts are counter-intuitive rather than
natural, and are the result of centuries of trial and error rather than easy
inferences from an a priori intuition.
My claim then is that a historical and sociological account of how numbers were actually constructed shows that Brouwer’s version of constructivism is not satisfactory, but this does not mean that the basic idea of
mathematical entities as human constructions should be abandoned. Rather
than being the subjective mental constructions of individual mathematicians, mathematical entities might be the social constructions of human



groups. This example shows the value for philosophy of mathematics of
some empirically based sociological considerations. The introduction of
this sociology makes some philosophical viewpoints implausible, while
suggesting new, and hopefully more realistic, approaches.

I began the paper by considering some of Lakatos’s arguments written in
the early 1970s against the view that the introduction of sociology could
be helpful for philosophy of mathematics. In this final section, I want to
consider a more recent line of argument, due to Azzouni, which supports a
similar conclusion. Azzouni does not argue directly against the use of sociology by philosophers of mathematics; rather, in chapter 6 of his [2006],
which is entitled: ‘The uniqueness of mathematics as a social practice’,
he argues that mathematical practice differs in a fundamental way from
other social practices. More specifically his claim is that, while in most
other social practices, conformity is assured by social pressures of various
kinds, this is not the case with mathematical practice. Instead conformity
or agreement in mathematical practice is produced, not by social pressure,
but in another way. This is how he puts his claim:
What seems odd about mathematics as a social practice is the presence of substantial conformity on the one hand, and yet, on the other,
the absence of (sometimes brutal) social tools to induce conformity
that routinely appear among us whenever behavior really is socially
constrained. Let’s call this ‘the benign fixation of mathematical practice.’ [2006, p. 129]

If this claim were true, then sociology would not be of much use in the
philosophy of mathematics, since agreement in mathematics would have
other, non-social, reasons or causes. Philosophers of mathematics would
better concern themselves with these other reasons or causes than introduce sociological considerations. Azzouni’s claim is therefore a challenge
to the views of this paper, and, accordingly, I will now expound it in detail
and subject it to criticism.
In fact Azzouni maintains that there are two ways in which mathematical practice differs from almost any other social practice. As he says:
There are two striking ways mathematical practice differs from just
about any other group practice that humans engage in. [2006, p. 124]

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5. Does Mathematical Practice Differ Fundamentally from Other
Social Practices?



The first of these two ways is the following:
It’s been widely observed, in contrast to other kinds of conformity
that really do have their source in social forces, that one finds in
mathematical practice nothing like the variability found in cuisine,
clothing, or metaphysical doctrine. [2006, p. 124]

What should strike you about ‘alternative mathematics’ . . . is that
such mathematics is mathematics as usual. [2006, p. 126]

But this does not seem an adequate reply. Let us compare a European banquet of 1500, eaten with knives and forks and following a certain procedure regarding the dishes etc., with a Chinese banquet of 1500, eaten with
chopsticks and following a very different procedure with very different
dishes. Would it be fair to say that both these events are just eating as
usual? Of course any cuisine at the end of the day involves humans eating, and so all cuisines do have something in common. However, this does
not preclude a great deal of variation, and indeed Azzouni lists ‘cuisine’
as an example of a standard social practice. Similarly all mathematical
practices have something in common, or we would not call the practice

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Against this it could be argued that there is in fact a great deal of variability
in mathematical practice. To begin with, one could consider the enormous
variability in the ways in which basic arithmetical computations have been
represented and performed. Some examples of this were given in the previous section. Curiously Azzouni does mention this case, writing ‘There
are examples of deviant computational practices: Babylonian fractions or
the one-two-many form of counting . . .’ [2006, p. 124]. He also mentions
‘. . . the development of alternative mathematics — intuitionism, for example, or mathematics based on alternative logics (e.g. paraconsistent logics)’
[2006, p. 126].
These are twentieth-century examples, but he also gives a reference (on
p. 126) to a pre-twentieth-century example, namely the different way in
which the calculus was handled by British and Continental European mathematicians during the eighteenth century and the early years of the nineteenth century. Here we had two different mathematical practices which
obviously had a social basis and which persisted for more than a hundred
years. One could also give many further examples. In the nineteenth century there was a period of divergence between those mathematicians who
accepted non-Euclidean geometry and those who did not. Then again Chinese mathematics differed a great deal from European mathematics in the
period say 1000–1500 CE. Is this so very different from the case of Chinese cuisine and European cuisine in the same period?
How does Azzouni reply to these difficulties? This is what he says:



Let’s turn to the second (unnoticed) way that mathematics shockingly
differs from other group practices. . . . What makes mathematics difficult is (i) that it’s so easy to blunder in; and (ii) that it’s so easy
for others (or oneself) to see — when they’re pointed out — that
blunders have been made. [2006, p. 126]

However the two characteristics, which allegedly here differentiate mathematics, apply just as much to other practices, which no one would deny are
based on conventions enforced socially. An example of such a practice is
the spelling of the English language. Much of this spelling is clearly highly
conventional and enforced socially. Indeed the same word is spelt ‘color’
in American English and ‘colour’ in British English. The school children
in the two countries are socially conditioned in different ways. However,
when it comes to English-language spelling, it is (i) very easy to blunder,
and (ii) very easy for others (or oneself) to see that blunders have been
made, when confronted with a standard dictionary. Obviously the writing of Chinese characters would constitute another similar (perhaps even
better) example. So Azzouni’s second way of distinguishing mathematics
from other social practices does not work any better than the first.
Let us now turn to Azzouni’s discussion of how what he calls ‘the
benign fixation of mathematical practice’ is in fact achieved. His thesis
is that, to achieve consensus in mathematical practice, we do not have to
resort to the kind of social conditioning needed to instil, for example, table
manners. Of course this is not at all obviously true. Children are subjected
to many years of social conditioning at school designed to make them do
their sums correctly. Moreover, in the past this social conditioning often
took quite brutal forms. Those who got their sums wrong were beaten. One
hopes that nowadays such brutal methods have been abandoned, but still
there can be little doubt that arithmetic is instilled by social conditioning
at school.
To this argument, it could be replied that while it may be true that the
rules of arithmetic are enforced on children partly by social conditioning, it
does not follow that the rules of arithmetic are social conventions. It could
be that the rules of arithmetic, in contrast to those of table manners, are

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‘mathematical’, but this does not preclude a great deal of variation. In
the past when communications were poor, social groups distant from one
another could continue for long periods doing mathematics in different
ways. Even in our age of instant communication, groups of mathematicians, such as constructivists, can maintain a mathematical practice different from the mainstream.
Let us now consider the second way in which Azzouni claims that
mathematical practices differ from other social practices. He describes it
as follows:



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fixed by considerations which have nothing to do with the social. Now here
it should be observed that the general character of most social practices is
not fixed entirely by social conventions. Most social practices are designed
to achieve some goal, and this places objective constraints on how the
practice can be organised. A social practice inculcated by social conditioning which altogether failed to achieve its aims would not be viable for
any long period. So, to take Azzouni’s own example of a standard social
practice, namely cuisine, there are obviously enormous variations in the
kind of food used, in how it is prepared, in how it is eaten, etc. However,
for any cuisine to be viable long-term, it must produce enough calories
to sustain human life. Moreover, if we require something more of a cuisine, namely that it should, as far as possible, produce healthy, long-lived
humans, then we have excellent grounds, which are not merely the result
of social conditioning, for preferring one cuisine to another. For example,
the English monks of the Middle Ages ate what was no doubt a delicious
cuisine which involved a great deal of cream and meat. Japanese monks
of the same period ate a very different cuisine, which was largely vegetarian in character, but prepared in a skilful fashion. Buddhist monasteries in
Japan of today often have restaurants where the visitor can sample this cuisine and it is indeed delicious. It would be easy to adapt children to either
cuisine through social conditioning. However, the long-term health effects
of the two cuisines are very different. Written records and the analysis of
the remains of the monks show that Japanese monks were slim and fit,
and relatively healthy and long-lived for the time. This was not true of the
English monks who were the pioneers of the present international obesity
crisis. English monks for the most part were very portly (as is revealed by
the effect of this on the bones of their legs), suffered from all the ailments
associated with a diet high in saturated fat, and were not very long-lived.
Now Azzouni points out [2006, p. 126] that the practice of British mathematicians in the eighteenth century regarding the calculus was inferior
to that of the Continental mathematicians and so eventually disappeared.
However, as we have just shown in the case of cuisine, it is perfectly possible in the case of other bona fide social practices, for one practice to prove
itself to be superior to another.
Let us now return to the question of how Azzouni explains what he
calls ‘the benign fixation of mathematical practice’ [2006, p. 129]. Of
course he rejects any explanation in terms of social conditioning, and so
has to consider other possibilities. The first view he considers [2006, §6.4,
pp. 129–130] is Platonism. Now if Platonism were indeed true it would
explain the benign fixation. Trained mathematicians would acquire the
ability to perceive through intuition the transcendental Platonic world of
mathematical objects, and, having seen the same things, they would naturally agree. As we have seen in Section 1 of this paper, Lakatos favoured
a kind of Platonism, and, for the Platonist, sociological considerations are



hardly necessary. Azzouni however, who is a nominalist, perforce rejects
this approach, and tries some different approaches. He finds it convenient
to distinguish between mathematics before about 1900, and mathematics
after about 1900 (contemporary mathematics).
As regards mathematics before about 1900, he has this to say:

So the agreement in mathematical practices is to be explained by the fact
that all humans have the principles of first-order classical (or Fregean)
logic hard-wired into their brains. It should be pointed out that this is not
the only non-social way of explaining the such agreement in mathematical
practices as did occur before 1900. An alternative explanation is obtained
by considering the aims of mathematical practices. One principal use of
mathematics was in physics, and it could be that only physical theories
based on classical logic were successful.
This alternative explanation overcomes a difficulty which Azzouni
notes in his own approach in terms of first-order classical logic being hardwired in the brain. This is how he puts the problem:
A complication that (potentially) mars this otherwise appealing view
of the implicit role of first-order logic in mathematics: The ‘logic’ of
ordinary language looks much richer than anything first-order predicate calculus can handle — notoriously, projects of canonizing the
logic of anything other than mathematics using (even enrichments
of) first-order predicate calculus have proved stunningly unsuccessful. . . . But it would be very surprising if the tacit logic of mathematics were different from that of ordinary language generally . . . [2006,
fn. 32, pp. 136–137]

However, there is quite strong evidence that the tacit logic of mathematics
is indeed different from that of everyday reasoning carried out in ordinary
language. This evidence arose from attempts made by those working in
artificial intelligence to represent typical kinds of everyday reasoning —
for example, the reasoning involved in choosing a train from a timetable
in order to arrive somewhere in time for a meeting. It turned out that this
kind of common-sense reasoning is best represented by a non-classical
logic, more specifically by the kind of non-monotonic logic involving
negation-as-failure, which is used in PROLOG. Conversely, even in such
a strange theory as quantum mechanics, the mathematics used is still

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The logical principles implicit in mathematical practice — until the
twentieth century, however — remained the same topic-neutral ones
(at least relative to mathematical subject matters). Such uniformity
of logical practice suggests, as does the uniformity of counting and
summation practices . . . an at least partially ‘hard-wired’ disposition
to reason in a particular way. [2006, p. 136]



Contemporary mathematics breaks away from the earlier practice
in two dramatic respects. First, it substitutes for classical logic . . .
proof procedures of any sort (of logic) whatsoever provided only that
they admit of the (in principle) mechanical recognition of completely
explicit proofs. . . . (p. 138) The second way that contemporary mathematics bursts out from the previous practice is that it allows pure
mathematics such a substantial life of its own that areas of mathematics can be explored and practiced without even a hope (as far as
we can tell) of empirical application. [2006, pp. 137–138]

I think Azzouni has here given an excellent characterisation of some of the
novel features of mathematics after about 1900. However, it also seems to
me that this characterisation goes against some of the theses for which
he argued a few pages earlier. If classical logic were really hard-wired
into the brain, then it would be very surprising that mathematical systems
which no longer use this logic could have been developed after 1900. On
the other hand, if classical logic was developed for a mathematics which
was designed to be useful in physics, then where the goal of being useful
in physics is dropped, it is not surprising that classical logic might also be
dropped. What Azzouni describes as allowing ‘pure mathematics . . . a substantial life of its own’ [2006, p. 138] clearly drops the goal of developing
mathematics which is useful for physics. This might well lead to dropping
classical logic as well, and indeed I think we can explain the emergence of
intuitionism in these terms.
There is one further point which Azzouni does not mention: post-1900
mathematics, as well as allowing pure mathematics a substantial life of
its own, also developed applications of mathematics to areas other than
physics, and this again might well lead to the use of logics different from
classical logic. This would explain the emergence of non-classical logics
in artificial intelligence.
9 A more detailed exposition of these claims together with detailed arguments for them
are to be found in [Gillies, 2011] — particularly Sections 3 and 4, pp. 181–189.

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standard mathematics based on first-order classical logic.9 If the logic of
everyday reasoning in ordinary language is not first-order classical logic,
then it is very unlikely that first-order classical logic is hard-wired into
the brain. Since first-order classical logic is used in the mathematics of
physics, it seems more likely than this way of reasoning was developed,
like the differential and integral calculus, in order to obtain empirically
successful theories in physics.
This point of view will be reinforced by considering what Azzouni has
to say about mathematics after about 1900 (contemporary mathematics).
He writes:



Azzouni, J. [2006]: Tracking Reason. Proof, Consequence and Truth. Oxford:
Oxford University Press.
Barrow, J.D. [1992]: Pi in the Sky. Counting, Thinking and Being. Penguin
Brouwer, L.E.J. [1905]: ‘Life, art and mysticism’, in [Brouwer, 1975], pp. 1–10.
——— [1907]: The Foundations of Mathematics. Doctoral Thesis. University of
Amsterdam. Reprinted in [Brouwer, 1975], pp. 11–101.
——— [1912]: ‘Intuitionism and formalism’, reprinted in [Brouwer, 1975],
pp. 123–138.
——— [1933]: ‘Volition, knowledge, language’, reprinted in [Brouwer, 1975],
pp. 443–446.
——— [1975]: Collected Works. Vol. 1 Philosophy and Foundations of Mathematics. A. Heyting, ed. Amsterdam: North-Holland.
Butterworth, B. [1999]: The Mathematical Brain. London: Macmillan. Papermac edition.
Galton, F. [1853]: The Narrative of an Explorer in Tropical South Africa.
London: John Murray.
Gillies, D.A., ed. [1992]: Revolutions in Mathematics. Oxford: Oxford University Press.
——— [2000]: ‘An empiricist philosophy of mathematics and its implications for the history of mathematics’, in E. Grosholz and H. Breger, eds.,
The Growth of Mathematical Knowledge, Synthese Library, pp. 41–58.

This is argued in more detail in [Gillies, 2000]. See particularly pp. 50–51.

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Let us now further consider systems of pure mathematics, which are
‘explored and practiced without even a hope (as far as we can tell) of
empirical application’ [Azzouni, 2006, p. 138]; such systems of pure mathematics are in effect metaphysical systems.10 Now one of Azzouni’s earlier
arguments for the uniqueness of mathematics as a social practice was that:
‘one finds in mathematical practice nothing like the variability found in
. . . metaphysical doctrine’ [2006, p. 124]. However, contemporary mathematics has turned metaphysical, and has developed a very great variety of
systems of pure mathematics, which are metaphysical in character. Once
the requirement of having some empirical application is dropped, there is
hardly any constraint on the diversity of mathematical systems which can
be developed.
My conclusion then is that Azzouni’s arguments for the uniqueness
of mathematics as a social practice fail. Quite to the contrary, the social
practice of mathematics has many features in common with other social
practices such as cuisine, clothing, or metaphysics. This makes it likely
that some sociological considerations will prove useful for philosophers
of mathematics.



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Dordrecht: Kluwer. Also available online at
gillies/guide to publications.htm
——— [2011]: ‘The empiricist view of logic’, in C. Cellucci, E. Grosholz, and
E. Ippoliti, eds., Logic and Knowledge, pp. 175–190. Newcastle upon Tyne:
Cambridge Scholars Publishing.
Ifrah, G. [1994a]: The Universal History of Numbers I. The World’s First
Number-Systems. David Bellos and E.F. Harding, trans. London: Harvill
Press, 2000.
——— [1994b]: The Universal History of Numbers II. The Modern NumberSystem. David Bellos, E.F. Harding, Sophie Wood, and Ian Monk, trans. London: Harvill Press, 2000.
Kuhn, T.S. [1962]: The Structure of Scientific Revolutions. Chicago: Chicago
University Press.
Lakatos, I. [1970]: ‘Falsification and the methodology of scientific research programmes’, reprinted in [Lakatos, 1978b], pp. 8–101.
——— [1971]: ‘History of science and its rational reconstructions’, reprinted in
[Lakatos, 1978c], pp. 102–138.
——— [1976]: ‘Understanding Toulmin’, in [Lakatos, 1978c], pp. 224–243.
Written c. 1973.
——— [1978a]: ‘The problem of appraising scientific theories: three approaches’,
in [Lakatos, 1978c], pp. 107–120. Written c. 1973.
——— [1978b]: Philosophical Papers, Volume I: The Methodology of Scientific
Research Programmes. J. Worrall and G. Currie, eds. Cambridge: Cambridge
University Press.
——— [1978c]: Philosophical Papers. Volume II: Mathematics, Science and
Epistemology. J. Worrall and G. Currie, eds, Cambridge: Cambridge University Press.
Lakatos, I., and A. Musgrave, eds. [1970]: Criticism and the Growth of Knowledge. Cambridge: Cambridge University Press.
Locke, J. [1690]: An Essay Concerning Human Understanding. Everyman’s
Library. London: J.M. Dent, 1961.
Marx, K. [1845]: ‘Theses on Feuerbach’, in Karl Marx and Frederick Engels.
Selected Works, pp. 28–30. London: Lawrence and Wishart, 1973.
Mehrtens, H. [1976]: ‘T.S. Kuhn’s theories and mathematics: A discussion
paper on the “new historiography” of mathematics’, Historia Mathematica 3,
297–320. Reprinted in [Gillies, 1992], pp. 21–41.
Toulmin, S. [1972]: Human Understanding, I: General Introduction and Part I.
Oxford: Oxford University Press.

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