Proof as a practice of mathematical pursuit in a cultural, socio-political and

intellectual context

Man-Keung Siu

Department of Mathematics

The University of Hong Kong

Abstract

Through examples we explore the practice of mathematical pursuit, in particular on

the notion of proof, in a cultural, socio-political and intellectual context. One objective

of the discussion is to show how mathematics constitutes a part of human endeavour

rather than stands on its own as a technical subject, as it is commonly taught in the

classroom. As a ‘bonus’, we also look at the pedagogical aspect on ways to enhance

understanding of specific topics in the classroom.

1. Prologue: Question 6 of the 29th IMO

Proof is, to some extent, as much an individualized activity as a social activity. It is an

individualized activity in that a breakthrough or an igniting spark arises from the mental

exertion of oneself, though sometimes aided or stimulated through the discussion with fellow

mathematicians. It is a social activity in that a proof has to pass the scrutiny of other

mathematicians in order to gain approval and acceptance by the mathematical community.

Hence, we will begin with an example this author experienced in person. This example

highlights the main function of a proof, which is to elucidate rather than just to verify.

After that, we will explore, through four examples (in Sections 3,4,5,6), the practice of

mathematical pursuit, in particular on the notion of proof, in a cultural, socio-political and

intellectual context.

Question 6 of the 29th International Mathematical Olympiad, held in Canberra in 1988,

reads:

“Let a and b be positive integers such that ab + 1 divides

a2 + b2. Show that
a2 + b2

ab + 1
is the square of an integer.”

A slick solution to this problem, offered by a Bulgarian youngster who received a special

award for it, starts by supposing that k =
a2 + b2

ab + 1
is not a square and rewriting the expression

as an equation.

a2 − kab + b2 = k, where k is a given positive integer (∗).

1



Note that any integral solution (a, b) of (∗) must satisfy ab ≥ 0, or else ab ≤ −1, and
a2 + b2 = k(ab + 1) ≤ 0, implying that a = b = 0 so that k = 0! Furthermore, since k is not
a square, we have ab > 0, that is, none of a or b is 0. Let (a, b) be an integral solution of

(∗) with a > 0 (and hence b > 0) and a + b smallest. We may assume a ≥ b by symmetry.
Regard (∗) as a quadratic equation with roots a and a′. Then a + a′ = kb and aa′ = b2 − k.
Hence a′ is also an integer and (a′, b) is an integral solution of (∗). Since b > 0, we have
a′ > 0. But

a′ =
b2 − k

a
≤ b

2 − 1
a

≤ a
2 − 1
a

< a,

so that a′ + b < a + b, contradicting the choice of (a, b)!. This proves that
a2 + b2

ab + 1
must be

the square of an integer.

Slick as the proof is, it also invites a couple of queries. (1) What makes one suspect

that
a2 + b2

ab + 1
is a square? (2) The argument by reductio ad absurdum should hinge crucially

upon the condition that k is not a square. In the proof this condition seems to have slipped

in casually so that one does not see what really goes wrong if k is not a square. More

pertinently, this proof by contradiction has not explained why
a2 + b2

ab + 1
must be a square,

even though it confirms that it is so. (For a discussion on the cognitive and didatic aspects

of students’ difficulty with proof by contradiction, see (Antonini, Mariotti, 2008).)

In contrast let us look at a less elegant solution, which is my own attempt. When I

first heard of the problem, I had a ‘false insight’ by putting a = N 3 and b = N so that

a2 + b2 = N 2(N 4 + 1) = N 2(ab + 1). Under the impression that any integral solution (a, b, k)

of k =
a2 + b2

ab + 1
is of the form (N 3, N, N 2) I formulated a strategy of trying to deduce from

a2 + b2 = k(ab + 1) the equality

[a − (3b2 − 3b + 1)]2 + [b − 1]2 = {k − [2b − 1]}{[a − (3b2 − 3b + 1)][b − 1] + 1}.

Were I able to achieve that, then I could have reduced b in steps of one until I got down to

the equation k =
a2 + 1

a + 1
for which a = k = 1. By reversing steps I would have solved the

problem. I tried to carry out this strategy while I was travelling on a train, but to no avail.

Upon returning home I could resort to systematic brute-force checking and look for some

actual solutions, resulting in a (partial) list shown below.

a 1 8 27 30 64 112 125 216 240 343 418 512 · · ·
b 1 2 3 8 4 30 5 6 27 7 112 8 · · ·
k 1 4 9 4 16 4 25 36 9 49 4 64 · · ·

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Then I saw that my ill-fated strategy was doomed to failure, because there are solutions

other than those of the form (N 3, N, N 2). However, not all was lost. When I stared at the

pattern, I noticed that for a fixed k, the solutions could be obtained recursively as (ai, bi, ki)

with

ai+1 = aiki − bi, bi+1 = ai, ki+1 = ki = k.
It remained to carry out the verification. Once that was done, all became clear. There is a

set of ‘basic solutions’ of the form (N 3, N, N 2) where N ∈ {1, 2, 3, · · ·}. All other solutions
are obtained from a ‘basic solution’ recursively as described above. In particular, k =

a2 + b2

ab + 1
is a square. I feel that I understand the phenomenon much more than if I just learn from

reading the slick proof.

2. Relevance to learning and teaching of mathematics in the classroom

The explanatory power of a proof, as exemplified in the example in Section 1, has long

been recognized and discussed at length by many authors. Instead of giving a list of ref-

erences, which is bound to be incomplete in view of the vast size of the relevant body of

literature (and many more that will be written), I would only mention one survey paper

(Hanna, 2000), two websites (see the two items at the end of the references) and three books

(Davis, Hersh, 1980; Hanna, 1983; Siu, 1990/2007/2008), with their numerous bibliographi-

cal references thereof.

Why then do I write this paper, realizing that it would be like adding one drop of water

to a huge ocean of existing works? What specific point of view do I try to offer in this paper?

As the title suggests, I like to explore the practice of mathematical pursuit known as a proof

in a cultural, socio-political and intellectual context. A broader message I like to convey is

that mathematics constitutes a part of human endeavour rather than stands on its own as a

technical subject, as it is commonly taught in the classroom.

In (Siu, 2006) one of the reasons given to account for a general hesitation of teachers

to integrate history of mathematics with the learning and teaching of mathematics in the

classroom is the concern that students lack enough knowledge on culture in general to ap-

preciate history of mathematics in particular. This is probably quite true, but we can look

at it from the reversed side. We can regard the integration of history of mathematics with

day-to-day learning and teaching as an opportunity to let students know more about culture

in general. In particular, proof is so much an important ingredient in a proper education

in mathematics that we can ill afford to miss such an opportunity in this regard. Although

the evolution of the standard of rigour or the epistemological aspect of mathematical proofs

3



(Lakatos, 1976; Rav, 1999)) are not our main focal points in this paper as far as learning

and teaching of mathematics in the classroom are concerned, they will invariably come into

the picture.

I will discuss four examples: (1) the influence of the exploratory and venturesome spirit

during the ‘era of exploration’ in the 15th and 16th centuries C.E. on the development of

mathematical practice in Europe, (2) the influence of the intellectual milieu in the period of

the Three Kingdoms and the Wei-Jin Dynasties from the 3rd to the 6th centuries C.E. in

China on the mathematical pursuit as exemplified in the work of Liu Hui (flourished in the

mid 3rd century C.E.), (3) the influence of Daoism on mathematical pursuit in ancient China

with examples on astronomical measurement and surveying from a distance. (4) the influence

of Euclid’s Elements in Western culture compared to that in China after its transmission

through the first Chinese translation by Matteo Ricci (1552-1610) and Xu Guang-Qi (1562-

1633) in 1607.

Example (1) touches on a broad change of mentality in mathematical pursuit, not just

affecting its presentation but more importantly bringing in an exploratory spirit. Example

(2) is about a similar happening, for a different reason, in the oriental part of the world,

with more emphasis on the aspect of argumentation. Example (3) concerns the possible

role religious, philosophical (or even mystical) teachings may play in mathematical pursuit.

Example (4) points to such kind of influence but in the reversed direction, namely, how the

thinking in mathematical pursuit may breed thinking in other areas of human endeavour.

As a ‘bonus’, these examples sometimes suggest ways to enhance understanding of specific

topics in the classroom.

3. ‘Era of exploration’

¿From the mid 15th century C.E. into the 16th century C.E. the European world saw the

emergence of a group of ‘ocean explorers’ (some would see them as ‘crusaders’ or ‘colonialists’

or even ‘pirates’, depending on one’s stand and view on history) who travelled to far-off shores

hitherto unheard of. Those who left their marks in history by such ventures include names

like Christopher Columbus (1451-1506), Vasco da Gama (1460-1524), Ferdinand Magellan

(1480-1521), Francis Drake (1540-1596), Walter Raleigh (1554-1618). Regardless of their

motive, one has to admire their exploratory and venturesome spirit.

This exploratory and venturesome spirit became a model and an inspiration for promoters

of modern science (Alexander, 2002). In his book Novum Organum (1620/2000) Francis

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Bacon wrote:

“We should also take into account that many things in nature have come

to light and been discovered as a result of long voyages and travels (which

have been more frequent in our time), and they are capable of shedding

new light in philosophy. Indeed it would be a disgrace to mankind if wide

areas of the physical globe, of land, sea and stars, have been opened

up and explored in our time while the boundaries of the intellectual

globe were confined to the discoveries and narrow limits of the ancients.”

(Bacon, 1620/2000, Book I, Section LXXXIV)

He also wrote in the same book:

“And therefore we should reveal and publish our conjectures, which make

it reasonable to have hope: just as Columbus did, before his wonderful

voyage across the Atlantic Sea, when he gave reasons why he was con-

fident that new lands and continents, beyond those previously known,

could be found; reasons which were at first rejected but were afterwards

proven by experience, and have been the causes and beginnings of great

things.” (Bacon, 1620/2000, Section XCII)

It may seem that mathematics, as a pure and abstract subject, would not fit in well

with this trend. Metaphorically speaking, mathematics rests on the sure stable ground of

Euclidean geometry while explorers should go out to the rough ocean to explore and to

discover the unknown world. However, even in mathematics change occurred in the 17th

century C.E., supplanted by philosophical consideration expressed in the words of Galileo

Galilei:

“Logic, it appears to me, teaches us to test the conclusiveness of an

argument already discovered and completed, but I do not believe that

it teaches us to discover correct arguments and demonstrations.” [The

translation is adopted from (Kline, 1977, p.118).]

and of René Descartes:

“I saw that, regarding logic, its syllogisms and most of its other precepts

serve more to explain to others what one already knows, or even, like the

art of Lully, to speak without judgement of those things one does not

know, than to learn anything new.” [The translation is adopted from

(Descartes, 1637/1968, p.40).]

5



One outcome is the bold venture to explain and to discover results in the unknown realm

of the infinite by the method of infinitesimals, for instance by Johannes Kepler (1571-1630)

in his Nova Stereometria Doliorum Vinariorum (1615), and the method of indivisibles, for

instance by Bonaventura Cavalieri (1598-1647) in his Geometria Indivisibilus Continuorum

Nova Quadam Ratione Promota (1635). In the former a geometric object is considered to

be made up of ‘very small’ objects of the same dimension, while in the latter a geometric

object is considered to be made up of objects of dimension one lower (Calinger, 1982/1995;

Mancosu, 1996). In fact, similar ideas already appeared in earlier works of mathematicians

in both the West (for instance, Archimedes in the 3rd century B.C.E.) and the East (for

instance, Liu Hui in the 3rd century C.E., Zu Chong-Zhi and his son Zu Geng in the late 5th

century C.E.). These examples provide stimulating and instructive didatical material for the

classroom (Calinger, 1982/1995; Shen, 1997; Siu, 1993; Wagner, 1978; Wagner, 1979).

The story on the computation of the volume of a sphere is particularly noteworthy, not

only because of the ingenious use of the method of indivisibles to compute the volume of

the closely related object ‘Mou-he-fang-gai’ (literally, it means “box with a closely fitted

square lid”), which is the common portion of two cylinders of equal diameter placed with

axes orthogonal to each other, but also because of the kind of intellectual integrity and

humbleness Liu Hui displayed. After sketching his brilliant idea on how to proceed, he said:

“I wish to give my humble reflections,

But fear that I will miss the correct principle;

I dare to let the doubtful points stand,

Waiting

For one who can expound them.”

[The translation is adopted from (Wagner, 1978)]

It reminds one of a saying from the late Russian mathematics educator, Igor Fedorovich

Sharygin (1937-2004):

“The life of mathematical society is based

on the idea of proof, one of the most highly

moral ideas in the world.”

The learning of proof does have its value in ‘moral education’ !

It is interesting to note that, after two centuries of exciting discoveries in calculus, the

subsequent development in the 19th century C.E. gradually reverted to a more conservative

6



style dominated by the ‘notorious’ (among generations of undergraduates!) “epsilon-delta”

analysis.

The pendulum may be swinging again. Since the latter part of the 20th century C.E.

the increasing power of computers and the increasing versatility of numerous software en-

able mathematicians to enter into another ‘era of exploration’. Some even start to question

whether the role of a proof should be reconsidered, leading to debates which are both philo-

sophical and controversial (Davis, 2006).

4. Intellectual milieu in China in the period from the 3rd to 6th centuries

The eminent British mathematician Godfrey Harold Hardy (1877-1947) once made the

comment (Hardy, 1940), “The Greeks first spoke a language which modern mathematicians

can understand; as Littlewood said to me once, they are not clever schoolboys or ‘scholar-

ship candidates’, but ‘Fellows of another college’.” He was speaking from a viewpoint that

holds the time-honoured axiomatic-deductive tradition inherited from the ancient Greeks,

exemplified in Euclid’s Elements, to be the only proper mode of a proof. Some authors

offer examples from other mathematical cultures to counterbalance this view (Chemla, 1996;

Chemla, 1997; Joseph, 1991/1994/2000; Siu, 1990/2007; Wilder, 1968/1978).

In (Siu, 1993) I described in details several specific examples from the commentaries of

Liu Hui on Jiuzhang Suanshu (The Nine Chapters on the Mathematical Art). Many of these

can be rendered, with the help of visual aids, into useful didactical material for the classroom.

One notable example is Problem 16 in Chapter 9 of the book, which says: “A right triangle

has sides of 8 steps and of 15 steps. What is the diameter of its inscribed circle?” The

method in the book gives the correct formula for the diameter, namely, d = 2ab/(a + b + c)

where c is the hypotenuse and a, b are the other two sides. In his commentary, Liu Hui

offered three different proofs. The first proof is by a colourful (literally, as the text indicates

pieces of different colours) method of dissection. (See (Siu, 1993, Figure 3) for a “proof

without words”.) The second proof uses knowledge about proportional quantities. The

third proof is most interesting from the viewpoint of the role of a proof, because Liu Hui was

probably looking for “consistency [of the theorem] with the body of accepted mathematical

results” (Hanna, 1983, p.70). (See (Siu, 1993, Section 3) for a detailed discussion on the

mathematics.)

7



At the beginning of his commentary Liu Hui wrote:

“I studied Jiuzhang [The Nine Chapters on the Mathematical Art] at an

early age and perused it when I got older. I see the separation of the

Yin and the Yang and arrive at the root of the mathematical art. In

this process of probing I comprehend its meaning. [......] Things are

related to each other through logical reasons so that like the branches

of a tree, diversified as they are, they nevertheless come out of a single

trunk. If we elucidate by prose and illustrate by pictures, then we may be

able to attain conciseness as well as comprehensiveness, clarity as well as

rigour. Looking at a part we will understand the rest.” [The translation

is adopted from (Siu, 1993).]

This passage not only indicates a different style in mathematical practice but also exhibits a

different mentality from that of the traditional school of Confucianism, which by the latter

part of the 1st century B.C.E. was made the orthodoxy belief of the Han Dynasty at the

expense of other schools of thought.

To put this trend in its historical perspective we should note that the 3rd century C.E., in

which Liu Hui flourished, fell into an exceptionally interesting period of Chinese history. The

four centuries, beginning with the collapse of the Han Dynasty in 220 and ending with the

establishment of the Sui Dynasty in 581, were a “prolonged period of disunity and confusion,

[...] marked by frequent warfare and political cleavage between a series of dynasties that ruled

in Central and South China, and another series that had control in the North” (Feng, 1948).

Although this prolonged period of disarray and strife was politically and socially a ‘dark

age’, it was also “an age in which, in several respects, we reach one of the peaks of Chinese

culture” (Feng, 1948). Ironically, the collapse in political and social order brought with it a

weakening of the orthodoxy belief, giving way to free and uninhibited thinking. The period

was known for a predilection for rhetoric and dialectic, characterized by a refined intellectual

activity referred to as qing-tan, which literally means “pure conversation”. According to the

historian Yu Ying-Shih, the intellectual milieu of this period was a result of a kind of “self-

awareness”, both as an individual and as a community, that was formed since the latter part

of the Late Han Dynasty (25-220) among the class of Shi1 (Yu, 1987, Chapter 6, Chapter

7). It is natural to propose that, in the area of science and mathematics, the predilection

1Shi is a rather peculiar but extremely important social class throughout the whole cultural history of

China. It is sometimes rendered in translation as ‘literati’, ‘scholar’, ‘scholar-official’, ‘intellectual’, but none

of these terms individually can capture a holistic meaning of the word.

8



for rhetoric and dialectic engaged under an atmosphere of free and uninhibited thinking

was conducive to the promotion of a notion of proof. (See also (Horng, 1982) for a lengthy

discussion on this thesis.)

5. Daoism and mathematical development in China

Daoism is a school of Chinese philosophy that came into being by the 4th century B.C.E..

Towards the end of the Late Han Dynasty (25-220) there was a related development as a

religion, usually referred to as daojiao. But in this section we refer mainly to the philosophical

aspect of Daoism. A central theme is the Dao (the Way) or the flow of forces of Nature by

which things come together and transform, which reflects a deep-seated Chinese belief that

change is a basic characteristic of things. The relationship between Daoism and Chinese

science and mathematics in the ancient and medieval times is a topic of scholarly investigation

by several authors. (See (Volkov, 1996a) for a survey as well as a new interpretation from

the author.)

In particular, the treatise Huainanzi (The Book of the Prince of Huai Nan) was a Daoist

book commissioned by Prince Liu An (179 B.C.E. - 122 B.C.E.), a grandson of the founding

emperor of the Han Dynasty (206 B.C.E. - 220 C.E.), in the 2nd century B.C.E.. It is a

compendium on different areas, one of which is astronomy. In Chapter 3 titled Tianwenxun

(Treatise on the Patterns of Heaven) we find the following problem on measuring the height

of heaven:

“To find the height of heaven (i.e. of the sun) we must set up two 10-

che gnomons and measure their shadows on the same day at two places

situated exactly 1000 li apart on a north-south line. If the northern one

casts a shadow of 2 che in length, the southern one will cast a shadow

1 and 9/10 che long. And for every thousand li southwards the shadow

diminishes by one cun. [In the Chinese system, 10 cun amount to 1 che].

At 20,000 li to the south there will be no shadow at all and that place

must be directly beneath the sun. (Thus beginning with) a shadow of

2 che and a gnomon of 10 che (we find that Southwards) for 1 che of

shadow lost we gain 5 che in height (of gnomon). Multiplying therefore

the number of li to the south by 5, we get 100,000 li, which is the height

of heaven (i.e. of the sun).” [The translation is adopted from (Needham,

1959).]

9



Phrased in modern notations, the calculation can be explained in the figure shown below.

Figure 1

y (decrease in length of shadow)

is a function of x (distance moved

by the gnomon), say y = f (x).

What is x that makes f (x) = 2?

That x should be L. If we know

what f (x) is, then we can calcu-

late L, hence H.

Let us try to find out what f (x) is. We know that
a

b − y =
H

L − (x − b + y) (1),
a

b
=

H

b + L
(2).

¿From (1) and (2) we obtain y =

(
a

H − a
)

x = αx, where α is a constant. When x =

1000, y = 0.1. Hence α = 0.0001, i.e. y = 0.0001x. [Note that x is measured in li and

y is measured in che.] When x = 20, 000(li), y = 2 (che), so there is no shadow. Hence,

L = 20, 000 (li).

H = (b + L)
a

b
=

(
2

180
+ 20000

) (
10

2

)
= 100, 000 +

1

18
(in li.)2

The calculation is based on an over-simplified model of ‘heaven and earth’, so it does

not measure the ‘height of heaven’. However, the same calculation can be used to measure

the height and distance of an inaccessible object. This method of using two gnomons for

measurement was explained in detail in Liu Hui’s Haidao Suanjing (Sea Island Mathematical

Manual) of the 3rd century C.E.. The same method was also explained by the Indian

mathematician Aryabhata in the early 6th century C.E.. In the West the instrument in

surveying known as cross-staff, believed to be invented in the beginning of the 14th century

C.E., relies on this same method. The way the answer was presented by Liu Hui, which was

explained by Yang Hui (flourished in the mid 13th century C.E.) in his Xugu Zhaiqi Suanfa

(Continuation of Ancient Mathematical Methods for Elucidating the Strange Properties of

Numbers) of 1275, is based on an elegant use of area computation as shown in the figure

below (Figure 1).

2A conversion from che to li accounts for the factor
1

180
.

10



Figure 2

EGF D = QGF B − QEDB
= N JP G − CKM E,

hence ad = b2(h − a) − b1(h − a)
= (b2 − b1)(h − a),

h =
ad

b2 − b1
+ a.

QEDB = CKM E,

hence a` = b1(h − a) = b1
(

ad

b2 − b1

)
,

` =
b1d

b2 − b1
.

(It is an easy exercise to compare h, ` with H, L to see that the answers agree.) When a

schoolboy of today faces this problem, very likely he will make use of similar triangles to set

up simultaneous equations in ` and h. The answers would come out to be the same, but Yang

Hui’s solution seems to be much more elegant. The solution in Huainanzi, arrived at through

yet another approach, is elegant in its own way, for it is a dynamic version using a functional

dependence, which is perhaps more akin to the thinking of change or transformation in

Daoism.

It seems that both Liu Hui and Yang Hui were unaware of the method explained in

Huainanzi, but the same problem and method appeared earlier in Zhoubi Suanjing (The

Arithmetical Classic of the Gnomon and the Circular Paths) of 100 B.C.E. and later in

another treatise written by a Daoist in 1230, namely Gexiang Xinshu (New Writing On

the Image of Alteration) of Zhao You-Qin (Volkov 1996b). This may be explained by the

historical happening in the early Han Dynasty. Prince Liu An, who commissioned the writing

of Huainanzi by convening a group of Daoist scholars around himself, was later forced to

commit suicide for treason. As a result, the book was banned; probably the proof using

functional dependence was also lost to the public except possibly within the Daoist circle.

6. Influence of Elements in Western culture and in Chinese culture

It is well-known that Euclid’s Elements exerts significant influence on western culture,

both as an exemplar of axiomatic approach and as an exemplar in logical proof (Grabiner,

1988). This mathematical classic of all times was transmitted into China through a col-

laboration in translation by the Italian Jesuit Matteo Ricci and the Chinese scholar, later

appointed to high-ranking officials in charge of various important duties in the imperial

court, Xu Guang-Qi of the Ming Dynasty (1368-1644), published in 1607 as Jihe Yuanben

11



(Siu, 1995/1996).

In an essay Jihe Yuanben Zayi (Discourse on the Jihe Yuanben) Xu Guang-Qi commented:

“The benefit derived from studying this book is many. It can dispel

shallowness of those who learn the theory and make them think deep.

It can supply facility for those who learn the method and make them

think elegantly. Hence everyone in this world should study the book....

Five categories of personality will not learn from this book: those who

are impetuous, those who are thoughtless, those who are complacent,

those who are envious, those who are arrogant. Thus to learn from this

book one not only strengthens one’s intellectual capacity but also builds

a moral base.” [The translation is by this author.]

Xu Guang-Qi felt rather disappointed when he saw that few people paid attention to the

translated text, but surmised that everybody would study it a hundred years later. However,

in 1681 Li Zi-Jin of the Qing Dynasty (1644-1911), said in the preface to Shuxue Yao (Key

to Mathematics) written by Du Zhi-Geng:

“Even those gentlemen in the capital who regard themselves to be erudite

scholars keep away from the book [Jihe Yuanben], or close it and do

not discuss its content at all, or discuss it with incomprehension and

perplexity.” [The translation is by this author.]

Even though Elements had little influence on mathematics in China, surprisingly it bore

fruit in another arena, exerting influence on Chinese liberals like Kang You-Wei (1858-1927)

and Tan Si-Tong (1865-1898), who were main figures in the futile attempt of the “Hundred-

Day Reformation Movement” of 1898 that ended in tragedy for many concerned. Little

would Xu Guang-Qi imagine that his somewhat over-optimistic prediction of the influence

of Euclid’s Elements came true in the political arena! A more detailed account of the

influence of Elements in China can be found in (Siu, 2007), written on the occasion of the

400th anniversary of the translation of Elements in China.

7. Conclusion

How would the message conveyed in this paper contribute to the learning and teaching

of proof? It would not yield specific tactics nor a comprehensive theory. But it serves to

remind us that, to make the subject more ‘humanistic’ so that students feel that it makes

good sense to spend time on it, mathematics is best studied along with its influence to and

12



from other human endeavour. Proof, as a characteristic component of the subject, shares

the same fate. As a ‘bonus’, in viewing proof in this light we may be able to pick up good

suggestions to enhance understanding of specific topics to make the learning of mathematics

a more interesting activity.

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