Mathematics is the study of patterns and relationships between numbers and shapes. While empirical evidence may be gathered, mathematical knowledge requires rigorous deductive proof based on agreed upon axioms and theorems. However, Gödel's incompleteness theorem showed that the axiomatic foundations of mathematics cannot be proven with absolute certainty from within the system. There is an ongoing debate around whether mathematical truths are discovered or invented by humans. Overall, mathematics relies on both deductive and empirical reasoning but cannot claim absolute certainty due its axiomatic foundations.

1. Area of Knowledge:
Mathematics

3. Lesson 1
The nature and importance
of Mathematics

4. Definitions
 How would you define Mathematics?
 Choose 3 words that describe Mathematics for you.

5. What is Mathematics?
 Dictionary: “The science of spatial and numerical
relationships”
 ToK Guide: “Mathematics is the study of patterns and
relationships between numbers and shapes.”
 Ivan Nomav: “What people spend eternity being
forced to do if they end up in hell.”
 Science Student: “A gate crasher to a Science party.”
 English Student: “An eleven letter word”
 Mathematics Student: “Everything”

6. How important is Mathematics?
 Read the Time article about the life of John
Nash.
 Does this article challenge you to believe that
there is more to the subject of Mathematics
than your school Mathematics curriculum
would suggest? Make a list of the reasons why
Mathematical knowledge is important.
 Euclid of Alexandria: “The laws of nature are
but the mathematical thoughts of God.” What
is the meaning and significance of this?

7. 1+ 5 Nature’s Beautiful Number : The Golden Section
2
Applications in nature, music, design, architecture, art.

9. An example of Mathematical knowledge
Starting in the red square, try to find the
correct path through the following maze,
moving horizontally, vertically or diagonally
In pairs, see if you can write the next square
in this sequence
If you manage to work it, reflect on how!

10. 32222112 22334411
1 22
122121 22
112233 2 21 11
13211311
312211 111221 1211 12311311
2211
11122233 11132132 31131211
13112221
3 11 131221
 Solution

11. Key question to discuss about the
nature of mathematics:
 Is Mathematics invented or discovered? Why
is this an important question?
 Discuss this in pairs.

12. “God created the integers, all the rest is
the work of man” Comment.

13. Mathematics in the beginning…
 If two dinosaurs joined two other dinosaurs in a
clearing, there would be four dinosaurs, even
though no humans were around to observe it, and
the beasts were too stupid to know it.
 If the four then had a race, medals could have been
awarded for first, second, third and fourth.
 The idea of cardinality, being able to place things
in order, existed and was discovered. But numbers
were invented to describe and manipulate
cardinality.
 What is the earliest evidence of human counting?

14.  There is a 35,000 year-old baboon’s thigh
bone discovered in the Lebombo Mountains
of Africa marked with 29 notches
representing a calendar stick.
 Is this evidence of counting?

15. One opinion:
 “I believe that mathematical reality lies
outside us, that our function is to discover
and observe it, and that the theorems which
we prove, and which we describe
grandiloquently as our ‘creations’ are simply
our notes of our observations.”
(G. H. Hardy, Apology)

16. A counter argument?
 “Did Shakespeare “discover” his sonnets? Surely all finite
sequences of English words “exist”, and Shakespeare
simply chose a few that he liked.”
 “A block of stone contains every possible statue that can be
carved from it. When a sculptor selects one of these statues,
it’s said to be an act of creativity. If mathematics already
exists, then so do all possible mathematics, including an
infinity of incorrect, worthless, boring, irrelevant, useless,
ridiculous, and incomprehensible mathematics. Shouldn’t
the finding of worthwhile mathematics be given the same
consideration as finding a work of art in a stone be called
creative?”

17. Knowledge issues:
 So, was ‘4 + 2 = 6’ created or discovered?
 Is it true or false?
 What if I define + to mean multiply?
 What if my symbol ‘4’ means something
different to someone else’s ‘4’?

18. When are the following true?
 10 + 10 = 20
 10 + 10 = 100
 10 + 10 = 8
 10 + 10 = 10

19. Answers…?
 10 + 10 = 20
 When using normal (decimal) arithmetic
 10 + 10 = 100
 When using binary arithmetic
 10 + 10 = 8
 When dealing with time – 10 o’clock add 10 hours is 8
o’clock
 10 + 10 = 10
 When adding 10 litres of water to a bucket that can only hold
10 litres of water

20. Therefore:
 The idea of cardinality (being able to measure
elements of a set i.e. numbers of dinosaurs)
existed and was discovered. But numbers
themselves were invented to describe and
manipulate cardinality.
 Who is to say 4 + 2 won’t equal 8 if someone
redefines the rules and is able to convince the
whole world of this new definition.

21. Knowledge questions:
 We need rules to define things, but who
defines these rules?
 To what extent are these mathematical
rules ‘knowledge’?
 List your own examples of mathematical
rules and reflect on the extent to which
these are examples of ‘knowledge’

23. BBC Radio Programmes
 If you have time, listen to the first 5 to 10
minutes of one or more of the following
 The number zero
 The number Pi
 Infinity ( and beyond )
 The imaginary number i

24. Homework
 Review your notes from today and read p.53-61 from
Alchin, and make any extra notes you need to in your
journal
 In your journal list some examples of mathematical
knowledge you have acquired and comment on how this
knowledge has been acquired. Has your knowledge been
acquired through an experiential, practical or propositional
method, or a combination?
 Why do some people learn mathematical knowledge very
easily and outperform their peers by years, whereas others
find it almost impossible to learn, however hard they try.
Describe your personal experience in this subject.

25. Lesson 2
Mathematics and knowledge
claims

27. Justifying a mathematical knowledge claim
 In groups discuss one piece of mathematical knowledge
you have acquired and try to justify this knowledge claim.
Is your knowledge based on experiential, practical or
propositional evidence?
 As a group, choose one of these knowledge claims to justify
to the rest of your class
 Reflect on how this differs from proof claims in other areas
of knowledge

28. What constitutes proof?
 Discuss the attempts to justify mathematical knowledge in
the following clip from ‘Ma and Pa Kettle’:
 http://www.youtube.com/watch?v=Bfq5kju627c
 What constitutes proof in relation to mathematical
knowledge? How does this compare to other areas of
knowledge?

29. What constitutes proof?
 Using the minimum number of colours, shade your map of
Africa so that no countries that share a common border have
the same colour.
 How many colours were required?
 Can you prove that this is the minimum number of colours
required for any map? Could there be a map that requires 5
colours? The States of mainland USA?
 Testing as many maps that we could, and finding that the
use of 4 colours was sufficient in each, would not constitute
proof for mathematicians – why?
 How is this different to the Scientific Method used to
establish a scientific theory? For example; from the atom
then to protons, electrons and neutrons, then to quarks and
smaller sub atomic particles.

30.  How does this story highlight the different use of empirical
evidence for the three individuals.
 Who has used induction and who has used deduction?
 Explain the similarities between this story and our inability
to prove the four colour map problem.

31. Knowledge claims: theorems and axioms
 The theorems used today are the accumulation of the
discoveries made by many cultures.
 Sometimes these discoveries have been lost and then found
again.
 Over the centuries, existing theorems lead to further
discoveries and the ever expanding field of mathematical
knowledge.
 Pythagoras and his brotherhood is thought to have started
formal proof around 500 BC.
 Euclid ( 330 BC ): Wrote a collection of 13 books called The
Elements, detailing his work, the work of the Pythagorean
Brotherhood and a compilation of the mathematical
knowledge to date.

32. Knowledge claims: theorems and axioms
 Based on axioms (postulates), self evident truths, and
deductive reasoning: rationalism.
 Using these axioms, a mathematician follows deductive
reasoning to formulate new theorems.
 While empirical evidence may be gathered to infer a
mathematical truth, on its own it is not sufficient for the
proof of this observation.
 Rigid deductive proof is required of new mathematical
knowledge before it can be claimed to be true.
 New proofs must be peer assessed by an authority from the
mathematical community before they can be claimed to be
true.

33. Undeniable Proof?
 “Archimedes will be remembered when Aeschylus is
forgotten, as languages die and mathematical ideas do not.
Immortality may be a silly word but probably a
mathematician has the best chance of whatever it may
mean.” G H Hardy
 “In most sciences one generation tears down what another
has built and what one has established another undoes.”
“In mathematics alone each generations adds a new story
to the old structure.” Hermann Hankel
 How might these quotes be used to assess how a
mathematician feels about knowledge claims in their
subject?

34. An example of mathematical proof:
180 degrees in a triangle
 Definition; 180 degrees in a
straight line. (Thank the
Babylonians for this. They Line 1
liked multiples of 60) a
k
y
 Euclid’s 5th postulate
(axiom) implies that if line b Line 2
x
1 and line 2 are parallel
then a+b=180 and
x+y=180

35.  Base angles in the triangle
must be a and y as they are on
the straight line and must
agree with the sum of 180
degrees as stated by the 5th
Postulate (Axiom)
 Now, angle sum in the Line 1
triangle is a+k+y a
k
y
 But, the straight angle at the
top of the triangle shows that
a+k+y=180 b a y x Line 2
 Therefore; angle sum in the
triangle is 180 degrees
 QED "quod erat
demonstrandum", which
means "which was to be
demonstrated (or proved)"

36. Using the previous proof for the triangle
b
c
y
a
x
z
 A diagonal of a quadrilateral divides the quadrilateral into 2
triangles.
 Angle sum of all triangles is 180 degrees. ( Proved earlier using
Euclid’s Axiom)
 Therefore, the internal angles of a quadrilateral sum to 2 x 180 =
360 degrees
 And one for you. How many degrees in a polygon with any
number of sides?

37. This would be fine except for…
 What about a triangle that is drawn on a curved surface?
 Pick three points on a disk and connect the points with
straight lines.
 Notice that the sum of the angles of this triangle is always
less than 180 degrees. This observation forced
mathematicians to redefine the use of Euclid’s Axioms and
then introduce a new field of mathematics called
Hyperbolic Geometry. (Non-Euclidean Geometry)

38. Case study: Fermat’s last theorem.

39.  Every Secondary school student has probably met
“Pythagoras’ theorem”, haven’t they? What does it state?
 Rational solutions for x +y =z
2 2 2
 Eg 32 + 4 2 = 52 , 52 +12 2 = 132
 Rather than squaring each number, the French
Mathematician Fermat tried raising each number to a
different power. The example below is nearly correct when
the numbers 6, 8 and 9 are cubed.
6 −
+1
=
333
89
 Fermat claims his new equation differs from Pythagoras'
original equation, which has infinitely many rational
solutions, to an equation which has no solutions for
integers n greater than 2, ie x y z
n
+= n n

40.  Fermat scribbled in his copy of Arithmetica. “ I have a truly
marvellous demonstration (Proof) of this proposition which
this margin is too narrow to contain.”
 Before he could demonstrate his proof, if it really did exist
as he often boasted about his mathematical talents, he was
to die several months later.
 And so the challenge began to prove this statement using
the existing axioms and theorems.
 Prize offered. Thousands of attempts throughout the world
over the next 300 years to find a proof.
 The “Winner” Andrew Wiles, 1995, after over 20 years of
work
 Wiles uses 20th century mathematical knowledge in his
proof. Did Fermat actually prove this theorem in the 17th
century?

41. Does Calvin need a TOK class?

42.  “God has a transfinite book with all the theorems and their
best proofs. You don’t really have to believe in God as long
as you believe in the book”.
 (Paul Erdos quoted in Bruce Schecter, My Brain Is Open: The Mathematical Journeys
of Paul Erdos
 What is the difference between believing in God and being
able to prove it and believing in Mathematics theorems and
being able to prove them? Discuss.

43. Mathematics; A knowledge system
without doubt?
 How do we choose the axioms underlying mathematics? Is
this an act of faith?
 Read Godel: Incompleteness Theorem Alchin pp 64-66
 While Godel showed that it may be impossible to prove that
all mathematical theorems can be proven using an axiomatic
system, this did not necessarily mean that these theorems
were false.
 If the axiomatic system, which appeared to be devoid of
doubt, cannot be proven to be infallible, imagine what might
happen if the man you are about to watch is thought of as the
mathematician who proves that one of the founding axioms
is not true.
 House of Cards

44. Plenary
 This is the end of this lesson. You should have some idea
about the following questions.
 Are all mathematical statements either true or false?
 Can a mathematical statement be true before it has been
proven?
 Is certainty possible in the axiomatic system that
underscores mathematical knowledge?
 What are the roles of empirical evidence and deductive
reasoning in establishing a mathematical claim?
 How does proof differ for mathematicians with say
scientific proof?

45. Homework: Respond to the following
 Review your notes from this lesson and read the rest of
chapter 4 from Alchin’s book making supplementary notes
where relevant
 Consider your own response to this question: What counts
as knowledge in ethics as compared to mathematics? To
what extent would you agree that knowledge claims are as
well-supported in ethics as in mathematics? Write down why
some people might agree with this belief and why others
might disagree. Make sure you refer to specific knowledge
claims in each of these two areas of knowledge.