The Nature of Mathematics
What is mathematics?
The mathematical paradigm:
Axioms
• fundamental
assumptions
Deductive
reasoning
Theorems
• syllogisms
• applying
axioms
• proven
arguments
• ‘mathematical
facts’
Axioms

The most fundamental assumptions
Assumed but unprovable

E.g: Euclidean Geometry

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It is possible to draw a straight line connecting any two points
A finite straight line may be extended without limit in either
direction
It is possible to draw a circle through a given point and with a
given centre
All right angles are equal to each other
There is just one straight line through a given point that is
parallel to a given line
What are the axioms for the natural
numbers?

Natural numbers are the non-negative whole numbers

What do you think some of the axioms governing these
might be:
According to Giuseppe Peano (1858-1932)
•
There is a natural number 0.
•
Every natural number a has a natural number
successor, denoted by S(a). Intuitively, S(a) is a+1.
•
There is no natural number whose successor is 0.
•
S is injective, i.e. distinct natural numbers have
distinct successors: if a ≠ b, then S(a) ≠ S(b).
•
If a property is possessed by 0 and also by the
successor of every natural number which
possesses it, then it is possessed by all natural
numbers. (This postulate ensures that the proof
technique of mathematical induction is valid.)
Deductive Reasoning (syllogism recap)

All IB subjects are awesome
Two Premises

TOK is an IB subject

Therefore TOK is awesome
Conclusion
Theorems – Mathematical Truths


Proven mathematical arguments – can be used to construct
more complex proofs
The following theorems come from reasoning based on the
• Given a + c = 180o
Euclidean axioms
1. Lines perpendicular to the same
line are parallel
•
•
Prove b = c
State which theorem(s) you are
applying
2. Two straight lines do not enclose
an area
3. The sum of the angles in a triangle
is 180o
4. The angles on a straight line sum
to 180o
ao bo
co
The Proof
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a + c = 180
a + b = 180
a+b=a+c
b=c
Theorem 4
Conjectures

A mathematical idea thought to be true but as yet
unproven
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Eg. Goldbach’s conjecture: every even number greater
than two can be expressed as the sum of two primes:
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4=2+2
6=3+3
…..complete this up to 24
How do you prove this?
Beauty and Intuition

2048 people enter a knock-out tennis tournament, how many
games will be played?

What is the sum of the integers from 1 to 200?

A piece of string is tied around the equator of a football. The
string is then loosened enough to create a 1 cm gap between
the string and the football all the way around. This takes about
6 cm extra string. How much extra string is needed to do this
to the Earth?
Certainty

How certain is mathematics? (Does 2+2 really equal 4?)
Is Maths Discovered or Invented?

In pairs: discuss your thoughts on the above question
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Pairs pair up and discuss your thoughts

Write your thoughts in the following format:
Thoughts/Evidence
For
Maths is discovered
Maths is invented
Thoughts/Evidence
Against
The Origin of Maths
How is it knowable?
Nature of proposition
Analytic
Proposition that is true by
definition
Synthetic
Proposition that is not analytic
A priori
A proposition that can be true
independent of experience
1.
4.
A posteriori
A proposition that cannot be
known to be true
independent of experience
2.
3.
Analytic a priori (Formalism)

True by definition
Knowable independent of experience

E.g. All Hungarian widows have suffered the loss of their husbands

2 + 2 = 4 because of the way we define 2-ness, plus-ness, equalsness and 4-ness
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But: Goldbach’s Conjecture – if it is true by definition, why can’t we
solve this….have we not defined ‘prime number’ properly or is
maths not in this box?
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To do: write a couple of examples of things that you think fit in this
box

Analytic a posteriori

True by definition
Knowable only through experience
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A contradiction in terms
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Synthetic a posteriori (Empiricism)

Not true by definition
Knowable only through experience
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E.g. Penang’s food is better than KL’s

Task: Write something else that fits in this box
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John Stuart Mill – maths as empirical generalisations
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We learn maths by counting apples (or durian or other culturally
appropriate fruit)
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Task: maths always starts at experience, but does it matter that it ends up
going beyond – does this undermine maths’ place in this box?
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Synthetic a priori (Platonism)
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Not true by definition
Knowable independent of experience
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Does anything fit here?
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Plato – able to learn about the nature of the universe through
reasoned thought alone
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If so, how is this possible?
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This suggests a harmony between the human mind and the universe
God?
Evolution?
Task: Discuss
So what do you think

Write a paragraph explaining where you think maths
comes from and your reasons for thinking so.