Mathematics and TOK
Exploring the Areas of Knowlege
Keith J. Devlin
“Mathematics is the abstract key
which turns the lock of the
physical universe.”
Reuben Hersch
“What kind of thing is a number?”
“What is mathematics? It’s neither
physical nor mental, it’s social. It’s part
of history. It’s like law, like religion, like
money, like all those other things which
are very real, but only as part of
collective human consciousness. That’s
what math is.”
Bertrand Russell
“Mathematics may be defined as the
subject in which we never know what we
are talking about, nor whether what we
are saying is true.”
“The mark of a civilized man is the ability
to look at a column of numbers and weep.”
Descartes
“To speak freely, I am convinced that
(mathematics) is a more powerful
instrument of knowledge than any other.”
“Math is an island of certainty in an ocean
of doubt.”
Albert Einstein
• “Mathematics is not waiting to be
discovered but instead exists as a
‘product of human thought, independent
of experience.’”
Video Clip
• Taken from “The Ascent of Man”
• A BBC Documentary from 1973
• J. Bronowski (Mathematician and
Narrator) looks at the interlocking of
numbers and nature in descriptions of
musical harmony and Pythagoras’
Theorem
TOK and Mathematics
• What is Math?
- The science of rigorous proof by
1.
2.
3.
4.
Deductive Reasoning
Exhaustive Proof
Proof by Contradiction
Proof by Mathematical Induction
TOK and Mathematics
What are the tools for proofs?
• Axioms – like premises (can be algebraic
such as 5 + 0 = 5 or geometric as in all
right angles are equal to one another).
Assumed to be true.
• Theorems– like conclusions. Can be used
to further other proofs.
TOK and Math: Proof
• What does it mean to prove something
mathematically?
• Mathematical Proof
– A collections of logically valid steps or
demonstrations that form an argument which
serves as a justification of a mathematical
claim. Steps within the argument normally
make the use of definitions, axioms,
properties and previous claims that are
consistent (theorems).
TOK and Math
• How do proofs come about?
– We identify problems (like experimentation,
develop a step by step procedure, and produce
a conjecture)
• Conjecture
– A conclusion made from a reasonable number
of individual cases which are nonetheless
insufficient to form substantial proof
– Leads to proofs which involves making sure
every possibility will work (unlike science)
Is Mathematics Invented or
Discovered?
Two types of opposing views:
1. Formalist: Mathematics is an abstract
activity governed by rules (like a logical
game such as chess)
2. Realist: Mathematics is fundamentally
describing the way the universe
actually works
Activity
1. Work in groups at your table.
2. Select either the formalist (invented) or
the realist (discovered) point of view.
3. Come up with 2-3 arguments why you
think mathematics is supported by this
point of view
4. Rely on your observations in
science/nature, games, math symbols and
language, math concepts (imaginary
numbers, quadratics, equations,
geometry,etc.).
TOK Math Questions to Consider
1.
Is infinity a number? Is it ever correct
to write x = infinity? Do some infinite
sets have more elements than others (i.e.
If there are an infinite amount of
distances between 1cm and 2cm on a
ruler, how many are there between 1 cm
and 10 cm?)
More TOK Questions to Consider
2.
How important is it to be exact? Discuss
this with regard to different scenarios in
mathematics, science, medicine,
architecture, etc.
More TOK Questions to Consider
3.
What is the difference between the
empty set, the number zero, and nothing
at all? Can there be such a thing as a
complete vacuum? Do you think we can
consider the universe to be the ultimate
universal set U containing everything?
More TOK Questions to Consider
4.
We must be very careful with brackets in
Mathematics. Are brackets in
Mathematics more important than
brackets used in other areas of
knowledge? (i.e. History, Literature,
Science, etc.)
More TOK Questions to Consider
5.
It is very important for the timing to be as
accurate as possible in the Olympic Games.
However, the same degree of accuracy is not
necessary when timing the cooking of a pot
of rice. So, how important is it to have
“exact” values? What do you understand by
an “exact value”? Is it more important to be
exact in Math & Science than in other Areas
of Knowledge?
More TOK Questions to Consider
6.
When conducting a survey how can you
ever be sure that the replies given are
true? In a survey the people questioned
could lie about the information they give.
How do both the type of questionnaire
and to whom it is distributed affect the
results of your survey?
More TOK Questions to Consider
7.
How do we know when two things are
related? Can two sets of data have a very
strong correlation and yet not be related?
Try to think of your own examples. Can
you find any examples in advertising?
More TOK Questions to Consider
8.
If mathematics is about following rules
and we know that 10/10 = 1 and 4/4 = 1,
then why does 0/0 not = 1 ?