```Math and TOK
Exploring the Areas of Knowledge
Keith J. Devlin
• “Mathematics is the abstract key which
turns the lock of the physical universe.”
Reuben Hersh
“What Kind of a Thing is a Number?”
• What is mathematics? It's neither physical nor
mental, it's social. It's part of culture, 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
• “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
• “Contrary philosophical positions view this
differently; mathematics is not waiting to be
discovered by instead exists as a ‘product of
human thought, independent of experience.’”
Math and TOK
• What do we want to accomplish by looking at
Math in a TOK way?
TOK and Math
• What is Math?
-The science of rigorous proof
-Axioms: Postulates, Self evident truths (??)
Math and TOK
• Does the fact that Math is built upon self
evident truths make this subject more or less
certain?
– Deductive reasoning
• Process using rules to arrive at a specific conclusion
drawn from general statements.
– Theorems
• statement proved on the basis of previously accepted
or established statements
The Pythagorean Theorem
Math and TOK
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1. All human beings are normal.
2. Socrates is a human being.
3. Therefore, Socrates is mortal.
1 & 2 are premises. 3 is the conclusion.
In mathematics axioms are like premises and
theorems are like conclusions.
Practical Math and TOK
• In groups of two create a math problem,
identify the axiom(s), be able to present the
deductive reasoning on the board, list out the
theorem(s), and identify a knowledge claim
imbedded within the problem
Hint: Euclid’s 5 axioms
• It shall be possible to draw a straight line
joining any two points.
• A finite straight line may be extended without
limit in either direction.
• It shall be possible to draw a circle with a
given center and through a given point.
• All right angles are equal to one another.
• There is just one straight line through a given
point which is parallel to a given line.
TOK and Math: Proof
• What does it mean to prove something
mathematically?
• Mathematical Proof
• a collection 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 use of definitions, axioms, properties and previous
claims that are consistent.
Buchberger’s Model: Conjecture
• Knowledge Spiral
– Identify Problem (phase
of experimentation)
– Develop algorithm or
step-by-step procedure
(phase of exactification)
– Conjecture (phase of
application)
– Conjecture = a
reasonable number of
individual cases which
are nonetheless
insufficient to form
substantial proof
– Quasi-empirical = having
a likeness to a scientific
method which requires
careful observation and
a gathering of relevant
factual support
Practical Exercise
• Proof vs. Conjecture
*In groups of 2 create two problems one that shows
proof is possible and another that gives an example
of conjecture.
Practical Questions
• Do axioms, theorems, deductive and inductive
reasoning, proofs, conjectures apply in real
everyday life?
• Is it actually real or do we make it real only in
our minds?
Math and TOK: Fibonacci Sequence
• Do we discover math or is it already there?
• 1,1,2,3,5,8,13,21,34,55
• Examples in nature
– Male bee
– Chambered nautilus
– Number of Flower petals
– Spiral pattern in broccoli
– Parthenon in Athens
TOK and Math
• How should we approach understanding
math?
-empirically
-analytically: true by definition
-synthetically through reason alone
Klein’s Quartic Curve
Escher’s Tessalations
The Droste Effect
TOK and Math
• How certain is math?
• How does it compare to other disciplines
when it comes to providing knowledge as
Plato defined (k=jtb)?
Math and TOK
• Is Math consistent?
-Riemannian Geometry
(playing with axioms)
-Problem of consistency
(you can’t make up the rules as you go)
-Godel’s incompleteness theorem
(it is impossible to prove that a formal
mathematical system is free from contradictions)
Real World Math Project:
Successfully approach some of the more abstract questions that naturally
arise through looking at this subject in a TOK fashion.
2. Decide on mathematical concept(s) to teach to a child that
concept has to be above the children’s natural abilities, and it
MUST attempt to prove your position concerning your
questions.)
3. Organize gathered information into an essay that you will
present to the class.
1.
2.
TOK Rubric
Evidence from experiment
3. Summarize how your group responded to the knowledge
claim(s) imbedded in the original question(s).
4. Any Questions????
TOK and Math: Questions for project
1. What does it mean to say that mathematics can be regarded
as a formal game devoid of intrinsic meaning? If this is the
case, how can there be such a wealth of applications in the
real world?
2. We can use mathematics successfully to model the real
world processes. Is this b/c we create mathematics to mirror
the world or b/c the world is intrinsically mathematical?
3. What do mathematicians mean by mathematical proof, and
how does it differ from good reasons in other areas of
knowledge?
4. Can math be characterized as a universal language? To what
extent then might math be different in different cultures?
How is math the product of human social interaction?
TOK and Math: Questions Continued
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Are all mathematical statements either true or false? Can a
mathematical statement be true before it has been proven? Is it correct
to say that math makes truth claims about non-existent objects? Explain
It has been argued that we come to know the number 3 through
examples. Does this support the existence of the number 3 and, by
extension, numbers in general? If so, what of number such a 0,-1, i, and
a trillion? If not, in what sense do numbers really exist?
What counts as understanding in math? Is it sufficient to get the right
answer to a problem and then say that one understands the concept and
it relevance to the world.
To what extent can math be beautiful? Can it be understood in a similar
fashion like the arts? If math has an element of elegance does this
jeopardize it level of certainty? Explain.
TOK and Math: Questions for panelists
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Is math built upon self evident truths?
Is math the only discipline that has self
evident truth?
How do we choose the axioms
underlying math? Is it an act of faith?
If math is discovered not invented do
we ever really have new mathematical
knowledge?
How does mathematical proof differ
from proof in other areas of
knowledge?
Can mathematics be characterized as a
universal language? Since languages
are determined in community what role
does community play in determining
mathematical certainty? Might this
mean that math is different from
culture to culture?
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Why do you think different cultures
value math in different ways? In what
ways does our American culture hinder
our understanding of math?
What counts as understanding in math?
Is it sufficient to get the right answer to
a problem and then say that one
understands the concept and it
relevance to the world.
What does it mean for math do be
beautiful?
Are numbers real tangible things that
exist outside our minds? If not them
how can math be applicable to the real
word?
Are there aspects of math that one can
choose whether or not to believe?
Does intuition play a role in Math?
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Create some of your own questions!!!
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