Transcript Mathematics in ToK - Missoula County Public Schools

Mathematics
in
ToK
Area of Knowledge 1: How do we apply
language, emotion, sensory perception,
and reason/logic to gain knowledge
through Mathematics?
THIS IS NOT MATH CLASS!!
…but rather, a unit designed to look at how we
arrive at knowledge, truth, and wisdom through
the use of mathematics. How we use math to
make sense of the world, and how far we can
trust its certainty.
What words come to mind when you
think about the term “mathematics”?
Where do you see mathematics in
everyday life?
May 3, 2013
• Blog!!
• EE to do—you should spend some time
researching this weekend. Also, update your
research question on Managebac!
• Need help with research direction, citation,
etc? Come to room 328 today at lunch!
• Do we need to update our calendar?
Galileo proposed that we find mathematics
everywhere in nature. Think of some examples.
Fibonacci series
The Fibonacci Sequence is the series of
numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
The next number is found by adding up the
two numbers before it.
Perception: The Golden ratio…is it
perfection?
This mask of the human face is based
on the Golden Ratio. The proportions
of the length of the nose, the position
of the eyes and the length of the
chin, all conform to some aspect of
the Golden Ratio.
Remember WHY we are attracted to
people…procreation! Does symmetry correlate
to a healthy mate?
http://www.intmath.com/numbers/math-ofbeauty.php
Where do you benefit from
mathematics?
Math Riddle Time!
How can you add eight 8's to get the
number 1,000? (only use addition)
Emotion
Think of the emotion experiments you did.
Where did math come into play in those?
Math and magic…
The Magic Gopher
Some Definitions of Math
• The science of rigorous proof.
• The study of patterns and relationships between
numbers and shapes.
• Patterns amidst the chaos.
• Right and wrong answers.
• The most accurate reflection of reality.
• The use of numbers an symbols as metaphors for
understanding even the incomprehensibly
abstract.
– What is a metaphor?
Why are some people drawn to mathematics as
an area of knowing? Conversely, why are some
people averse to mathematics?
Similarly, do you believe that being ‘good’ at
mathematics is inborn, intuitive, or learned?
Why do we study mathematics? What’s the
point?
Is, as it says in the gold-packet, mathematics a
retroactive science? (meaning, the math hasn’t
yet been discovered for some of the problems
we attempt to solve) (p. 148)
Some possible answers…from real
people like you
• In math, there is a right and a wrong.
• Other subjects are too relative.
• Math requires you to ignore context and operate on a
purely abstract level.
• Math is useful.
• You can use numbers and statistics in an argument or
as evidence and it’s more convincing.
• Failure and success are equally important teachers.
• Numbers are merely symbols that can stand for
anything; they are not things or people. They are not
concrete.
Imperialism in mathematics
Imperialism: “my way is better than your way.”
If you can’t express something in mathematical
symbols then it has no intellectual value. Agree
or disagree?
Do you believe that applied mathematics came
first or that pure mathematics came first? Why?
Math Riddle Time!
How many two cent stamps are there in a
dozen?
Euclid
• Father of geometry (on a plane).
• “The pursuit of knowledge is an end in itself.”
Euclid’s formal reasoning was in
syllogism
(Begin with) AXIOMS

Premise
(Use)
DEDUCTIVE REASONING Premise
(Arrive at) THEOREMS

Conclusion
Why not use inductive reasoning?
What’s an axiom?
• Basic assumption, self-evident truths, used to
create firm foundations of understanding on
which to build new ideas. (19th century)
• Required to be: consistent, independent, simple,
and fruitful. (Review pg. 190 in text for
explanations)
• Current: axioms are not ‘self-evident’ truths, but
assumptions premises, definitions, or givens at
the base of a mathematical system.
Euclid’s Axioms
1. It shall be possible to draw a straight line
joining any two points.
2. A finite straight line may be extended
without limit in either direction.
3. It shall be possible to draw a circle with a
given center and through a given point.
4. All right angles are equal to one another.
5. There is just one straight line through a given
point which is parallel to a given line.
And theorems…
• Lines perpendicular to the same line re
parallel.
• Two straight lines do not enclose an area.
• The sum of the angles of a triangle is 180
degrees.
• The angles on a straight line sum to 180
degrees.
To what extent do you think the geometric
paradigm can be applied to other areas of
knowledge? What are the strengths and
limitations of applying this type of knowledge to
other areas?
Non-Euclidian Geometry (19th century
onward)
• Riemannian geometry: what if the surface on
which we work is a curve, not a plane?
• The reverse of Euclid’s axioms cannot be
disproved based on the curvature of space.
• Einstein used Riemannian geometry.
• Math game break.
Correspondence and Coherence in
Mathematics
Correspondence: accurately explains what exists.
Coherence: axioms used as foundations are logically
consistent.
Then, one can manipulate ideas in a process of
‘pure’ thinking, creating new knowledge. (?)
How do correspondence and coherence lead to
consistency in mathematics?
5/8/10
Homework…
1) Blog is due Saturday
2) Bring a statistic to class that you read/saw; take
note of the source and purpose. Write it down.
3) N. Science—not due until next Monday.
1)
2)
3)
4)
5)
Book: 221-255—Notes
Gold packet
Vocab
1 KI
1 Quote
• I will be collecting comp books again the week
before finals. You will have them with you over
the summer.
• Start looking at the “Recommended Further
Reading” list on the blog. You will need to pick
one book and read it over the summer. Keep an
informal reading log in your comp book. At the
beginning of next semester you will be
responsible for sharing with the class what you
read and learned.
• Final Presentation assignment coming soon!
Invented or Discovered?
Platonists: believe math is discovered; it exists in a
realm we cannot fully comprehend. Plato’s
arguments:
1. Mathematics is more certain than perception
2. mathematics is timelessly true (you CAN step
in the same river twice)
*What does this remind you of from our unit on
perception? Scientific realism? Phenomenalism?
Criticism
• Too much mysticism when dealing with an
infinite number or mathematical possibilities.
If mathematical objects have an idealized
existence, how can we (as physical beings)
comprehend that they even exist, let alone
allow them to make sense?
Formalist
• Math is invented by man to help us make
sense of reality.
• Math consists of man-made definitions,
axioms, and theorems.
• The “perfect” circle and “perfect” line (by
definition) do not exist. It is the idea of these
things that we use in mathematics.
Euclid: formal systems are suggested to us by reality in
response to practical problems, then turn out to be a useful
way of describing reality.
Einstein: Mathematical systems are invented, but it is a matter
of discovery which of the various systems apply to reality. You
can invent any formal system and prove theorems from
axioms with complete certainty; however, once you ask which
system applies to the world, you are faced with an empirical
question which can only be answered on the basis of
observation. Thus, the Riemannian geometry is a better
descriptor of physical space.
Proofs and conjectures
• In a proof, a theorem is shown to follow
logically from axioms.
• A conjecture is a hypothesis that may not
necessarily be true.
Goldbach’s Great Unproven
Conjecture
Every even number is the sum of two primes.
• You can test something 1,000,000 times but it
is still a relatively small ratio of tested to nontested when taking infinity into account.
– How far do you have to go before you can say
something is proven?
– When does a conjecture become an axiom?
Is Descartes’ statement, “I think therefore I am,”
a theorem, an axiom, or neither?
Creativity, Intuition, Beauty, Elegance
How are these words associated with
mathematics?
When is intuition helpful and harmful in
mathematics?
Math or Art words?
Symmetry
Proportion
Sequence
Frequency
Medium
The universal language
Are mathematical concepts something that extend
beyond the way human begins make sense of the world?
Film: Contact (1997)
Hollywood and Mathematics…emotion/passion
Film/Play: Proof
Film: Good Will Hunting
But remember…
Mathematics is somewhat reliant on being
explained in a non-mathematical language and
classification systems.
To test how different it can be, take a moment
and briefly jot down the definition of “to add”.
Math Riddle Time!
As I was going to St. Ives
I met a man with seven wives.
Each wife had seven sacks,
Each sack had seven cats,
Each cat had seven kits;
Kits, cats, sacks and wives,
How many were going to St. Ives?
Intuition
• Should math that we trust intuitively be put to
the test?
• Is any formal system free from contradiction?
• What are the local, global, and universal
implications of math we intuitively trust?
Mathematics and Certainty
Analytic propositions: true by definition
Synthetic Propositions: every propositions that
is not analytic.
SO: All propositions are either analytic or
synthetic.
A priori: a proposition that is true independent
of experience.
A posteriori: cannot be known independent of
experience.
SO: all true propositions can be known a priori
or pa posteriori.
Review: pg. 197--201 in text
• Mathematics as empirical?
• Mathematics as analytic?
• Mathematics as synthetic a priori?
• Re-read these pages and discuss.
3 Philosophies on Math and certainty
1. Math truths are empirical generalizations
based on a number of experiences.
2. Math is analytic: true by definition.
3. Math gives us knowledge independent of
experience.
Which do you agree with the most?
Then there was Godel
Who believed that any system of logic is, by its nature,
incomplete.
Godel’s incompleteness theorem: (1931) It is impossible to
prove that a formal mathematical system is free from
contradiction. Godel did not prove that maths contains
contradictions, but that we cannot be certain that it doesn’t. It
is always possible that one day we will find a contradiction;
and one small contradiction in a formal system would destroy
the system.
What does he mean by this? How does this apply to ideas in
reason/logic?
Statistics Revisited
Rhetorical device
Right and wrong
Upon reading a statistic…keep in mind…
Where is mathematics in…
Aesthetics
History
Human Sciences
Natural Sciences
Ethics
Spirituality/theology
Imagination/Memory
So again…
• What’s the point of studying mathematics?
• Where will you see or use mathematics in
your future career?
• What are some of the global, local, and
universal implications of the mathematics you
are currently using in your classes?
Math Riddle answers
1. 888 + 88 + 8 +8 +8
2. 12. A dozen of anything is still twelve.
3. One. The narrator