Transcript Chapter 5 Mathematics - East Irondequoit Central School

Chapter 5
Mathematics
Lesson 1 Math and Deduction
What is this saying?
Mathematical knowledge claims
• Sound and hard to argue with because
they are based on logical deduction
• Different than knowledge in some other
areas of knowledge because there is a
possibility of proving claims completely
• Are objective claims which anyone who
understands math can agree on
Logical deduction
• The foundation on which mathematics is
built
• Deduction can be defined as: making
conclusions based on premises known to
be true
• Mathematics proves itself through
deduction
Logic
• Logic is the science of
correct reasoning
• Reasoning is any
argument in which certain
assumptions of premises
are stated, and then
some other conclusion or
fact necessarily follows.
• Logic sometimes
called the science of
necessary inference
Aristotle 384-322 B.C.
Math and metaphysics
• The logic behind mathematics is also viewed as being a
member of the branch of philosophy known as
metaphysics
• Metaphysics is the study and the description of the
nature of reality
• Math in the west originated as a branch of philosophy. It
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attempted to describe and understand the nature of
reality
The assumption that reasoning is the best way at getting
to the nature of reality is a philosophical assumption
which may not necessarily be true in all or any
circumstances
Subjects and Predicates
• A subject is considered an individual
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phenomenon or entity, such as a tree or a bird.
A predicate is an attribute of the subject, such
as the tree being big or the bird being grey.
There are agreed upon rules in math regarding
the use of subjects and predicates
Fundamental principles regarding
subjects and predicates
• Identity: Everything is what it is and acts
accordingly
• Non-contradiction: It is impossible for
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something to both be and not be. A given
predicate cannot both belong and not belong
to a given subject in a given respect at a given
time. Contradictions do not exist.
Either-or: Everything must either be or not
be. A given predicate either belongs or does
not belong to a given subject in a given
respect at a given time.
Syllogisms
• A syllogism is the
basic Aristotelian
unit of reasoning.
• A an indisputable
conclusion reached
based on premises
known to be true
Examples of Syllogisms
• Some A is B
• All B is C
• Therefore, some A is C
• The conclusion is indisputable
Examples of Syllogisms
• All cats that live in Sweden have six toes
• Cindy the cat lives in Sweden
• Therefore, Cindy the cat has six toes
• The conclusion is indisputable
Examples of Syllogisms
• Premise 1: a2 + b2 = c2 in right triangles
• Premise 2: a = 6, b = 8 in the right
triangle ABC
• Therefore, C = 10
• The conclusion is indisputable
Paradoxical Logic
• Math is based on
logic
• But logic can prove
the impossible
• What does this say
about math?
• Is it reflective of
reality?
When will Jessie catch the Zombie?
• If Jesse is to catch up to
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the zombie, first, Jesse
must pass the spot where
the zombie started
This spot will be called
point A. But when Jesse
gets to point A, the
Zombie will have moved
on to point B
Jesse Owens. Winner of four Olympic gold medals
in the 1936 Berlin Olympics
When will Jessie catch the Zombie?
• Then to catch up, Jesse must
get to point B before he can
actually get to the zombie. So,
he gets to point B
• But when he gets there, the
zombie has moved on to point
C. Now, to catch up to the
zombie Jesse must first pass
through point C
• When he gets to point C
though, the Zombie has already
moved on to point D, and so on
and so on for all eternity.
A zombie. Winner of nothing. Loves brains
So when will Jessie catch the
Zombie?
• NEVER!!!!
• The distance between them is getting smaller
and smaller, but the fact remains that the
zombie will always be moving and as soon as
Jessie gets to any point (point n) the zombie will
have moved and the zombie will be at a new
point.
Absurdity?
• The paradox may be an absurdity but the
logic is sound.
• What does this say about math since math
is based on logic and logic can prove the
impossible?
Lesson 2 Does Mathematics Even
Exist?
Mathematics is a game played according to certain simple rules with meaningless
marks on paper.
--David Hilbert
Does math have anything to do
with reality?
• Does math even exist?
• Is it only a construct of the human mind?
• Just because it is used does it prove that it
is real?
• Would math exist without human culture?
• Is math logical nonsense?
Mathematic realism or Platonism
• According to Plato the ideal
world was a place that was
made up of numbers and
mathematical relationships
• Mathematicians, such as Euclid
and Pythagoras had discovered
how the world was made
• According to the mathematic
realist, math is something
discovered. It is something that
exists and awaits discovery
Is it here that math resides?
However…
• If there is a mathematical reality somewhere,
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where exactly is this reality?
How and where do all of these mathematical
entities exist?
Is it a separate world? Is it an internal world?
Perhaps nothing is mathematically related to
anything else unless we, human beings, say they
are related and explain that relationship with the
language of numbers.
Is Math an Arbitrary Game?
• As far as the laws of mathematics refer to reality,
they are not certain; and as far as they are
certain, they do not refer to reality --Albert Einstein
• If mathematics is an arbitrary creation, the
mathematical concepts and rules are not
something that correspond to reality in any way,
but instead are simply created and given meaning
by the people that create them.
• Perhaps math is arbitrary exactly like language.
It is possible that math was created to
accommodate the needs of people?
• Positive numbers were needed to facilitate
the organization of societal interaction
• Much later other complex elements like
zero were added
• Axiomatic understanding
changed at will. Logic
itself can be changed
Why is there no “zero” bead on this Chinese abacus?
The arbitrariness continues
• Other abstract components continue to be
added to mathematics as needed. These
are things such as:
• Negative numbers
• Irrational numbers
• Imaginary numbers
Mathematics changes and evolves
• With the needs of society
• As society becomes more complex, so to
does the demands put on mathematics
• New mathematics are developed to deal
with the needs
• Does this mean math exists?
• That’s a good question.
Lesson 3 How Do We Know What
We Know in Math?
Faith
• Not necessarily the first basis for
knowledge claims in mathematics, but it
still plays a role
• An axiom in math is a “fact” that is
assumed to be true
• Axioms lead to assumptions that are
based on faith in the truth of the axiom
Proof
• Proof means there has to be 100 percent
certainty that what is being claimed is
really the case
• The certainty comes through deductive
reasoning
• Proof in mathematics is also based upon
consensus
Authority
• The complexity of much mathematical
thought means the “consumers” of
mathematics can only trust in the
authority of the mathematical experts
Authority contd.
• Formulas are like recipes.
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Most people can use them
but do not fully
understand why they work
On the consumer of math
level, knowledge comes to
a great degree from
following the authority of
the math book.
Pure mathematics
• Pure mathematics is mathematics which is done
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purely for the sake of doing math (like abstract
algebra)
The aim is not to apply the knowledge to a real
world setting
“Knowledge” produced in pure mathematics
often has little or no importance to everyday life.
it may be argued that no real knowledge is
actually produced at all as long as there is no
practical application to real life settings.
Pure mathematics contd.
• As far as pure mathematicians
are concerned, pure
mathematics does produce
knowledge because it explores
the boundaries of mathematics
and pure reason
• It produces knowledge about
reasoning and how
mathematical structures
function
Perhaps Rodin’s thinker is pondering the boundaries of
mathematics and pure reason.
Applied mathematics
• Applied mathematics is all about
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application to real world settings.
Used in such professions as
engineering, economics, statistical
analysis, and computer science
Applied mathematics is using
mathematical knowledge to do
something
In applied mathematics,
knowledge is gained at a
pragmatic level
This astronaut is sure glad someone did
the math right at Houston.
Lesson 4 Paradoxes
Impossible images, like this one from M.C. Escher
are types of visual paradoxes.
What is a paradox?
• A paradox is a statement or a group of
statements that do (or seem to) lead to a
contradiction
• They lead to a situation that defies logic
and intuition
• They “prove” the impossible
Why of interest for math and TOK?
• Paradoxes attack the power of reasoning
• Mathematics is seen by many people as a
pure science and an expression of logic
perfected
• At the same time though, that same logic
can prove the most absurd claims to be
true
• What does this say about reason?
Paradox one: 2=1
• Let x and y be equal, non•
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zero quantities
x=y
Add x to both sides
2x = x + y
Take 2y from both sides
2x − 2y = x − y
Factorize
2(x − y) = x − y
Divide out (x − y)
2=1
That is an odd elephant!
Burdian’s ass
• The point to be made by
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this paradox is that
reason is not necessarily
the best tool when
considering the choices
made during life
Can you think of areas
where reason will not
provide the answers?
Houston, we have a problem!
Zeno’s dichotomy
• To get the plate of
brains bill must get from
point A to point B
• But, before he gets to
point B from point A,
the zombie must first
reach the halfway point
between the two points
The zombie formerly known as Bill.
Zeno’s dichotomy contd.
• However, when reaching the
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halfway point, another halfway
point comes into existence
which he must reach
However, to his despair, upon
reaching the new halfway
point, yet another halfway
point is created, and again
and again and again….
The zombie is doomed to
failure, because he always has
to reach a halfway point
before he reaches the final
destination.
The zombie formerly known as Bill.
The Monty Hall Paradox
• Here is how it works: Behind one of the
three doors below is a fancy red sports
car, behind the other two are goats. You
get whatever it is behind the door. If you
pick the door with the sports car, you drive
away in luxury. If you pick the door with
the goat, well, you will probably need
some hay. So make your pick.
The Monty Hall Paradox
• Behind one of these doors is a fancy red sports
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car.
Behind the other two is a goat
Pick the door you think the car is behind.
The Monty Hall Paradox
• For the sake of the example let’s say you pick door
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number two
The game show host is just about to open door number
two, but then he presents you with an offer.
He says, I’ll open one of the doors (which he does and a
goat is revealed)
He then makes you the offer that you may switch doors
from the door you picked to the other door
• If you switch doors, are your chances of winning
increased?
The Monty Hall Paradox
• Obviously the car is not behind door number three
• Before door number two is opened you may switch from
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door number two to door number one
What do you choose? Will your chance of winning
increase?
Believe it or not your chances of
winning are increased by switching
doors. Here is why:
• In the first pick you choose goat number 1. The
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game host picks the other goat. Switching will
win the car
In the first pick you choose goat number 2. The
game host picks the other goat. Switching will
win the car
In the first pick you choose the car. The game
host picks either of the two goats. Switching will
lose
So, by switching, your chances of winning
actually increase from 1/3 to 2/3
Omnipotence paradox
• If a god is truly all
powerful, then it
should be able to
do anything. Well
then, can this god
create a stone so
big that he/she/it
can not move it?
Is this the big stone?
The reasoning runs this way:
• Either this omnipotent god can create a stone it cannot
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lift or it cannot create a stone which it cannot lift
If this god can create a stone which it cannot lift, then
there is one thing it cannot do; namely lift the stone it
just created
If this god cannot create a stone which it cannot lift,
then there is one thing it cannot do; namely create such
a large stone
Therefore there is at least one task this god cannot
perform
Omnipotent means that this being can do anything
Subsequently, this god is not omnipotent
Concluding thoughts about logic
and reason
• Logic and reason is not always consistent and is not
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always the answer to the nature of reality
Reasoning and rationale can play tricks on us and make
us believe in the impossible since the impossible
sometimes seems so rationally likely
Mathematics is bound by the rules of logic. As has been
shown through several examples, the rules of logic are
not always the best way to get at the nature of reality
Therefore math can not always be the best way to
understand the nature of reality
There are many sides to every issue and knowledge is
not as straightforward as it seems at first glance