Transcript MA110lecture02_Spring09 - University of South Alabama

MA 110: Finite Math
Lecture 1/14/2009
Section 1.1
Homework: 5, 9-15, 21-22 (56 BP)
Today We’ll Learn
1. How math is unique as a subject.
2. The basis of mathematical thought: logical
(valid) deduction and the syllogism.
3. The difference between validity and truth.
4. Deduction verses Induction.
Mathematics
.. is a way of “knowing”.
“Epistemology”
“Epistemology”
The theory of knowledge.
What do we know?
Isn’t it neat that we know that we know?
How do we know if we (really) know??
Some problems with knowing:
1. Illogical conclusions / bad reasoning.
2. Reasoning and logic cannot answer everything.
1.
2.
3.
Ethical questions.
The great ‘why’ questions.
We trust that we know, but we can’t know that we know.
“Chuang-Tzu once dreamed he was a butterfly. When he
awoke, he no longer knew if he was a butterfly dreaming he
was a man, or a man who had dreamed he was a butterfly.”
Class discussion: what are different
ways that we know things?
(1)
(2)
(3)
(4)
(5)
(6)
Four commonly accepted
ways of knowing:
(1)
(2)
(3)
(4)
Reason (logic)
Perception (senses, induction)
Language (stories/authority)
Emotion (intuition/inspiration/revelation)
From http://www.acperesearch.net/knowing.html
Four commonly accepted
ways of knowing:
(1)
(2)
(3)
(4)
Reason (logic) …………………Mathematics
Perception (senses, induction)
Language (stories/authority)
Emotion (intuition/inspiration/revelation)
From http://www.acperesearch.net/knowing.html
Four commonly accepted
ways of knowing:
(1)
(2)
(3)
(4)
Reason (logic) …………………Mathematics
Perception (senses, induction) ………Science
Language (stories/authority)
Emotion (intuition/inspiration/revelation)
From http://www.acperesearch.net/knowing.html
Four commonly accepted
ways of knowing:
(1)
(2)
(3)
(4)
Reason (logic) …………………Mathematics
Perception (senses, induction) ………Science
Language (stories/authority) ……Humanities
Emotion (intuition/inspiration/revelation)
From http://www.acperesearch.net/knowing.html
Areas of Knowledge
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Mathematics
Natural Science
Human Sciences
History
Arts
Ethics
Spirituality
Areas of Knowledge
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Mathematics
Natural Science
Human Sciences
History
Arts
Ethics
Spirituality
Objective
Subjective
Areas of Knowledge
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Mathematics
Natural Science
Human Sciences
History
Arts
Ethics
Spirituality
Objective
Subjective
Mathematics
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Most objective.
Entirely rational.
Logical.
Universal validity.
What is meant be universal validity?
The validity…
• Doesn’t depend upon a specific context.
• Doesn’t depend upon knowing anything –
doesn’t even depend upon knowing that we
know something.
• Instead of asserting that something is true,
mathematics very much describes only what
would be true if certain other things are true.
What is meant be universal validity?
The validity…
• Doesn’t depend upon a specific context.
• Doesn’t depend upon knowing anything –
doesn’t even depend upon knowing that we
know something.
• Instead of asserting that something is true,
mathematics very much describes only what
would be true given certain other things are
true.
What is meant be universal validity?
The validity…
• Doesn’t depend upon a specific context.
• Doesn’t depend upon knowing anything –
doesn’t even depend upon knowing that we
know something.
• Instead of asserting that something is true,
mathematics very much describes only what
would be true assuming certain other things are
true.
Example
• Two bags contain coins.
• Maybe the blue bag contains 10 coins.
Maybe not.
• Maybe the red bag contains 5 coins.
Maybe not.
• In mathematics, it doesn’t matter.
• Mathematics says:
Example
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IF the blue bag contains exactly 10 coins
AND
IF the red bag contains exactly 5 coins
THEN both bags contain 15 coins.
A conclusion that is
guaranteed given the
assumptions.
• Uh oh! You combine both bags and instead
have only 10 coins.
• What happened?
• Was the previous argument invalid?
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IF the blue bag contains exactly 10 coins
AND
IF the red bag contains exactly 5 coins
THEN both bags contain 15 coins.
Still true: universal validity
no matter how many coins
the bags actually have.
This is the basis of mathematics: based on a set
of assumptions (things we assume to be true
for the sake of argument), what conclusions
can we make that have to be true, if those
assumptions are true?
Mathematics
Aristotle
384BC – 322 BC
Aristotle
• Together with Plato, and Socrates (Plato's teacher),
Aristotle is one of the most important founding
figures in Western philosophy. He was the first to
create a comprehensive system of Western
philosophy, encompassing morality and aesthetics,
logic and science, politics and metaphysics.
• All aspects of Aristotle's philosophy continue to be
the object of active academic study today.
• Aristotle defined the syllogism.
Syllogism
Premises: Sentences that are assumed to be true
for the sake of argument.
Syllogism: Premises are combined that lead to a
conclusion.
Valid syllogism: The premises correctly lead to
the conclusion in the sense that the conclusion
has to be true given the assumption.
Truth verses Validity
Validity:
For any given set of premises, if the
conclusion of an argument is guaranteed, the
argument is valid. (If the conclusion in not
guaranteed the conclusion is invalid.)
Truth
A conclusion is true if it happens to be the
case in the ‘real world’.
Truth verses Validity
A valid argument can lead to a false conclusion
if the premises are false.
An invalid argument can lead to a true
conclusion
just by accident.
BUT
A valid argument always leads to a true
conclusion
if the premises are true.
Truth verses Validity
A valid argument can lead to a false conclusion
if the premises are false.
Women are bad drivers. False premise
Danica Patrick is a woman.
Therefore, Danica Patrick is a bad driver.
Valid argument, but false.
Truth verses Validity
An invalid argument can lead to a true
conclusion
just by accident.
Some cats are black.
This is a cat.
Therefore, this cat is black.
Invalid argument, but true.
Truth verses Validity
A valid argument always leads to a true
conclusion
if the premises are true.
All cats are mammals.
This is a cat.
Therefore, this cat is a mammal.
Valid argument, premises true,
Conclusion true.
Deductive Reasoning
Deductive Reasoning – from the general to
the specific.
If something is true for the general, then it is
true for a specific instance.
Example: All cats are mammals. If this is a cat,
then it is a mammal.
“All TREES”
Deduction: from the
general to the specific
A little plug for science…
In the real world, how do you
know anything in general?
“TREES”
Induction: from the
specific to the general
Inductive Reasoning
Inductive Reasoning – from the specific to
the general.
If something is true in every observed
instance, it may be true in the next instance.
Example: The sun will rise tomorrow morning.
Areas of Knowledge
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Mathematics
Natural Science
Human Sciences
History
Arts
Ethics
Spirituality
…Deductive Reasoning
…Inductive Reasoning
(listed from most objective to most subjective)
Areas of Knowledge
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Mathematics
Natural Science
Human Sciences
History
Arts
Ethics
Spirituality
…Deductive Reasoning PROOF
…Inductive Reasoning EVIDENCE
(listed from most objective to most subjective)
Deductive or Inductive?
(Proof or evidence?)
1
I was walking through the park and saw 14 dogs.
Every dog had a tail. Therefore I conclude that
all dogs have tails.
Deductive or Inductive?
(Proof or evidence?)
2
I know that stars are bodies of hot gas. Our sun is
a star. Therefore, our sun is a body of hot gas.
Deductive or Inductive?
(Proof or evidence?)
3
During rush hour Bob watches the cars at a busy
intersection. He notices that of the 48 people
who did not use their turn signals when
turning, 75% were driving Fords. He
concludes, people who drive Fords are more
likely to not use turn signals.
Deductive or Inductive?
(Proof or evidence?)
4
The senator has supported every tax increase so
far so I am sure he will support this one.
Deductive or Inductive?
(Proof or evidence?)
5
-2 cubed is -8 so I conclude that not all numbers,
when cubed, give a positive number.
Inductive reasoning is only as good as
the last n observations.
Rare and beautiful event.
Watch out for it in 2012.
Transit of Venus
Inductive reasoning is only as good as
the last n observations.
• First transit in 3rd Millenium:
June 8th, 2004
• Second transit:
June 6th, 2012
Transit of Venus
Inductive reasoning is only as good as
the last n observations.
• First transit in 3rd Millenium:
June 8th, 2004
• Second transit:
June 6th, 2012
Induction: transit
every 8 years?
Transit of Venus
Inductive reasoning is only as good as
the last n observations.
• First transit in 3rd Millenium:
June 8th, 2004
• Second transit:
June 6th, 2012
• Third transit:
December, 2117
Transit of Venus