ma 110: finite math dr. maria byrne instructional laboratory 0345 when you enter, pick up a quiz and...
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MA 110: Finite MathDr. Maria Byrne
Instructional Laboratory 0345
When you enter, pick up a quiz and complete it at
your desk.
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MA 110: Finite MathDr. Maria Byrne
Instructional Laboratory 0345
Lecture 8/20/2008
Homework: 1.1: 4,8,12,16,20
Due next Monday 8/25
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• When
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“Epistemology”
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“Epistemology” The theory of knowledge.
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“Epistemology” The theory of knowledge.
What do we know?
Isn’t it cool that we know that we know?
How do we know if we (really) know??
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Some problems with knowing:
• Illogical conclusions / bad reasoning.
• Reasoning and logic cannot answer everything. – Ethical questions.
– The great ‘why’ questions.
• “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.”
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Class discussion: what are different ways that we know things?
(1)
(2)
(3)
(4)
(5)
(6)
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Four commonly accepted ways of knowing:
(1) Reason (logic)
(2) Perception (senses, induction)
(3) Language (stories/authority)
(4) Emotion (intuition/inspiration/revelation)
From http://www.acperesearch.net/knowing.html
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Four commonly accepted ways of knowing:
(1) Reason (logic) …………………Mathematics
(2) Perception (senses, induction)
(3) Language (stories/authority)
(4) Emotion (intuition/inspiration/revelation)
From http://www.acperesearch.net/knowing.html
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Four commonly accepted ways of knowing:
(1) Reason (logic) …………………Mathematics
(2) Perception (senses, induction) ………Science
(3) Language (stories/authority)
(4) Emotion (intuition/inspiration/revelation)
From http://www.acperesearch.net/knowing.html
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Four commonly accepted ways of knowing:
(1) Reason (logic) …………………Mathematics
(2) Perception (senses, induction) ………Science
(3) Language (stories/authority) ……Humanities
(4) Emotion (intuition/inspiration/revelation)
From http://www.acperesearch.net/knowing.html
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Areas of Knowledge
• Mathematics• Natural Science• Human Sciences• History• Arts• Ethics• Spirituality
(listed from most objective to most subjective)
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Areas of Knowledge
• Mathematics
• Natural Science
• Human Sciences
• History
• Arts
• Ethics
• Spirituality
Objective
Subjective
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Areas of Knowledge
• Mathematics
• Natural Science
• Human Sciences
• History
• Arts
• Ethics
• Spirituality
Objective
Subjective
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Areas of Knowledge
• Mathematics
• Natural Science
• Human Sciences
• History
• Arts
• Ethics
• Spirituality
Objective
Subjective
SEE BLACKBOARD
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Mathematics
• Most objective.
• Entirely rationale.
• Universal validity.
• LOGICAL
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Mathematics
• Most objective.
• Entirely rationale.
• Universal validity.
• LOGICAL
“logike”
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Mathematics
• Most objective.
• Entirely rationale.
• Universal validity.
• LOGICAL
Aristotle384BC – 322 BC
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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.
• From Wikipedia. SEE BLACKBOARD
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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: Bachelors are unmarried men. If Tom is a bachelor, he is a man.
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“TREES”
Deduction: from the general to the specific
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In the real world, how do you know anything in general?
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“TREES”
Induction: from the specific to the general
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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.
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SEE BLACKBOARD
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Areas of Knowledge
• Mathematics …Deductive Reasoning• Natural Science …Inductive Reasoning• Human Sciences• History• Arts• Ethics• Spirituality
(listed from most objective to most subjective)
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Areas of Knowledge
• Mathematics …Deductive Reasoning• Natural Science …Inductive Reasoning• Human Sciences• History• Arts• Ethics• Spirituality
(listed from most objective to most subjective)
PROOF
EVIDENCE
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Deductive or Inductive?(Proof or evidence?)
I was walking through the park and saw 14 dogs. Every dog had a tail. Therefore I conclude that all dogs have tails.
1
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Deductive or Inductive?(Proof or evidence?)
I know that any positive or negative real number, when squared, gives a positive number. I know that zero, when squared, gives zero. Therefore, no real number, when squared, gives a negative number.
2
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Deductive or Inductive?(Proof or evidence?)
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.
3
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Deductive or Inductive?(Proof or evidence?)
The senator has supported every tax increase so far so I am sure he will support this one.
4
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Deductive or Inductive?(Proof or evidence?)
-2 cubed is -8 so I conclude that not all numbers, when cubed, give a positive number.
5
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Inductive reasoning is only as good as the last n observations.
Transit of Venus
Rare and beautiful event.
Watch out for it in 2012.
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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
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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
Induction: transit every 8 years?
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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
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SEE BLACKBOARD
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Women are bad drivers. Dr. Byrne is a woman. Therefore, Dr. Byrne is a bad driver.
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The following argument is valid:
Women are bad drivers. Dr. Byrne is a woman. Therefore, Dr. Byrne is a bad driver.
But false.
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The following argument is valid:
Women are bad drivers. False premiseDr. Byrne is a woman. Therefore, Dr. Byrne is a bad driver.
But false.
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Consider the following argument:
Fluoride may not be safe for consumption. Drinking water in Mobile contains fluoride.
Therefore, drinking water in Mobile may not be safe for consumption.
Valid?True?
You Decide…
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Consider the following argument:
Some doctors are women. Some women are mothers.
Therefore, some doctors are mothers.
Valid?
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Consider the following argument:
Some A are B. Some doctors are women.Some B are C. Some women are mothers.
Therefore, some doctors are mothers. Therefore, some A are C.
Valid?
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Consider the following argument:
Some A are B. Some cats are black.Some B are C. Some black things are televisions.
Therefore, some A are C. Some cats are televisions.
Not Valid