MA 110: Finite MathDr. Maria Byrne

Instructional Laboratory 0345

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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

• When

“Epistemology”

“Epistemology” The theory of knowledge.

“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??

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.”

Class discussion: what are different ways that we know things?

(1)

(2)

(3)

(4)

(5)

(6)

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

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

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

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

Areas of Knowledge

• Mathematics• Natural Science• Human Sciences• History• Arts• Ethics• Spirituality

(listed from most objective to most subjective)

Areas of Knowledge

• Mathematics

• Natural Science

• Human Sciences

• History

• Arts

• Ethics

• Spirituality

Objective

Subjective

Areas of Knowledge

• Mathematics

• Natural Science

• Human Sciences

• History

• Arts

• Ethics

• Spirituality

Objective

Subjective

Areas of Knowledge

• Mathematics

• Natural Science

• Human Sciences

• History

• Arts

• Ethics

• Spirituality

Objective

Subjective

SEE BLACKBOARD

Mathematics

• Most objective.

• Entirely rationale.

• Universal validity.

• LOGICAL

Mathematics

• Most objective.

• Entirely rationale.

• Universal validity.

• LOGICAL

“logike”

Mathematics

• Most objective.

• Entirely rationale.

• Universal validity.

• LOGICAL

Aristotle384BC – 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.

• From Wikipedia. SEE BLACKBOARD

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.

“TREES”

Deduction: from the general to the specific

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.

SEE BLACKBOARD

Areas of Knowledge

• Mathematics …Deductive Reasoning• Natural Science …Inductive Reasoning• Human Sciences• History• Arts• Ethics• Spirituality

(listed from most objective to most subjective)

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

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

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

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

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

Deductive or Inductive?(Proof or evidence?)

-2 cubed is -8 so I conclude that not all numbers, when cubed, give a positive number.

5

Inductive reasoning is only as good as the last n observations.

Transit of Venus

Rare and beautiful event.

Watch out for it in 2012.

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

Transit of Venus

Induction: transit every 8 years?

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

SEE BLACKBOARD

Women are bad drivers. Dr. Byrne is a woman. Therefore, Dr. Byrne is a bad driver.

The following argument is valid:

Women are bad drivers. Dr. Byrne is a woman. Therefore, Dr. Byrne is a bad driver.

But false.

The following argument is valid:

Women are bad drivers. False premiseDr. Byrne is a woman. Therefore, Dr. Byrne is a bad driver.

But false.

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…

Consider the following argument:

Some doctors are women. Some women are mothers.

Therefore, some doctors are mothers.

Valid?

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?

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