MA 110: Finite Math Lecture 1/14/2009 Section 1.1 Homework: 5, 9-15, 21-22 (56 BP)

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. Ethical questions.2. The great ‘why’ questions.

3. 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) Reason (logic)

(2) Perception (senses, induction)

(3) Language (stories/authority)

(4) Emotion (intuition/inspiration/revelation)

From http://www.acperesearch.net/knowing.html


(1) Reason (logic) …………………Mathematics

(2) Perception (senses, induction)






(2) Perception (senses, induction) ………Science






(2) Perception (senses, induction) ………Science

(3) Language (stories/authority) ……Humanities



Areas of Knowledge

• Mathematics

• Natural Science

• Human Sciences

• History

• Arts

• Ethics

• Spirituality

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

Mathematics

• Most objective.

• Entirely rational.

• Logical.

• 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…



• Instead of asserting that something is true, mathematics very much describes only what would be true given certain other things are true.


The validity…



• 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

• 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?

• 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

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.

• 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.)

TruthA conclusion is true if it happens to be the case in the ‘real world’.

Truth verses ValidityA 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 ValidityA valid argument can lead to a false conclusion

if the premises are false.

Women are bad drivers. Danica Patrick is a woman. Therefore, Danica Patrick is a bad driver.

Valid argument, but false.

False premise

Truth verses ValidityAn 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 ValidityA 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

In the real world, how do you know anything in general?

A little plug for science…

“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

• 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


I know that stars are bodies of hot gas. Our sun is a star. Therefore, our sun is a body of hot gas.

2


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


The senator has supported every tax increase so far so I am sure he will support this one.

4


-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.


• First transit in 3rd Millenium:

June 8th, 2004

• Second transit:

June 6th, 2012

Transit of Venus


• First transit in 3rd Millenium:

June 8th, 2004

• Second transit:

June 6th, 2012

Transit of Venus

Induction: transit every 8 years?


• First transit in 3rd Millenium:June 8th, 2004

• Second transit:June 6th, 2012

• Third transit:December, 2117 Transit of Venus

Documents

MA 110: Finite Math Lecture 1/14/2009 Section 1.1 Homework: 5, 9-15, 21-22 (56 BP)