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Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

Dec 25, 2015

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Page 1: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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MaThOKYosef Karasik

Page 2: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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Bertrand Russell

• "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without the gorgeous trappings of painting or music.”

Page 3: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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Reasoning as a WOK

Find a function for:• X=1 Y=1• X=2 Y=2• X=3 Y=3• X=4 Y=4• X=5 Y=125

Page 4: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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Inductive reasoning

• Is reasoning that takes specific information and makes a broader generalization that is considered probable, allowing for the fact that the conclusion may not be accurate.

• Jesse and the divas

Page 5: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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Get to work!

• Draw 3 different triangles on paper and measure the sum of their angles (as accurate as possible) using your protractors.

• What results did you get?

Page 6: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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Questions• Can we come up with a conjecture for the

sum of angles in a triangle based on your measurements? What will that conjecture be?

• (KQ) Under what conditions can we ignore results that come as contradictions to the hypothesis?

Page 7: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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Deductive reasoning

• Is a type of reasoning which goes from general to specific. Deductive reasoning is based on premises and if the premises are true, then the reasoning will be valid.

• Come up with a different proof for the sum of angles in a triangle.

Page 8: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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Inductive or deductive?

• Jennifer leaves for school at 7:00 a.m. and is on time. Jennifer assumes, then, that she will always be on time if she leaves at 7:00 a.m.

• Snakes are reptiles and reptiles are cold-blooded; therefore, snakes are cold-blooded.

• All observed women in one area wear high heels, so all women must wear high heels.

• All dogs are mammals. Spot is a dog. Spot is a mammal.• Robert is a teacher. All teachers are nice. Therefore, it can

be assumed that Robert is nice.

Page 9: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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Questions• Were there any assumptions (premises) in

your proof?

• Can you prove them as well?

• Axioms – a self-evident truth that requires no proof. A universally accepted principle or rule.

Page 10: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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The postulates of geometry

• You can only draw one line between two points.• There's 360° around a circle• Two lines can intersect at ONLY one point• A line segment has ONLY one midpoint• An angle can only have one bisector• Any geometric shape can be moved without

changing its shape• And more…

Page 11: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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Proof

• What methods of proving are there?Direct proofProof by inductionProof by contrapositionProof by contradictionProof by constructionProof by exhaustionOthers…

Page 12: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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Further explore

• What makes a proving method valid?

• What makes a sound proof?

• Can one proof method be better then the other?

Page 13: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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Balloons!

• Draw triangles on balloons and measure the angles.– Calculate the sum of the three angles

• Measure side lengths of any right angle triangle.

Page 14: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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WHAT THE…!?!?!?

• Why not as expected?

• Does size of balloon make a difference?

• What if you were inside the balloon?

• How many right angles can the triangle contain?

• How can you decide which one is the hypotenuse?

Page 15: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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The postulates stop working!

• They work for planes, but not curved surfaces

• How about earth then? Is it flat?

Page 16: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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Spherical Geometry

• Redefine “straight line”, “circle”, “Plane”

• Lobachevsky’s Geometry

Page 17: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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Further explore

• What role perception plays in mathematical understanding?

• Does context affect mathematical understanding?

Page 18: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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Is Zero Nothing?

• What do you think of when you see O? 0?

• Do they bring the same or different things to your mind?

• History of the symbol (quill pen)

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A few questions

• What remains when 7 is taken away from 7?• A place value symbol as in 402.6 , giving value

to other digits around it but having no value itself.

• The number that causes great difficulties if you try to divide by it 12/0=?– Undefined?– Infinity?

• What is ?

Page 20: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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Does 2=1?

Page 21: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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More of nothing

• What happened to the certainty of Math?

• Nothing, emptiness, lack of anything, absence, void, black, darkness, oblivion… and therefore the embodiment of evil!

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A bit of history

• Zero as absence in ancient times.– Ancient Greeks and Romans– Babylonians– Chinese traders

• Zero as a number, entity, presence

• A shift from NAME to SYMBOL

Page 23: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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Magic number

• Zero counts the totality of what isn’t there

• It is the gateway between the positive and negative numbers

• A number without qualities

Page 24: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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Zero or nothing? Presence or absence?

• {} null set – called ϕ

• {0} the set containing 0

• So how about {ϕ}?

Page 25: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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In art

• Nothing is? Black(absence of color) or White(presence of every color)?

Page 26: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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In Music

• John Cage

“The material of music is sound and silence. Integrating these is composing. I have nothing to say and I’m saying it”

He sat there turning pages from time to time, then stood up and bowed

Page 28: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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In Literature

• George Gershwin – in Porgy and Bess

“I got plenty of nothin’, nothin’s plenty for me”

• Lewis Carroll – Alice in conversation with the king– A: I see nobody on the road– K: I only wish I had such eyes, to be able to see

nobody, and at such a distance too

Page 29: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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The church

• God exists!

• Demonstrating that something can come from nothing if nothing is acted on by an infinite being – hence the act of creation can take place.

Page 30: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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Zero and infinity – a love story

• A young child when asked about what is infinity: “The biggest number ever was thousand thousand thousands”

• What about thousand thousand thousands plus 1?

• “Well”, she said, “I was very close wasn’t I?”• Geometric series – adding infinite numbers

and end up with finite sum.

Page 31: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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Nonsense Rhyme

• As I was going up the stair, I met a man who wasn’t there.

He wasn’t there again today. I wish, I wish

he’d stay away.

Page 32: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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Knowledge questions for you (you’re welcome!)

• Can ethics (and others areas of knowledge) be reduced to equations? (George Price)

• Does mathematics provide us with more practicable knowledge than less quantitative data?

• Are those who understand mathematics better equipped to deal with the modern world? (algorithms)

• What role does mathematics play in our knowledge acquisition? Do those who understand math have a clearer understanding of the world?

Page 33: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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More KQs

• To what extent does our perception of ordinary life depend on statistical knowledge? What are the dangers associated with this in terms of providing us with a realistic view of the world?

• To what extent is our perception of the world shaped by the way a ‘base 10 approach’ is instilled in us from an early age?

• Correlation VS causation – To what extent interpretation of statistics affects our understanding.

• Why does it have very little effect to tell people that they can’t statistically beat the odds in a casino?

Page 34: Ma T h OK Yosef Karasik. Bertrand Russell "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without.

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Thanks for nothing