math tok presentation - jchase.com tok presentation.pdf · liberal education glimpsing the mind of...
TRANSCRIPT
![Page 1: Math TOK Presentation - jchase.com TOK Presentation.pdf · Liberal Education Glimpsing the mind of God. ... Questions? Title: Microsoft PowerPoint - Math TOK Presentation Author:](https://reader031.vdocument.in/reader031/viewer/2022021823/5b35fb277f8b9a6b548dd2ce/html5/thumbnails/1.jpg)
MATHEMATICAL THINKINGA guest lecture by Mr. Chase
![Page 2: Math TOK Presentation - jchase.com TOK Presentation.pdf · Liberal Education Glimpsing the mind of God. ... Questions? Title: Microsoft PowerPoint - Math TOK Presentation Author:](https://reader031.vdocument.in/reader031/viewer/2022021823/5b35fb277f8b9a6b548dd2ce/html5/thumbnails/2.jpg)
Is mathematics invented or discovered?
Aristotle Plato
![Page 3: Math TOK Presentation - jchase.com TOK Presentation.pdf · Liberal Education Glimpsing the mind of God. ... Questions? Title: Microsoft PowerPoint - Math TOK Presentation Author:](https://reader031.vdocument.in/reader031/viewer/2022021823/5b35fb277f8b9a6b548dd2ce/html5/thumbnails/3.jpg)
Is mathematics invented or discovered?
Poll!
Options:
1. Invented2. Discovered3. Unresolvable4. I don’t know
![Page 4: Math TOK Presentation - jchase.com TOK Presentation.pdf · Liberal Education Glimpsing the mind of God. ... Questions? Title: Microsoft PowerPoint - Math TOK Presentation Author:](https://reader031.vdocument.in/reader031/viewer/2022021823/5b35fb277f8b9a6b548dd2ce/html5/thumbnails/4.jpg)
invented!
“Newton and Leibniz invented Calculus.”
our number system
conventions and symbols
And if you think mathematics is discovered: if a mathematical theory goes undiscovered, does it truly exist?
![Page 5: Math TOK Presentation - jchase.com TOK Presentation.pdf · Liberal Education Glimpsing the mind of God. ... Questions? Title: Microsoft PowerPoint - Math TOK Presentation Author:](https://reader031.vdocument.in/reader031/viewer/2022021823/5b35fb277f8b9a6b548dd2ce/html5/thumbnails/5.jpg)
discovered!
Is prime or composite?
no contradictions
Are there an infinite number of “twin primes”?
arbitrary notation
math is like science—it’s true, regardless of whether we discover it or not.
![Page 6: Math TOK Presentation - jchase.com TOK Presentation.pdf · Liberal Education Glimpsing the mind of God. ... Questions? Title: Microsoft PowerPoint - Math TOK Presentation Author:](https://reader031.vdocument.in/reader031/viewer/2022021823/5b35fb277f8b9a6b548dd2ce/html5/thumbnails/6.jpg)
Correct answer…
discovered!
![Page 7: Math TOK Presentation - jchase.com TOK Presentation.pdf · Liberal Education Glimpsing the mind of God. ... Questions? Title: Microsoft PowerPoint - Math TOK Presentation Author:](https://reader031.vdocument.in/reader031/viewer/2022021823/5b35fb277f8b9a6b548dd2ce/html5/thumbnails/7.jpg)
Is this always true? Aren’t you dying for a proof?
Is 9 1 always divisible by 8?
There exist two people in DC with the exact same number of hairs on their heads. Why?
![Page 8: Math TOK Presentation - jchase.com TOK Presentation.pdf · Liberal Education Glimpsing the mind of God. ... Questions? Title: Microsoft PowerPoint - Math TOK Presentation Author:](https://reader031.vdocument.in/reader031/viewer/2022021823/5b35fb277f8b9a6b548dd2ce/html5/thumbnails/8.jpg)
Mathematics is a queen of science.Carl Friedrich Gauss
![Page 9: Math TOK Presentation - jchase.com TOK Presentation.pdf · Liberal Education Glimpsing the mind of God. ... Questions? Title: Microsoft PowerPoint - Math TOK Presentation Author:](https://reader031.vdocument.in/reader031/viewer/2022021823/5b35fb277f8b9a6b548dd2ce/html5/thumbnails/9.jpg)
what mathematicians have to say…
The mathematician does not study pure mathematics becauseit is useful; he studies it because he delights in it and he delightsin it because it is beautiful.
Jules Henri Poincaré
Wherever there is number, there is beauty.Proclus
It is impossible to be a mathematician without being a poet in soul.
Sofia Kovalevskaya
![Page 10: Math TOK Presentation - jchase.com TOK Presentation.pdf · Liberal Education Glimpsing the mind of God. ... Questions? Title: Microsoft PowerPoint - Math TOK Presentation Author:](https://reader031.vdocument.in/reader031/viewer/2022021823/5b35fb277f8b9a6b548dd2ce/html5/thumbnails/10.jpg)
what mathematics are we free to invent?
the symbols and conventions we choose are arbitrary.
FORMALISM
Mathematics is a game played according to certain simple rules with meaningless marks on paper.
David Hilbert
![Page 11: Math TOK Presentation - jchase.com TOK Presentation.pdf · Liberal Education Glimpsing the mind of God. ... Questions? Title: Microsoft PowerPoint - Math TOK Presentation Author:](https://reader031.vdocument.in/reader031/viewer/2022021823/5b35fb277f8b9a6b548dd2ce/html5/thumbnails/11.jpg)
the field axioms.Closure of under addition and multiplication
For all a, b in F, both and are in (or more formally, and are binary operations on ).
Associativity of addition and multiplicationFor all , , and in , the following equalities hold:
and .
Commutativity of addition and multiplicationFor all and in , the following equalities hold:
and .
Existence of additive and multiplicative identity elementsThere exists an element of , called the additive identity element and denoted by 0, such that for all in , 0 . Likewise, there is an element, called the multiplicative identity element and denoted by 1, such that for all in , 1 . To exclude the trivial ring, the additive identity and the multiplicative identity are required to be distinct.
Existence of additive inverses and multiplicative inversesFor every in , there exists an element in , such that 0. Similarly, for any
in other than 0, there exists an element in , such that 1. (The elements and are also denoted and / , respectively.) In other words,
subtraction and division operations exist.
Distributivity of multiplication over additionFor all , and in , the following equality holds: .
![Page 12: Math TOK Presentation - jchase.com TOK Presentation.pdf · Liberal Education Glimpsing the mind of God. ... Questions? Title: Microsoft PowerPoint - Math TOK Presentation Author:](https://reader031.vdocument.in/reader031/viewer/2022021823/5b35fb277f8b9a6b548dd2ce/html5/thumbnails/12.jpg)
Can we break or change the rules?
YES.
groupring
domain
skew field
Abelian group
![Page 13: Math TOK Presentation - jchase.com TOK Presentation.pdf · Liberal Education Glimpsing the mind of God. ... Questions? Title: Microsoft PowerPoint - Math TOK Presentation Author:](https://reader031.vdocument.in/reader031/viewer/2022021823/5b35fb277f8b9a6b548dd2ce/html5/thumbnails/13.jpg)
David Hilbert Kurt Gödel
Epic math battlesProve the
thing!I want to create a
formal system in which we can
prove all statements.
You can’t prove the
thing!In every
formal system, there must be
unprovable statements.
![Page 14: Math TOK Presentation - jchase.com TOK Presentation.pdf · Liberal Education Glimpsing the mind of God. ... Questions? Title: Microsoft PowerPoint - Math TOK Presentation Author:](https://reader031.vdocument.in/reader031/viewer/2022021823/5b35fb277f8b9a6b548dd2ce/html5/thumbnails/14.jpg)
Axioms: it is raining outside.
if it is raining, I will take an umbrella.
Statements: I will take an umbrella.
It is not raining outside.
I will take my pet hamster as well.
Silly example
Provably true.
Provably false.
Undecidable
![Page 15: Math TOK Presentation - jchase.com TOK Presentation.pdf · Liberal Education Glimpsing the mind of God. ... Questions? Title: Microsoft PowerPoint - Math TOK Presentation Author:](https://reader031.vdocument.in/reader031/viewer/2022021823/5b35fb277f8b9a6b548dd2ce/html5/thumbnails/15.jpg)
Math is useful
But…WHY is it useful?
It’s like a gorgeous painting that also functions as a dishwasher!
Ben Orlin
![Page 16: Math TOK Presentation - jchase.com TOK Presentation.pdf · Liberal Education Glimpsing the mind of God. ... Questions? Title: Microsoft PowerPoint - Math TOK Presentation Author:](https://reader031.vdocument.in/reader031/viewer/2022021823/5b35fb277f8b9a6b548dd2ce/html5/thumbnails/16.jpg)
Why study math?
Liberal Education
Glimpsing the mind of God
![Page 17: Math TOK Presentation - jchase.com TOK Presentation.pdf · Liberal Education Glimpsing the mind of God. ... Questions? Title: Microsoft PowerPoint - Math TOK Presentation Author:](https://reader031.vdocument.in/reader031/viewer/2022021823/5b35fb277f8b9a6b548dd2ce/html5/thumbnails/17.jpg)
In summary…Math is different. It allows certain knowledge.
![Page 18: Math TOK Presentation - jchase.com TOK Presentation.pdf · Liberal Education Glimpsing the mind of God. ... Questions? Title: Microsoft PowerPoint - Math TOK Presentation Author:](https://reader031.vdocument.in/reader031/viewer/2022021823/5b35fb277f8b9a6b548dd2ce/html5/thumbnails/18.jpg)
Questions?