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Mathematics in the Modern World Mathematics in Our World Joel Reyes Noche, Ph.D.
[email protected] Department of Mathematics Ateneo de Naga University Council of Deans and Department Chairs of Colleges of Arts and Sciences Region V Conference and Enrichment Sessions on the New General Education Curriculum Ateneo de Naga University, Naga City September 2, 2017–September 3, 2017
Mathematics in the Modern World The Nature of Mathematics
Mathematics in Our World
Mathematics in Our World Mathematics is a useful way to think about nature and our world
Learning outcomes I
Identify patterns in nature and regularities in the world.
I
Articulate the importance of mathematics in one’s life.
I
Argue about the nature of mathematics, what it is, how it is expressed, represented, and used.
I
Express appreciation for mathematics as a human endeavor.
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Nature by Numbers A short movie by Crist´ obal Vila, 2010
Original (3:44): https://vimeo.com/9953368 Alternative soundtrack (4:04): https://vimeo.com/29379521
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Fibonacci sequence (Vila, 2016)
The Fibonacci sequence “is an infinite sequence of natural numbers where the first value is 0, the next is 1 and, from there, each amount is obtained by adding the previous two.”
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Fibonacci spiral (Vila, 2016)
Circular arcs connect the opposite corners of squares in the Fibonacci tiling.
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Golden spiral (Golden spiral in rectangles, 2008)
r = ϕ2θ/π where θ is in radians and ϕ =
√ 1+ 5 2
is the golden ratio
Mathematics in the Modern World The Nature of Mathematics
Mathematics in Our World
Comparison of spirals (Approximate and true Golden Spirals, 2009)
Quarter-circles in green, golden spiral in red, overlaps in yellow
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Nautilus spiral (Vila, 2016)
Mathematics in Our World
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Vila’s mistake (Vila, 2016)
Vila admits he made a mistake in the animation for the Nautilus shell. (It is neither a Fibonacci spiral nor a golden spiral.)
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Golden rectangle (Vila, 2016)
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Golden ratio (Vila, 2016)
Mathematics in Our World
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Golden angle (Vila, 2016)
Mathematics in Our World
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Golden angle and arrangement of sunflower seeds (Vila, 2016)
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Fibonacci numbers and arrangement of sunflower seeds (Vila, 2016)
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Fibonacci numbers and arrangement of sunflower seeds (continuation) (Vila, 2016)
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Delaunay triangulation and Voronoi diagram (Vila, 2016)
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Delaunay triangulation and Voronoi diagram (continuation) (Vila, 2016)
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Delaunay triangulation and Voronoi diagram (continuation) (Vila, 2016)
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Delaunay triangulation and Voronoi diagram (continuation) (Vila, 2016)
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Delaunay condition (Vila, 2016; Delaunay triangulation, 2017)
A Delaunay triangulation for a set of points in a plane is a triangulation such that no point is inside the circumcircle of any triangle.
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Dragonfly wings (Vila, 2016)
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Relationship between golden ratio and Fibonacci sequence (Vila, 2016; Golden ratio, 2017)
F0 = 0 F1 = 1 Fn = Fn−1 + Fn−2 for n > 1, n ∈ Z ϕn − (1 − ϕ)n ϕn − (−ϕ)−n √ √ = 5 5 Fn+1 =ϕ lim n→∞ Fn
Fn =
Mathematics in the Modern World The Nature of Mathematics
Mathematics in Our World
Sample exam questions
1. Let ϕ =
1 2
1+
√ 1 5 . Show that ϕ = ϕ2 − 1 = + 1. ϕ
2. Let φ =
1 2
1−
√ 1 5 . Show that φ = φ2 − 1 = + 1. φ
3. Explain why√you √ is defined to be think the golden ratio ϕ = 12 1 + 5 and not φ = 21 1 − 5 .
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Mathematics in the Modern World The Nature of Mathematics
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Mathematics is a useful way to think about nature (Stewart, 1995, p. 19)
Whatever the reasons, mathematics definitely is a useful way to think about nature. What do we want it to tell us about the patterns we observe? There are many answers. We want to understand how they happen; to understand why they happen, which is different; to organize the underlying patterns and regularities in the most satisfying way; to predict how nature will behave; to control nature for our own ends; and to make practical use of what we have learned about our world. Mathematics helps us to do all these things, and often it is indispensable.
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Sample reading assignment Read Stewart (1995) and be ready to answer the following discussion questions. I
Which sentence or paragraph in the book is your favorite? Why?
I
Is there any statement or point of view in the book that you disagree with?
I
How would you summarize each of the nine chapters (in one, two, or three sentences per chapter)?
I
How does Stewart differentiate the external aspects of mathematics from the internal aspects of mathematics?
I
What term does Stewart use to describe his dream of an effective mathematical theory of form and the emergence of pattern?
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Sample assignment Using Stewart (1995) as your reference, write an essay of around 250 to 350 words answering exactly one of the following questions. I I
I I
How did Stewart explain why numbers from the Fibonacci series appear when some features in plants are counted? Leonhard Euler got into an argument with Daniel Bernoulli because their solutions to the one-dimensional wave equation differed. How did Stewart explain the argument’s resolution? How did Stewart use Poincar´e’s concept of a phase space to explain why tides are predictable but weather is not? How did Stewart use coupled oscillator networks to model animal gaits?
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Is math discovered or invented? A TED-Ed Original lesson by Jeff Dekofsky, 2014
(5:11): https://youtu.be/X xR5Kes4Rs
See also http://ed.ted.com/lessons/ is-math-discovered-or-invented-jeff-dekofsky.
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Philosophy of mathematics (Philosophy of Mathematics, 2017)
“Mathematical realism [...] holds that mathematical entities exist independently of the human mind. Thus humans do not invent mathematics, but rather discover it [...].” One form of mathematical realism is Platonism. “Mathematical anti-realism generally holds that mathematical statements have truth-values, but that they do not do so by corresponding to a special realm of immaterial or non-empirical entities.” One form of mathematical anti-realism is formalism.
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Platonism “Mathematical Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging.” (Philosophy of Mathematics, 2017)
(Richard Hamming, 2013)
“Very few of us in our saner moments believe that the particular postulates that some logicians have dreamed up create the numbers—no, most of us believe that the real numbers are simply there and that it has been an interesting, amusing, and important game to try to find a nice set of postulates to account for them.” (Hamming, 1980, p. 85)
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Formalism “Formalism holds that mathematical statements may be thought of as statements about the consequences of certain string manipulation rules.” (Philosophy of Mathematics, 2017)
“Mathematics, according to David Hilbert (1862-), is a game played according to certain simple rules with meaningless marks on paper.” (Stabler, 1935, p. 24) (David Hilbert, 2017)
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Ultrafinitism “[C]onstructivism involves the regulative principle that only mathematical entities which can be explicitly constructed in a certain sense should be admitted to mathematical discourse. [...] Finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps. [...] Ultrafinitism is an even more extreme version of finitism, which rejects not only infinities but finite quantities that cannot feasibly be constructed with available resources.” (Philosophy of Mathematics, 2017) “What is completely meaningless is any kind of infinite, actual or potential. So I deny even the existence of the Peano axiom that every integer has a successor. [...] The phrase ‘for all positive integers’ is meaningless. [...] Similarly, Euclid’s statement: ‘There are infinitely many primes’ is meaningless.” (Zeilberger, 2001, p. 5) (Doron Zeilberger, 2007)
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Constructive and nonconstructive proofs
Theorem There exist irrational numbers a and b such that ab is rational.
Nonconstructive proof. √ Consider
√
2
. If it is rational,√ then the proof is complete. If it is √ 2 √ not rational, then take a = 2 and b = 2 so that ab = 2. 2
Constructive proof (sketch). Take a =
√
2 and b = 2 log2 3.
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Mathematics in the Modern World References
Approximate and true Golden Spirals. (2009, August 29). In Wikimedia Commons, the free media repository. Retrieved April 20, 2017, from https://upload.wikimedia.org/wikipedia/commons/ thumb/a/a5/FakeRealLogSpiral.svg/ 640px-FakeRealLogSpiral.svg.png David Hilbert. (2017, February 14). In Wikimedia Commons, the free media repository. Retrieved April 22, 2017, from https://upload.wikimedia.org/wikipedia/commons/7/79/ Hilbert.jpg Delaunay triangulation. (2017, February 27). In Wikipedia, The Free Encyclopedia. Retrieved April 20, 2017, from https://en.wikipedia.org/w/index.php?title=Delaunay triangulation&oldid=767721079 Doron Zeilberger. (2007, December 13). In Wikimedia Commons, the free media repository. Retrieved April 22, 2017, from https://upload.wikimedia.org/wikipedia/commons/thumb/ e/e9/Doron Zeilberger %28circa 2005%29.jpg/ 365px-Doron Zeilberger %28circa 2005%29.jpg
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Mathematics in the Modern World References
Golden ratio. (2017, March 21). In Wikipedia, The Free Encyclopedia. Retrieved April 20, 2017, from https://en.wikipedia.org/w/ index.php?title=Golden ratio&oldid=771516527 Golden spiral in rectangles. (2008, January 27). In Wikimedia Commons, the free media repository. Retrieved April 19, 2017, from https://upload.wikimedia.org/wikipedia/commons/7/70/ Golden spiral in rectangleflip.png Hamming, R. W. (1980). The unreasonable effectiveness of mathematics. American Mathematical Monthly, 87, 81–90. Philosophy of Mathematics. (2017, February 6). In Wikipedia, The Free Encyclopedia. Retrieved April 22, 2017, from https://en.wikipedia.org/w/index.php?title=Philosophy of mathematics&oldid=763957978 Richard Hamming. (2013, August 7). In Wikimedia Commons, the free media repository. Retrieved April 22, 2017, from https://upload.wikimedia.org/wikipedia/en/0/08/ Richard Hamming.jpg
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Mathematics in the Modern World References
Stabler, E. R. (1935). Interpretation and comparison of three schools of thought in the foundations of mathematics. The Mathematics Teacher, 28, 5–35. Stewart, I. (1995). Nature’s numbers: The unreal reality of mathematics. New York, NY: BasicBooks. Vila, C. (2016, September). Nature by numbers. The theory behind this movie. Retrieved April 18, 2017, from http://www.etereaestudios.com/docs html/nbyn htm/ about index.htm Zeilberger, D. (2001, November 27). “Real” analysis is a degenerate case of discrete analysis. Retrieved April 22, 2017, from http://www.math.rutgers.edu/~zeilberg/mamarim/ mamarimPDF/real.pdf
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