Mathematics In The Modern World - Mathematics In Our
Mathematics in the Modern World1/35Mathematics in the Modern WorldMathematics in Our WorldJoel Reyes Noche, Ph.D.jnoche@gbox.adnu.edu.phDepartment of MathematicsAteneo de Naga UniversityCouncil of Deans and Department Chairs of Colleges of Arts and Sciences Region VConference and Enrichment Sessions on the New General Education CurriculumAteneo de Naga University, Naga CitySeptember 2, 2017–September 3, 2017
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldMathematics in Our WorldMathematics is a useful way to think about nature and our worldLearning outcomesIIdentify patterns in nature and regularities in the world.IArticulate the importance of mathematics in one’s life.IArgue about the nature of mathematics, what it is, how it isexpressed, represented, and used.IExpress appreciation for mathematics as a human endeavor.2/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldNature by NumbersA short movie by Cristóbal Vila, 2010Original (3:44): https://vimeo.com/9953368Alternative soundtrack (4:04): https://vimeo.com/293795213/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldFibonacci sequence(Vila, 2016)The Fibonacci sequence “is an infinite sequence of naturalnumbers where the first value is 0, the next is 1 and, from there,each amount is obtained by adding the previous two.”4/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldFibonacci spiral(Vila, 2016)Circular arcs connect the opposite corners of squares in theFibonacci tiling.5/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our World6/35Golden spiral(Golden spiral in rectangles, 2008)r ϕ2θ/π where θ is in radians and ϕ 1 52is the golden ratio
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldComparison of spirals(Approximate and true Golden Spirals, 2009)Quarter-circles in green, golden spiral in red, overlaps in yellow7/35
Mathematics in the Modern WorldThe Nature of MathematicsNautilus spiral(Vila, 2016)Mathematics in Our World8/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldVila’s mistake(Vila, 2016)Vila admits he made a mistake in the animation for the Nautilusshell. (It is neither a Fibonacci spiral nor a golden spiral.)9/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldGolden rectangle(Vila, 2016)10/35
Mathematics in the Modern WorldThe Nature of MathematicsGolden ratio(Vila, 2016)Mathematics in Our World11/35
Mathematics in the Modern WorldThe Nature of MathematicsGolden angle(Vila, 2016)Mathematics in Our World12/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldGolden angle and arrangement of sunflower seeds(Vila, 2016)13/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldFibonacci numbers and arrangement of sunflower seeds(Vila, 2016)14/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldFibonacci numbers and arrangement of sunflower seeds(continuation) (Vila, 2016)15/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldDelaunay triangulation and Voronoi diagram(Vila, 2016)16/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldDelaunay triangulation and Voronoi diagram(continuation) (Vila, 2016)17/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldDelaunay triangulation and Voronoi diagram(continuation) (Vila, 2016)18/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldDelaunay triangulation and Voronoi diagram(continuation) (Vila, 2016)19/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldDelaunay condition(Vila, 2016; Delaunay triangulation, 2017)A Delaunay triangulation for a set of points in a plane is a triangulationsuch that no point is inside the circumcircle of any triangle.20/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldDragonfly wings(Vila, 2016)21/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our World22/35Relationship between golden ratio and Fibonacci sequence(Vila, 2016; Golden ratio, 2017)F0 0F1 1Fn Fn 1 Fn 2 for n 1, n Zϕn (1 ϕ)nϕn ( ϕ) n 55Fn 1 ϕlimn FnFn
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldSample exam questions1. Let ϕ 121 15 . Show that ϕ ϕ2 1 1.ϕ2. Let φ 121 15 . Show that φ φ2 1 1.φ3. Explain why you is defined to be think the golden ratioϕ 12 1 5 and not φ 21 1 5 .23/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldMathematics is a useful way to think about nature(Stewart, 1995, p. 19)Whatever the reasons, mathematics definitely is auseful way to think about nature. What do we want it totell us about the patterns we observe? There are manyanswers. We want to understand how they happen; tounderstand why they happen, which is different; toorganize the underlying patterns and regularities in themost satisfying way; to predict how nature will behave; tocontrol nature for our own ends; and to make practicaluse of what we have learned about our world.Mathematics helps us to do all these things, and often itis indispensable.24/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldSample reading assignmentRead Stewart (1995) and be ready to answer the followingdiscussion questions.IWhich sentence or paragraph in the book is your favorite?Why?IIs there any statement or point of view in the book that youdisagree with?IHow would you summarize each of the nine chapters (in one,two, or three sentences per chapter)?IHow does Stewart differentiate the external aspects ofmathematics from the internal aspects of mathematics?IWhat term does Stewart use to describe his dream of aneffective mathematical theory of form and the emergence ofpattern?25/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldSample assignmentUsing Stewart (1995) as your reference, write an essay of around250 to 350 words answering exactly one of the following questions.IIIIHow did Stewart explain why numbers from the Fibonacciseries appear when some features in plants are counted?Leonhard Euler got into an argument with Daniel Bernoullibecause their solutions to the one-dimensional wave equationdiffered. How did Stewart explain the argument’s resolution?How did Stewart use Poincaré’s concept of a phase space toexplain why tides are predictable but weather is not?How did Stewart use coupled oscillator networks to modelanimal gaits?26/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldIs math discovered or invented?A TED-Ed Original lesson by Jeff Dekofsky, 2014(5:11): https://youtu.be/X xR5Kes4RsSee also vented-jeff-dekofsky.27/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldPhilosophy of mathematics(Philosophy of Mathematics, 2017)“Mathematical realism [.] holds that mathematical entities existindependently of the human mind. Thus humans do not inventmathematics, but rather discover it [.].” One form ofmathematical realism is Platonism.“Mathematical anti-realism generally holds that mathematicalstatements have truth-values, but that they do not do so bycorresponding to a special realm of immaterial or non-empiricalentities.” One form of mathematical anti-realism is formalism.28/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldPlatonism“Mathematical Platonism is the form of realism that suggests thatmathematical entities are abstract, have no spatiotemporal orcausal properties, and are eternal and unchanging.” (Philosophy ofMathematics, 2017)(Richard Hamming, 2013)“Very few of us in our saner moments believe thatthe particular postulates that some logicians havedreamed up create the numbers—no, most of usbelieve that the real numbers are simply there andthat it has been an interesting, amusing, and important game to try to find a nice set of postulatesto account for them.” (Hamming, 1980, p. 85)29/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldFormalism“Formalism holds that mathematical statements may be thoughtof as statements about the consequences of certain stringmanipulation rules.” (Philosophy of Mathematics, 2017)“Mathematics, according to David Hilbert(1862-), is a game played according to certainsimple rules with meaningless marks on paper.”(Stabler, 1935, p. 24)(David Hilbert, 2017)30/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldUltrafinitism“[C]onstructivism involves the regulative principle that only mathematicalentities which can be explicitly constructed in a certain sense should beadmitted to mathematical discourse. [.]Finitism is an extreme form of constructivism, according to which amathematical object does not exist unless it can be constructed from naturalnumbers in a finite number of steps. [.]Ultrafinitism is an even more extreme version of finitism, which rejects not onlyinfinities but finite quantities that cannot feasibly be constructed with availableresources.” (Philosophy of Mathematics, 2017)“What is completely meaningless is any kind of infinite,actual or potential. So I deny even the existence of thePeano axiom that every integer has a successor. [.]The phrase ‘for all positive integers’ is meaningless. [.]Similarly, Euclid’s statement: ‘There are infinitely manyprimes’ is meaningless.” (Zeilberger, 2001, p. 5)(Doron Zeilberger, 2007)31/35
Mathematics in the Modern WorldThe Nature of MathematicsMathematics in Our WorldConstructive and nonconstructive proofsTheoremThere exist irrational numbers a and b such that ab is rational.Nonconstructiveproof. 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 2.2Constructive proof (sketch).Take a 2 and b 2 log2 3.32/35
Mathematics in the Modern WorldReferencesApproximate and true Golden Spirals. (2009, August 29). In WikimediaCommons, the free media repository. Retrieved April 20, 2017,from ral.svg.pngDavid Hilbert. (2017, February 14). In Wikimedia Commons, the freemedia repository. Retrieved April 22, 2017, /7/79/Hilbert.jpgDelaunay triangulation. (2017, February 27). In Wikipedia, The FreeEncyclopedia. Retrieved April 20, 2017, fromhttps://en.wikipedia.org/w/index.php?title Delaunaytriangulation&oldid 767721079Doron Zeilberger. (2007, December 13). In Wikimedia Commons, thefree media repository. Retrieved April 22, 2017, /thumb/e/e9/Doron Zeilberger %28circa 2005%29.jpg/365px-Doron Zeilberger %28circa 2005%29.jpg33/35
Mathematics in the Modern WorldReferencesGolden 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 771516527Golden spiral in rectangles. (2008, January 27). In Wikimedia Commons,the free media repository. Retrieved April 19, 2017, /7/70/Golden spiral in rectangleflip.pngHamming, R. W. (1980). The unreasonable effectiveness ofmathematics. American Mathematical Monthly, 87, 81–90.Philosophy of Mathematics. (2017, February 6). In Wikipedia, The FreeEncyclopedia. Retrieved April 22, 2017, fromhttps://en.wikipedia.org/w/index.php?title Philosophyof mathematics&oldid 763957978Richard Hamming. (2013, August 7). In Wikimedia Commons, the freemedia repository. Retrieved April 22, 2017, /Richard Hamming.jpg34/35
Mathematics in the Modern WorldReferencesStabler, E. R. (1935). Interpretation and comparison of three schools ofthought in the foundations of mathematics. The MathematicsTeacher, 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 thismovie. Retrieved April 18, 2017, fromhttp://www.etereaestudios.com/docs html/nbyn htm/about index.htmZeilberger, D. (2001, November 27). “Real” analysis is a degenerate caseof discrete analysis. Retrieved April 22, 2017, fromhttp://www.math.rutgers.edu/ zeilberg/mamarim/mamarimPDF/real.pdf35/35
The Nature of Mathematics Mathematics in Our World 2/35 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 natu
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