READING GUIDES

Prologue and Chapter 1

  The purpose of these reading guides is to help you while reading the chapters in Devlin.  You should of course read the entire chapter carefully, but this guide may assist you in seeing what issues and questions you should especially consider during your reading.  In addition, it can help you with what parts you might focus on more and what parts you might focus on less.  You’ll note that I don’t specifically tell you not to focus on any section this time (I will for some subsequent chapters), so don’t simply ignore any of the reading even if I don’t explicit mention it.

Prologue

·        According to Devlin, math was the study of what, when?  And how did it continue to change over time?

·        Why is abstraction and abstract notation important in mathematics?  And how does this impact us all from “access” to parts of math?

·        Note the parallels Devlin makes between math and music, literature and art.  What do you or don’t you agree with?

Chapter 1

·        You should read the entire chapter fairly closely with the understanding that there may be a few pieces that are particularly challenging, depending on your mathematical background.  For those parts, get as much from the reading as you can.

·        Historical development of the concept of number.

·        Greek mathematics, especially geometry and “proof.”

·        Pythagoreans and the discovery of irrational numbers.

·        Some of the ideas of prime numbers.  In particular, try to understand Euler’s proof that there exists an infinite number of primes (p. 20).  Also, think about the various theorems and conjectures that are stated in the book about prime numbers.

·        Fundamentals of modular arithmetic.

·        Use of primes in encryption codes.

·        Basic ideas and history of Fermat’s Last Theorem.  We will probably watch a video on the ultimate proof of this theorem late in the semester.

Chapter 2

·        You should read the entire chapter fairly closely with the understanding that there may be a few pieces that are particularly challenging, depending on your mathematical background.  For those parts, get as much from the reading as you can.  I highlight below those sections we’ll discuss more fully in class, and those we will not deal with much at all.

·        Greek Logic & Venn Diagrams – we will deal with these topics extensively in class, so you should read these sections carefully a few times.

·        Boole’s Logic – this is an algebraic analysis of Aristotelian (Greek) Logic, but we will not do much with this in class.

·        Propositional Logic & Truth Tables – we will deal with these topics extensively in class, so read these sections carefully a few times.

·        We won’t deal with much of the rest of the Chapter explicitly in class.  But while you are doing the reading of the rest of the Chapter, try to focus on the following:

• Get a sense of the key players and the history and development of reasoning and proof in mathematics through history
• Get a sense of what the axiomatic method is
• For a possible future discussion, it may be helpful to have a sense of the construction and notation of sets (see the right hand column in the box on p. 57)
• Try to understand as best you can the argument of Russell’s paradox (see end of p. 58 through p. 59)
• Try to understand the impact Russell’s paradox and Gödel’s Theorem had on our beliefs in the axiomatic method
• Study of language is one “real-world” application of this pretty abstract notion of an axiomatic system in mathematics

Chapter 3

·        There are many interesting paradoxes related to infinity.  Get a good feeling of the one presented on Achilles and the tortoise.

·        Understand what an infinite series is – as the sum of an infinite number of terms.

·        There are two MAIN areas of focus in Calculus – differentiation and integration.  Ideas of differentiation are on pages 81-94.  Get a sense of what the derivative is about as a “rate of change” and how it might be used.  Integration is often seen as “accumulation” and is dealt with on pages 94-97.  Although a lot of this reading is a bit technical from time to time, get what you can about what these two main concepts in Calculus are and how they might be used. 

·        Read carefully the historical information about the development of Calculus and two key figures in its development (pages 84-85).

·        For those of you interested in music, Fourier Analysis (page 89) has important applications to sound (and other) wave forms.

·        Chapter 1 focused primarily on the natural (or counting) numbers.  There are lots of other important sets of numbers, including the real numbers (pages 97-98) and complex numbers (pages 98-101).

·        Recall we talked about the Fundamental Theorem of Arithmetic in Chapter 1.  In this Chapter, they present two other “fundamental theorems”.  What are they?  What do they basically say?  [Hint:  See pages 97 & 100.]

·        You can simply skim the last section on “Analytic Number Theory”.  

Chapter 4

·        You should read most of the chapter fairly closely and understand the general ideas presented in it.

·        Carefully read the sections on Euclid, his attempt to carefully axiomitize “plane” geometry and the work he did in The Elements.

·        We will spend some time in class on the Platonic solids (pp. 112-115), so read these ideas carefully.

·        Get a sense of what the three “Classic Problems” are about.

·        You may never have heard that there is any other geometry than the standard Euclidean (“plane”) geometry you learned about in junior or senior high school.  So read the section on Non-Euclidean Geometry carefully and get a sense of how these other geometries are developed and generally what they are.  We will spend some time talking about these in class.

·        Get a good sense of what Projective Geometry is about, as we will spend time talking about this in class.

Chapter 5

·        What examples of transformations does Devlin give (p. 146)?  Can you explain each of the five in your own words?

·        Do you see that if you rotate the snowflake on p. 146 by 60, 120, 180, or any multiple of 60 degrees, it will always look the same?  This is why we consider these to be rotational symmetries of the snowflake.

·        If you do the following to the triangle on p. 149, it will look exactly the same:

1.      Do nothing (Identity transformation)

2.      Rotate the triangle by 120 degrees (obviously an example of a rotation)

3.      Rotate the triangle by 240 degrees

4.      Flip the triangle around the line X (this is an example of a reflection)

5.      Flip the triangle around the line Y

6.      Flip the triangle around the line Z

·        What one transformation is equivalent to doing the following two transformations in order:  Do (2) first (Devlin calls this transformation v) then (4) next (Devlin calls this transformation x).

·        Who was Galois and what are 2 or 3 interesting facts about him?

·        We will not cover Sphere Packing in any detail in class.  Skim (which means you DO need to have read it) this section and get a general idea of the discussion, but you will not be responsible for the details of this section.

·        Why was Kepler interested in sphere packing?

·        What is the connection between pomegranates, snowflakes, honeycombs, shipping oranges, and error correcting codes?

·        What is the general idea presented on pp. 163-164?

·        What is the general idea of the last section beginning on page 165?

·        Which regular polygons can tile the plane completely (p. 165)?

·        Which regular polyhedron can fill all of 3-D space (p. 170)?

·        Who cares about symmetry, sphere packing, wallpaper patterns, or tiling?  What are some “practical examples” given by Devlin regarding these mathematical topics?