Review of Introduction to Mathematical Thinking by Keith Devlin

This is a review of the book Introduction to Mathematical Thinking by Keith Devlin.  This is a paperback book of length x + 92, i.e. 102 pages.  This is a paperback that can fit into a large boxy like extra pocket on some men’s shorts.  It fits into mine, I just checked.

This is a charming and concise book that gives an introduction to proofs and logic.   You can read a page or two here or there on the train, coffee shop or while taking a break during a walk.  For a moment or two you can look at some topic and get an explanation of it.  You can then do some exercises on it.

If one compares  Devlin to other introductions to mathematical proof they tend to be longer and bulkier.  So you couldn’t haul them around this way so easily.

If you don’t understand something in this book, there may be alternative and possibly better explanations elsewhere.   One can compare some of the explanations in the text with those in these web pages, the associated twitter account and in the series Pre-Algebra New Math Done Right Peano Axioms and now Geometry of Addition.

For mathematical induction, these works and these webpages may help those struggling with the idea.  The Devlin approach to mathematical induction starts with the sum of the first n integers.  This is done in some detail.

Nonetheless, rewriting the steps using labels Devlin introduces, A(1), A(n) are left for the reader without the answer.  Some readers who are hung up on this will find this a real block.  This is a problem with most books trying to explain proofs. They still give you a lot to do at points and if you can’t do it, you are blocked.

Truth tables are presented in exercises and some students may find this a block. They can go elsewhere for explanations and then come back to Devlin’s book.  No doubt, he intends his videos and other materials to bridge this gap in the Coursera course.

In my texts, I go to elaborate lengths to clearly identify each case in mathematical induction proofs and also use a technique I call BaHIC.  The standard approaches usually skip over articulating the conclusion step.  But I find that labeling this as a step and summarizing the conclusion helps clarify the logic.

  1. Base Step
  2. Hypothesis Step
  3. Induction Step
  4. Conclusion Step.

A more elaborate version is

  1. Set Up Step
  2. Base Step
  3. Hypothesis Step
  4. Induction Step
  5. Conclusion Step.
  1. In this approach, we set up a set M of natural numbers for which the formula or theorem is true.
  2. We show that 0 is in M or 1 is in M as appropriate.
  3. We assume that n is in M.
  4. We show that n is in M implies n’ is in M.
  5. We conclude that all n natural numbers are in M.

Here n’ = n+1 is the successor of n.

These proofs are done with this structure and labeling over and over in the texts I have written.

Lance and Rips indicate that pre-service math teachers find the algebra of mathematical induction proofs the hardest part to learning to do the proofs.  Devlin’s book has this defect.  (I discuss the Lance Rips work more in my book on Peano Axioms.)

In my texts, by contrast, I have no proofs with algebraic formulas above linear.  Despite this I have many examples of proofs. These include proving that addition is associative and commutative.  But it also includes simpler proofs that go from right additive identities to left ones.

Right Additive Identities are

x+0 = x

x+y’ = (x+y)’.

Left ones are

0+y = y

x’+y = (x+y)’

Going from right to left are induction proofs.  They don’t involve any complicated algebraic formulas.  Thus they are ideal for teaching mathematical induction. Those are in my Peano Axioms book, the first volume in my series.

Devlin’s approach is not really innovative, but it is convenient.  So for those for whom it doesn’t work, they have to look elsewhere.  This is the MOOCPIT version of MOOCs.

In the proof of the sum of first n integers, Devlin does not use the n’ notation, thus making the proof clumsier than it needs to be.  Moreover, at the induction step he is very clumsy in requiring unneeded algebra.

In contrast, one can do it more efficiently with the n’ notation.

Lemma sum 1 to n = nn’/2

Set up Step

Let M be the set of n such that the formula is true.  M is restricted to natural numbers, i.e. 0, 1,2,… are the candidate numbers.  We try to show they are the numbers that work.

Base Step


0 = 0 * 0’/2 = 0

Hypothesis Step

sum 1 to n = nn’/2

Induction Step

Add n’ to each side

(sum 1 to n) + n’ = nn’/2 + n’

(sum 1 to n) + n’ = sum 1 to n’

nn’/2 + n’ = nn’/2 + 2n’/2 = (nn’+2n’)/2 = (n+2)n’/2

= (n’+1)n’/2 = (n’)’*n’/2

This is the induction step.

Conclusion Step

M contains 0 by the Base Step.  If M contains n, then it contains n’ by the induction step we proved.  By Peano Axiom 5, the Principle of Mathematical Induction, M contains all natural numbers.

The conclusion is not something you can derive from ordinary logic of finite cases.  It is an Axiom of Proof.  This was not made as clear as it could have been by Devlin.

A way to make such proofs easier I call the Principle of Green Induction.

Green Set up.  The number n is called green if the formula

is true for n.

Base Case.  Show that n=0 is green.

Hypothesis Step.  Assume n is green.

Induction Step.  Show that if n is green then n’ is green.

Conclusion Step All n are green.

This language makes it easier to remember and to talk about a mathematical induction proof.  I discussed this method earlier in these webpages and in some tweets.

The books I have written on Peano Axioms are an extensive set of examples of mathematical induction without any quadratic or higher algebra formulas.   So the student who wants practice or doesn’t get it can follow these.  These are also the most extensive and slow paced introduction to the Peano Axioms.

=First Book

Buy Pre-Algebra New Math Done Right Peano Axioms at this vendor

Buy Pre-Algebra New Math Done Right Peano Axioms at an alternative vendor

Pre-Algebra New Math Done Right Peano Axioms is a textbook on order and addition of natural numbers. The natural numbers are 0,1,2, etc. They are sometimes called whole numbers.

The Peano Axioms are basic statements about the natural numbers. For example they state that each natural number is followed by a unique natural number. One of the Peano Axioms is the axiom of induction, which is a method of proof and can be used with suitable care as a method of definition of sets and functions.

Using the Peano Axioms, addition is defined as a function with two inputs. The definition of addition is a recursive definition. It is also called definition by
induction. This requires special care to set up.

From the Peano Axioms we can prove that addition is commutative and associative. These proofs rely on mathematical induction. The idea of proving the commutative and associative laws of addition using mathematical induction is due to Grassmann in 1861. This was developed further by Dedekind in 1888 and Peano in 1889. Since then many texts and university lecture notes have explained these axioms and done the proofs of the addition laws and those of multiplication.

These treatments usually go very fast and are considered college level. This text has a slow pace on proofs by mathematical induction and on the proofs of the laws of addition using mathematical induction.

The book has 178 Examples, 463 Exercises, 98 definitions, 45 Lemmas with proofs, references to over 96 webpages, 58 chapters, and is 391 pages when formatted as pdf with Latex.

Typical treatments of the Peano Axioms cover the same material as in the book in about 20 pages or less. They usaully have few examples and the few exercises are as difficult as the theory.

In contrast, this text has many examples too trivial for the current texts on the Peano Axioms to cover. Building at a very slow pace with many numerical examples, the reader is taken through the Peano Axioms themselves, simple consequences, order of natural numbers, and simple identities used to prove the properties of addition. This build up includes many very simple proofs by mathematical induction.

There are no quadratic or higher algebraic formulas in the book. Complicated algebra formulas are the main stumbling block to learning mathematical induction. None are in the book, yet there are many worked out proofs of simple relationship using mathematical induction and simple problems for students to do.

The book culminates with proofs of the commutative and associative laws of addition. These are done in detail. These are the most detailed explanations of these proofs available.

The goal is to make the Peano Axioms and their use to define addition and prove its basic laws available to a much wider audience and at a lower than college level. This is intended to be an easy way for students, parents,
teachers, educators, and others to learn this basic part of math.

New Math in the 1960s did not focus in this way on defining and proving the properties of addition using the Peano Axioms. This was a mistake. This text rectifies that mistake by focusing on defining addition and proving its properties.

The book also covers the basics of set theory. It covers some parts of order of natural numbers. These are often not given much attention in treatments of the Peano Axioms. They are however of critical importance to understand the nature of addition and counting. Place value addition is reviewed using letters for each digit and carry.

This text in gives versions of the Peano Axioms in terms of number lines. This leads to a definition of addition using two number lines. This is valuable for students, teachers and parents to both learn this approach and to then be able to better explain addition at lower grade levels. Pairs of bead strings are done the same way.

==Second Book which is in 10 parts, 4 published and 6 being processed in next couple days.

Pre-Algebra New Math Done Right Geometry of Addition

This is a continuation from:

Pre-Algebra New Math Done Right Peano Axioms

This book is on the geometry of Peano Axiom addition. This book contains a new definition of addition in the Peano Axiom framework. This is called pitch line addition. Let x’ = x+1 and ‘y = y-1. The new definition is

x+0 = x

x’ + ‘y = x+y

Lines of constant x+y are defined prior to defining addition. These are called pitch lines. Their properties are proven. This is new material to math.

The book contains no pdf, jpeg, or pngs. It is pure html.

Entire book is approximately 240 pages of material when formatted as a pdf. The entire book contains a total of 270 exercises, 119 lemmas, 30 note fields, 6 web url note fields, 161 examples, and 54 definitions. This is a total of 640 fields. This is in addition to the text. It is organized into 33 chapters. Fields and chapters are renumbered within each part of the overall work.

The first volume Peano Axioms has 178 Examples, 463 Exercises, 98 definitions, 45 Lemmas with proofs, references to over 96 web pages, 58 chapters, and is 391 pages when formatted as pdf with Latex. 178 examples + 463 exercises + 98 definitions + 45 lemmas + 96 webpages equals 880. Added to the 640 fields above is 1520 fields. This is in addition to the text not in such fields.

Total examples is 178 + 161 = 339 examples. 463 + 270 = 733 Exercises. 54 + 98 = 152 definitions. 119+45 = 164 lemmas. 96+6 = 102 web page references. 58+33 = 91 chapters. 391 + 240 = 631 pages.

There are no quadratic or higher algebra formulas. Students complain about complicated algebra in learning how to do proofs by mathematical induction. This material is thus the ideal for learning mathematical induction proofs.

These totals compare to treatments of 20 pages on university web pages on the Peano Axioms and Peano Addition. Some may contain more, but usually these go beyond just the Peano Axioms and Peano Addition. Those mostly are repeats of each other. Those tend to be theoretical and without starter examples and starter exercises. They also tend to duplicate each other. Thus in terms of distinct and unique material on the Peano Axioms and Peano Addition, the bulk of it is apparently contained in these materials.

The book is divided into ten parts. These are not of equal length. The pricing approximately reflects the length of each part. So short parts are priced at the minimum allowed. Overall the sum of the prices of the parts is fair for the sum of the lengths.

Ten parts of book

Geometry of Addition Introduction

Addition bead shift definition and diagrams

Pitch line drills

Addition geometry fundamentals

Lattice square induction

Pitch line lemmas

Y-intercept function

Pitch Line induction

Addition Commutative

Addition Associative

=Parts Posted at e-vendor

Part 1 Introduction

Part 2 Bead Shift Diagrams

Part 3 Pitch Line Drills

Part 4 Addition Geometry Core


About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
This entry was posted in Devlin Mathematical Thinking Coursera, Easy Algebra Math Induction Proofs, Keith Devlin Stanford, Mathematical Induction, Mathematical Induction BaHIC, Mathematical Thinking, MOOC, MOOCPIT. Bookmark the permalink.

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