Inductive Discovery in Number Theory

If one has no specific knowledge about a mathematical question, the only way to proceed is inductively, that is, by examining several example situations and trying to guess at what is going on. What is the nature of this inductive process?

How can we collect relevant observations, examine and compare them, notice some tentative and fragmented regularities, and after a series of hesitations and blunders eventually succeed in " combining the scattered details into an apparently meaningful whole"?

In trying to evaluate a proposed hypothesis to cohere our inductive work, how do the number of special cases verified, the preciseness of our hypothesis, and the presence or absence of rival conjectures lend support or suspicion upon our guesses?

How can questions in number theory shed light on how to work at and solve mathematical questions in general?

We will discuss the nature of inductive discovery in number theory while also reviewing and discussing participants efforts to solve 8 example problems about the sum of four squares. Our objective is to practice and thereby to more deeply understand the inductive process which is at the heart of exploring any mathematics question.
This topic will focus on a short 15 page passage (pp. 59-75 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n77/mode/2up)) and 9 example problems (examples #7-15 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n91/mode/2up)) in Chapter 4 "Induction in the Theory of Numbers" in George Pólya's 1954 classic Mathematics and Plausible Reasoning: Induction and Analogy in Mathematics (you can read the book for free on-line at https://archive.org/details/Induction_And_Analogy_In_Mathematics_1_ ; the "Read On-Line" format is excellent or you can download a PDF (https://archive.org/download/Induction_And_Analogy_In_Mathematics_1_/Induction_And_Analogy_In_Mathematics_1_.pdf) or if you click the "i" widget you can get copies in plain text, daisy, epub, or kindle formats).

http://press.princeton.edu/images/j1735.gif

Although the reading for this session is light (only 13 pages). The example problems are challenging and will require substantial time and effort to solve. Solving the example problems is not necessary to participate, but participants should spend at least a few hours trying to solve each of the 8 problems.

Preparation: Please prepare for the discussion with the following two steps:

• Read the 15 page passage "Induction in the Theory of Numbers" on pp. 59-75 which can be found through this link (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n77/mode/2up).

• Spend some time trying to solve the nine (9) example problems #7-15 on pp. 72-73 which can be found through this link (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n91/mode/2up) (Note: Table II is on pp. 74-75 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n93/mode/2up)). Unless you already thoroughly understand examples #1-2 on pp. 70-71 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n89/mode/2up), you will find that working on them and perhaps checking their answers in the back of the book is important background for understanding problems #7-15.

Note: since some of the problems are challenging, it is recommended that you plan to attempt the harder ones repeatedly over the course of several days. If you do get stuck on a problem, move on to the other problems in the book: the problems are all related and one of them may give you some ideas about how to proceed. Also feel free to post questions to the event page so we can discuss anything you are struggling with before the meetup.

To get a printable copy of the text, you may want to download the PDF of the whole book (https://archive.org/download/Induction_And_Analogy_In_Mathematics_1_/Induction_And_Analogy_In_Mathematics_1_.pdf).

Agenda.

• Detailed discussion of pp. 63-68 sections 4.3-4.6 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n81/mode/2up) about 4u and the sum of four squares.

• Detailed discussion of pp. 68-70 sections 4.7-4.8 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n87/mode/2up) about the nature of inductive discovery.

• Detailed discussion of your attempts at solving the 9 example problems #7-15 on pp. 70-71 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n91/mode/2up). We will help anyone who got stuck and review your successes as well. Bring any questions you may have.

Interspersed with discussing the text and the problems or with any remaining time at the end we will also explore the following broader questions:

• What insights did working on these questions in the theory of numbers give into the project of Pólya's book to explore the principles of plausible reasoning and (scientific) induction (see sections 1-4 in the Preface (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n7/mode/2up) and Chapter 1 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n21/mode/2up))?

• In reading the text and working out the problems in Chapter 4 did you find supporting evidence for Pólya's claim on p. 7 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n25/mode/2up) that "A conjectural general statement becomes more credible if it is verified in a new particular case"? Did you find supporting evidence for his claim on p. 22 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n41/mode/2up) that "A conjecture becomes more credible by the verification of any new consequence"? Or his other claim on p. 22 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n41/mode/2up) "A conjecture becomes more credible if an analogous conjecture becomes more credible"?

• What if anything did you learn about the importance and the interplay of the roles of generalization, specialization and analogy (see Chapter 2 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n31/mode/2up)) in reading the text, working out the example problems, and/or reflecting on your efforts?

• What if anything did you learn about plausible reasoning by reading the text, working out the example problems, and/or reflecting on your efforts?

• What if anything did you learn about better understanding the art of problem-solving in reading the text, working out the example problems, and/or reflecting on your efforts?

• What if anything did you learn about the challenges in identifying relationships among integers and sums of their squares in reading the text, working out the example problems, and/or reflecting on your efforts?

https://upload.wikimedia.org/wikipedia/commons/f/f3/Straight_line.png

Newcomers to Pólya's book can get some useful but not required background by reading an additional 35 pages: the Preface (pp. v-x (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n7/mode/2up), pay particular attention to sections 1-4), the very important Hints to the Reader (pp. xi-xii (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n13/mode/2up)), Chapter 1 (pp. 3-11 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n13/mode/2up), pay special attention to the examples 9-14 on pp. 9-11 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n27/mode/2up)), Chapter 2 (pp. 12-30 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n31/mode/2up), pay special attention to examples 5, 7, 10, 11, 18-20, and 46). You should consider and compare the end of section 1.3 on p. 7 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n25/mode/2up) and section 2.7 on pp. 21-22 (https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n39/mode/2up). Reading this material will help you understand and assess Pólya's project to explore plausible reasoning, the nature of induction, and their relationship to mathematical problem-solving. Understanding the perspective of the book may help you in working on the example problems. In addition some of our discussion questions will explore this broader context of the book.

https://upload.wikimedia.org/wikipedia/commons/f/f3/Straight_line.png

This is the second event in a three part exploration of the theory of numbers with Pólya. The first event was on 23 July "Exploring The Theory of Numbers with Pólya" (https://www.meetup.com/MathCounts/events/vwqvtlyvkbfc/). The final event will be on 24 September "Exploring Integer Decompositions into Sums of Squares" (https://www.meetup.com/MathCounts/events/vwqvtlyvmbgc/).