Mathematics And The World's Most Famous Maths Problem: The Riemann Hypothesis

By Jeffrey Todd

DESCRIPTION OF BOOK

This book takes an analytical look at mathematics and provides some important suggestions. It also points towards a better way of thinking. Hopefully, this will induce the reader to be a better thinker, at least mathematically.

The book also ambitiously tackles the question of the Riemann hypothesis which represents the most important and deepest problem in mathematics, providing the possible solutions to this important problem. Importantly, the book aims to guide the reader towards greater clarity of thought and a greater understanding of the Riemann hypothesis by carefully explaining what the hypothesis is about. As there is evidently a lot of mystery surrounding the Riemann hypothesis, the book should help to clear up this mystery.

 

 

 

 

JEFFREY TODD, PH.D., is a professor and an author of many books. He is also an expert on the prime numbers and has published several important papers on the prime numbers in mathematics journals.

• Language: English
• Publisher: Jeffrey Todd
• Release date: Oct 9, 2015
• ISBN: 9781519996619

Jeffrey Todd

JEFFREY TODD, PH.D., is a professor and an author of many books. He is also an expert on the prime numbers and has published several important papers on the prime numbers in mathematics journals.

Book preview

MATHEMATICS AND THE WORLD’S MOST FAMOUS MATHS PROBLEM: THE RIEMANN HYPOTHESIS

––––––––

JEFFREY TODD

Copyright © 2015 by Jeffrey Todd

––––––––

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publishers. 

MATHEMATICS AND THE WORLD’S MOST FAMOUS MATHS PROBLEM: THE RIEMANN HYPOTHESIS 

––––––––

PREFACE

Mathematics is an exact science which is much utilized in human affairs, e.g., in engineering, physics, finance, business, et al. However, on account of its somewhat abstract nature making it relatively difficult to grasp one either loves mathematics (if one is prepared to face its challenges) or loathes it (if one is fearful of facing its challenges). Whether one loathes it or not one cannot live without mathematics. It is therefore important to accept it on the best terms possible.

This book analyzes the various aspects of mathematical reasoning, even providing suggestions for improvement, and discusses the most famous mathematics problem in the world, viz., the Riemann hypothesis, offering some important solutions.

Jeffrey Todd, PhD

CONTENTS

1.  Introduction

2.  The Problem With Mathematics

3.  The Basics Of Mathematics

4.  The Logical Aggregation Of Symbols And Numbers

5.  On Mathematical Manipulation

6.   Axioms In Mathematics

7.   The Problem Of Infinity

8.  What Is Mathematical Reasoning About?

9.   Indepth Look At Mathematical Reasoning

10. Further Look At Mathematical Reasoning

11. The Achilles Heel Of Mathematics?

12. Inconsistency In Mathematics

13. More On Inconsistency In Mathematics

14. Why The Riemann Hypothesis?

15. Solving The Riemann Hypothesis

16. More On Solving The Riemann Hypothesis

17. Afterthought On The Riemann Hypothesis

18. Conclusion

Bibliography

1  INTRODUCTION

Mathematics is the world of the abstract. It concerns pure numbers, ideal forms, theorems and algebraic formulas. However, the roots of mathematics and numbers lie in the real world which is a combination of people, animals, plants and other aspects of nature. Numbers were first conceived in relation to things and for a long time concerned the concrete. The earlier civilizations conceived of mathematics as a tool for dealing with real objects and their quantities, such as weighing grain, measuring fields and counting animals. The ancient Babylonians, Egyptians, Indians and other pre-Classical peoples however managed to build the foundations of arithmetic, geometry, algebra and number theory.

The mathematical genius Carl Friedrich Gauss called mathematics the queen of the sciences, which many of the greatest minds in history have agreed with. Scientists have always regarded mathematics as the purest and most profound form of truth and beauty in the universe regardless of whether they approached the world from a religious or scientific point of view. The ancient Greeks regarded mathematics as the foundation of the cosmos. John Dee, the Elizabethan mathematician and magician, thought of mathematics as the ultimate tool of the Creator employed in the distinct creation of all creatures by order, and most absolute number, brought from nothing to the formality of their being and state. Galileo Galilei, the great Italian mathematician and scientist, opined that the book (of the universe) cannot be understood unless one first learns to comprehend the language ... in which it is written. It is written in the language of mathematics ....

Modern mathematics covers at least 30 different fields, from the common such as geometry and algebra to the esoteric such as combinatorics, which is the field of mathematics concerned with the problems of selection, arrangement and order, and topology, which is the field of mathematics concerned with continuity and sometimes called rubber sheet geometry. Mathematics has now become highly specialized and much complex, with large portions of it hopelessly out of the reach of the non-specialist. It is evident that a lot of people consider themselves much ignorant of mathematics, having forgotten most of it after leaving school, while a significant number suffers from an active fear of mathematics. But mathematics is around us all the time, whether we like it or not, manifesting itself in activities such as dividing a cake into equal portions, counting the change, looking at the clock or comparing sizes.

In the nineteenth and twentieth centuries, mathematics entered strange new fields such as statistics, game theory, computer mathematics and chaos theory. Mathematicians have also since come to grips with the concept of infinity. Mathematics has laid the foundation for the computer revolution, with computers in turn helping discover the strange world of chaos where nothing can be determined though nothing is random.

Mathematics can be a very challenging subject, with many professional mathematicians, as well as amateur mathematicians, trying to crack the many unsolved problems, the most important among them being evidently the Riemann hypothesis, which is a problem relating to the distribution of the prime numbers. Why is the Riemann hypothesis important? Having a better understanding of the way the prime numbers are distributed will make the prediction or forecast of their appearance a surer thing. It should be noted that though the appearance of the prime numbers seems random, when viewed en masse or on a very large scale their appearance is orderly, which explains why the prime counting function, via the prime number theorem, is able to provide a more and more accurate estimate of the number of primes less than a given quantity the further and further down the number line we go. The last few chapters in the book will provide more details, including solutions, on the very important Riemann hypothesis.    

2  THE PROBLEM WITH MATHEMATICS

Mathematics is considered the most objective science compared to the other sciences such as physics, biology or economics. But it is not without its drawbacks for those who are its practitioners, i.e., mathematicians, being human, could interpret mathematical ideas in their own personal, subjective ways. There could be disputes about mathematical ideas wherein egos could be at play. In other words, there is also disagreement or conflict in mathematics, like any other human activity.

In mathematics, which is regarded as an exact science where logic or reasoning is concerned, there is a need to convince others that a mathematical statement is true or valid by utilizing other proven statements (lemmas) or evident assumptions (axioms) in the reasoning process which may run to hundreds of pages. The reasoning and explanation has to be water-tight in order for the statement to be accepted as a theorem, a proven fact. The whole reasoning process or proof (as it is called in mathematical terms) must make the truth or validity of the mathematical statement evident or obvious to everyone, i.e., the explanation has to be clear and convincing, leaving no room for doubt or uncertainty. Such a laborious way to achieve certainty for a mathematical statement which may just require one sentence to describe reflects the limitation of the human intellect, i.e., the human brain is inefficient. A super-intelligent alien race may grasp the truth or validity of a mathematical statement (or any other statements) right away without the need of reasoning or proof perhaps due to a superlative intuition, not forgetting that logic or reasoning itself depends on intuition, the feeling of correctness, of certainty. The following example should make this point clear. Very intelligent students pick up things very quickly without the need of the teacher explaining much. For the dull students the teacher has to go more slowly, explain many times, use many examples or analogies, and yet these students may not understand the teacher. The point is that if mathematical reasoning is convoluted and hard or even impossible to comprehend how is the mathematician able to meet the important objective of convincing others of his reasoning? Shouldn’t mathematical reasoning be simpler, less convoluted and easily comprehended in order to meet this important objective?  In fact, clear and easily comprehended