The Theory of Mental Arithmetic

By Doctor Stephen Taylor

This book introduces the fundamental concepts behind a new form of calculation, "Circlemaths". Being one of the original books written by Dr Taylor the actual techniques ("COT tables" and so forth) are largely outdated, however what is of tremendous interest is the clear no-nonsense approach to number along with insights into the philosophy which produced the number theory. Not many philosophical approaches can be utilized to produce simple arithmetical results, but this one does. The link between philosophy and number becomes clear as you read.

• Language: English
• Publisher: Robert Taylor
• Release date: Apr 6, 2019
• ISBN: 9780463794494

Doctor Stephen Taylor

Dr Taylor (1926 - 2008) was a New Zealand medical practitioner who grew tired of anesthetics being routinely used at childbirth. He developed a program which resulted in all those participating having smooth, drug free natural births. His theories on birth developed into a theory of mind and a philosophy which he then successfully applied to ordinary arithmetic showing that the theory "added up" literally. The arithmetic is called Circlemaths and opens the door to understanding philosophy, the mind, and funnily enough, birth. He has written many books on all of these subjects. He is also well known for his 40 day fast against the Vietnam war in the 1970's held in Albert Park.

Book preview

PREFACE

This book is 'idea intensive' and ranges from arithmetic to philosophy. For that reason this Preface will take the form of a planning guide to help you to find your way through.

First, and in general, it should be stated that the book as a whole is a follow-on, explaining the circular counting line theory, which stands behind the new Zeta mental arithmetic. The first two books, The SCIENCE OF MENTAL ARITHMETIC, and its smaller supplement, SPACE AGE ARITHMETIC, set out the method of that new arithmetic.

Each book is designed to stand alone, and can be read independently of the others though each contributes additionally to the subject.

This is the THEORY book, but taking the arithmetic and its theory together, then behind both stands the philosophy or more distant theory from which the whole set out, and to which, hopefully, it will return, that the debt owed to the philosophy will be repaid.

WHERE TO LOOK FOR WHAT

The INTRODUCTION studies the mental origin of 0 and 1. This is to establish the basis of arithmetic, for the whole subject pirouettes on those two slender, 0 and 1 pins! And the true nature of 0 and 1, as they concern us here, is as ideas. The Introduction therefore belongs to the philosophical side. So if philosophy is not your thing skip the Introduction and come back to it later as your interest may direct.

For the surprising style of the arithmetic,

see Robert's Illustrative Example on page 7.

Do not be alarmed if the figuring in that example seems to you like something from out of an intergalactic spaceship. That number mix, believe it or not, is more rational, more simple and easier to follow and do, than any arithmetic you have met in the past. And one day it will be universally recognized and accepted!

Chapter 1

Details some of the logical absurdities in present day arithmetic. Our name for it is The Hidden Metaphysics.

Chapter 2.

Sets out the simple rules of arithmetic. A pedestrian, but very necessary chapter. It gives only what is correct and essential and lets the contradiction at the heart of arithmetic peep through.

Chapters 3, 4 and 5:

The -1x-1 problem, first raised in 'THE SCIENCE,' is answered, and in the process a surprising truth behind all our calculations emerges.

Chapters 6 and 7:

The question arises, How can we wring RESULTS out of the new circular counting lines? It is quite easy, and in these chapters it is explained.

What is a ‘cradle' in ordinary arithmetic? Find this in APPENDIX 1. It also has something to say about arithmetic in the cradle!

APPENDIX 2 is a ground-plan sketch of objective or scientific knowledge with accent on its genesis.

APPENDIX 3 is a philosophical outline originally planned for the Introduction, but later condensed and retained, with excerpts from the original, as this Appendix.

For those who have not yet learned the 'Zeta,' or mental arithmetic technique and do not have access to the first two books, but wish to learn the method first, even before making a study of the circle theory, APPENDIX 4 is the place to start. Then go on to Chapters 6 and 7.

THE COLOUR CODE

This greatly facilitates description, as well as work with the method and understanding of it. It can be briefly summarized as follows;

BLUE means 10-circle (10 or 100 or 1000 etc.)

RED means 9-circle (9 or 99 or 999 etc.)

GREEN means 11-circle and its numbers.

The colour reminds us to which circle sequence the different numbers belong. Find it all at Chapters 6 and 7.

KEY TO THE CROSS-REFERENCING SYSTEM

This has been designed to allow cross-references to be found quickly and accurately. Main chapter paragraphs are numbered. The lead-in paragraphs introducing each chapter are marked by Roman numerals. The boxed excerpts in Appendix 3, are identified by alphabet letters.

The sign; §

is used to mean ‘paragraph.’

So, (§4), means, ‘paragraph 4.’

And, (§4,15ff) means, ‘paragraphs 4 and 15, and further forward.’

We put the chapter number in front of the paragraph sign, so;

(3§4,15), means, ‘chapter 3, paragraphs 4 and 15.'

In.

stands for INTRODUCTION

App.

stands for APPENDIX

G

stands for GLOSSARY

If no chapter number is given the paragraph is in the same chapter. An occasional reference may be to a place further ahead in the book. For quick location of the exact reference margin indicators have been inserted at the text point-of-entry. These help to tie the reference system together and have been omitted only on a few occasions where no purpose would be served.

Although the books in the Series, namely;

The Science of Mental Arithmetic,

Space Age Arithmetic,

The Theory of Mental Arithmetic

have been designed to 'stand alone,’ they all contribute to a better grasp of the subject. Some trans-book cross-references have therefore been inserted to assist anyone making a study of the whole subject. Thus, (Sc3§20) is a reference to, 'The Science of Mental Arithmetic,' Chapter 3, paragraph 20. But such book-to-book cross-references are not essential to the understanding of the text.

AN ILLUSTRATIVE EXAMPLE

by Robert M. Taylor

Mathematics is peculiar in that it demands logic or it refuses to yield correct answers.

This mental arithmetic does yield correct answers. The reason it may not appear to be logical is due to unfamiliarity. Which is not surprising, because it is actually based on an entirely different form of logic to that with which we are all familiar.

In this demonstration no results are explained. Explanations of the techniques used may be found later in this book. In the event of the sums appearing in any way confusing or difficult, attention should be drawn to the following analogy:—

Consider the case of a learner driver who finds it difficult to drive down the street at the normal speed limit. He is not only dealing with the ever changing traffic conditions, but also with learning the basic techniques of driving.

How can one possibly change gears and steer at the same time? he might say.

It all seems so confusing.

And yet years later he may happily drive down the very same street chatting to a friend, not a care in the world.

No difficulty.

He changes gear automatically, almost unconsciously. He has mastered the techniques of driving and only has to deal with one thing - the road.

Similarly with the logic of this new arithmetic.

There is a tendency to try to come to terms with the totally unfamiliar techniques while at the same time trying to deal with the actual computations.

This makes for hard work, yet once the techniques are mastered and one has only the computations left to deal with, it can be seen that they are very simple.

So I suggest you get comfortable, sit back and take it easy on yourself, and suffice for now to merely check out that the actual numerical steps involved are simple, that they do lead to the correct answers and that the style of the calculation is, to say the least, surprising!

I hope you'll enjoy following the seemingly crazy path of the numbers as they scramble around and join in a mad race to their goal!

I think you will!

WE CHOOSE SOME FIGURES

not too hard, not too easy

88 x 92

Because 8 x 2 is 16, we know that the answer will end in 6

Now

8+8 is 16, and 1+6 is 7

and

9+2 is 11, and 1+1 is 2

7x 2 is 14

and 14 is 1+4 which equals 5

Now we do four standard things with our result which so far is 5:

We...

Multiply it by 11;

5x 11 =55

Take it from 99;

99 - 55 = 44

Halve it;

44/ 2 is 22

Take it from 99 again

99 - 22 is 77.

This 77 is called, Big Zeta, (App.4§ 1)

Our answer ends in 6, and we must take this off our 77.

77 - 6 is 71.

Till now the procedure has been straightforward!!??

So now it will go straightbackwards!!

Stop and reverse!!

The reverse of 71 is 17!

Now, our answer seems to be 17, and ends in 6, i.e., 176. But this seems too small. We know that, 88 x 92 is roughly, 90 x 90,

which is 8100

So we guess our answer should be in the eight thousand range. Taking our guessed '8' for eight thousand out of the 17,

we 'unfold" our final answer:

176 becomes (8 + 09) 6 i.e., answer 8096

- whew - a lot of steps, yes.

But each is simple and fast

By running up many steps one can climb a very high mountain. It is the speed and ease with which these many small steps can be taken which makes the process suitable for mental arithmetic. There are no carries. It is a mental dance forgetting as you go all but the running total which turns into the answer! In practice, the time to do this sum is measured in seconds.

SAME SUM, DIFFERENT APPROACH

88 x 92

Just to be different we will take each number off 99 which seems to be a popular number in this method for some reason:

99-88 is 11

99 - 92 is 7

Our sum is. now: 11 x 7

and11x7=77

This 77 is called, Big Zeta,

a figure we have seen before!

We complete the sum as before:-

77-6 is 71, spins to 17, ends in 6.

Remove our guessed 8 thousand from 17, to get

Answer: 8096

ONCE AGAIN, SAME SUM, DIFFERENT APPROACH.

88 x 92

9+2 = 11, and 1+1 = 2

92 has turned into 2

Now here is a short-cut:

Make a short cut down the middle of the 88.

We don't need two of them. Throw one away!

That leaves 8.

remember 92 is 2

So, 8 x 2 is 16, and 1+6 = 7

We cut the 88 in haif, so to be fair

we should 'shake double' our 7 7 becomes 77

This is Big Zeta.

We complete the sum as before:-

77 - 6, is 71, spins to 17, ends in 6.

Remove our guessed 8 thousand from 17, to get

Answer:- 8096

Both numbers are even and relatively close together

so the following method suits:

88 x 92

We note that in the line;

[88] 89 90 91 [92]

the 88 and the 92 are both 2 away from their mean, which is 90.

So,

90 x 90 = 8100

and,

2x2=4

and,

8100 less 4, is 8096.

Answer!

Children taught this arithmetic from the earliest will have two, rather than one, complementary forms of logic to face the world.

Just as in language, there are many ways of saying the same thing (different words can express the same idea), so in mathematics there are different alternative pathways to the same result.

Picking the best path for each sum is a question of skill.

Adults who learn a second language must sink most of their efforts into translation, whereas children who learn two languages from birth do not need to translate. They can think in both languages. As their attention is freed from the task of translation they can be creative in the second language.

Similarly, children who are taught the two arithmetics from birth will be able to think number. Freed from the need to translate between bases they will become mathematically creative. Certainly they will be able to mentally add, subtract, multiply and divide numbers into the millions.

We are entering a computerized future. Understanding number they will be able to do more with computers than we ever could.

When children learn the arithmetic they also learn the logic behind it, and the logic can be applied to any subject (and has already been applied to a few with startling results). The potential exists for a new physics, a new chemistry, a new psychology and so on!

INTRODUCTION

The Ultimate Foundations of Arithmetic

i) Before plunging into the arithmetic proper we will, in this Introduction, look at the derivation of O and 1 from their mental origins. This is an essential step before arithmetic just as it is essential before building to prepare one's site, establish exact foundations and assemble the building materials. [7§xi]

ii) In 'mental origins,’ we are looking at the nature of mind and that is an altogether different subject from arithmetic. It would be quite in order therefore to begin with the arithmetic at Chapter 1 and come back to this Introduction later as your interest may direct. Meantime we will spare a sideways glance at the new medium for the sake of orientation.

iii) From the point of view of understanding the number sequences of the new arithmetic which comes later, it is not really required either that we put forward, or the reader acquire, a consistent comprehensive and balanced view of the mind and mental function. Our attitude is rather that the reader may wish to obtain at least an idea of the lines on which we are thinking when we refer to the mental origins of arithmetic's founding concepts. For that reason the following has been penned.

iv) 0 and 1 are the foundations of arithmetic. Can we just accept them as 'given' as we accept the ground under our feet? We may presume that it is solid, but a good architect will test the ground thoroughly on which his building is to be erected, and prepare his foundations with the utmost care. [2§32] [4§1] [App.3§18]

v) Yet ordinary science does not test its ground! It accepts its material as given and by that token labels it as objective (App.3§BoxF). It also accepts the reasoning mind which examines these materials as equally given, and it never asks who or what does the giving.

vi) The need is rather to make the most careful site preparation and to concentrate all one's planning ability on the foundations if the building is going to rest secure.

vii) Given 0 and 1 the whole of arithmetic is given, because using these two numeral workhorses alone the binary system is established capable of all arithmetical computations as every computer student knows. It is critical therefore to determine from whence come the 0 and 1. [App.3§18]

viii) Absolute science requires that all elements be logically derived, hence our need to derive the 0 and 1. Then, in the following chapters, the promise made in our earlier books to solve the -1 x -l equation to everyone's satisfaction will be made good.

ix) But we will solve it not on the basis of a conventional approach but within the context of an absolute arithmetic which cannot itself 'get free' without liberating the genie of absolute science.

So in this Introduction our concern will be to examine the nought and one basis of arithmetic and at the same time give some idea of what the term, ‘absolute' means in relation to arithmetic, science and knowledge as a whole.

The Absolute Conception

x) Absolute arithmetic is simply ordinary arithmetic in an absolute context. It will do everything that ordinary arithmetic will do and more, because it retains the conventional non-absolute arithmetic within it as a special case. The successful resolution of the -1 x -l problem will lead the way in the later chapters but only because it happens to be a suitable entry point for the discussion.

xi) In