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MODELS TO ILLUSTRATE

THE FOUNDATIONS OF

MATHEMATICS

BY

C. ELLIOTT

Price 2s. 6d. net

EDINBURGH

PUBLISHED BY LINDSAY & CO.. 17 BLACKFRIARS STREET

1914

MODELS TO ILLUSTRATE THE

FOUNDATIONS OF MATHEMATICS

MODELS TO ILLUSTRATE

THE FOUNDATIONS OF

MATHEMATICS

BY

C. ELLIOTT

Price 2s. 6d. net

EDINBURGH

PUBLISHED BY LINDSAY & CO., 17 BLACKFRIARS STREET

1914

CONTENTS.

PAGE

INTRODUCTION - - - - - - 1

CHAPTER I.

THE MEANING OF CORRESPONDENCE.

SECTION

1. The Words used in Mathematics - - - 6

2. Some Undefined Words - - - - 8

3. "Likeness" and "Position" op Things in a Class 10

4. The Meaning op Correspondence - - - 11

5. Illustrations of Correspondence - - - 13

6. Symbols - - - 19

7. Classification of Correspondences - - 22

8. Change and Order - - - - - 24

9. Direct and Inverse Correspondence - - 26

10. Numbers - - - 28

11. Relative Position - - - - - 29

12. Counting and Measurement - - - 30

13. Multiplication and Addition of Correspondences 31

14. Illustrations of Multiplication - - - 34

15. Powers and Roots of Correspondences - - 35

16. Choice of an Intermediate Class - - 37

17. Tabulation of the Correspondence of Classes

TO AN " Intermediate Class " - - - 37

Summary of Chapter I. - - - - 38

VI

CHAPTER II. -MULTIPLEXES.

SECTION PAGE

1. Duplex Classification - - - .42

2. Symbolical Representation of Things in a Duplex 42

3. Triplex Classification - - - - 43

4. QuADRUPLEX Classification - - - 44

Summary of Chapter II. - - - .44

CHAPTER III. SPACES.

section

\. Illustrations of Ordered Multiplexes - - 46

2. "Pure" Geometry - - - - - 47

3. Terms used in "Pure" Geometry - - 48

4. Four-dimensional Geometry - - - 50

5. Classificatory Meaning of some Common Terms - 52

6. Difference of Ordinary Position - - 53

7. Notation for Numbers - - - - 56

Summary of Chapter III.

section

L Two Languages of Geometry - - - 58

2. The Part played by Experiment in Ordinary

Geometry - - - - - 60

3. The Part played by Correspondence in Arith-

metic AND Geometry - - . - 61

4. General Relations of Algebra with Geometry 63

vu

CHAPTER IV.

CORRESPONDENCE OF OPERANDS TO FUNCTIONS.

section page

1. Correspondence of Operands to Functions which

ARE in no way connected WITH EACH OTHER - 65

2. Correspondence of Operands to Functions which

ARE connected BY HAVING A ClASS IN CoMMON 66

3. Meaning op One -to -two Correspondence, and

Notation employed - - - - 67

4. ONE-TO-ri Correspondences - - - - 69

5. Classification of One-to-t^ Correspondences - 69

6. The Case where all the Quantities consist of

THE same Operands - - - - 70

7. Such Correspondences may, or may not, be

" Permutative " - - - - - 71

8. The Permutative and Non-permutative Forms

OF A Table - - - - - 72

9. Notations used for a Division Table - - 73

10. A Multiplication Table of Functions is also a

Multiplication Table of Super-Functions - 74

11. The Multiplication Table of Numbers - - 75

12. Use of the Word " Substitution " - - 77

13. The " Associative " Property - - - 79

14. The Meaning of a "Group" - - - 80

15. The Addition Table of Numbers - - - 81

16. The "Distributive" Property - - - 83

17. The " Inversor " is distributive over a Product 85

18. A "Tensor" is distributive over a Product - 87

19. The Meaning of the "Square Root" of Minus One 88

Summary of Chapter IY. - - - - 91

Vlll

CHAPTER v. MULTIPLE CORRESPONDENCE.

SECTION PAGE

L Meaning of Multiple Correspondence - - 94

2. Notation for Multiple Operands - - 96

3. Notations for Multiple Correspondence - 97

4. Multiple Numbers - - - 98

5. Some Illustrations of Multiple Correspondence 101

6. "Forms" in Elementary Algebra , - - 101

7. Correspondences of " Forms " - - - 102

8. Artificial " Forms " . . . . 104

9. Multiple Groups - - - - - 105

10. The Common Use of " Addition " and " Multi-

plication " - - - - - 105

11. Meaning of the Cube Root of One - - 107

12. Representation of Multiple Operands by Single

Symbols - - - 108

13. Case where a Multiple Operand is a Multiplex 109

Summary of Chapter Y. - - - - 110

INDEX .. - -.. 113

MODELS TO ILLUSTRATE

THE FOUNDATIONS OF MATHEMATICS.

INTRODUCTION.

There are no fixed and unalterable beginnings of mathematics.

As progress is made in mathematical science, it becomes necessary

from time to time to alter the presentation of the rudiments, and

to view subjects which have found a place in elementary mathe-

matics for centuries under a new aspect. Were this not to be

done, the difficulties which a sudent meets with in attaining the

point of view of advanced workers would continually increase, and

would form a hindrance to further improvement.

The present writer considers that some existing difficulties may

be overcome, at least in part, by the introduction of a new kind of

practical work into schools. And though the reader may think

that it is not feasible to do as proposed here, and make it form part

of the groundwork of mathematical teaching, still it may be found

of use as throwing fresh light upon the other methods employed

to-day. The advantages or otherwise of any one view of mathe-

matics can scarcely be realised until the attempt has been made

to develop the subject from that one standpoint, so that, even if

judged a failure, the endeavour may not be valueless to others.

This work supplements both some matters of everyday experi-

ence and certain aspects of ordinary elementary mathematics, and

therefore is not an absolute novelty, but serves to emphasise some

already existing features of mathematical teaching. On that

account it is hoped that the following description may be intelli-

gible even without the actual models, some examples of which, how-

ever, the reader is strongly advised to construct for himself.

They are intended to illustrate some modern views upon the

Foundations of Mathematics, and to show that the " abstract "

character of that subject does not forbid any attempt to bring

A

2

elementary teaching up to date in that direction. The importance

of the ideas which can be illustrated by models in this way is fairly

generally recognised, and it is hoped that the form in which they

are here presented will make them more available for schools than

they have hitherto been. The objection that such a method would

be too abstract may be met by the fact that the ideas under

discussion are visualised by the models, and that numerous

examples are given, drawn from everyday experience.

A clear account of the development of pure or abstract mathe-

matics will be found in a recent book, the Fundamental Concepts

of Algebra and Geometry,* by Prof. J. W. Young, to which frequent

references will be made. I have tried to make the description

self-contained, so that possession of Prof. Young's work may not be

absolutely necessary, but, for those who have not read that or some

similar book, the remarks in the next three paragraphs will perhaps

not convey much meaning, and may be passed over.

In that work it will be found that pure mathematics is regarded

as consisting of attempts to deduce propositions " by the methods

of Formal Logic " from postulates, or axioms, about terms of which

the meaning is intentionally left undefined. These postulates or

axioms are not looked upon as truths necessarily self-evident to the

mind, nor as experimental facts, but merely as assumptions (p. 38).

Calling such a series of logically connected propositions an abstract

mathematical system, then mathematics as a whole is defined as

consisting of all such systems together with all their concrete

applications (p. 221).

This definition, however, the author points out, is not to be

taken to imply that Formal Logic is the chief method of mathe-

matical discovery. ^^ Imaginatio7i, geometric intuition, experi-

m,entation, analogies sometim,es of the vaguest sort, and judicious

guessing, these are the instruments continually employed in m.athe-

matical research " (p. 221).

Yet the definition does seem to imply that a training in logic

is absolutely necessary to comprehend the modern view of pure

mathematics, and, as a consequence, that " the points of view to be

developed in these lectures, and the results reached, are not directly

of use in elementary teaching'^ (p. 7).

* Macmillan, 1911, price 7s. net.

3

The present writer considers that, by viewing mathematics from

a slightly different standpoint, less importance may be ascribed to

abstract reasoning, and more to observation and description, and

therefore the elements of modern pure mathematics become more

accessible to beginners.

The kind of postulation referred to above is of course to be

found only in modern works ; in ancient text-books, such as that of

Euclid, it exists in an obscure form which does not permit of

accurate deductions according to modern standards. An examina-

tion of the nature of the modern form shows that what are

postulated are classificatory relationships (a term which the models

described later are intended to explain), and it is those which lend

themselves to "formal reasoning."

The special point of view which has influenced the following

treatment of elementary work is that from which mathematics is

regarded as the science of classification. Such a statement as that

the propositions of both Algebra and Geometry are among the im-

plications of the same set of axioms {Fundamental Concepts, p. 183)

we would translate by saying that the fundamental ideas of both can

be illustrated by the same set of classificatory models. Or, again,

the distinction drawn between pure and applied science {Funda-

mental Concepts, p. 54*) we would interpret by saying that the

student of mathematics and physics must either be engaged in

studying classification in general, that is, in arriving at the pro-

positions of pure mathematics, or in obtaining by practical exercises

the experimental data necessary for the application of his classi-

ficatory knowledge, or in making the application itself.

The pure mathematician is assumed to be really engaged in the

investigation and description of classificatory relationships, and

* " If we adopt the point of view of Peanoand Russell, aXlpure mathe-

matics is abstract. Any concrete representation of such an abstract science

is then a branch of applied mathematics. Geometry, for example, as a branch

of pure mathematics, consists, then, simply of the formal logical implications

of a set of assumptions. Whenever we think of geometry as describing pro-

perties of the external world in which we live, we are thinking of a branch

of applied mathematics in the same sense in which analytical mechanics is a

branch of applied mathematics. We need not quibble over this distinction.

The important thing is to recognise that there exists an abstract science under-

lying any branch of mathematics, and that the study of this abstract science is

essential to a clear understanding of the logical foundations."

4

hence the view is taken that models of classifications might play a

part in the beginning of pure mathematical teaching, and be

supplemented by tabulations of symbols, standing for classifications

of the things symbolised. Such models are easy to construct, so

easy that some readers may think they scarcely deserve the name

of models, but, owing chiefly to the fact that mathematical language

is somewhat redundant and defective, their explanation in simple

terms is at present not free from difficulties.

No attempts at formal reasoning will be found therefore in the

following pages, which are devoted solely to the explanation of

those ideas relating to classification which play a prominent part in

elementary mathematics, when looked at from the particular stand-

point chosen. But although the deduction of propositions from a

set of axioms, or unproved propositions, is avoided, yet the deri-

vation of mathematical terms from one another has had some

attention, in order to lay emphasis upon the necessity of beginning

our explanations with a set of undefined terras.

Of the ideas to be explained, two of the most important,

namely, that of a correspondence or function, and that of a multi-

plex, lend themselves readily to illustration by classificatory models,

and therefore the opinion is set forth, though naturally with

diffidence, that these ideas should, with some of their developments,

be introduced at a very early stage, and form part of the ground-

work of mathematical teaching. One difficulty met with is that it

does not seem to be possible to employ such important words as

Addition and Multiplication, the definition of which naturally

follows that of Correspondence, throughout in only one sense, without

somewhere coming in conflict with existing usage. The employment

of those and some other words has, in course of time, been extended

in a somewhat arbitrary and unsystematic manner, which puts a

certain difficulty in the way of adhering to any one definition.

A model of a classification is intended to show those features

which are important, and to omit all others ; it is a set of things

showing clearly those resemblances and differences alone, which are

essential features of the type under discussion. The observation of

likenesses and differences is a necessary preliminary to classifica-

tion, and, since the differences with which we first become familiar

are those of shape, size, colour, markings, and a few more, these

are naturally to be utilised in the artificial production of classi-

fications for elementary teaching. Flat pieces of cardboard can

be made to show a sufficient variety of differences from each other

to serve for such artificial productions, and oifer the advantage that

scissors, and a paint-box, and brush are the only tools necessary.

It is, of course, not suggested that the usual school practical

work in paper-folding, in the construction and measurement of

diagrams and models, in the plotting of graphs, and so on, should

be neglected, but only that the language and ideas based upon

this work need to some extent to be revised, and, especially, the

distinction between pure and applied mathematics placed in a

clearer light. As already said, to adopt the plan suggested here is

not to introduce absolute novelties, but to develop an already

existing feature of mathematical teaching. For the classificatory

ideas to be explained here are already studied in schools, but

almost solely through applications, with which they are usually

almost inextricably mixed. Therefore the beginner has at present

to face two kinds of difficulties simultaneously, namely, those

belonging to the body of classificatory ideas, and those inherent in

some application. The aim of this book is to assist him by isolating

the former from the latter.

This object requires more repetition of the ordinary elements of

algebra and geometry than perhaps the reader would expect or

desire in a work professing to deal with models, but it also involves

a certain detachment from what are usually regarded as the main

subjects of mathematical interest. For, although the mathematician

is concerned with correspondences, and with multiplexes, and their

related ideas, he is not exclusively concerned with the corre-

spondences of numbers to numbers, nor with multiplexes of things

differing in ordinary position. Thus the exclusive attention paid

to numbers in school arithmetics, and to geometric points (in the

common sense) in school geometries may actually disqualify to some

extent such books from serving as an introduction to the ideas of

modern mathematics.

In conclusion, it may be mentioned that a selection of the

classificatory models referred to above was exhibited at the Inter-

national Congress of Mathematicians held at Cambridge in 1912.

0. Elliott.

CHAPTER I.

THE MEANING OF CORRESPONDENCE.

Section 1. The Words used in Mathematics.

Among other aspects mathematics may be viewed as a special

language invented for special needs, and it possesses a set of words

of its own, just as do music, or cricket, or botany.

Broadly speaking, the meaning of a new word may be explained

in two ways, namely, by showing the thing or action for which it

stands, or by the method used in an English dictionary, that of

explaining the word by means of other words. If these other

words are not understood, recourse must again be had to the

dictionary. But, since the number of words in the dictionary is

finite, we must, in the end, either use the words to be explained,

in their own explanation, or be content to leave them unexplained,

except by the first method, which might be called the laboratory or

workshop method. Th^s necessity of leaving some words undefined

occurs, of course, in writings of every description, and not only in

mathematics.

In order, therefore, to explain the words actually used in

mathematics, the teacher must make a choice of words, the meaning

of which he considers to be already known to his students, or of

which he can easily demonstrate the meaning, and build his system

of definitions upon these.

It has been customary for the two main branches of elementary

mathematics, namely. Arithmetic and Algebra on the one hand and

Geometry on the other, to be developed with distinct systems of

words, and for no great attention to be paid to the list of words

which it is necessary to leave undefined in each system.

E very-day experience, and practical exercises in school, are

supposed to sufficiently demonstrate the meaning of the elementary

terms employed in the two branches.

Now it is natural to ask whether mathematical science as a

whole could not be developed from but one set of words, and the

question has been answered with some success as the result of

modern research on the Foundations of Mathematics. It cannot,

however, be expected that the old words employed in the branches

treated separately will serve this new purpose. An entirely fresh

set must be chosen, and the old words employed in Arithmetic or

Geometry defined in terms of the new ones. It may, however, be

expected that such a unification of the branches of mathematics

will obviate the need for so many distinct terms by showing that

the same concepts occur in what, from the older standpoint, were

different departments, and that therefore the same words, and not

distinct ones, may be used to express them.

Instead of such words as length, breadth, magnitude, direction,

and the names of numbers, used in the elements of the old branches

of mathematics, those of the new set (which, with their derived

words, can be made to define the old ones) include such as class,

belonging to a class, sub-class, correspondence, duplex, etc. The

meanings of some of these words, which are used as undefined words

in modern work, are considered in the paragraphs below, one object

of this book being to show that these, too, like the older ones, can

have their meaning illustrated both by school practical exercises,

and by examples from every-day life.

The same remark applies to many of the words defined by

means of them, which, on that account, could perhaps be equally

well taken as undefined terms.

It may also be noticed that the teacher is under no compulsion

to use only practical experience as a guide to the meaning of his

undefined words, and only verbal explanations for his defined

words. On the contrary, there seems to be no good reason why, in

the beginning of mathematics, the former words should not be

accompanied by verbal explanations, and the latter by practical

illustrations. Nor is it to be expected that axioms and deductions

should, in a school course, follow immediately upon the intro-

duction of a few words. The importance of the fundamental ideas

relating to classification is so great that it may be better to dwell

upon their explanation and illustration, rather than to provide

exercises in reasoning.

Section 2. Some Undefined Words.

(a) The first we shall take is respect, or kind of difference.

Various kinds of difference between things will already be known

to the student, for example, differences in ordinary position, in size,

shape, taste, hotness, etc., and school practical work in science

consists largely in adding to their number. Among the differences

a knowledge of which is gradually acquired are density, refractive

index, specific heat, and many others. Therefore when, in what

follows, bodies are spoken of as differing in one, two, three, or more

respects, it will be assumed that a sufficiently clear meaning is

attached to the word respect. The possibility of giving importance

to the word has been brought about by the introduction of

elementary practical science into schools.

(6) Class. Things form a class when they agree in at least

one respect or circumstance, and differ from each other in at least

one respect. Thus, the chairs in this room form a class, because

they agree in the circumstance that they are all chairs in the room,

and they differ in position. To make clear the nature of what is

meant by " class," we can construct model or artificial classes, the

members of which agree in some given respect, as colour, and differ

in some given respect, as shape. Pieces of cardboard properly

coloured and shaped will serve this purpose. It is generally

supposed that we are unable to distinguish different bodies at all,

if they occupy the same position, and therefore, in whatever other

respects the members of any class differ, they must differ in

ordinary position as well. For example, the pieces of cardboard

referred to must differ from each other in position, obviously, as

well as in shape. For the sake of brevity in speaking of any class,

the respect in which its members are alike will be referred to as A,

and the respect in which they differ as D, so that, in the model

suggested as an example, A is colour, and D is shape. It will be

seen that the meaning of the word class above has been explained

by making use of the word "respect," previously selected as an

undefined term.

A peculiarity of our mental capabilities may be noticed while

considering classes, namely, that we are able to group unlike things

into an entirely arbitrary class, and remember the individuals thus

placed together. The fact that this mental process has been applied

9

to them constitutes the respect A in which the members of the

arbitrary class agree. The grouping is arbitrary in the sense that,

though the person who made it can state at once what objects out

of a number are, or are not, in the class, no mere examination of the

objects would enable anobher person to do so, as he would be unable

to detect any property A, serving to distinguish the members of

the class from the rest.

(c) Sub-class. One class is said to be a sub-class of another

when all its members belong also to that other. Since they belong

to it, they must agree in the respect A of the other, or they would

not belong to it at all, but with regard to the respect D of the

larger class it seems advisable to consider at least two cases. For

the members of the sub-class may differ in D and agree in some

third respect, which marks them off from the rest of the class, or

they may agree in D, but differ in a third respect, D'. Both kinds

of sub-class should be illustrated by artificial classes, containing

sub-classes. As already said, the members of a class may be

arbitrarily grouped together, and, similarly, a sub-class may be

constructed by making an arbitrary selection of members from

a class.

(d) Classification. A class may have many sub-classes, and any

sub-class may, in its turn, have its own sub-divisions. To sort a

class of things is to distribute them into sub-classes, and the result-

ing distribution is called a classification of the class with which we

began. In illustration may be shown collections of various kinds,

or such a case as the classification of families by nationality, town,

street, and house, may be referred to. If it is to boys' drawing,

modelling, carpentering, and counting that teachers look to supply

a knowledge of the older undefined terms, it is perhaps boys' col-

lections of things which may help to play the same part for the

undefined terms now under consideration.

It has already been said, that one view which might be taken

of mathematics is, that it is a special language invented for special

needs. Of these special needs one of the most important is that of

describing different kinds of classification. We might suppose a

person to collect so many kinds of things, as to become virtually a

UC-NRLF

$B m? no

/^

MODELS TO ILLUSTRATE

THE FOUNDATIONS OF

MATHEMATICS

BY

C. ELLIOTT

Price 2s. 6d. net

EDINBURGH

PUBLISHED BY LINDSAY & CO.. 17 BLACKFRIARS STREET

1914

MODELS TO ILLUSTRATE THE

FOUNDATIONS OF MATHEMATICS

MODELS TO ILLUSTRATE

THE FOUNDATIONS OF

MATHEMATICS

BY

C. ELLIOTT

Price 2s. 6d. net

EDINBURGH

PUBLISHED BY LINDSAY & CO., 17 BLACKFRIARS STREET

1914

CONTENTS.

PAGE

INTRODUCTION - - - - - - 1

CHAPTER I.

THE MEANING OF CORRESPONDENCE.

SECTION

1. The Words used in Mathematics - - - 6

2. Some Undefined Words - - - - 8

3. "Likeness" and "Position" op Things in a Class 10

4. The Meaning op Correspondence - - - 11

5. Illustrations of Correspondence - - - 13

6. Symbols - - - 19

7. Classification of Correspondences - - 22

8. Change and Order - - - - - 24

9. Direct and Inverse Correspondence - - 26

10. Numbers - - - 28

11. Relative Position - - - - - 29

12. Counting and Measurement - - - 30

13. Multiplication and Addition of Correspondences 31

14. Illustrations of Multiplication - - - 34

15. Powers and Roots of Correspondences - - 35

16. Choice of an Intermediate Class - - 37

17. Tabulation of the Correspondence of Classes

TO AN " Intermediate Class " - - - 37

Summary of Chapter I. - - - - 38

VI

CHAPTER II. -MULTIPLEXES.

SECTION PAGE

1. Duplex Classification - - - .42

2. Symbolical Representation of Things in a Duplex 42

3. Triplex Classification - - - - 43

4. QuADRUPLEX Classification - - - 44

Summary of Chapter II. - - - .44

CHAPTER III. SPACES.

section

\. Illustrations of Ordered Multiplexes - - 46

2. "Pure" Geometry - - - - - 47

3. Terms used in "Pure" Geometry - - 48

4. Four-dimensional Geometry - - - 50

5. Classificatory Meaning of some Common Terms - 52

6. Difference of Ordinary Position - - 53

7. Notation for Numbers - - - - 56

Summary of Chapter III.

section

L Two Languages of Geometry - - - 58

2. The Part played by Experiment in Ordinary

Geometry - - - - - 60

3. The Part played by Correspondence in Arith-

metic AND Geometry - - . - 61

4. General Relations of Algebra with Geometry 63

vu

CHAPTER IV.

CORRESPONDENCE OF OPERANDS TO FUNCTIONS.

section page

1. Correspondence of Operands to Functions which

ARE in no way connected WITH EACH OTHER - 65

2. Correspondence of Operands to Functions which

ARE connected BY HAVING A ClASS IN CoMMON 66

3. Meaning op One -to -two Correspondence, and

Notation employed - - - - 67

4. ONE-TO-ri Correspondences - - - - 69

5. Classification of One-to-t^ Correspondences - 69

6. The Case where all the Quantities consist of

THE same Operands - - - - 70

7. Such Correspondences may, or may not, be

" Permutative " - - - - - 71

8. The Permutative and Non-permutative Forms

OF A Table - - - - - 72

9. Notations used for a Division Table - - 73

10. A Multiplication Table of Functions is also a

Multiplication Table of Super-Functions - 74

11. The Multiplication Table of Numbers - - 75

12. Use of the Word " Substitution " - - 77

13. The " Associative " Property - - - 79

14. The Meaning of a "Group" - - - 80

15. The Addition Table of Numbers - - - 81

16. The "Distributive" Property - - - 83

17. The " Inversor " is distributive over a Product 85

18. A "Tensor" is distributive over a Product - 87

19. The Meaning of the "Square Root" of Minus One 88

Summary of Chapter IY. - - - - 91

Vlll

CHAPTER v. MULTIPLE CORRESPONDENCE.

SECTION PAGE

L Meaning of Multiple Correspondence - - 94

2. Notation for Multiple Operands - - 96

3. Notations for Multiple Correspondence - 97

4. Multiple Numbers - - - 98

5. Some Illustrations of Multiple Correspondence 101

6. "Forms" in Elementary Algebra , - - 101

7. Correspondences of " Forms " - - - 102

8. Artificial " Forms " . . . . 104

9. Multiple Groups - - - - - 105

10. The Common Use of " Addition " and " Multi-

plication " - - - - - 105

11. Meaning of the Cube Root of One - - 107

12. Representation of Multiple Operands by Single

Symbols - - - 108

13. Case where a Multiple Operand is a Multiplex 109

Summary of Chapter Y. - - - - 110

INDEX .. - -.. 113

MODELS TO ILLUSTRATE

THE FOUNDATIONS OF MATHEMATICS.

INTRODUCTION.

There are no fixed and unalterable beginnings of mathematics.

As progress is made in mathematical science, it becomes necessary

from time to time to alter the presentation of the rudiments, and

to view subjects which have found a place in elementary mathe-

matics for centuries under a new aspect. Were this not to be

done, the difficulties which a sudent meets with in attaining the

point of view of advanced workers would continually increase, and

would form a hindrance to further improvement.

The present writer considers that some existing difficulties may

be overcome, at least in part, by the introduction of a new kind of

practical work into schools. And though the reader may think

that it is not feasible to do as proposed here, and make it form part

of the groundwork of mathematical teaching, still it may be found

of use as throwing fresh light upon the other methods employed

to-day. The advantages or otherwise of any one view of mathe-

matics can scarcely be realised until the attempt has been made

to develop the subject from that one standpoint, so that, even if

judged a failure, the endeavour may not be valueless to others.

This work supplements both some matters of everyday experi-

ence and certain aspects of ordinary elementary mathematics, and

therefore is not an absolute novelty, but serves to emphasise some

already existing features of mathematical teaching. On that

account it is hoped that the following description may be intelli-

gible even without the actual models, some examples of which, how-

ever, the reader is strongly advised to construct for himself.

They are intended to illustrate some modern views upon the

Foundations of Mathematics, and to show that the " abstract "

character of that subject does not forbid any attempt to bring

A

2

elementary teaching up to date in that direction. The importance

of the ideas which can be illustrated by models in this way is fairly

generally recognised, and it is hoped that the form in which they

are here presented will make them more available for schools than

they have hitherto been. The objection that such a method would

be too abstract may be met by the fact that the ideas under

discussion are visualised by the models, and that numerous

examples are given, drawn from everyday experience.

A clear account of the development of pure or abstract mathe-

matics will be found in a recent book, the Fundamental Concepts

of Algebra and Geometry,* by Prof. J. W. Young, to which frequent

references will be made. I have tried to make the description

self-contained, so that possession of Prof. Young's work may not be

absolutely necessary, but, for those who have not read that or some

similar book, the remarks in the next three paragraphs will perhaps

not convey much meaning, and may be passed over.

In that work it will be found that pure mathematics is regarded

as consisting of attempts to deduce propositions " by the methods

of Formal Logic " from postulates, or axioms, about terms of which

the meaning is intentionally left undefined. These postulates or

axioms are not looked upon as truths necessarily self-evident to the

mind, nor as experimental facts, but merely as assumptions (p. 38).

Calling such a series of logically connected propositions an abstract

mathematical system, then mathematics as a whole is defined as

consisting of all such systems together with all their concrete

applications (p. 221).

This definition, however, the author points out, is not to be

taken to imply that Formal Logic is the chief method of mathe-

matical discovery. ^^ Imaginatio7i, geometric intuition, experi-

m,entation, analogies sometim,es of the vaguest sort, and judicious

guessing, these are the instruments continually employed in m.athe-

matical research " (p. 221).

Yet the definition does seem to imply that a training in logic

is absolutely necessary to comprehend the modern view of pure

mathematics, and, as a consequence, that " the points of view to be

developed in these lectures, and the results reached, are not directly

of use in elementary teaching'^ (p. 7).

* Macmillan, 1911, price 7s. net.

3

The present writer considers that, by viewing mathematics from

a slightly different standpoint, less importance may be ascribed to

abstract reasoning, and more to observation and description, and

therefore the elements of modern pure mathematics become more

accessible to beginners.

The kind of postulation referred to above is of course to be

found only in modern works ; in ancient text-books, such as that of

Euclid, it exists in an obscure form which does not permit of

accurate deductions according to modern standards. An examina-

tion of the nature of the modern form shows that what are

postulated are classificatory relationships (a term which the models

described later are intended to explain), and it is those which lend

themselves to "formal reasoning."

The special point of view which has influenced the following

treatment of elementary work is that from which mathematics is

regarded as the science of classification. Such a statement as that

the propositions of both Algebra and Geometry are among the im-

plications of the same set of axioms {Fundamental Concepts, p. 183)

we would translate by saying that the fundamental ideas of both can

be illustrated by the same set of classificatory models. Or, again,

the distinction drawn between pure and applied science {Funda-

mental Concepts, p. 54*) we would interpret by saying that the

student of mathematics and physics must either be engaged in

studying classification in general, that is, in arriving at the pro-

positions of pure mathematics, or in obtaining by practical exercises

the experimental data necessary for the application of his classi-

ficatory knowledge, or in making the application itself.

The pure mathematician is assumed to be really engaged in the

investigation and description of classificatory relationships, and

* " If we adopt the point of view of Peanoand Russell, aXlpure mathe-

matics is abstract. Any concrete representation of such an abstract science

is then a branch of applied mathematics. Geometry, for example, as a branch

of pure mathematics, consists, then, simply of the formal logical implications

of a set of assumptions. Whenever we think of geometry as describing pro-

perties of the external world in which we live, we are thinking of a branch

of applied mathematics in the same sense in which analytical mechanics is a

branch of applied mathematics. We need not quibble over this distinction.

The important thing is to recognise that there exists an abstract science under-

lying any branch of mathematics, and that the study of this abstract science is

essential to a clear understanding of the logical foundations."

4

hence the view is taken that models of classifications might play a

part in the beginning of pure mathematical teaching, and be

supplemented by tabulations of symbols, standing for classifications

of the things symbolised. Such models are easy to construct, so

easy that some readers may think they scarcely deserve the name

of models, but, owing chiefly to the fact that mathematical language

is somewhat redundant and defective, their explanation in simple

terms is at present not free from difficulties.

No attempts at formal reasoning will be found therefore in the

following pages, which are devoted solely to the explanation of

those ideas relating to classification which play a prominent part in

elementary mathematics, when looked at from the particular stand-

point chosen. But although the deduction of propositions from a

set of axioms, or unproved propositions, is avoided, yet the deri-

vation of mathematical terms from one another has had some

attention, in order to lay emphasis upon the necessity of beginning

our explanations with a set of undefined terras.

Of the ideas to be explained, two of the most important,

namely, that of a correspondence or function, and that of a multi-

plex, lend themselves readily to illustration by classificatory models,

and therefore the opinion is set forth, though naturally with

diffidence, that these ideas should, with some of their developments,

be introduced at a very early stage, and form part of the ground-

work of mathematical teaching. One difficulty met with is that it

does not seem to be possible to employ such important words as

Addition and Multiplication, the definition of which naturally

follows that of Correspondence, throughout in only one sense, without

somewhere coming in conflict with existing usage. The employment

of those and some other words has, in course of time, been extended

in a somewhat arbitrary and unsystematic manner, which puts a

certain difficulty in the way of adhering to any one definition.

A model of a classification is intended to show those features

which are important, and to omit all others ; it is a set of things

showing clearly those resemblances and differences alone, which are

essential features of the type under discussion. The observation of

likenesses and differences is a necessary preliminary to classifica-

tion, and, since the differences with which we first become familiar

are those of shape, size, colour, markings, and a few more, these

are naturally to be utilised in the artificial production of classi-

fications for elementary teaching. Flat pieces of cardboard can

be made to show a sufficient variety of differences from each other

to serve for such artificial productions, and oifer the advantage that

scissors, and a paint-box, and brush are the only tools necessary.

It is, of course, not suggested that the usual school practical

work in paper-folding, in the construction and measurement of

diagrams and models, in the plotting of graphs, and so on, should

be neglected, but only that the language and ideas based upon

this work need to some extent to be revised, and, especially, the

distinction between pure and applied mathematics placed in a

clearer light. As already said, to adopt the plan suggested here is

not to introduce absolute novelties, but to develop an already

existing feature of mathematical teaching. For the classificatory

ideas to be explained here are already studied in schools, but

almost solely through applications, with which they are usually

almost inextricably mixed. Therefore the beginner has at present

to face two kinds of difficulties simultaneously, namely, those

belonging to the body of classificatory ideas, and those inherent in

some application. The aim of this book is to assist him by isolating

the former from the latter.

This object requires more repetition of the ordinary elements of

algebra and geometry than perhaps the reader would expect or

desire in a work professing to deal with models, but it also involves

a certain detachment from what are usually regarded as the main

subjects of mathematical interest. For, although the mathematician

is concerned with correspondences, and with multiplexes, and their

related ideas, he is not exclusively concerned with the corre-

spondences of numbers to numbers, nor with multiplexes of things

differing in ordinary position. Thus the exclusive attention paid

to numbers in school arithmetics, and to geometric points (in the

common sense) in school geometries may actually disqualify to some

extent such books from serving as an introduction to the ideas of

modern mathematics.

In conclusion, it may be mentioned that a selection of the

classificatory models referred to above was exhibited at the Inter-

national Congress of Mathematicians held at Cambridge in 1912.

0. Elliott.

CHAPTER I.

THE MEANING OF CORRESPONDENCE.

Section 1. The Words used in Mathematics.

Among other aspects mathematics may be viewed as a special

language invented for special needs, and it possesses a set of words

of its own, just as do music, or cricket, or botany.

Broadly speaking, the meaning of a new word may be explained

in two ways, namely, by showing the thing or action for which it

stands, or by the method used in an English dictionary, that of

explaining the word by means of other words. If these other

words are not understood, recourse must again be had to the

dictionary. But, since the number of words in the dictionary is

finite, we must, in the end, either use the words to be explained,

in their own explanation, or be content to leave them unexplained,

except by the first method, which might be called the laboratory or

workshop method. Th^s necessity of leaving some words undefined

occurs, of course, in writings of every description, and not only in

mathematics.

In order, therefore, to explain the words actually used in

mathematics, the teacher must make a choice of words, the meaning

of which he considers to be already known to his students, or of

which he can easily demonstrate the meaning, and build his system

of definitions upon these.

It has been customary for the two main branches of elementary

mathematics, namely. Arithmetic and Algebra on the one hand and

Geometry on the other, to be developed with distinct systems of

words, and for no great attention to be paid to the list of words

which it is necessary to leave undefined in each system.

E very-day experience, and practical exercises in school, are

supposed to sufficiently demonstrate the meaning of the elementary

terms employed in the two branches.

Now it is natural to ask whether mathematical science as a

whole could not be developed from but one set of words, and the

question has been answered with some success as the result of

modern research on the Foundations of Mathematics. It cannot,

however, be expected that the old words employed in the branches

treated separately will serve this new purpose. An entirely fresh

set must be chosen, and the old words employed in Arithmetic or

Geometry defined in terms of the new ones. It may, however, be

expected that such a unification of the branches of mathematics

will obviate the need for so many distinct terms by showing that

the same concepts occur in what, from the older standpoint, were

different departments, and that therefore the same words, and not

distinct ones, may be used to express them.

Instead of such words as length, breadth, magnitude, direction,

and the names of numbers, used in the elements of the old branches

of mathematics, those of the new set (which, with their derived

words, can be made to define the old ones) include such as class,

belonging to a class, sub-class, correspondence, duplex, etc. The

meanings of some of these words, which are used as undefined words

in modern work, are considered in the paragraphs below, one object

of this book being to show that these, too, like the older ones, can

have their meaning illustrated both by school practical exercises,

and by examples from every-day life.

The same remark applies to many of the words defined by

means of them, which, on that account, could perhaps be equally

well taken as undefined terms.

It may also be noticed that the teacher is under no compulsion

to use only practical experience as a guide to the meaning of his

undefined words, and only verbal explanations for his defined

words. On the contrary, there seems to be no good reason why, in

the beginning of mathematics, the former words should not be

accompanied by verbal explanations, and the latter by practical

illustrations. Nor is it to be expected that axioms and deductions

should, in a school course, follow immediately upon the intro-

duction of a few words. The importance of the fundamental ideas

relating to classification is so great that it may be better to dwell

upon their explanation and illustration, rather than to provide

exercises in reasoning.

Section 2. Some Undefined Words.

(a) The first we shall take is respect, or kind of difference.

Various kinds of difference between things will already be known

to the student, for example, differences in ordinary position, in size,

shape, taste, hotness, etc., and school practical work in science

consists largely in adding to their number. Among the differences

a knowledge of which is gradually acquired are density, refractive

index, specific heat, and many others. Therefore when, in what

follows, bodies are spoken of as differing in one, two, three, or more

respects, it will be assumed that a sufficiently clear meaning is

attached to the word respect. The possibility of giving importance

to the word has been brought about by the introduction of

elementary practical science into schools.

(6) Class. Things form a class when they agree in at least

one respect or circumstance, and differ from each other in at least

one respect. Thus, the chairs in this room form a class, because

they agree in the circumstance that they are all chairs in the room,

and they differ in position. To make clear the nature of what is

meant by " class," we can construct model or artificial classes, the

members of which agree in some given respect, as colour, and differ

in some given respect, as shape. Pieces of cardboard properly

coloured and shaped will serve this purpose. It is generally

supposed that we are unable to distinguish different bodies at all,

if they occupy the same position, and therefore, in whatever other

respects the members of any class differ, they must differ in

ordinary position as well. For example, the pieces of cardboard

referred to must differ from each other in position, obviously, as

well as in shape. For the sake of brevity in speaking of any class,

the respect in which its members are alike will be referred to as A,

and the respect in which they differ as D, so that, in the model

suggested as an example, A is colour, and D is shape. It will be

seen that the meaning of the word class above has been explained

by making use of the word "respect," previously selected as an

undefined term.

A peculiarity of our mental capabilities may be noticed while

considering classes, namely, that we are able to group unlike things

into an entirely arbitrary class, and remember the individuals thus

placed together. The fact that this mental process has been applied

9

to them constitutes the respect A in which the members of the

arbitrary class agree. The grouping is arbitrary in the sense that,

though the person who made it can state at once what objects out

of a number are, or are not, in the class, no mere examination of the

objects would enable anobher person to do so, as he would be unable

to detect any property A, serving to distinguish the members of

the class from the rest.

(c) Sub-class. One class is said to be a sub-class of another

when all its members belong also to that other. Since they belong

to it, they must agree in the respect A of the other, or they would

not belong to it at all, but with regard to the respect D of the

larger class it seems advisable to consider at least two cases. For

the members of the sub-class may differ in D and agree in some

third respect, which marks them off from the rest of the class, or

they may agree in D, but differ in a third respect, D'. Both kinds

of sub-class should be illustrated by artificial classes, containing

sub-classes. As already said, the members of a class may be

arbitrarily grouped together, and, similarly, a sub-class may be

constructed by making an arbitrary selection of members from

a class.

(d) Classification. A class may have many sub-classes, and any

sub-class may, in its turn, have its own sub-divisions. To sort a

class of things is to distribute them into sub-classes, and the result-

ing distribution is called a classification of the class with which we

began. In illustration may be shown collections of various kinds,

or such a case as the classification of families by nationality, town,

street, and house, may be referred to. If it is to boys' drawing,

modelling, carpentering, and counting that teachers look to supply

a knowledge of the older undefined terms, it is perhaps boys' col-

lections of things which may help to play the same part for the

undefined terms now under consideration.

It has already been said, that one view which might be taken

of mathematics is, that it is a special language invented for special

needs. Of these special needs one of the most important is that of

describing different kinds of classification. We might suppose a

person to collect so many kinds of things, as to become virtually a

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