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INTRODUCTION - - - - - - 1



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




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



\. 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.


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



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


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




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



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



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.


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."


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.


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

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


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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Online LibraryC ElliottModels to illustrate the foundations of mathematics → online text (page 1 of 11)