Hobson, E.eI.            Math. QA 7. H68








On the Infinite and t;he Infinitesimal in Mathemratical Analysis.*
Presidential Address, by E. W. HOBSON, Sc.D., F.R.S.,
November 13th, 1902.
MR. PRESIDENT,
In the days of our forefathers, when an unsuccessful politician
had reached the end of his career, it was customary to grant him one
last privilege, that of delivering an address upon topics chosen by
himself to the assembled multitude on Tower Hill. Although my
conscience acquits me of having been guilty during my period of office
of conduct traitorous to the interests of our Society, I avail myself of
the correspondcling privilege accorded by our custom to a retiring
President.
The remark that the nineteenth century has been an age of unexampled progress in all branches of science has been so often made
as to have become a commonplace.   The remark is true in a preeminent degree of our own department of science. As is known to you
ail, at no earlier time has a more rapid development taken place in
all parts of mathematical science, involving the creation of entirely
new branches and of new and powerful general methods. However,
it is not in the main of these new developments and of the extensions
made in our science in the outward direction that I propose to speak
this evening. In the past century, and perhaps especially during the
second half of it, the attention of mathematicians has been devoted in
an unusual degree to a critical examination of the foundations of the
various branches of mathematical thought. In analysis, geometry,
and mechanics a close scrutiny has been made of the fundamental
assumptions and concepts. This scrutiny has resulted not only in a
large measure of restatement of the base principles of these departments of science, but has also powerfully reacted on the methods of
procedure within these departments, and has suggested new and
fruitful lines of research. Although outside criticism of the founda* Ordered by the Council to be printed a aa pamphlet and distributed to Meinber.,
to avoid the delay that would be occasioned by waiting for the appropriate issue of
Proceedings,




2            Dr. E. W. Hobson on the Infinite and     [Nov. 13,
tions of mathematics has at all times been abundant, the work of
underpinning the edifice of our science has been for the most part
carried out by the same workmen who have been engaged in the
general work of the structure, and especially in building new wings.
There are times when it is appropriate to draw attention in general
terms to the critical side of sorne part of our science, and I think it
will be adinitted that a Presidential address is such an occasion. I
have accordingly chosen as the subject of my discourse this evening
" The Infinite and the Infinitesimal in Mathematical Analysis."  It
will be found that my intention to speak of critical rather than constructional results admits of some considerable exceptions. These
exceptions will, however, illustrate the fact that pertinent criticism of
fundamentals almost invariably gives rise to new construction. On
such a subject as that I have chosen, I cannot hope to have anything
essentially new to say; but, nevertheless, I venture to hope it may
not be profitless, if I state as explicitly as I can, what seems to me to
be the trend of thought in this connexion at present prevailing among
mathematicians as the result of the labours of some of the most distinguished of their number during the last half century. I am
strengthened in this view by my knowledge of the fact that many
British inathematicians, absorbed as they rightly are in the technique
of their science, and in the work of applying it to the quantitative
description of natural phenomena, have not yet fully appreciated the
results of recent movements in mathematical thought in this connexion.
In some of the text-books in common use in this country, the symbol oo
is still used as if it denoted a number, and one in all respects on a
par with the finite numbers. The foundations of the integral calculus
are treated as if Riemann had never lived and worked. The order in
which double limits are taken is treated as immaterial, and in many
other respects the critical results of the last century are ignored. It
would, however, be unjust not to recognize the fact that a great improvement in these respects has been shown in some of the most
recent of our text-books.
Essentially connected as views about the infinite and the infinitesimal are with the most fundamental notions on which analysis is
based, with the concepts of number and magnitude, with the notions
of continuity and discreteness, with the doctrine of limits in all the
various forms in which it has appeared, with the nature of the ideal
objects with which mathematical thought operates, these ideas have
been a. subject of unceasing controversy since the very commencement
of abstract thought-a controversy which has by no means ceased at




1902.]    the Infiinitesimal in Mathematical Analysis.        3
the present time. The fact that these fundamentals lie on the
border-line across which mathematics passes into the wider region of
philosophy has brought it about that in all ages philosophical
thinkers as well as mathematicians, before as well as after the two
classes ceased to be identical, have occupied themselves with the
attempt to introduce clearness into the doctrine concerning them.
The kind of judgments which are made in mathematical thinking,
forming as they do a class which in certain aspects are of a comparatively simple character, have at all times formed a kind of
touchstone on which epistemologists have tested their general theories
of knowledge.
To attempt to give, even in outline, a history of thought upon the
subjects of the infinite and the infinitesimal, involving as it would the
task of tracing the history of the various theories of the infinitesimal
calculus, would be altogether beyond the scope of such a discourse as
the present one. In order, however, to make clear what has been
the precise effect of the more recent movements of thought in this
order of iceas, it will be necessary for me to take a brief glance at
the mode in which the subject presented itself at various times to
thinkers confronted with the ordinary problems of mathematical
analysis.
How, then, did the problems of analysis present themselves to the
earliest mathematicians? What were the elements with which those
mathematicians bad to work? The two notions of number and of
magnitude with which they had to operate in problems of a geometrical
or kinematical character, have points of resemblance and also points
of difference. Both number and magnitude appear by their very
nature to be unlimited in two directions: there is no greatest
number or magnitude, and (excluding zero) no smallest one. A set
of numbers or of magnitudes may be conteniplated, each one of which
is definite and finite, and yet the set contains numbers or magnitudes
which are greater than any particular number or magnitude which
we may choose to assign. A similar possibility holds as regards
smallness. A symbol to which are assigned successively the increasing
or diminishing values of the numbers or magnitudes in the set contemplated, is said to become in the one case indefinitely great, in the
other case indefinitely small. At any particular stage the symbol
represents a finite number or magnitude, but the absence of a limit
is designated by the phrase " becoming indefinitely great, or small."
The indefinitely great thus described is the potentially infinite, das
uneigentlich Tnendliche, and expresses the mere absence of upper limit




4            Dr. E. W. Hobson on the Infinite and     [Nov. 13,
to a variable. In this form, as expressing a mere potentiality, the
infinite and the infinitesimal seem so inevitable a necessity of thought
as hardly to give rise to differences of opinion, except, perhaps, upon
matters of language. But when it is conceived that these mere
potentialities pass into actualities, thatfixed numbers or magnitudes
exist whicl are infinite or infinitesimal, that the merely indefinitely
great becomes an actual infinite, or the merely indefinitely small
becomes an actual infinitesimal, the region of serious controversy lias
been reached-a controversy which is still proceeding, and about the
modern aspect of which I shall have some remarks to make later on.
In respect of the actually infinite, there have been exhibited at
different times and by various thinkers the extremes of faith and of
scepticism; there have been believers and sceptics, critics and
freethinkers, idealists and empiricists. The infinite of mathematics
has at times been treated with that familiarity which is bred of
innocent inappreciation. Bold generalizations have been made in
which rules applicable to the finite were uncritically and unconditionally extended to the indefinitely great, as if that represented an
actuality necessarily subject to the same rules of operation as the
finite. At other times, the desire to remain on what was felt to be the
firmer ground of empirical knowledge has led almost to a denial
of all validity to the conceptions of the infinite and the infinitesimal,
and to all processes involving their use. It is noteworthy that both
these attitudes of mind have at different times been of direct advantage to science, and that the most opposite tendencies in regard to
this order of ideas have led to the advancement of knowledge.
One of the principal forms in which an indefinitely great number of
operations occurs is that of infinite series, which were introduced in the
seventeenth century. The mathematicians of the eighteenth century
used these series freely, without troubling themselves much as to
questions of convergence. Early in the nineteenth century came a
rude awakening. In a letter written by Abel*' in 1826 we read:
"Divergent series are in toto an invention of the Devil, and it is a
disgrace that any one should venture to found on them the smallest
demonstration. One can get out of them anything one likes when
one employs them, anid it is they which have produced so many difficulties and so many paradoxes." And further: " I have become prodigiously attentive to all that; for, if one excepts the cases of the
* See Abel's correspondence, p. 16, in the volume Niels Heitrikc Abel: Mémnorial
publié a l'occasion du centenaire de sa naissance,




1902.]    the Infinitesimal in Mathematical Analysis.        5
most extreme simplicity-for example, geometric series-there is
scarcely in the whole of mathematics a single infinite series of which
the sum is determined in a rigorous manner. In other words, all that
is most important in mathematics is without foundation. Most of
the things are exact, that is true, anc it is extraordinarily surprising.
I am trying to find out the reason." In our time, now that the use
of divergent series has been to a large extent placed upon a sound
mathematical basis, the work of Poincaré, Stieltjes, Borel, and
others, has given us an answer to the question that puzzled Abel.
-To the Greeks, and to later thinkers, magnitude, as given by the
intuition of space, time, and especially of motion, appeared to present
itself essentially as a continuum, the intuitional or sensuous continuum. On the other hand, number (and it must be recollected that
the Greeks only knew rational numbers) appeared to be essentially
discrete. Fractional numbers arose historically from the necessity
for the representation of the sub-divisions of a unit magnitude into
equal parts. The Greek discovery of the existence of magnitudes
which are incommensurable with a given unit, by exhibiting the inadequacy of such discrete numbers for the complete representation of
prima facie continuous magnitude, served to emphasize the distinction
contained in the antinomy of the continuous and the discrete.
In order completely to envisage the problem of analysis as it presented itself to the minds of mathematicians from the earliest commencement of the attempts to deal with geometrical and kinematical
problems numerically, we must take into account that peculiarity of
the human mind in virtue of which it is in general unable to deal
with an object of thought as a whole, but is obliged to consider it
piecemeal, dividing it up into some kind of elements, taking account
of these, and reconstructing the object mentally by a process of
synthesis. This inability to grasp a scheme of relations at once as a
whole, involved as it is in our essentially discursive modes of apprehension, leads to the necessity of dividing up a geometrical figure, or
a portion of time, into parts regarded as elements of the whole, dealing separately with these, and of obtaining final judgments as to the
integral properties of the figure or the motion, by means of a process
of summation. This necessity of mathematical method led directly
to a discussion of the nature of the elements of which magnitudes
were to be regarded as made up: Could, for example, the straight line
be legitimately regarded as made up of points? If so, of how iany?
Ought it not rather to be regarded as made up of infinitesimal
elements, each of which possess all the properties of the finite




6            Dr. E. W. Hobson on the Infinite and     [Nov. 13,
length? Ought such elements to be regarded as fixed or as essentially
in a state of flux? Such were the questions which inevitably presented themselves as soon as men began to investigate geometrically
or analytically the properties of curves and surfaces, to determine
areas and volumes.
Again, in order to deal with a geometrical figure, not only hiad the
figure to be divided up into elements, but qualitative changes in the
figure had to be introduced: for example, a curve without corners
had to be replaced by a rectilineal polygon with corners, if its leiigth
was to be found. Here we have the origin of the method of limits, in
its geometrical and its arithmetical forms, and here we come across
the central difficulty of the mode in which a limit was regarded as
being actually attained. A limit, which appeared only as the unattainable end of a process of indefinite regression, and to which
unending approach was made, had, by some process inaccessible to
the sensuous imagination, to be regarded as actually reached; the
chasm which separated the limit from the approaching magnitudes
had in some mysterious way to be leapt over; the attainment of a
numerical limit, and an actual qualitative change in a geometrical
figure were to be regarded as somehow taking place simultaneously
as the result of a process which contained no principle of termination
within itself.
The germ of the methods of the infinitesimal calculus appears in
a geometrical form in the method of exhaustions, employed by the
Greek geometers. It was by this method that Archimedes showed
that the area of the surface of a sphere is four times that of a great
circle, by which he expressed the area of the surface of a right cone,
and solved other problems of a similar nature. We have an example
of this method in the proof in Euclid XII. 2, that the circumferences
of circles are as their diameters; in this case the quantity to be determined is trapped in between two sets of polygons, the one circumscribed to, and the other inscribed in, the circle; as the number of
sides of the polygons is increased the space between the two sets of
polygons is exhausted; the proof that the required result is obtained
is then carried out by the method of reductio ad absurdzm. This
method would, in the cases in which it can be carried out, leave
nothing to be desired as regards rigour, provided the existence of the
limit is a priori admitted.
It will be observed that the Greeks did not deem it necessary to
define the length of a circle, or other curve; that every curve has a
length, and every surface an area, was taken by them to be a truth




1902.]    the Infinitesimal in Mathematical Analysis.         7
obvious from intuition. Naturally they were not led by intuition to
contemplate the existence of not-rectifiable curves.
In the method of indivisibles employed by Cavallieri, Pascal,
Roberval, and others, straight lines are regarded as made up of an
infinite number of points, surfaces as made up of lines, and volumes
of surfaces. This method was applied with considerable success
before the introduction by Newton and Leibniz, of the methods of the
Infinitesimal Calculus, but it appears to have been regarded, by some
at least of those who employed it, in the light of a shortened mode of
procedure in which the method of exhaustions is used with an
abbreviated form of language, rather than as a method, the principles
of which, when taken literally, were to be regarded as rigorous.
Thus Pascal writes: "J'ai voulu faire cet avertissement, pour montrer
que tout ce qui est démontré par les véritables règles des indivisibles,
se démontrera aussi à la rigueur et à la manière des anciens; et
qu'ainsi l'une de ces méthodes ne diffère de l'autre qu'en la manière
de parler; ce qui ne peut blesser les personnes raisonnables quand on
les a une fois averties de ce qu'on entend par là."
The infinitesimal calculus, in the form which was devised by
Leibniz, has usually been regarded as the art of employing infinitesimal quantities as auxiliaries for the purpose of finding the
relations between certain quantities of which the existence is assumed.
In order to find the relations between certain quantities, some of
which are constant, others variable, the system is imagined as having
arrived at a determinate state regarded as fixed; this state is then
compared with other states of the same system, which are regarded
as continually approaching the first state, so as to differ arbitrarily
little from it. These other states of the system are regarded only as
auxiliary systems introduced to facilitate the comparison between
the parts of the fixed one. The differences of corresponding quantities
in all these systems can be regarded as arbitrarily small, without
changing the quantities which define the fixed state, and the relations
between which are to be found; these differences are the infinitesimals,
and unity divided by one of these infinitesimals was regarded as
giving rise to an infinite quantity. The question as to the true nature
of these infinitesimals gave rise to almost endless discussions; the
views which have been maintained with regard to them fall under
three main heads. By some, the infinitesimals have been regarded
as fixed objects, having a real existence and in a state of rest outside
the ordinary realm of magnitude, two finite magnitudes which differ
by an infinitesimal being regarded as equal to one another. A second




8            'Dr. E. W. Hobson ot the Infinite and    [Nov. 18,
view as regards infinitesimals is that they are ordinary magnitudes
essentially in a state of motion towards zero. This conception of
magnitudes continually in a state of flux has been sarcastically
described by P. Du Bois Reymond as follows:-" As long as the book
is closed there is perfect repose, but as soon as I open it there commences a race of all the magnitudes which are provided with the
letter d towards the zero limit."  The third view as to the nature of
the infinitesimals is that they  are simply ordinary magnitudes
too small to be perceived by the senses, and possessing thus only a
relative smallness. This view is that of those mathematicians who
regard a geometrical point as simply an object whose size is too
small to be perceived by the means at our command; a line as a
volume of which two of the dimensions are insensible; and so on.
The empiricists of this school refuse to idealize objects of perception
which form the subject of calculations, by bringing them under exact
abstract definitions; the calculus thus regarded is an approximative
system in which the results make no claim to absolute exactness,
but only to freedom from errors which are observable. Apart altogether from the difficulties as to the true nature of the differentials,
it will be observed that in the Liebnizian calculus the existence of
the magnitudes between which the relations are in any special
problem to be found is regarded as a priori known, or, in other
words, no doubt is admitted as to the existence of the limit; this it
has in common with the Greek method of exhaustions, of which it is
essentially a translation into a more analytical and convenient form.
In the method of limits devised by Newton, and employed in a
different form by later writers, infinitesimals are not employed singly,
but the ratio of two quantities at the moment when they vanish is
contemplated, and forms what was later known as the differential
coefficient.  These  "ghosts of departed quantities,"  as Bishop
Berkeley derisively designated them, whose ratio at the moment of
their disappearance is the quantity dealt with, present very much the
same kind of difficulty as in the Leibnizian form of the calculus.
No criterion was obtained for the determinacy of such an ultimate
ratio, whose existence was regarded as obvious from intuition.
In that form of the Newtonian calculus known as the method of
fluxions, the appeal to intuition was made more cogent by representing the vanishing ratio in the form of a velocity; that a moving
point has at every instant necessarily a definite velocity was apparently hardly doubted until comparatively recently. We now know
that such a velocity has no such unconditional existence as was
supposed.




1902.]     the Infinitesimal in Mathematical Analysis.          9
Speaking of the method of vanishing ratios, Lagrange writes:
"Cette méthode a le grand inconvénient de considérerles quantités, dans
l'état où elles cessent, pour ainsi dire, d'être quantités; car quoiqu'on
conçoive toujours bien le rapport de deux quantités, tant qu'elles
demeurent finies, ce rapport n'offre plus à l'esprit une idée claire et
précise, aussitôt que ces termes deviennent l'un et l'antre nuls à la
fois."  This clear perception on the part of Lagrange of the difficulty
at the root of the method of limits or of differential coefficients, was
doubtless a determining factor in deciding him to embark upon his
great attempt to place the calculus upon a basis independent of the
idea of infinitesimals or of their ratios. The title of his great work,
' Théorie des fonctions analytiques, contenant les principes du calcul
différentiel, dégagés de toute considération  d'infiniment petits,
d'évanouissans, ce limites et de fluxions, et réduits à l'analyse
algébrique des quantités finies," contains the most concise statement
of his aim. Although his attempt was in principle a failure, his idea
of making Taylor's series the cardinal form by which functions are to
be represented must be regarded as containing the germ of the theory
of analytical functions which was developed with so much success at
a later period.
In the various forms of the infinitesimal calculus to which I have
referred, a crucial difficulty is that of the existence of the limit. That
this difficulty is no merely imaginary one, but indicates a real gap
in the logical basis of the systems, receives an a posteriori confirmation
from the discoveries made in the latter half of the nineteenth century,
that special restrictions in the nature of the functions employed are
necessary for the validity of the ordinary processes of the calculus.
The   exhibition  by Weierstrass and others, of continuous nondifferentiable functions, the resulting investigations of the restrictive
conditions over and above that of continuity which are necessary for
the existence of a differential coefficient, Riemann's investigation of
the conditions of integrability of a function, the various theorems discovered as to the conditions of the reversibility of the order of double
limits, all indicate that the existence of a limit cannot be presumed
apart from  all restrictive conditions.  The failure of the older
analysis to exhibit the existence and nature of such restrictive conditions is a clear proof of defectiveness in the logical basis of that
analysis. That the earlier mathematicians were usually able to
obtain correct results by means of their methods, is due to the fact that
the functions with wbich they operated were of a comparatively
simple character; in point of fact, almost all the functions which are




10          Dr. E. W. Hobson on the Infinite and      [Nov. 13,
required for the investigation of the problems arising from ordinary
intuition satisfy the restrictive conditions first brought to light in our
day.
I now corne to consider the changes which have been brought about
in the point of view of mathematicians with respect to the matters I
have discussed, as the result of the critical efforts of recent times.
In the first place, the notion of number, integral or fractional, has
been placed upon a basis entirely independent of measurable magnitude, and pure analysis is regarded as a scheme which deals with
number only, and has, per se, no concern with measurable quantity.
Analysis thus placed upon an arithmetical basis is characterized by
the rejection of all appeals to our special intuitions of space, time,
and motion,* in support of the possibility of its operations. It is a
very significant fact that the operation of counting, in connexion
with which numbers, integral and fractional, have their origin, is
the one, and only absolutely exact, operation of a mathematical
character which we are able to undertake upon the objects which we
perceive; this is due to the fact that the operation is of a highly
abstract character, since in counting objects, all special qualitative
or quantitative peculiarities of the objects counted are treated as
irrelevant.  On the other hand, all operations of the nature of
measurement which we can perform in connexion with the objects
of perception contain an esssential element of inexactness, corresponding to the approximative character of our sensuous intuition.
The theory of exact measurement in the domain of the ideal objects
of  abstract geometry is not immediately   derivable from  intuition, but is now usually regarded as requiring for its development a previous independent investigation   of the nature and
relations of number.  The relations of number having been developed on an independent basis, the scheme is applied by the help
of the principle of congruency, or other equivalent principle, to the
representation of extensive or intensive magnitude. In any such
theory of measurement the non-arithmetical conception of a unit
is involved. Those departments of science, including geometry, in
which abstract measurement is applied are thus regarded as fields
of application for analysis; but they do not directly contribute
towards the development of pure analysis, although they may, no
doubt, suggest to it problems for treatment in accordance with its
* It is not intended here to prejudge the questions as to the part which intuition
may have in the formation of the concepts of number.




1902.]    the InfiJnitesimal in Mathematical Analysis.      11
own principles. This complete separation of the notion of number,
especially fractional number, from that of magnitude, involves, no
doubt, a reversal of the historical anc psychological orders. It is,
however, no uncommon occurrence that the logical order of a subject
should be very different from the historical order in which the concepts of the subject have arisen. Is it not an essential part of our
scientific procedure that, in our conceptual schemes, factors are
separated from one another, which intuitionally appeared in combination?
The  so-called  arithmetization of analysis is,. and lias been,
accepted in somewhat various degrees by different mathematicians.
The extreme arithmetizing school, of which, perhaps, Kronecker
was the founder, ascribes reality, whatever that may mean,
to integral numbers only, and regards fractional numbers as
possessing only a derivative character, and as being introduced
only for convenience of notation.  The ideal of this school is that
every theorem of analysis should be interpretable as giving a relation between integral numbers only.  The validity and feasibility
of this ideal I cannot here discuss. Some mathematicians, on the
other hand, like P. Du Bois Reymond, while using to a large extent
the ideas and methods of arithmetical analysis, appear still to regard the notion of continuous magnitude as a necessary part of the
foundations of the subject.
The true ground of the difficulties of the older analysis as
regards the existence of limits, and in relation to the application to measurable quantity, lies in its inadequate conception
of the domain of number, in accordance with which the only
numbers really defined were rational numbers.  This inadequacy
has now been removed by means of a purely arithmetical definition
of irrational numbers, by means of which the continuum of real
numbers has been set up as the domain of the independent variable
in ordinary analysis. This definition has been given in the main
in three forms-one by Heine and Cantor, the second by Dedekind,
and the third by Weierstrass. Of these the first two are the simplest
for working purposes, and are essentially equivalent to one another;
the difference between them is that, whilst Dedekind defines an
irrational number by means of a section of ail the rational numbers,
in the Heine-Cantor form of definition a selected convergent aggregate of such numbers is employed. The essential change introduced
by this definition of irrational numbers is that, for the scheme
of rational numbers, a new scheme of numbers is substituted, in




12           Dr. E. W. Hobson on the Infinite and    [Nov. 13,
which each number, rational or irrational, is defined and can be
exhibited in an indefinitely great number of ways, by means of a
convergent aggregate of rational numbers.  In this continuum  of
real numbers the notion of number is, as it were, raised to a
different plane. By this conception of the domain of nuniber the
root difficulty of the older analysis as to the existence of a limit is
turned, each number of the continuum being really defined in such
a way that it itself exhibits the limit of certain classes of convergent sequences.  It would, of course, be futile to define a
number by means of a convergent aggregate, were it not shown-as
has been, in fact, done-that tle ordinary operations of arithmetic
car be defined for such numbers in such a way as to be in agreement with the ordinary scheme of operations for the rational
numbers taken on the lower plane. It should be observed that the
criterion for the convergence of an aggregate is of such a character
that no use is made in it of infinitesimals, definite finite numbers
alone being used in the tests.  The old attempts to prove the
existence of limits of convergent aggregates were, in default of a
previous arithmetical definition of irrational number, doomed to
inevitable failure. It could not, for example, in general be shown
that an unending decimal formed according to prescribed rules
possessed a limit, since it was clearly impossible to infer from the
existence and properties of a set of rational numbers, the existence
of a nunlber which itself is in general not rational, and was therefore undefined within the domain of operation. A considerable part
of the newer analysis consists of putting the criterion for the convergence of an aggregate into various forms suitable for application
in various classes of cases. The convergence of an aggregate having
in any given case been established by the application of one or
other of such derivative rules, the aggregate itself defines the limit.
In all such proofs the only statements made are as to relations of
finite numbers, no such entities as infinitesimals being recognized
or employed. Such is the essence of the e proofs with which we
are familiar. In such applications of analysis-as, for example, the
rectification of a curve-the length of the curve is defined by the
aggregate formed by the lengths of a proper sequence of inscribed
polygons. The length is'not regarded as something whose existence
is a prior known. In case the aggregate is not convergent, the curve
is regarded as not rectifiable. If it can be shown that the lengths of
these inscribed polygons form a convergent aggregate which is independent of the particular choice of the polygons of the sequence,




1902.]    the Infinitesimal in Mathematical Analysis.      13
the curve is rectifiable, its length being defined by the number given
by the aggregate.
The older analysts regarded the domain of the real variable, or of
a set of real variables, as the continuum given by our intuition of
space, tinme, and motion; this continuum was usually accepted uncritically as a notion completely given by intuition and hardly
capable of further analysis; however, those points of the continuum
which could be represented by number (rational number) formed
only a discrete aggregate, and thus the variable had to pass through
values which were not definable as numbers. This intuitive notion
of the continuum  appears to have as its content the notion of
unlimited divisibility, the facts that, for instance, in the linear continuum we can within any interval PQ find a smaller one P'Q',
that this process may be continued as far as the limits of our perception allow, and that we are unable to conceive that even beyond the
limits of our perception the process of divisibility in thought can
come to an end. However, the modern discussions as to the nature
of the arithmetic continuum have made it clear that this property
of unlimited divisibility, or connexity, is only one of the distinguishing characteristics of the continuum, and is insufficient to mark it off
from other domains which have the like property. The aggregate of
rational numbers, or of points on a straight line corresponding to such
numbers, possesses this property of connexity in common with the
continuum, and yet is not continuous; between any two rational
numbers another pair can be found, and this process may be continned until we obtain an arbitrarily small interval. The other
property of an aggregate which is characteristic of a continuum, is
that of being, in the technical language of the theory of aggregates
(MIengenlehre) perfect: the meaning of this is that all the limits of
convergent sequences of numbers or points belonging to the aggregate
themselves belong to the aggregate; and, conversely, that every
number or point of the aggregate can be exhibited as the limit of
sucha sequence. The aggregate of rational numbers does not possess
this property of being perfect, since the limit of a sequence of such
numbers does not necessarily belong to the aggregate.  That the
aggregate of rational numbers is not perfect, or even closed, is the
root defect of that aggregate, which led to the difficulty as regards
the existence of limits, in the older analysis. The two properties
of connexity and of perfection are regarded as the necessary and
sufficient characteristics of a continuum; it is remarkable that in
analysis the latter property of a continuum, which was not brought




14          Dr. E. W. Hobson on the Ilfinite and      [Nov. 13,
to light by those who took the intuitive continuum as a sufficient
basis, is in some respects the more absolutely essential property for
the domain of a function which is to be submitted to the operations
of the calculus.  It has in fact been shown 'that many of the
properties of functions, such as continuity, differentiability, are
capable of precise definition when the domain of the variable is not
a continuum, provided, however, that domain is perfect; this has
appeared clearly in the course of recent investigations of the properties
of non-dense perfect aggregates, and of functions of a variable whose
domain is such an aggregate.
The arithmetical continuum having been defined and explored, it
is then postulated that on a straight line there exists one point, and
one only, corresponding to each number of the arithmetical continuum, and that no other points exist on the straight line; this
fixing of the point contents of a straight line amounts to an exclusion
of the contemplation of fixed infinitesimal lengths. Similarly, it is
postulated that in three-dimensional space there exists one point,
and one only, corresponding to each specification of three coordinates
of the point by means of numbers, and that the points whose existence is thus postulated exhaust the space. The arithmetizing school
thus regard the nature of the geometrical continuum as being cleared
up and described by means of the previously defined arithmetical
continuum; this is, of course, a reversal of the traditional view.
The view I have   sketched of the philosophy of the continuum
does not meet with the universal acceptance of mathematicians, as an
adequate scheme, at the present time. As an example of a rival
scheme I may briefly touch upon the one propounded by Veronese.
He develops the notion of the abstract linear continuum from the
intuitive side, and traces the consequence of supposing that on a
straight line two intervals PQ, P'Q' can co-exist siich that the
smaller P'Q' is so small compared with PQ that no integer n can be
found which will make n.P'Q' exceed PQ, thus rejecting what is
known as the axiom of Archimedes. This amounts to the affirmation
of the existence of fixed infinitesimal lengths, and of fixed infinite
lengths, on the straight line. In this scheme, when a unit length is
chosen on the straight line, Dedekind's section of rational points is
made, not by a single point, as in the Cantor-Dedekind scheme, but
by an infinitesimal length, that is, by a length which is infinitesimal
relative to the scale of measurement chosen. Veronese contemplates
the existence of an indefinite series of scales in the linear continuum,
such that each unit is infinite compared with one belonging to a




1902.]    the Ifinitesimal in Mathematica, Analysis.       15
lower scale, and is infinitesimal compared with a unit belonging to a
higher scale; he then proceeds to introduce a scheme of infinite and
infinitesimal numbers which will suffice for the complete representation of points of the straight line. On this view, the DedekindCantor continuum, when represented on the straight line, is only a
relative continuum, that is, relative to the particular scale employed
in the representation; the absolute continuum would require for its
representation an indefinite series of infinite and infinitesimal
numbers. As to the validity of Veronese's scheme, that is, as to its
consistency with a logical theory of magnitude, I do not propose to
express any opinion; the matter has been a subject of considerable
controversy. Assuming, however, its validity as a possible scheme,
it does not affect the validity of the Cantor-Dedekind scheme; the
comparative simplicity of the latter would indicate it as the natural
basis for analysis, and for the applications to the measurement of
magnitude. One of the most interesting results on the speculative
side of abstract science, which has been obtained in the nineteenth
century, is that it is possible to set up two or more conceptual
systems, each self-consistent, but contradictory with one another,
each of which provides a sufficient representation of the facts of perception; the most striking example of this has been in geometry,
where it has been shown that, under a certain limitation, Euclidean,
hyperbolic, and elliptic geometry may each afford a sufficient representation of the properties of figures in perceptual space. We are
entitled to postulate the existence of whatever points we choose upon
that ideal object, the line of geometry, provided our scheme does not
contradict itself, and, further, provided the ideal object thus constituted affords an adequate representation of the concrete lines which
we perceive in the external world. Between two such schemes intuition can make no choice, and in abstract science we make that
choice between them which is dictated by considerations of simplicity
and of suitability for the special purpose on hand.
The question as to the legitimacy of the use of infinite numbers,
that is, not merely of the use of a variable which is regarded as becoming indefinitely great, but of numbers which are actually infinite
and to be regarded as capable of entering into relations, is a matter
which has been discussed by philosophical thinkers from the time of
Aristotle onwards.  The balance of opinion seems to have been
decidedly against the validity of the conception of such numbers; in
support of this negative view, Aristotle himself, Locke, Descartes,
Spinoza, ard Leibniz may be quoted. The grounds of the objection




16           Dr. E. W. Hobson on the Infinite and       [Nov. 13,
to the introduction of such numbers may in the main be reduced to
three heads. First, it is said that a number is, by its very nature,
finite: this is supported by the plea that all actual operations of
coLnting and measuring are performed upon finite aggregates or
finite magnitudes; to refute this view, it may be urgec that the introduction of infinite numbers, if it can be made at all, will justify
itself by a proof of the capability of such numbers for the representation and characterization of non-finite aggregates; in fact, it may
be held that the objection contains a petitio principii.  Secondly, it
has been widely held that a scheme of infinite numbers represents an
endeavour to make distinctions and determinations within the infinite;
whereas the true infinite admits of no determination. If the infinite
be identified with the all-eiibracing absolute of idealistic philosophy,
it will probably be admitted that such an absolute admits of no distinctions, for " omnis determinatio est negatio "; however, the question
arises whether a domain may not exist which, though not finite, is
still not to be regarded as engulfed in the absolute, and which therefore may still in some measure admit of definition and determination,
and which may require a special non-finite system of number for the
specification of its characteristics; such an intermediate domain has
been named by Cantor the "transfinite " or " superfinite." Thirdly,
it has been urged that finite numbers would be unable to maintain
themselves as against infinite ones; that the finite and its relations
would be absorbed in the infinite, and could enter into no relations
with it: the value of this objection can be estimated a posteriori only,
if and when a system   of infinite or transfinite numbers has been
actually defined and the nature of its connexion with the finite
brought to light.  That mathematicians still shrink from leaving
what they regard as the firm ground of the finite based upon experience is illustrated by a remark in an introductory passage in
Tannery's work: Introduction à la Théorie des Fonctions d'une Variable
réelle. He writes, " On peut constituer entièrement l'analyse avec la
notion de nombre entier et les notions relatives à l'addition des
nombres entiers; il est inutile de faire appel à aucun autre postulat, à
aucune autre donnée de l'expérience, la notion de l'infini dont il ne
faut pas faire mystère en mathématiquese réduit à ceci, après chaque
nombre entier il y'a un autre."  However sufficient the restriction
to the merely indefinitely great, here indicated, may be for the more
ordinary purposes of analysis, provided, however, that an exploration of the properties of the continuum of real numbers is not carried
too far, I hope to be able to show, as clearly as possible in the brief




1902.]    the Lifinitesimal in Maitheneatical Atnalysis.  17
space at my disposal, that the introduction by Cantor of systems of
transfinite numbers is justified by the primary necessities of our
analytical system; it may be justified in point of utility by the
numerous applications which are being made, both in analysis and in
geometry, of the conceptions and results of the theory of aggregates,
to express the characteristics of which these transfinite numbers are
required. No mathematician will wish to make a mystery of the
infinite in analysis; mathematics has nothing to do with mysteries
except to endeavour to remove thein. It is to be remarked that the
introduction into analysis of the transfinite numbers was historically
by no means the result of a purely speculative tendency to explore the
unknown and mysterious, and certainly did not arise from any taste
on the part of their inventor for " tricks to show the stretch of human
brain "; their introduction arose principally out of the necessities
of investigations connected with the peculiarities of Fourier's series
and of the functions representable by such series. Cantor writes:"Zu dem Gedanken, das Unendlichgrosse nicht bloss in der Form
des unbegrenzt Wachsenden und in der hiermit eng zusammmenhangenden Form der im siebenzehnten Yahrhundert zuerst eingefihrten convergenten unendlichen Reihen zu betrachten, sondern es
auch in der bestimmten Form des Vollendetunendlichen mathematisch
durch Zahlen zu fixiren, bin ich fast wider meinen Willen, weil im
Gegensatz zu mir werthgewordenen Traditionen, durch den Verlauf
vieljahriger wissenschafticher Bemiihungen und Versuche logisch
gezwungen worden."
The first real breach in the infinite —one which established a true
line of cleavage-was made when Cantor showed that the aggregate of
rational numbers is enumerable, whereas the aggregate of real numbers,
rational and irrational, is unenumerable. This denotes that a (1, 1)
correspondence can be established between the rational numbers in
any given interval, and the aggregate of positive integers, whereas no
such correspondence can be established between the numbers of a
continuum and the aggregate of integral numbers. Thus the rational
numbers can be counted and the irrational numbers cannot be counted.
All the rational numbers in any interval can be arranged in a definite
order (not of magnitude), so that one of them stands first and each parWicular number has its assigned place. No such arrangement can be
made when the irrational numbers of the interval are taken into
account. This far-reaching result brings out in a strong light the
difficult nature of the conception of the continuum as a given totality.
If it be asked in what sense can the numbers of the continuum be




18          Dr. E. W. Hobson on the Infinite and     [Nov. 13,
considered as forming a given or determinate aggregate, we must contrast this aggregate with that of the rational numbers or with that
of the integral numbers. These latter are not, of course, given in
the sense that we can exhaustively exhibit them by means of symbols
on a sheet of paper. We could only do that in the case of a finite
aggregate; but they are given in the sense that we can say of any
particular number where it is to be found in a regularly arranged
scheme. On the other hand, the aggregate of all real numbers is not
given in the same sense; no ule, and no set of rules, can be given by
which we could obtain successively all the numbers of the aggregate,
so that each particular number would necessarily appear in the
course of the procedure; and this is a consequence of the unenumerable
character of the aggregate. The aggregate of real numbers can be
regarded as given only in the sense that every possible real number
that we may choose to define by means of an analytical process
belongs to the aggregate. This somewhat negative conception of its
determinacy is an essential characteristic of the unenumerable
aggregate. How far the mathematicians of the future will rest
satisfied with this conception of the arithmetic continuum, time alone
can decide.
When we count a fiinte number of objects we take them in some
definite order, and establish a correspondence between them and the
ordinal numbers. The last ordinal. number employed, we call the
ordinal number, or simply the number of the collection. When we
take into account the fundamental property that this number is independent of the order in which the objects are counted, we identify
this number with the cardinal number of the collection. Thus, in
dealing with finite aggregates, the distinction between the ordinal
and the cardinal number, though logically existent, may be practically
disregarded. This, however, is no longer the case when we deal with
non-finite aggregates; here cardinal and ordinal numbers must be
kept quite distinct, and their properties must be developed on
different lines. The theorem that the ordinal number of an aggregate
is unaltered by changing the order of the elements of the aggregate
no longer holds.
In order to exhibit the way in which transfinite ordinal numbers
are required when we deal with non-finite aggregates, I propose to
refer to a well known paradox, that of Achilles and the tortoise, which
in various forms has afforded an interesting exercise to logicians.
Achilles goes ten times as fast as the tortoise, and the latter has ten
feet start. When Achilles has gone ten feet the tortoise is one foot in




1902.]    the Infinitesimal in Mathematical Analysis.         19
front of himn; when Achilles has gone one foot further the tortoise is
o ft. in front; when Achilles has gone yo ft. further the tortoise
is 6TO ft. in front; and so on, without end; therefore Achilles will
never catch the tortoise. The fallacy, of course, lies in the surreptitious transcending of the convergent process when the word
" never " is used in the conclusion. Let us indicate the successive
positions of Achilles referred to, by the ordinal numbers 1, 2, 3,...
suffixed to the letter A, so that Ai, A1, A~,... represent the
positions of Achilles mentioned in the paradox.     These points
AP, A.2, A3,... have a limiting point, which represents the place
B2           3 _ __B34 B4 o__
AI          "I 2 -3 -l 4 Ao      Ac>+,  A,+2 A,2
where Achilles actually catches the tortoise. This limiting point is
not contained in the set of points A, A,, A3,...; if we wish to represent it, we must introduce a new symbol o, and denote the point by
A". This symbol w represents Cantor's first transfinite ordinal
number. It does not occur in the series 1, 2, 3,..., but is preceded by
all these numbers, and yet there is no number immediately preceding
it; it is the first of a new series of numbers.
I may now, perhaps, be allowed to tamper with this classical
paradox to the extent of supposing that there is a second tortoise
moving at the same rate as the first, and ten feet in front of it, and of
supposing that we wish to represent the positions of Achilles when
he is 1 ft., - ft., 1 - ft.,... behind the second tortoise. To represent
these positions we naturally take A,,+, A,,,2 A,+3,...; the place where
Achilles catches the second tortoise will be denoted by A+L,  or A,>.
These numbers, w + 1, o +2, w + 3,..., w2, are the transfinite ordinal
numbers immediately following w. It thus appears that, as w +
succeeds w, the two cannot be regarded as identical; thus, w +1 > w.
If now we had commenced by denoting the positions of Achilles when
he was 100, 10, 1, -o,,... ft. behind the first tortoise by
B1, B2, B8..., so that A,^ and Bl,,, represent the same point, we see
that the place where the tortoise is caught would still have to be
represented by B,>; it is, consequently, necessary to distinguish
betwe n w+l1 and 1+ o, and to write 1+w -=. The two relations
o -1 +-, w + 1 > w, enable us to illustrate the extent to which a finite
number is able to maintain itself against a transfinite one.
The above perennially instructive example of a limiting process at
once throws light upon the relation of the unending process to the
limit, and upon the necessity for the introduction of transfinite




20          Dr. E. W. Hobson on the Infinite and      [Nov. 13,
numbers for the representation of the limit which is not itself contained within the region of the convergent process. If we could
imagine that we had no independent knowledge of the position and
existence of the point at which Achilles overtakes the tortoise, we
should be in the position of the olcer analyst or geometer in face of
most of the problems which he solved. In the above paradox we
have, by artificially involving ourselves in a convergent process,
placed the limiting point, and all points beyond it, outside the reach
of the process itself; for the representation of this point and the
points beyond it we have to commence anew with a fresh series of
ordinal numbers. The only reason why, in ordinary life, transfinite
numbers are unnecessary is because we do not make use of such convergent processes. It would be easy for me to arrange artificially a
series of points of tiie during the delivery of the present address
which would be such that the moment of the termination of my
address could only be represented by a transfinite number; higher
transfinite numbers would be required to denote the times of all subsequent events this evening.
Although, for the reason I have indicated, transfinite numbers are
unnecessary for the purposes of ordinary life, this is by no means the
case in certain departments of mathematical analysis, where we are
in many cases compelled to make use of convergent processes. It
appears that a region which from one point of view belongs to the
finite, may from another point of view  belong to the transfinite,
and it is frequently just this latter point of view which the exigencies
of analysis compel us to adopt; hence arises the necessity for, and the
justification of, the use of transfinite ordinal numbers.
I propose, as an illustration of the use of transfinite ordinal
numbers, to give a simple means which I have devised for their
systematic representation by a set of points on a given finite segment
of a straight line.  On the straight line AB  let us denote by
AB
P0, P1, P, P,... those points at which the expression log Ap-, where
k is a fixed number greater than unity, has the values 0, 1, 2, 3,...
A                             B
the point P0 coincides with A, and the point B can only be represented
by P". Now take any one of the segments PP,.P,+, this we may for
convenience represent on an enlarged scale; denote by Q,0, Qrl, Q,2,...




1902.]   the Infinitesimal in Mathenzatical A1nalyss'.    21
P,. P, ~
the points on P,.P,.+1, at which logk. Qp'- takes the values 0,1, 2,3,...;
QPr+l
__ _ _  1  Q2............  Q,.r
P.                            P ".+1
thus P,.+l can only be represented by Q,.". Suppose this to have been
done with every segment of AlB, and now imagine all the points Q
to be marked on AB, and to be numbered from left to right;
in P oP we shall have    0, 1, 2, 3,..., w,
in P1P2,+, w,     w+2,..., w2,
in PP3,,        W 2+1, w2+2,..., w3;
the point B can be represented by ww or w2. If now we proceed to
take each segment Q,.s Qs+,,,  and to divide this in a similar manner
at points B for which log, Qi- '  h4 bas the values 0, 1, 2, 3,..., and
at points R for which log Q,s. 
then imagine all the points R thus obtained to be marked on the
original straight line AB, and numbered as before, from left to right,
it will be seen that all the numbers 2p + wq + r will be required, and
that the point B can be represented by o3. The points P0, P1,..., P.
will have for their ordinal numbers 0, w2, wt2,...., 3; the point Q,-. will
be numbered wr- + ws; the finite numbers are all used up in the first
sub-segment of AB. By proceeding in a similar manner to further
sub-division, we may exhibit on AB the ordinal numbers
wPp, + w ~  p_- +... -po,
and the point B will then be represented by wo'~'. The distance from
A of the point represented by any of these numbers can be easily
expressed.
I turn now to the subject of the transfinite cardinal numbers.
Every two finite aggregates between the elenlents of which a (1, 1)
correspondence can be established have the same cardinal number,
and the cardinal number of an aggregate is independent of the
order in which the objects are arranged.  The extension of this
notion of cardinal number to non-finite aggregates leads to the
conception of the power (Mdchtigkeit, _puissance) of an infinite
aggregate, which power is represented by a transfinite cardinal
number. Two infinite aggregates between the elements of which a
(1, 1) correspondence can be established have the same power or
cardinal number. Thus, the aggregate of all rational numbers,
or of rational numbers in a given interval, has the same power or




22          Dr. E. W. Hobson on the Infinite and     [Nov. 13,
cardinal number as the aggregate of integral numbers. This may
be denoted by a, and is the first transfinite cardinal number. As I
have before mentioned, the continuum of real numbers cannot be
placed in (1, 1) correspondence with the integra] numbers, and has,
therefore, a power or cardinal nuniber different from a; this is
usually denoted by c. It is a surprising fact that a continuum of
two, three, or any number of dimensions lias the same power c as
that of the linear continLum-L i.e., a (1, 1) correspondence exists
between all points in an area or in a volume and the points in a
straight line, or in a finite segment of a straight line.  It was, in
fact, until recently supposed that all known aggregates have either
the power of the aggregate of natural numbers or else that of the continuum. It lias, however, now been shown that the aggregate of all
possible functions of a variable x of which the domain is a continLuous interval (a, b) has a power higher than that of the continuum;
this higher cardinal number is denoted by f. It should, however,
be observed that, if the function is restricted to be analytic, the
aggregate of such functions then has the power c of the continuum.
The numerous attempts which have been made to prove that a and c
are consecutive cardinal numbers-that is to say, that no aggregate
exists whose power exceeds a and is less than c-have hitherto been
unsuccessful.  This remains, for the present, as a hiatus in the
theory of transfinite cardinal numbers.  I have here been able to
touch only the fringe of the subject of transfinite numbers; but my
object has been to indicate, as clearly as is possible in the necessarily
brief space I have allotted to them, how they necessarily arise when
we try to investigate the peculiarities of non-finite aggregates. The
profound study of these numbers and their relations, and especially
of their connexion with the theory of the types of ordered aggregates, which has been made by G. Cantor, who, I am proud to remember, has been added to our list of foreign members during my
term of office, has resulted in the creation of a veritable arithmetic
of the infinite, which seems to be destined to have an ever-increasing
range of application in analysis and geometry.
The place which the conception of the infinite occupies in the
various schemes of geometry, especially the manner in which infinite elements are adjoined to the finite elements, is a subject on
which many interesting remarks might be made.    This, however,
does not belong to the subject of my discourse this evening, and
would, in any case, better be left for treatment by some one more
competent to discuss it than I can claim to be.




1902.]    the Infinitesicmal in Mathemnatical Analysis.     23
In the minds of many men who are engaged in the active work of
assisting in the progressive development of science, there is a certain
impatience with what they are apt to regard as a hypercritical
attitude towards fundamental concepts; this feeling of impatience
exists perhaps in exceptionally large measure in the English mind,
whose genius is in a preponderating degree directed towards the
concrete, and upon which, to a considerable extent, purely abstract
questions seem to exercise a peculiar repulsion. However, taken on
the whole, the impulse towards clear thinking, which leads men to
make an ever renewed dissection of the fundamentals of science, and
to an ever renewed attempt to state fundamental principles in a form
which shall satisfy more nearly the canons of logical thought, is an
ineradicable tendency of the human mind, and I, for one, cannot but
regard its presence as one of the conditions which are in the long run
necessary to render possible the progress of scientific knowledge.
Even from the point of view of those who regard mathematics as
existing exclusively for the purposes of physical research, it is a
short-sighted policy to discourage that free development of mathematics on its abstract side, which is probably a necessary stage in
the process of sharpening the tools which mathematics provides for
the use of the physicist.
Even in the remarks on one aspect of our science which I have
made this evening, I think it has been apparent that criticism, and
even erroneous criticism, is not infrequently the parent of construction. It may well be true that perfect intellectual transparency with
regard to the fundamentals of any branch of knowledge is an unattainable ideal. May it not even be the case that a perfect comprehension of anything would involve a perfect comprehension of
everything?  Are there not those with us who assert that an
analysis of the abstract creations of the human nind inevitably,
when pushed far enough, leads to contradiction, and that this is a
necessary consequence of the divorce of these ideal objects from
reality? However this may be, the thinkers of each age must do
what they can with the possibilities open to them, in faith that,
however far short of what is completely satisfactory to the intellect may be the results to which they are led, the efforts of their
generation in the direction of criticism, as well as of construction,
may be found by those who come after them     not to have been
entirely fruitless.
Printed by C. F. Hodg'son & Son, 2 Newton Street, High Holborn, W.C,