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V; Number and Its Developments

Number and Its Developments

Arithmetic the Science of Number taken in the general sense of the term now employed by Mathematicians is the most advanced and also the most purely abstract department of Natural Science. Abstract Arithmetic is usually spoken of as a formal Science like Logic; and it may therefore perhaps occasion some surprise that I should speak of it as a department of Natural Science. However it resembles other branches of Natural Science in the fact that it was in connection with physical experience that it took its origin. Its earliest development as a Science consisted in setting up a conceptual scheme for the representation of certain aspects of the physical world. In this respect it does not differ generically from other branches of Natural Science; although the stages by which it became formal and deductive were much more rapid than in those departments which deal with what may perhaps be described as more complex and less superficial aspects of the world of physical phenomena. I have already suggested that every branch of Natural Science when it reaches a sufficiently advanced stage of development tends to become purely formal and deductive. The Science of Geometry has reached this stage of development and the same thing may be said of Mechanics. It is unnecessary and would be outside the scope of these lectures to enter into any discussion of the questions whether or how far what are known as the truths of Arithmetic embody a priori knowledge and may thus be regarded as logically necessary truths presupposed as a precondition of experience and whether this accounts for the kind of apodictic certitude which we attach to the knowledge of simple arithmetical relations. Whatever views be held as regards the much debated subject of the philosophy of Arithmetic the fact remains that the knowledge of the individual has its origin as had that of the race in physical experience; although it may of course be maintained that the function of physical experience was simply to awaken and make explicit conceptions already present in the mind in a latent form. In any case the part played by physical experience in the development of arithmetical concepts in an explicit form affords I think sufficient justification for reckoning Arithmetic amongst the Natural Sciences. I have spoken of Arithmetic including the higher Mathematical Analysis which is a development of more elementary Arithmetic as the most advanced branch of Natural Science; this must be taken to mean that it has more completely than in the case of any other department with the possible exception of Geometry which I shall consider in the next lecture reached the stage in which it consists of the detailed development of the implications of a purely conceptual scheme; no further recourse being required to observation or experiment in order to test its range of applicability to describe certain aspects of the physical world.

Whilst Arithmetic in the wide sense of the term is the most advanced and the most abstract branch of Science its rudimentary parts are far more popularly understood and used in applications than is the case with any other branch of Natural Science. The grocer when he weighs out and sells his goods makes use of conceptions which were developed only by a long process of evolution; when he enters his receipts in his books the notation he uses is a warrant of the great importance of notation in a formal Science and embodies one of the triumphs of our race as a mode of economizing thought. The concepts of Arithmetic in their more elementary form or in the higher developments to which they attain in Mathematical Analysis pervade all departments of Natural Science and all the Mechanical arts. The Philosopher in his reflections on spatial and temporal relations on number and quantity on matter and motion is in a region of thought in which the boundary between his own domain and that of the Mathematical Analyst is difficult to delimit with precision. The Epistemologist has always been accustomed to consider Mathematical knowledge as a kind of touchstone on which to test his theories of the nature of knowledge. The dominant views in some departments of philosophical thinking have been notably influenced by the results of recent Mathematical research and may not improbably be in future further modified from the same quarter. The universality of Arithmetic in the Natural Sciences consists of the fact that numbers or variables which are interpreted numerically enter into every conceptual scientific scheme that has reached a stage characterized by extreme precision of statement. For Arithmetic in the developed form of Mathematical Analysis provides the very language in which the precise descriptions contained in such schemes are clothed.
The concepts of unity and of number or degree of plurality were in their formation occasioned by physical perceptions.. The precise mode in which these concepts were formed is a matter for psychologists to discuss and determine. As formal categories they lie at the base of the Science of Arithmetic and consequently of Mathematical Analysis which is now regarded as essentially no more than abstract Arithmetic carried to a higher stage of development by the help of certain postulations concerned with the domain of the infinite or transfinite. The notion of unity the product of mental activity in relation with an environment is the form under which an object is subsumed when it is the object of attention. Thus unification is the result of an act of attention which involves a differentiation of the presentational continuum. A physical object brought under this category of unity may for all other purposes be recognized as possessing any degree of complexity. It is sufficient in order that the object may be subsumed under the form of unity that is regarded as a single object that it be so far distinct within the presentational continuum as to be recognized as discrete and identifiable. What external marks are necessary that an object may be so recognized as discrete is a matter for the judgment of the mind which performs the act of unification. There is a large degree of arbitrariness limited only by the powers of perception of the individual mind in this act of unification or of apprehending an object under the form of unity; no special or uniquely definable physical characteristics of the object are essential for this purpose but only some sufficient degree of differentiation of the object from its environment.
The notion of plurality is involved when attention is paid either successively or simultaneously to objects each of which is subsumed under the form of unity. This notion at first indefinite takes the form of definite plurality when a collection or aggregate of objects is attended to. Such a group or collection is then regarded both as a single whole to which unity is attributed and also as a definite plurality consisting of a set of objects each of which is regarded for the particular purpose as one. The single objects which compose the collection need not possess any parity as regards size weight or any other special quality but may be of the most diverse characters although in practice they usually have some similarity of nature which forms the ground of their being treated as a collection. In any case a certain logical parity is ascribed to them in virtue of the fact that each one of them is subsumed under the form of unity. The fact has recently been pointed out and illustrated by Prof. James Ward1 that the earliest conception of number as a degree of plurality probably arose as the result of immediate intuition of the differing qualitative characters of very small groups of objects. Thus a pair of objects can be intuitively recognized as qualitatively different from a group of three objects without recourse to the process of counting. This immediate intuition of the number of objects in a group is facilitated when the objects are arranged in some recognizable pattern. It is probable that not only human beings but also higher animals possess the power of discerning intuitively this qualitative characteristic of a very small group of objects and of recognizing that something is changed in a group originally say of three objects when one of them has disappeared. It is clear however that this avenue to the concept of number is of very limited scope. All further development of the concept was made in connection with the process of counting or tallying. For this process the two notions of order and correspondence are requisite.
In virtue of the notion of order relative rank is assigned to each object in a collection so that the collection becomes an ordered aggregate. In actual counting the order is usually assigned to the objects during the process itself as an order in time and this may be done in an arbitrary manner. The order of the elements in an aggregate may however be assigned in a manner dependent on their sizes weights or other qualities or in a manner dependent on their relative spatial positions. Order may however be regarded as an abstract conception independent of any particular mode of ordering; for an aggregate to be an ordered one it is necessary and sufficient that each object of the aggregate be recognized to possess a certain rank in virtue of which it is definite as regards any two of the objects which may be selected which of the two has the lower and which the higher rank. The notion of correspondence underlies the process of tallying or that of counting on the fingers. The objects of one aggregate are regarded as standing in a logical relation with those of another aggregate of such a character that a definite element of one aggregate is regarded as corresponding to a definite element of the other aggregate. The correspondence between two aggregates is complete or (1 1) when to each object of either of the aggregates there corresponds one object and one only of the other aggregate1. The number or degree of plurality of an aggregate can then be defined as the concept of the quality of plurality which the aggregate has in common with all aggregates with which it can be placed in complete correspondence. Thus a number is the concept of the quality which the members of a family of similar aggregates have in common. The number although not characteristic of a plurality is still in formal Arithmetic regarded as one of the integral numbers. If in counting an aggregate the process is stopped before the aggregate is completely counted we may regard the part counted as a section of the aggregate; such a section can then be taken to be an aggregate having a particular number. The sections of an aggregate together with the aggregate itself have an order assigned to them the same as that of the objects of the aggregate itself. Thus the numbers of the sections are themselves ordered and we thus conceive the numbers 1 2 3 etc. to be arranged in a definite order usually called their natural order.
A number which specifies the degree of plurality of an aggregate is called a cardinal number but a number which specifies the rank of a particular object in an aggregate is called an ordinal number Cardinal and ordinal numbers are distinguished from one another in that their descriptive functions are different. It has been maintained by some writers on the foundations of Arithmetic that the notion of an ordinal number is logically prior to that of a cardinal number. This does not however seem to be necessarily the case; either concept can be employed in a systematic treatment of the subject as fundamental the other being then regarded as derivative.
Before Arithmetic can be considered to be a developed Science the further step must be made of the introduction of a scheme of relations between the numbers dependent on the operations of addition and multiplication with the inverse operations of subtraction and division. There has been a very prolonged discussion amongst philosophers as to whether the judgment expressed by such a proposition as that 7 + 5 = 12 is an analytic judgment in the sense that the truth of the proposition can be deduced from an analysis of the concepts of the three numbers or whether that judgment is synthetic in the sense that some further extraneous knowledge is required to warrant the judgment. There can however be no doubt that historically and in the individual the explicit knowledge of such relations in simple cases was empirical; being derived from actual counting of the combined aggregate when two aggregates are amalgamated into one. The general conceptual scheme of relations involving operations arose as a generalization of knowledge obtained from such physical experience. It seems certain that the fundamental notions that I have specified must have been possessed by primitive man in an implicit form long before the notion of abstract number reached an explicit and developed form. But the earliest records we possess of ancient peoples those of the Egyptians the Babylonians and the Chinese show that they possessed arithmetical knowledge that had already attained a very considerable degree of development.
The origin of fractional numbers is doubtless to be ascribed to the necessities arising in connection with measurement. The division of an object into equal parts and the representation of one or more of such equal parts was the empirical origin of the notion of a fraction. But in our present theories of Arithmetic the concept of number both integral and fractional is taken to be independent of any conceptions relating to measurement. That operation is now regarded as requiring the application of Arithmetic but as not connected with the foundations of the subject the necessary empirical basis of which rests exclusively upon the operation of counting. A fraction as for example 3⁄5 may be regarded as representing the operation of counting three objects each of which belongs to an aggregate of five objects no assumption of equality of the objects in respect of size or other quality being requisite. On this basis the theory of the Arithmetical operations involving fractional numbers can be made to rest. The more purely formal theory of fractional numbers as usually expounded regards each one as defined by a pair of integers forming a single object a couple; formal laws are then postulated as to the relations between such couples forming the basis of the scheme of operations involving them.
It is very important to remark that the operation of counting conceptual objects in which the integral numbers are employed and the extension of that process in which fractional numbers are employed are both free from that element of approximativeness which appertains to every operation of actual measurement. The number of an aggregate of objects is an exact description of that aggregate in a certain respect but number as applied to the measurement of extensive magnitude represents the measure only subject to the inexactitude inherent in our sensuous perception even when refined instruments are employed. This unique peculiarity of the application of the conceptual scheme of Arithmetic to the perceptual domain depends upon the fact that the operation of unification is independent of any special physical properties of the object to which it is applied and involves only the mental operation of general differentiation of the object from the environment. Measurement on the other hand is directly concerned with one at least of such physical properties. The application of the conceptual scheme of Arithmetic for descriptive purposes in the physical domain so long as we confine that application to the original purpose for which it exists has a certain absolute exactitude which does not appertain to any other conceptual scheme in its application to the concrete and which no longer appertains to Arithmetic when it is applied to represent measures of physical magnitudes.
In all schemes of a symbolic character such as written language symbolic logic or Arithmetic the facility with which the scheme works depends very largely upon the choice of a suitable and simple notation. In the case of Arithmetic the notation which we employ and which is of Indian origin represents perhaps the greatest labour-saving invention that has ever been made. The notation for numbers is systematic in the sense that all integral numbers can be denoted on a uniform plan by the employment in the decimal scale which we use of ten distinct symbols. This number of symbols could be reduced to two if we employed the dyad scale. The principle that the place which a digit occupies in the set of digits used to represent a number indicates the mode in which it is interpreted to represent a multiple of a power of ten is the crucial point in the principle upon which the notation is founded. It is a remarkable fact that neither the Greeks nor the Romans were in possession of a systematic notation for numbers. An attempt to carry out even a simple addition or multiplication in which the numbers are represented by Roman numerals is the simplest path to a conviction of the vast importance of the great Indian invention which renders arithmetical operations practicable in accordance with uniform rules. There is an important invention included in our system of notation which appears to have been introduced later than other parts of the system. This is the use of a symbol denoting zero which is employed to indicate the absence of a particular power of ten in the representation of a number as the sum of multiples of powers of ten together with a digit which is one of the first nine numbers or is itself zero.
The most ancient account of the Arithmetic of the Egyptians is contained in the Papyrus of Ahmes and is entitled Direction for attaining a knowledge of all secret things. In this arithmetic both integral and fractional numbers appear but the notation is not a systematic one. Only fractions with unity as numerator are employed such a fraction being denoted by the integer which represents the denominator but with a dot placed over it. The only exception is the fraction 2⁄3 for which a special sign was employed. Other fractions are expressed as the sums of fractions all of which have unity for their numerators and a table is given in the Papyrus for expressing fractions in this manner. Ahmes also dealt with some problems involving arithmetical and geometrical progressions. A Babylonian table has been discovered in which the first sixty square numbers are given and also some cube numbers. In this table a hexadecimal system of notation is employed in which the place in order represents a power of 60. This was in essence a systematic notation and so far an anticipation of the invention of a later age but the Babylonians do not seem to have employed a zero to represent vacant units. Although the Greeks possessed no systematic arithmetical notation and no sign for zero they managed to perform arithmetical operations of some difficulty. For example the greatest Greek Mathematician Archimedes in his discussion of the quadrature of the circle inscribed a regular polygon of seventy-two sides in a circle and obtained a good approximation to the ratio of a side to the diameter of the circle. To do this he had to extract the square roots of several large numbers to a sufficient degree of accuracy for his purpose. The nature of the method he employed in performing these operations has been a subject of considerable discussion in our time. It has frequently been said that the ancient Greeks were great Geometers but poor calculators. It is certain that their command of Arithmetic was seriously hampered by the unsystematic character of the notation they employed. The Greek Mathematicians even the arithmetician Diophantus did not take the important step of introducing negative numbers in order to remove the impossibility of carrying out the operation of subtraction of a greater number from a lesser. This step was however taken by the Indians; the Indian Astronomer Aryabhata employed distinct names for positive and negative numbers which denote respectively possession and debt. Even the representation of positive and negative numbers by segments of a straight line in opposite directions was known to the Indians.
As a result of the introduction of difference of sign we possess what is known as the ordered aggregate of rational numbers a conceptual arithmetic scheme in which the operation of subtraction is in all cases a possible one. We have here an example of the extension of the domain of number in order to extend the scope of a formal operation and thereby to increase the utility of the conceptual scheme in regard to its applications to the description of magnitudes. As we shall see this extension is but one example of several extensions of the domain of number made in accordance with the same general principle. The inadequacy of rational numbers for the complete representation of all ideal magnitudes was discovered by the Greeks. A rigorous theory of the ratios of incommensurable magnitudes the discovery of the existence of which is attributed to Pythagoras was given by Euclid in the tenth book of his treatise on Geometry. A valid proof was given by the Pythagoreans that the ratio of the length of the diagonal of a square to that of a side cannot be exactly represented by any rational number. On the formal side the restrictions which hold within the domain of rational numbers upon the possibility of operations such as the extraction of roots of numbers pointed in the direction of an extension of the conception of number of such a character that in an extended domain of number such operations would no longer be impossible when both rational and irrational numbers are recognized as falling within the extended domain.
The modern theory of the aggregate of real numbers arose out of the exigency of the requirement of a complete theory of ideal magnitudes and out of the limitations which exist in the domain of rational numbers as regards arithmetical operations within that domain. The term “real number” is a somewhat unfortunate one but it arose historically from an attempt to distinguish between these numbers and the so-called imaginary numbers of which I shall presently speak. The theory of the aggregate of real numbers as developed in a complete form by Cantor Dedekind and Weierstrass involves a postulation related to the conception of the infinite. The integral numbers 1 2 3... form an ordered sequence which has a first term the number 1 but no last term. The very principle of the sequence is that any particular term is succeeded by another term and thus the existence of a last term would be in contradiction with this principle. The sequence is thus an example of the unending or indefinitely great. The postulation is made that nevertheless the sequence can be regarded as a single object for thought having definite properties. Considered as an aggregate of distinct objects that of the integral numbers this single object the aggregate is infinite or transfinite. In the form given to this postulation by Cantor the sequence is regarded as defining a new ordinal number ω which is not identical with any of the finite ordinal numbers 1 2 3... but is of higher rank than any of them and is not immediately preceded by any one of them. This new number ω is called the first transfinite ordinal number and is made by Cantor the starting point of a new series of transfinite ordinal numbers. The more general form of the postulate to which I have referred is that if any sequence of objects P1 P2 P3... is so defined by means of a finite set of rules that a definite object of the sequence corresponds to each integer of the sequence 1 2 3... then the unending sequence Pl P2 P3... may be regarded as a single definite object with definite properties. The justification for making this postulation is two-fold. In the first place it must not lead to contradiction when the logical consequences of the scheme based upon it are scrutinized and in the second place the scheme which involves this postulation must be of utility as a conceptual scheme for the application of number in a general theory of magnitude.
For the purpose of defining a real number which is not rational that particular kind of unending sequence is employed which is known as a convergent sequence of rational numbers. If we carry out for example the steps of the process of extracting the square root of the number 2 we obtain successive sections of a convergent sequence. The rational numbers which form the elements of the sequence cannot of course be exhibited exhaustively but we possess a set of rules which suffice for the calculation of any one of the numbers by means of a definite procedure involving in each case a finite number of applications of the rules. In this sense the sequence is regarded in accordance with the postulate as a single definite object defined by a norm or set of rules although it cannot be completely exhibited as in the case of a finite set of objects. We observe that when we have carried out a large number of steps of the process we have determined a rational number represented by a decimal with a large number of digits and this differs from any of the later rational numbers which may be determined by a decimal in which a corresponding large number of places are occupied by zero. This is a particular example of a convergent sequence the rational numbers successively obtained representing a set of continually closer approximations to the object which we regard as defined by the sequence namely the irrational number √2. In the general case every convergent sequence of rational numbers is taken to define a real number and as in the case of √2 this real number is not necessarily a rational number. The establishment of the complete theory of real numbers involves a detailed investigation which fixes the precise theory of identity and of relative order of the real numbers so defined. It also establishes the essential fact that the arithmetical operations which express relations between rational numbers can all be extended with enlarged scope to the case of real numbers.
The result of this investigation is that we have before us an aggregate that of the real numbers positive zero and negative in which any two of the real numbers have a definite relation of rank specifying their relative order in the aggregate. This aggregate has two properties of capital importance. In the first place every convergent sequence of which the elements are real numbers itself defines a real number known as the limit of the sequence. This is expressed by the statement that the domain of real numbers is closed. The domain of rational numbers is not closed in this sense of the term since a convergent sequence of rational numbers does not necessarily define or have as its limit a rational number. The second property is that every real number can be exhibited in an indefinite number of ways as a sequence of real numbers. This property may be described as that of convexity and also appertains to the aggregate or domain of the rational numbers. Both these properties in one of which the domain of rational numbers is lacking are essential to the fitness of the aggregate of real numbers for the purpose of the complete conceptual representation of linear magnitudes.
The question has frequently been asked how can Number which is essentially discrete be employed for the complete representation of magnitude which is essentially continuous? In considering the answer which should be given to this question it is necessary to scrutinize the meaning which can be assigned to the expression “continuous.” As in all other such cases we must consider both the perceptual and the conceptual meanings which may be attached to the term continuum. On the perceptual side in which we are considering actual measurement of a magnitude the notion of continuity or absence of gaps implies that between any two magnitudes of the kind considered there exist other magnitudes and that the process of continual contemplation of magnitudes filling up the gaps between any pair can be carried out to an extent limited only by our powers of perception even when instruments of the utmost precision are employed for the purpose. The notion of continuity regarded in this way gives rise to what may be described as the sensible continuum of magnitudes and containing as it does an inherent element of indefiniteness dependent on the approximative character of our sensuous perceptions it can only be raised into the position of a precise conceptual scheme by a process of abstraction and idealization. For the representation of all actual measurements the rational numbers are sufficient and can be applied to represent any actual magnitude to any required degree of approximation the degree of approximation which it is worth while to aim at being dependent on an estimate of the inevitable errors which the mode of observation entails. The exigencies of our method of representing aspects of the perceptual world by ideally exact conceptual schemes necessitate however the development of a theory of measurement in which the characteristic properties of a conceptual continuum are assigned by definitions and postulations. That the employment of rational numbers only is insufficient for the purposes of such a conceptual scheme was known as we have before remarked as early as the time of the Greek Mathematicians. The modern theory of the aggregate of real numbers or of the arithmetic continuum as it is now called has been devised as a scheme sufficient for the purpose of denoting the magnitudes in the conceptual continuum which is taken to be the idealization of the notion of the sensible continuum of perception. The aggregate of rational numbers since it is not closed is not a conceptual continuum in the sense in which the term is applied to the aggregate of real numbers. It has been charged against Mathematicians that in setting up such a scheme as the arithmetic continuum they have introduced an unnecessary complication in view of the fact that rational numbers suffice for the representation to any required degree of approximation of all sensibly continuous actual magnitudes; that in fact an instrument has been created of an unnecessary degree of fineness for the purposes to which it is to be applied. The answer to the charge is that Mathematical Analysis which is based upon the arithmetic continuum and essentially consists of operations involving numbers would become unworkable as a conceptual scheme or at least much more cumbrous if the conception of irrational numbers were excluded from it. The results of operations involving rational numbers constantly lead to irrational numbers without which the operations would be impossible if their effects are to be regarded as definite. But in order to appreciate the full weight of this answer it is necessary to consider the great generalization of Arithmetic which is made when variables are introduced which denote unspecified numbers. The passage is then made from the primitive form of Arithmetic to Algebra in which the formal operations of Arithmetic are represented as relations between sets of unspecified numbers represented by non-numerical symbols. The result of an algebraic operation expressed by general formulae such for instance as the simple case of the solution of a quadratic equation would not always be interpretable when special numerical values are assigned to the symbols if the only admissible numbers were rational ones.
Without the employment of the conception of irrational numbers the function of Mathematical Analysis would be degraded to that of determining only approximate results of the operations it employs and in consequence its technique would have indefinitely greater complication of such a character that at least in its more abstruse operations it would break down or lead to results which contained a margin of error difficult to estimate.
I have described in general terms the gradual extension of the concept of number commencing with integral numbers proceeding to fractional numbers then to negative numbers and thus attaining to the aggregate of rational numbers and lastly by the employment of an ontological postulate the extension to the arithmetic continuum which is capable of describing adequately relations of magnitude in the conceptual domain of continuous magnitude. There is however one further extension of the conception of number essential for the purposes of general Mathematical Analysis which has very frequently been a stumbling-block for non-Mathematicians who have founded upon it the charge that it consists of a species of jugglery with symbols that from their very nature are meaningless. Allude to the introduction of complex or so-called imaginary numbers. The operation of determining a square root of a negative number is not a possible one within the domain of real number since the square of every real number is a positive number. On a principle similar to that by which the domain of numbers was extended by the introduction of positive and negative signs in order that the operation of subtraction might become always a possible one in the enlarged domain the domain of real number is further extended so as to become one in which every number whether positive or negative has a square root. The new domain is then that of what is known as complex number. The existence subject to the law of contradiction of a new number whose square is −1 is postulated; this is usually denoted by i. It can then be shown that in a new domain in which each number is the sum of a real number and of i multiplied by a real number where either of the real numbers may be any number belonging to the arithmetic continuum it is possible to postulate a consistent scheme of relations of operations. This scheme is of such a character that the laws of operations are in formal agreement with the laws which hold for operations which involve only real numbers. This extension of the domain of real numbers to that of complex numbers has the advantage that it involves a notably enlargement of the scope of algebraical processes. For example every quadratic equation has solutions which would not be the case if the existence of real numbers only be admitted and thus a much greater degree of generality is introduced into Arithmetic and Mathematical Analysis generally by the extension. Moreover the boldness of the Mathematicians who ventured upon this extension has its reward in the fact that the new set of numbers the complex numbers are applicable to the specification of the positions of points in a plane as is illustrated by the well-known Argand diagram. The theory of functions of a complex variable has become a most important branch of Analysis indispensable for many purposes among which are applications to abstract or Mathematical Physics. The popular prejudice against the use of the number i or √(−1) and of the whole system of complex numbers is based on the ground that √(−1) represents an impossible operation. When the matter is regarded aright there is no justification for this prejudice. What is a possible operation or what is an impossible one does not depend upon any absolute criterion of possibility but upon the characteristics of the domain in which operations are carried out; the possibility or impossibility is in fact relative to a particular domain. So long as the domain was that of signless number the operation of subtraction was not always a possible one for example 3 − 5 represented an impossible operation and could only be taken to represent an "imaginary" number in relation to the domain. Similarly the operation of extracting a square root of a negative number is only impossible within the domain of real number; it becomes a possible operation within the enlarged domain of complex number. The so-called imaginary numbers have just the same conceptual reality as the so-called real numbers provided the domain of such numbers has been defined in accordance with a precise scheme of definitions and postulations subject to the law of contradiction. The validity of the scheme having been justified its utility is the justification for its actual construction.
In the time of the Greek Mathematicians and also during the progressive period of Mathematical thought which commenced in the sixteenth century and has lasted to the present time the conceptions relating to the infinite and infinitesimal together with the related conception of a limit have almost continuously occupied the attention of Mathematicians and Philosophers. Much of the confusion of ideas on these matters which lasted from the time of the origin of the Differential Calculus created by Newton and Leibniz was due to a failure to distinguish with sufficient clearness of outline between the conceptual and the perceptual sides of the measurement of magnitudes and to the uncritical acceptance of notions derived from sensuous intuition as sufficient for the basis of a rigorous conceptual scheme. On the subject of the infinite and the infinitesimal the views expressed have frequently presented a diversity akin to that which has been exhibited in relation to general Philosophy. There have been all shades of believers sceptics pragmatists and finitists. Although it is a popular belief that Mathematics is of such a character as to leave no room for differences of opinion it is a fact that even at the present time there exist differences of opinion amongst Mathematicians about the foundations of the Science and more especially about matters in which the notion of the infinite is involved. In the earlier presentations of the Differential and Integral Calculus frequently called the Infinitesimal Calculus theories concerning infinitesimal magnitudes played a large part. The notion of an infinitesimal was frequently complicated with ideas about motion derived uncritically from spatial and temporal intuition. Resulting from the clarification of the conceptions of the foundations of Arithmetic and Mathematical Analysis resting upon a purely arithmetical basis which has taken place during the last half century the conception of infinitesimal numbers has been excluded from ordinary Arithmetical Analysis as an unnecessary conception in the scheme. Every number regarded for the purposes of the Calculus as existent belongs to the arithmetic continuum and is therefore finite if it be not the single number zero. The term infinitesimal is no longer used to denote a number or a magnitude; when it is used at all it is employed to describe a process of change and even for that purpose it is better avoided so as not to give rise to misunderstandings.
The fundamental conception of a limit to which I have already referred in connection with the definition of an irrational number emerged first amongst the Greeks in a geometrical form embodied in the method of exhaustions. It is remarkable that their conception of the nature of a limit attained a standard of rigour greater than that which obtained amongst modern Mathematicians before our own time. This is exemplified in the proof given by Euclid in the eleventh book of his Elements that the circumference of a circle varies as its radius. Archimedes applied the method of limits in a rigorous manner to various problems of what we now call integration such as the determination of areas and volumes. The advantages which the methods of Leibniz and of Newton had over that of Archimedes were of a practical kind as they had a form which made them readily applicable to calculations whereas the geometrical form in which the Greek method was clothed together with the absence of a convenient arithmetical scheme made the application of the Greek method to particular problems decidedly cumbersome. But in the matter of soundness of theory the method of Archimedes was superior to those of Leibniz and Newton. The construction of the arithmetic continuum was an absolutely necessary requisite for a rigorous theory of limits and thus for the foundation on a logical basis of the Differential and Integral Calculus. In default of such a construction when the notion of magnitude as directly given by sensuous intuition was taken as part of the basis of the doctrine of limits the frequent endeavours that were made to prove that every convergent sequence of numbers necessarily converges to a limit were doomed to inevitable failure.
That the contemplation of the infinite in some form is indispensable to Mathematicians arises from the fact that even comparatively simple problems such as those of the determination of the length of a curvilinear arc or of the area enclosed by a curve can in no case be solved by employing a finite number of the operations of arithmetic except when the curve consists of segments of straight lines. An approximation to the magnitude of an area can be found by division of the area into a sufficiently large number of rectangles leaving a small undetermined part of the area out of account. By increasing the number of rectangles indefinitely in a suitable manner the measure of the area required is exhibited as the limit of the sequence of numbers determined by the approximations. Thus the magnitude to be determined is only obtainable as the limit of a sequence involving an indefinitely great set of arithmetical operations. The Integral Calculus is concerned with methods of calculating the limits defined in such a manner. It is clear moreover that the very conception of an area or length as having a definite magnitude is dependent upon the concept of a limit and that it is only defined by and is dependent upon the existence of a definite limit to a sequence. The notion of the gradient or differential coefficient of one variable with respect to another is one which is essentially dependent upon the conception of a limit and the gradient only has a precise meaning when a rigorous theory of limits has been previously established. The older idea that a gradient may be regarded as the limiting ratio of vanishing quantities and which was justly derided by Bishop Berkeley affords a striking example of the hazy conceptions of fundamental matters with which Mathematicians for a long period of time contented themselves. The abstract conceptual schemes of an advanced character employed for the purpose of representing measurable physical processes are very frequently expressed in the form of relations between the gradients of various magnitudes with respect to other magnitudes or to the time-variable. These relations known as differential equations are of fundamental importance in connection with such abstract schemes of representation; and the mathematical theory concerned with the determination of the values of variables under certain conditions in terms of the time-variable is consequently for the purposes of theoretical physics of great importance. In the theory of differential equations we have one of the most important examples of the fact that the abstract theories of Pure Mathematics provide the means for utilizing conceptual scientific schemes for the purpose of representing in their quantitative aspect a great variety of physical phenomena.
The fact that Mathematical methods are in a very large class of cases unable to deal with objects or with processes except by breaking them up into parts and increasing indefinitely the number of those parts is a significant example of a limitation imposed upon us by what appears to be a definite characteristic of our modes of apprehension. We appear to be unable to grasp some of the relations of a whole without breaking it up as it were atomistically and then proceeding to reconstruct the whole by a synthetic process which is confined to a continual approach to the whole along the path of an endless regress which by its very nature is such that the whole is never actually reached within the process although a scrutiny of the laws of the regress may enable us to obtain a knowledge of the relations of the whole.
The mathematical theory of the numerically infinite and especially the developments of Georg Cantor and his school in the theory of infinite aggregates of objects have aroused considerable interest in the ranks of Philosophers. Some of them such as for example Josiah Royce have suggested that the results of the theory of transfinite numbers may throw light upon questions of General Philosophy such as the fundamental problem of the One and the Many. I cannot enter here into a discussion of the extension of the theory of the transfinite into the more general region of thought but I would suggest that extreme caution should be exercised in attempting to extend results of such a theory as that of transfinite aggregates to a domain wider than its original one. The theory has been created for a special purpose that of dealing with certain aspects of the numerically infinite and its constructions and results are all dependent upon a set of postulations and definitions which it has been the aim of investigators to make as precise in character as possible. The most careful scrutiny of the meaning to be attached to the terms employed in any extensions of the theory such as those to which I have alluded is of the last importance; otherwise there is a serious danger of falling into grave errors in setting up theories in which vague analogies involving the surreptitious use of such terms as the infinite in a sense different from that in which they are employed in the Mathematical theory take the place which should be occupied by a critically explored foundation.