Mathematics in the History of Thought

The science of Pure Mathematics, in its modern developments, may claim to be the most original creation of the human spirit. Another claimant for this position is music. But we will put aside all rivals, and consider the ground on which such a claim can be made for mathematics. The originality of mathematics consists in the fact that in mathematical science connections between things are exhibited which, apart from the agency of human reason, are extremely unobvious. Thus the ideas, now in the minds of contemporary mathematicians, lie very remote from any notions which can be immediately derived by perception through the senses; unless indeed it be perception stimulated and guided by antecedent mathematical knowledge. This is the thesis which I proceed to exemplify.

Pure mathematics, in its modern developments, may claim to be the most original creation of the human spirit. Music is another contender for that position. But let us set aside all rivals and consider the grounds for making this claim about mathematics. The originality of mathematics lies in the connections it reveals between things. Without human reason, these connections would be extremely hard to see. Thus, the ideas in the minds of contemporary mathematicians are far removed from any notions that come directly from the senses. The exception is when perception is stimulated and guided by prior mathematical knowledge. This is the argument I will now illustrate.

Suppose we project our imaginations backwards through many thousands of years, and endeavour to realise the simple-mindedness of even the greatest intellects in those early societies. Abstract ideas which to us are immediately obvious must have been, for them, matters only of the most dim apprehension. For example take the question of number. We think of the number ‘five’ as applying to appropriate groups of any entities whatsoever—to five fishes, five children, five apples, five days. Thus in considering the relations of the number ‘five’ to the number ‘three,’ we are thinking of two groups of things, one with five members and the other with three members. But we are entirely abstracting from any consideration of any particular entities, or even of any particular sorts of entities, which go to make up the membership of either of the two groups. We are merely thinking of those relationships between those two groups which are entirely independent of the individual essences of any of the members of either group. This is a very remarkable feat of abstraction; and it must have taken ages for the human race to rise to it. During a long period, groups of fishes will have been compared to each other in respect to their multiplicity, and groups of days to each other. But the first man who noticed the analogy between a group of seven fishes and a group of seven days made a notable advance in the history of thought. He was the first man who entertained a concept belonging to the science of pure mathematics. At that moment it must have been impossible for him to divine the complexity and subtlety of these abstract mathematical ideas which were waiting for discovery. Nor could he have guessed that these notions would exert a widespread fascination in each succeeding generation. There is an erroneous literary tradition which represents the love of mathematics as a monomania confined to a few eccentrics in each generation. But be this as it may, it would have been impossible to anticipate the pleasure derivable from a type of abstract thinking which had no counterpart in the then-existing society. Thirdly, the tremendous future effect of mathematical knowledge on the lives of men, on their daily avocations, on their habitual thoughts, on the organization of society, must have been even more completely shrouded from the foresight of those early thinkers. Even now there is a very wavering grasp of the true position of mathematics as an element in the history of thought. I will not go so far as to say that to construct a history of thought without profound study of the mathematical ideas of successive epochs is like omitting Hamlet from the play which is named after him. That would be claiming too much. But it is certainly analogous to cutting out the part of Ophelia. This simile is singularly exact. For Ophelia is quite essential to the play, she is very charming,—and a little mad. Let us grant that the pursuit of mathematics is a divine madness of the human spirit, a refuge from the goading urgency of contingent happenings.

Suppose we imagine ourselves going back many thousands of years. Let us try to understand how simply even the greatest minds in those early societies thought. Abstract ideas that seem immediately obvious to us must have been only dimly understood by them. Take the question of number, for example. We think of the number ‘five’ as applying to any group of five things: five fish, five children, five apples, five days. So when we consider how the number ‘five’ relates to the number ‘three,’ we think of two groups of things. One group has five members and the other has three. But we set aside all thought of the particular things that make up either group, or even what kinds of things they are. We think only of the relationships between the two groups that do not depend on the individual nature of any member. This is a remarkable feat of abstraction, and it must have taken humanity ages to achieve it. For a long time, people will have compared groups of fish with one another to see how many each contained. They will have done the same with groups of days. But the first person to notice the likeness between a group of seven fish and a group of seven days made a notable advance in the history of thought. He was the first person to entertain a concept belonging to pure mathematics. At that moment, he could not have foreseen the complexity and subtlety of the abstract mathematical ideas waiting to be discovered. Nor could he have guessed that these ideas would fascinate so many people in every generation to come. There is a mistaken literary tradition that portrays the love of mathematics as an obsession confined to a few eccentrics in each generation. But whatever the case, no one could have anticipated the pleasure to be found in a kind of abstract thinking unknown in society at the time. Thirdly, the enormous future effects of mathematical knowledge must have been even more completely hidden from those early thinkers. They could not have foreseen its effects on people's lives, their daily activities, their habitual thoughts, or the organization of society. Even now, our grasp of the true place of mathematics in the history of thought is very uncertain. I would not go so far as to say that writing a history of thought without a deep study of each era's mathematical ideas is like leaving Hamlet out of the play named after him. That would be claiming too much. But it is certainly like cutting the part of Ophelia. The comparison is remarkably exact. Ophelia is essential to the play, very charming, and a little mad. Let us admit that the pursuit of mathematics is a divine madness of the human spirit, a refuge from the relentless pressure of events as they happen.

When we think of mathematics, we have in our mind a science devoted to the exploration of number, quantity, geometry, and in modern times also including investigation into yet more abstract concepts of order, and into analogous types of purely logical relations. The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities. So that for example, no mathematical truths apply merely to fish, or merely to stones, or merely to colours. So long as you are dealing with pure mathematics, you are in the realm of complete and absolute abstraction. All you assert is, that reason insists on the admission that, if any entities whatever have any relations which satisfy such-and-such purely abstract conditions, then they must have other relations which satisfy other purely abstract conditions.

When we think of mathematics, we think of a science devoted to exploring number, quantity, and geometry. In modern times, mathematics also explores more abstract concepts of order and similar kinds of purely logical relationships. The point of mathematics is that it always leaves out particular examples and even particular kinds of things. For example, no mathematical truths apply only to fish, only to stones, or only to colors. As long as you are dealing with pure mathematics, you are in the realm of complete and absolute abstraction. All you assert is that reason requires us to accept the following: if any things have relationships that meet certain purely abstract conditions, they must also have other relationships that meet other purely abstract conditions.

Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about. So far is this view of mathematics from being obvious, that we can easily assure ourselves that it is not, even now, generally understood. For example, it is habitually thought that the certainty of mathematics is a reason for the certainty of our geometrical knowledge of the space of the physical universe. This is a delusion which has vitiated much philosophy in the past, and some philosophy in the present. This question of geometry is a test case of some urgency. There are certain alternative sets of purely abstract conditions possible for the relationships of groups of unspecified entities, which I will call geometrical conditions. I give them this name because of their general analogy to those conditions, which we believe to hold respecting the particular geometrical relations of things observed by us in our direct perception of nature. So far as our observations are concerned, we are not quite accurate enough to be certain of the exact conditions regulating the things we come across in nature. But we can by a slight stretch of hypothesis identify these observed conditions with some one set of the purely abstract geometrical conditions. In doing so, we make a particular determination of the group of unspecified entities which are the relata in the abstract science. In the pure mathematics of geometrical relationships, we say that, if any group of entities enjoy any relationships among its members satisfying this set of abstract geometrical conditions, then such-and-such additional abstract conditions must also hold for such relationships. But when we come to physical space, we say that some definitely observed group of physical entities enjoys some definitely observed relationships among its members which do satisfy this above-mentioned set of abstract geometrical conditions. We thence conclude that the additional relationships which we concluded to hold in any such case, must therefore hold in this particular case.

Mathematics is thought that abstracts completely from any particular example of what it discusses. This view of mathematics is far from obvious. In fact, we can easily see that it is not generally understood even now. For example, people habitually think that the certainty of mathematics makes our geometrical knowledge of physical space certain. This is a delusion that has corrupted much philosophy in the past and some philosophy today. Geometry is a particularly urgent test case. There are different possible sets of purely abstract conditions for relationships among groups of unspecified entities. I will call these geometrical conditions. I give them this name because they are generally similar to conditions we believe apply to geometrical relationships among things we directly observe in nature. Our observations are not quite precise enough to tell us the exact conditions governing the things we encounter in nature. But with a slight stretch of hypothesis, we can identify these observed conditions with one set of purely abstract geometrical conditions. In doing so, we give a particular identity to the unspecified entities that are the relata, or related things, in the abstract science. In the pure mathematics of geometrical relationships, we say that any group of entities may have any relationships among its members. If those relationships satisfy this set of abstract geometrical conditions, then certain additional abstract conditions must also apply to them. But when we turn to physical space, we speak of a particular observed group of physical entities. We say that certain observed relationships among its members do satisfy that set of abstract geometrical conditions. We therefore conclude that the additional conditions that must hold in any such case must also hold in this particular case.

The certainty of mathematics depends upon its complete abstract generality. But we can have no à priori certainty that we are right in believing that the observed entities in the concrete universe form a particular instance of what falls under our general reasoning. To take another example from arithmetic. It is a general abstract truth of pure mathematics that any group of forty entities can be subdivided into two groups of twenty entities. We are therefore justified in concluding that a particular group of apples which we believe to contain forty members can be subdivided into two groups of apples of which each contains twenty members. But there always remains the possibility that we have miscounted the big group; so that, when we come in practice to subdivide it, we shall find that one of the two heaps has an apple too few or an apple too many.

The certainty of mathematics depends on its complete abstract generality. But we cannot know in advance with certainty that the things we observe in the concrete universe fit our general reasoning. Take another example from arithmetic. Pure mathematics gives us a general abstract truth: any group of forty things can be divided into two groups of twenty. We are therefore justified in concluding that a particular group of apples we believe contains forty apples can be divided into two groups of twenty. But we may always have miscounted the large group. So when we divide the apples in practice, we may find that one of the two piles has one apple too few or too many.

Accordingly, in criticising an argument based upon the application of mathematics to particular matters of fact, there are always three processes to be kept perfectly distinct in our minds. We must first scan the purely mathematical reasoning to make sure that there are no mere slips in it—no casual illogicalities due to mental failure. Any mathematician knows from bitter experience that, in first elaborating a train of reasoning, it is very easy to commit a slight error which yet makes all the difference. But when a piece of mathematics has been revised, and has been before the expert world for some time, the chance of a casual error is almost negligible. The next process is to make quite certain of all the abstract conditions which have been presupposed to hold. This is the determination of the abstract premises from which the mathematical reasoning proceeds. This is a matter of considerable difficulty. In the past quite remarkable oversights have been made, and have been accepted by generations of the greatest mathematicians. The chief danger is that of oversight, namely, tacitly to introduce some condition, which it is natural for us to presuppose, but which in fact need not always be holding. There is another opposite oversight in this connection which does not lead to error, but only to lack of simplification. It is very easy to think that more postulated conditions are required than is in fact the case. In other words, we may think that some abstract postulate is necessary which is in fact capable of being proved from the other postulates that we have already on hand. The only effects of this excess of abstract postulates are to diminish our aesthetic pleasure in the mathematical reasoning, and to give us more trouble when we come to the third process of criticism.

When evaluating an argument that applies mathematics to particular facts, we must always keep three processes clearly separate. First, we must examine the mathematical reasoning itself for simple mistakes, such as lapses in logic. Every mathematician knows from bitter experience how easy it is to make a small error when first working out a line of reasoning. Yet that small error can make all the difference. But once a piece of mathematics has been revised and examined by experts for some time, a simple error is almost negligible. The next process is to establish all the abstract conditions that the reasoning assumes to be true. This means identifying the abstract premises from which the mathematical reasoning begins. Doing so is quite difficult. In the past, some remarkable oversights have been accepted by generations of the greatest mathematicians. The chief danger is overlooking a condition that we naturally assume to be true, but that may not always hold. We may introduce such a condition without stating it. An opposite oversight is also possible. It does not lead to error, but it does make the reasoning less simple. It is easy to think that more conditions must be assumed than are actually required. In other words, we may think an abstract postulate is necessary when it can actually be proved from the postulates we already have. These extra abstract postulates only reduce our aesthetic pleasure in the mathematical reasoning and give us more trouble during the third process of criticism.

This third process of criticism is that of verifying that our abstract postulates hold for the particular case in question. It is in respect to this process of verification for the particular case that all the trouble arises. In some simple instances, such as the counting of forty apples, we can with a little care arrive at practical certainty. But in general, with more complex instances, complete certainty is unattainable. Volumes, libraries of volumes, have been written on the subject. It is the battle ground of rival philosophers. There are two distinct questions involved. There are particular definite things observed, and we have to make sure that the relations between these things really do obey certain definite exact abstract conditions. There is great room for error here. The exact observational methods of science are all contrivances for limiting these erroneous conclusions as to direct matters of fact. But another question arises. The things directly observed are, almost always, only samples. We want to conclude that the abstract conditions, which hold for the samples, also hold for all other entities which, for some reason or other, appear to us to be of the same sort. This process of reasoning from the sample to the whole species is Induction. The theory of Induction is the despair of philosophy—and yet all our activities are based upon it. Anyhow, in criticising a mathematical conclusion as to a particular matter of fact, the real difficulties consist in finding out the abstract assumptions involved, and in estimating the evidence for their applicability to the particular case in hand.

The third kind of criticism checks whether our abstract assumptions apply to the particular case in question. All the trouble arises in checking a particular case. In some simple cases, such as counting forty apples, a little care can give us practical certainty. But in general, complete certainty is impossible in more complex cases. Volumes, libraries of volumes, have been written on the subject. Rival philosophers battle over it. There are two distinct questions. First, we observe particular things. We must make sure that the relations between them actually meet certain precise abstract conditions. There is plenty of room for error here. Science’s precise methods of observation are all devices for limiting such mistaken conclusions about directly observed facts. But another question arises. The things we directly observe are almost always only samples. We want to conclude that the abstract conditions that apply to the samples also apply to all other entities that, for some reason, seem to us to be of the same kind. This reasoning from the sample to the whole class is called Induction. The theory of Induction is the despair of philosophy, yet all our activities are based on it. In any case, when we examine a mathematical conclusion about a particular matter of fact, the real difficulties are identifying the abstract assumptions involved and assessing the evidence that they apply to the case at hand.

It often happens, therefore, that in criticising a learned book of applied mathematics, or a memoir, one’s whole trouble is with the first chapter, or even with the first page. For it is there, at the very outset, where the author will probably be found to slip in his assumptions. Farther, the trouble is not with what the author does say, but with what he does not say. Also it is not with what he knows he has assumed, but with what he has unconsciously assumed. We do not doubt the author’s honesty. It is his perspicacity which we are criticising. Each generation criticises the unconscious assumptions made by its parents. It may assent to them, but it brings them out in the open.

Therefore, when criticizing a scholarly book or paper on applied mathematics, one’s whole problem is often with the first chapter, or even the first page. For that is where the author will probably slip in assumptions at the very start. Moreover, the problem is not what the author says, but what the author leaves unsaid. Nor is it what the author knows they have assumed, but what they have assumed unconsciously. We do not doubt the author’s honesty. We are criticizing the author’s ability to see clearly. Each generation criticizes the unconscious assumptions of its parents. It may agree with those assumptions, but it brings them into the open.

The history of the development of language illustrates this point. It is a history of the progressive analysis of ideas. Latin and Greek were inflected languages. This means that they express an unanalyzed complex of ideas by the mere modification of a word; whereas in English, for example, we use prepositions and auxiliary verbs to drag into the open the whole bundle of ideas involved. For certain forms of literary art,—though not always—the compact absorption of auxiliary ideas into the main word may be an advantage. But in a language such as English there is the overwhelming gain in explicitness. This increased explicitness is a more complete exhibition of the various abstractions involved in the complex idea which is the meaning of the sentence.

The history of language illustrates this point. It is a history of the gradual analysis of ideas. Latin and Greek were inflected languages. In those languages, a simple change to a word expresses a complex set of ideas without analyzing them. English, by contrast, uses prepositions and auxiliary verbs to bring the whole set of ideas into the open. For some forms of literary art, packing auxiliary ideas into the main word may be an advantage. But that is not always true. A language like English gains enormously in explicitness. This greater explicitness shows more fully the various abstractions involved in the complex idea that a sentence expresses.

By comparison with language, we can now see what is the function in thought which is performed by pure mathematics. It is a resolute attempt to go the whole way in the direction of complete analysis, so as to separate the elements of mere matter of fact from the purely abstract conditions which they exemplify.

By comparing pure mathematics with language, we can now see its role in thought. Pure mathematics is a determined attempt to carry analysis as far as possible. Its aim is to separate matters of fact from the purely abstract conditions they illustrate.

The habit of such analysis enlightens every act of the functioning of the human mind. It first (by isolating it) emphasizes the direct aesthetic appreciation of the content of experience. This direct appreciation means an apprehension of what this experience is in itself in its own particular essence, including its immediate concrete values. This is a question of direct experience, dependent upon sensitive subtlety. There is then the abstraction of the particular entities involved, viewed in themselves, and as apart from that particular occasion of experience in which we are then apprehending them. Lastly there is the further apprehension of the absolutely general conditions satisfied by the particular relations of those entities as in that experience. These conditions gain their generality from the fact that they are expressible without reference to those particular relations or to those particular relata which occur in that particular occasion of experience. They are conditions which might hold for an indefinite variety of other occasions, involving other entities and other relations between them. Thus these conditions are perfectly general because they refer to no particular occasion, and to no particular entities (such as green, or blue, or trees) which enter into a variety of occasions, and to no particular relationships between such entities.

The habit of analyzing in this way illuminates everything the human mind does. First, it isolates and emphasizes our direct aesthetic appreciation of the content of experience. This means grasping what an experience is in itself, in its own particular essence, including its immediate concrete values. This is a matter of direct experience, which depends on a subtle sensitivity. Next, we abstract the particular entities involved. We consider them in themselves, apart from the particular experience in which we are perceiving them. Finally, we grasp the absolutely general conditions that are satisfied by the particular relations among those entities in that experience. These conditions are general because they can be expressed without referring to those particular relations or the particular entities involved. The conditions might hold for an indefinite variety of other experiences, involving other entities and other relations among them. Thus, these conditions are perfectly general because they refer to no particular experience. Nor do they refer to any particular entities, such as green, blue, or trees, which enter into a variety of experiences. Nor do they refer to any particular relations among such entities.

There is, however, a limitation to be made to the generality of mathematics; it is a qualification which applies equally to all general statements. No statement, except one, can be made respecting any remote occasion which enters into no relationship with the immediate occasion so as to form a constitutive element of the essence of that immediate occasion. By the ‘immediate occasion’ I mean that occasion which involves as an ingredient the individual act of judgment in question. The one excepted statement is,—If anything out of relationship, then complete ignorance as to it. Here by ‘ignorance,’ I mean ignorance; accordingly no advice can be given as to how to expect it, or to treat it, in ‘practice’ or in any other way. Either we know something of the remote occasion by the cognition which is itself an element of the immediate occasion, or we know nothing. Accordingly the full universe, disclosed for every variety of experience, is a universe in which every detail enters into its proper relationship with the immediate occasion. The generality of mathematics is the most complete generality consistent with the community of occasions which constitutes our metaphysical situation.

There is, however, a limit to the generality of mathematics. The same qualification applies to all general statements. We can make only one statement about a remote occasion that has no relationship with the immediate occasion and therefore forms no part of its essence. By the ‘immediate occasion’ I mean the occasion that includes the individual act of judgment in question. The one exception is: If anything is unrelated, then we are completely ignorant of it. By ‘ignorance,’ I mean ignorance. Therefore, no advice can be given on how to expect or treat it, in ‘practice’ or otherwise. Either we know something of the remote occasion through the cognition that is itself an element of the immediate occasion, or we know nothing. Accordingly, the full universe, as revealed through every variety of experience, is a universe in which every detail has its proper relationship with the immediate occasion. The generality of mathematics is the fullest generality consistent with the community of occasions that constitutes our metaphysical situation.

It is further to be noticed that the particular entities require these general conditions for their ingression into any occasions; but the same general conditions may be required by many types of particular entities. This fact, that the general conditions transcend any one set of particular entities, is the ground for the entry into mathematics, and into mathematical logic, of the notion of the ‘variable.’ It is by the employment of this notion that general conditions are investigated without any specification of particular entities. This irrelevance of the particular entities has not been generally understood: for example, the shape-iness of shapes, e.g., circularity and sphericity and cubicality as in actual experience, do not enter into the geometrical reasoning.

Particular entities need these general conditions to enter into any occasions. But many types of particular entities may need the same general conditions. These conditions extend beyond any one set of particular entities. This is why the idea of the ‘variable’ enters mathematics and mathematical logic. By using variables, we can investigate general conditions without specifying any particular entities. The irrelevance of particular entities has not been generally understood. For example, the shape-iness of shapes, e.g., circularity, sphericity, and cubicality as we actually experience them, plays no part in geometrical reasoning.

The exercise of logical reason is always concerned with these absolutely general conditions. In its broadest sense, the discovery of mathematics is the discovery that the totality of these general abstract conditions, which are concurrently applicable to the relationships among the entities of any one concrete occasion, are themselves inter-connected in the manner of a pattern with a key to it. This pattern of relationships among general abstract conditions is imposed alike on external reality, and on our abstract representations of it, by the general necessity that every thing must be just its own individual self, with its own individual way of differing from everything else. This is nothing else than the necessity of abstract logic, which is the presupposition involved in the very fact of interrelated existence as disclosed in each immediate occasion of experience.

Logical reasoning always deals with these completely general conditions. In the broadest sense, the discovery of mathematics is the discovery of a pattern with a key to it. The pattern connects all the general abstract conditions that apply at once to relationships among the entities in any one concrete occasion. This pattern of relationships among general abstract conditions applies equally to external reality and to our abstract representations of it. It does so because everything must necessarily be its own individual self, with its own individual way of differing from everything else. This is simply the necessity of abstract logic. Abstract logic is assumed in the very fact that things exist in relation to one another, as each immediate occasion of experience reveals.

The key to the pattern means this fact:—that from a select set of those general conditions, exemplified in any one and the same occasion, a pattern involving an infinite variety of other such conditions, also exemplified in the same occasion, can be developed by the pure exercise of abstract logic. Any such select set is called the set of postulates, or premises, from which the reasoning proceeds. The reasoning is nothing else than the exhibition of the whole pattern of general conditions involved in the pattern derived from the selected postulates.

The key to the pattern is this: a selected set of general conditions can all occur in the same occasion. From that set, pure abstract logic can develop a pattern involving an infinite variety of other general conditions. These other conditions also occur in the same occasion. Any such selected set is called the set of postulates, or premises, from which the reasoning proceeds. The reasoning simply reveals the whole pattern of general conditions involved in the pattern derived from the selected postulates.

The harmony of the logical reason, which divines the complete pattern as involved in the postulates, is the most general aesthetic property arising from the mere fact of concurrent existence in the unity of one occasion. Wherever there is a unity of occasion there is thereby established an aesthetic relationship between the general conditions involved in that occasion. This aesthetic relationship is that which is divined in the exercise of rationality. Whatever falls within that relationship is thereby exemplified in that occasion; whatever falls without that relationship is thereby excluded from exemplification in that occasion. The complete pattern of general conditions, thus exemplified, is determined by any one of many select sets of these conditions. These key sets are sets of equivalent postulates. This reasonable harmony of being, which is required for the unity of a complex occasion, together with the completeness of the realisation (in that occasion) of all that is involved in its logical harmony, is the primary article of metaphysical doctrine. It means that for things to be together involves that they are reasonably together. This means that thought can penetrate into every occasion of fact, so that by comprehending its key conditions, the whole complex of its pattern of conditions lies open before it. It comes to this:—provided we know something which is perfectly general about the elements in any occasion, we can then know an indefinite number of other equally general concepts which must also be exemplified in that same occasion. The logical harmony involved in the unity of an occasion is both exclusive and inclusive. The occasion must exclude the inharmonious, and it must include the harmonious.

Logical reason grasps the complete pattern implied by a set of postulates. Its harmony is the most general aesthetic property arising from things existing together in the unity of a single occasion. Wherever there is a unity of occasion, an aesthetic relationship is established among the general conditions involved in that occasion. We grasp this aesthetic relationship by exercising reason. Whatever falls within the relationship is exemplified in that occasion. Whatever falls outside the relationship is excluded from being exemplified in that occasion. The complete pattern of general conditions exemplified in an occasion is determined by any one of many select sets of those conditions. These key sets are sets of equivalent postulates. A complex occasion must have a rational harmony of being in order to have unity. In that occasion, everything involved in its logical harmony must also be fully realized. Together, these requirements are the primary principle of metaphysical doctrine. They mean that things must be rationally related in order to exist together. Thus thought can penetrate every occasion of fact. By understanding its key conditions, thought can see the whole complex pattern of its conditions. In short, suppose we know something perfectly general about the elements in any occasion. We can then know an indefinite number of other equally general concepts that must also be exemplified in that occasion. The logical harmony involved in the unity of an occasion both excludes and includes. The occasion must exclude the inharmonious, and it must include the harmonious.

Pythagoras was the first man who had any grasp of the full sweep of this general principle. He lived in the sixth century before Christ. Our knowledge of him is fragmentary. But we know some points which establish his greatness in the history of thought. He insisted on the importance of the utmost generality in reasoning, and he divined the importance of number as an aid to the construction of any representation of the conditions involved in the order of nature. We know also that he studied geometry, and discovered the general proof of the remarkable theorem about right-angled triangles. The formation of the Pythagorean Brotherhood, and the mysterious rumours as to its rites and its influence, afford some evidence that Pythagoras divined, however dimly, the possible importance of mathematics in the formation of science. On the side of philosophy he started a discussion which has agitated thinkers ever since. He asked, ‘What is the status of mathematical entities, such as numbers for example, in the realm of things?’ The number ‘two,’ for example, is in some sense exempt from the flux of time and the necessity of position in space. Yet it is involved in the real world. The same considerations apply to geometrical notions—to circular shape, for example. Pythagoras is said to have taught that the mathematical entities, such as numbers and shapes, were the ultimate stuff out of which the real entities of our perceptual experience are constructed. As thus boldly stated, the idea seems crude, and indeed silly. But undoubtedly, he had hit upon a philosophical notion of considerable importance; a notion which has a long history, and which has moved the minds of men, and has even entered into Christian theology. About a thousand years separate the Athanasian Creed from Pythagoras, and about two thousand four hundred years separate Pythagoras from Hegel. Yet for all these distances in time, the importance of definite number in the constitution of the Divine Nature, and the concept of the real world as exhibiting the evolution of an idea, can both be traced back to the train of thought set going by Pythagoras.

Pythagoras was the first person to grasp the full scope of this general principle. He lived in the sixth century BC. Our knowledge of him is incomplete. But we know some things that establish his greatness in the history of thought. He insisted that reasoning should be as general as possible. He also recognized that numbers could help us describe the conditions involved in the order of nature. We also know that he studied geometry. He discovered the general proof of the remarkable theorem about right-angled triangles. He founded the Pythagorean Brotherhood, and mysterious rumors spread about its rites and influence. These facts suggest that Pythagoras saw, however dimly, how important mathematics might be in the development of science. In philosophy, Pythagoras started a discussion that has troubled thinkers ever since. He asked, ‘What is the status of mathematical entities, such as numbers for example, in the realm of things?’ The number ‘two,’ for example, is in some sense free from the passage of time. It is also free from the need to have a position in space. Yet it is part of the real world. The same is true of geometrical concepts, such as circular shape. Pythagoras is said to have taught that mathematical entities, such as numbers and shapes, were the basic material of reality. According to this view, the real things we perceive are made from them. Stated so boldly, the idea seems crude and indeed silly. But Pythagoras had undoubtedly hit upon a philosophical idea of considerable importance. The idea has a long history, has influenced thinkers, and has even entered Christian theology. About a thousand years separate the Athanasian Creed from Pythagoras. About two thousand four hundred years separate Pythagoras from Hegel. Yet despite these long intervals, two ideas can both be traced back to the line of thought Pythagoras started. One is the importance of a definite number in the makeup of the Divine Nature. The other is the idea that the real world displays the evolution of an idea.

The importance of an individual thinker owes something to chance. For it depends upon the fate of his ideas in the minds of his successors. In this respect Pythagoras was fortunate. His philosophical speculations reach us through the mind of Plato. The Platonic world of ideas is the refined, revised form of the Pythagorean doctrine that number lies at the base of the real world. Owing to the Greek mode of representing numbers by patterns of dots, the notions of number and of geometrical configuration are less separated than with us. Also Pythagoras, without doubt, included the shape-iness of shape, which is an impure mathematical entity. So to-day, when Einstein and his followers proclaim that physical facts, such as gravitation, are to be construed as exhibitions of local peculiarities of spatio-temporal properties, they are following the pure Pythagorean tradition. In a sense, Plato and Pythagoras stand nearer to modern physical science than does Aristotle. The two former were mathematicians, whereas Aristotle was the son of a doctor, though of course he was not thereby ignorant of mathematics. The practical counsel to be derived from Pythagoras, is to measure, and thus to express quality in terms of numerically determined quantity. But the biological sciences, then and till our own time, have been overwhelmingly classificatory. Accordingly, Aristotle by his Logic throws the emphasis on classification. The popularity of Aristotelian Logic retarded the advance of physical science throughout the Middle Ages. If only the schoolmen had measured instead of classifying, how much they might have learnt!

The importance of an individual thinker depends partly on chance. It depends on what happens to his ideas in the minds of those who come after him. In this respect, Pythagoras was fortunate. His philosophical ideas reach us through the mind of Plato. Plato’s world of ideas is a refined and revised form of the Pythagorean doctrine that number lies at the foundation of the real world. The Greeks represented numbers as patterns of dots. As a result, they saw number and geometric shape as less separate than we do. Pythagoras also undoubtedly included the shape-iness of shape, which is an impure mathematical entity. Today, Einstein and his followers say that physical facts, such as gravitation, should be understood as expressions of local features of space and time. In doing so, they follow the pure Pythagorean tradition. In a sense, Plato and Pythagoras are closer to modern physical science than Aristotle is. Plato and Pythagoras were mathematicians, while Aristotle was a doctor’s son. Of course, that did not mean Aristotle knew nothing about mathematics. The practical advice to draw from Pythagoras is to measure, and so express quality in terms of quantity determined by numbers. But the biological sciences were then, and remained until our own time, overwhelmingly concerned with classification. Accordingly, Aristotle’s Logic emphasizes classification. The popularity of Aristotelian Logic slowed the advance of physical science throughout the Middle Ages. If only the schoolmen had measured instead of classifying, how much they might have learned!

Classification is a halfway house between the immediate concreteness of the individual thing and the complete abstraction of mathematical notions. The species take account of the specific character, and the genera of the generic character. But in the procedure of relating mathematical notions to the facts of nature, by counting, by measurement, and by geometrical relations, and by types of order, the rational contemplation is lifted from the incomplete abstractions involved in definite species and genera, to the complete, abstractions of mathematics. Classification is necessary. But unless you can progress from classification to mathematics, your reasoning will not take you very far.

Classification is a halfway point between the immediate concreteness of an individual thing and the complete abstraction of mathematical concepts. Species take account of specific character, while genera take account of generic character. But we can relate mathematical concepts to facts of nature through counting, measurement, geometric relations, and types of order. By doing so, our reasoning moves beyond the incomplete abstractions of particular species and genera to the complete abstractions of mathematics. Classification is necessary. But unless you can move from classification to mathematics, your reasoning will not take you very far.

Between the epoch which stretches from Pythagoras to Plato and the epoch comprised in the seventeenth century of the modern world nearly two thousand years elapsed. In this long interval mathematics had made immense strides. Geometry had gained the study of conic sections and trigonometry; the method of exhaustion had almost anticipated the integral calculus; and above all the Arabic arithmetical notation and algebra had been contributed by Asiatic thought. But the progress was on technical lines. Mathematics, as a formative element in the development of philosophy, never, during this long period, recovered from its deposition at the hands of Aristotle. Some of the old ideas derived from the Pythagorean-Platonic epoch lingered on, and can be traced among the Platonic influences which shaped the first period of evolution of Christian theology. But philosophy received no fresh inspiration from the steady advance of mathematical science. In the seventeenth century the influence of Aristotle was at its lowest, and mathematics recovered the importance of its earlier period. It was an age of great physicists and great philosophers; and the physicists and philosophers were alike mathematicians. The exception of John Locke should be made; although he was greatly influenced by the Newtonian circle of the Royal Society. In the age of Galileo, Descartes, Spinoza, Newton, and Leibniz, mathematics was an influence of the first magnitude in the formation of philosophic ideas. But the mathematics, which now emerged into prominence, was a very different science from the mathematics of the earlier epoch. It had gained in generality, and had started upon its almost incredible modern career of piling subtlety of generalization upon subtlety of generalization; and of finding, with each growth of complexity, some new application, either to physical science, or to philosophic thought. The Arabic notation had equipped the science with almost perfect technical efficiency in the manipulation of numbers. This relief from a struggle with arithmetical details (as instanced, for example, in the Egyptian arithmetic of B. C. 1600) gave room for a development which had already been faintly anticipated in later Greek mathematics. Algebra now came upon the scene, and algebra is a generalisation of arithmetic. In the same way as the notion of number abstracted from reference to any one particular set of entities, so in algebra abstraction is made from the notion of any particular numbers. Just as the number ‘5’ refers impartially to any group of five entities, so in algebra the letters are used to refer impartially to any number, with the proviso that each letter is to refer to the same number throughout the same context of its employment.

Nearly two thousand years passed between the era from Pythagoras to Plato and the seventeenth century of the modern world. During this long interval, mathematics made immense advances. Geometry gained the study of conic sections and trigonometry. The method of exhaustion came close to anticipating integral calculus. Above all, Asian thinkers contributed Arabic numerical notation and algebra. But the progress was technical. Throughout this long period, mathematics never recovered its role in shaping philosophy after Aristotle displaced it. Some old ideas from the Pythagorean-Platonic era lived on. They can be traced among the Platonic influences that shaped the early development of Christian theology. But philosophy received no fresh inspiration from the steady advance of mathematical science. In the seventeenth century, Aristotle’s influence was at its lowest, and mathematics regained its earlier importance. It was an age of great physicists and great philosophers. Physicists and philosophers alike were mathematicians, with the exception of John Locke. Even he was greatly influenced by the Newtonian circle of the Royal Society. In the age of Galileo, Descartes, Spinoza, Newton, and Leibniz, mathematics was a major influence on the formation of philosophical ideas. But the mathematics that now came to prominence was very different from that of the earlier era. It had become more general. It had begun its almost incredible modern career of building ever subtler generalizations upon one another. With each increase in complexity, mathematics found a new application in either physical science or philosophical thought. Arabic notation had made the manipulation of numbers almost perfectly efficient. Mathematicians no longer had to struggle with arithmetic details as they had before. Egyptian arithmetic from 1600 BC gives one example of that struggle. This relief made room for a development already faintly anticipated in later Greek mathematics. Algebra now appeared on the scene. Algebra is a generalization of arithmetic. The idea of number is abstracted from any particular set of things. In the same way, algebra abstracts from the idea of particular numbers. The number ‘5’ refers equally to any group of five things. Similarly, letters in algebra can refer equally to any number. But each letter must refer to the same number throughout the same context.

This usage was first employed in equations, which are methods of asking complicated arithmetical questions. In this connection, the letters representing numbers were termed ‘unknowns.’ But equations soon suggested a new idea, that, namely, of a function of one or more general symbols, these symbols being letters representing any numbers. In this employment the algebraic letters are called the ‘arguments’ of the function, or sometimes they are called the ‘variables.’ Then, for instance, if an angle is represented by an algebraical letter, as standing for its numerical measure in terms of a given unit, Trigonometry is absorbed into this new algebra. Algebra thus develops into the general science of analysis in which we consider the properties of various functions of undetermined arguments. Finally the particular functions, such as the trigonometrical functions, and the logarithmic functions, and the algebraic functions, are generalised into the idea of ‘any function.’ Too large a generalisation leads to mere barrenness. It is the large generalisation, limited by a happy particularity, which is the fruitful conception. For instance the idea of any continuous function, whereby the limitation of continuity is introduced, is the fruitful idea which has led to most of the important applications. This rise of algebraic analysis was concurrent with Descartes’ discovery of analytical geometry, and then with the invention of the infinitesimal calculus by Newton and Leibniz. Truly, Pythagoras, if he could have foreseen the issue of the train of thought which he had set going would have felt himself fully justified in his brotherhood with its excitement of mysterious rites.

This use first appeared in equations, which are ways of asking complicated arithmetic questions. In equations, the letters representing numbers were called ‘unknowns.’ But equations soon suggested a new idea: a function of one or more general symbols. These symbols were letters representing any numbers. Used this way, the algebraic letters are called the ‘arguments’ of the function, or sometimes the ‘variables.’ For example, an algebraic letter can represent an angle by standing for its numerical measure in a given unit. In this way, Trigonometry becomes part of this new algebra. Algebra thus develops into the general science of analysis. Analysis considers the properties of various functions of undetermined arguments. Finally, particular functions, such as trigonometric, logarithmic, and algebraic functions, are generalized into the idea of ‘any function.’ Too broad a generalization leads to mere barrenness. A broad generalization limited by a useful particular feature is the fruitful idea. For example, the idea of any continuous function introduces the limiting condition of continuity. This fruitful idea has led to most of the important applications. The rise of algebraic analysis took place alongside Descartes’ discovery of analytical geometry. It was followed by Newton and Leibniz’s invention of the infinitesimal calculus. Truly, if Pythagoras could have foreseen the results of the train of thought he had set in motion, he would have felt fully justified in founding his brotherhood, with its exciting mysterious rites.

The point which I now want to make is that this dominance of the idea of functionality in the abstract sphere of mathematics found itself reflected in the order of nature under the guise of mathematically expressed laws of nature. Apart from this progress of mathematics, the seventeenth century developments of science would have been impossible. Mathematics supplied the background of imaginative thought with which the men of science approached the observation of nature. Galileo produced formulae, Descartes produced formulae, Huyghens produced formulae, Newton produced formulae.

My point is that the idea of functions dominated abstract mathematics. This dominance was reflected in the natural order as mathematically expressed laws of nature. Without these advances in mathematics, the scientific developments of the seventeenth century would have been impossible. Mathematics provided the imaginative framework scientists brought to their observations of nature. Galileo produced formulas, Descartes produced formulas, Huyghens produced formulas, Newton produced formulas.

As a particular example of the effect of the abstract development of mathematics upon the science of those times, consider the notion of periodicity. The general recurrences of things are very obvious in our ordinary experience. Days recur, lunar phases recur, the seasons of the year recur, rotating bodies recur to their old positions, beats of the heart recur, breathing recurs. On every side, we are met by recurrence. Apart from recurrence, knowledge would be impossible; for nothing could be referred to our past experience. Also, apart from some regularity of recurrence, measurement would be impossible. In our experience, as we gain the idea of exactness, recurrence is fundamental.

Consider the idea of periodicity as an example of how abstract mathematics affected the science of the time. In everyday life, we can easily see that things recur. Days recur, as do the phases of the moon and the seasons. Rotating bodies return to their former positions. Heartbeats and breaths recur. Recurrence is all around us. Without recurrence, knowledge would be impossible, because we could not relate anything to our past experience. Without some regularity in recurrence, measurement would also be impossible. As we develop an idea of exactness through experience, recurrence is fundamental.

In the sixteenth and seventeenth centuries, the theory of periodicity took a fundamental place in science. Kepler divined a law connecting the major axes of the planetary orbits with the periods in which the planets respectively described their orbits: Galileo observed the periodic vibrations of pendulums: Newton explained sound as being due to the disturbance of air by the passage through it of periodic waves of condensation and rarefaction: Huyghens explained light as being due to the transverse waves of vibration of a subtle ether: Mersenne connected the period of the vibration of a violin string with its density, tension, and length. The birth of modern physics depended upon the application of the abstract idea of periodicity to a variety of concrete instances. But this would have been impossible, unless mathematicians had already worked out in the abstract the various abstract ideas which cluster round the notions of periodicity. The science of trigonometry arose from that of the relations of the angles of a right-angled triangle, to the ratios between the sides and hypotenuse of the triangle. Then, under the influence of the newly discovered mathematical science of the analysis of functions, it broadened out into the study of the simple abstract periodic functions which these ratios exemplify. Thus trigonometry became completely abstract; and in thus becoming abstract, it became useful. It illuminated the underlying analogy between sets of utterly diverse physical phenomena; and at the same time it supplied the weapons by which any one such set could have its various features analysed and related to each other.²

In the sixteenth and seventeenth centuries, the theory of periodicity became fundamental to science. Kepler discovered a law connecting the major axes of planetary orbits with the time each planet took to complete its orbit. Galileo observed the periodic vibrations of pendulums. Newton explained sound as the result of periodic waves of compression and expansion passing through the air. Huyghens explained light as the result of transverse waves of vibration in a subtle ether. Mersenne connected the period of a violin string’s vibration with its density, tension, and length. Modern physics was born through the application of the abstract idea of periodicity to a variety of concrete cases. But this would have been impossible if mathematicians had not already worked out the abstract ideas surrounding periodicity. Trigonometry began with the study of how the angles of a right triangle relate to ratios between its sides and hypotenuse. Then, influenced by the newly discovered mathematical science of the analysis of functions, it broadened. Trigonometry came to study the simple abstract periodic functions that these ratios exemplify. In this way, trigonometry became completely abstract, and by becoming abstract, it became useful. It revealed the underlying similarities among sets of entirely different physical phenomena. At the same time, it supplied the tools for analyzing and relating the different features of any such set.²

Nothing is more impressive than the fact that, as mathematics withdrew increasingly into the upper regions of ever greater extremes of abstract thought, it returned back to earth with a corresponding growth of importance for the analysis of concrete fact. The history of the seventeenth century science reads as though it were some vivid dream of Plato or Pythagoras. In this characteristic the seventeenth century was only the forerunner of its successors.

Nothing is more striking than what happened as mathematics moved into ever more abstract realms of thought. As it did, mathematics also returned to earth, becoming increasingly important for analyzing concrete facts. The history of seventeenth-century science reads like a vivid dream of Plato or Pythagoras. In this respect, the seventeenth century was only a forerunner of the centuries that followed.

The paradox is now fully established that the utmost abstractions are the true weapons with which to control our thought of concrete fact. As the result of the prominence of mathematicians in the seventeenth century, the eighteenth century was mathematically minded, more especially where French influence predominated. An exception must be made of the English empiricism derived from Locke. Outside France, Newton’s direct influence on philosophy is best seen in Kant, and not in Hume.

The paradox is now fully established: the most abstract ideas are our best tools for guiding our thinking about concrete facts. Mathematicians were prominent in the seventeenth century. As a result, the eighteenth century was mathematically minded, especially where French influence was strongest. English empiricism derived from Locke was an exception. Outside France, Newton’s direct influence on philosophy is most evident in Kant, not Hume.

In the nineteenth century, the general influence of mathematics waned. The romantic movement in literature, and the idealistic movement in philosophy were not the products of mathematical minds. Also, even in science, the growth of geology, of zoology, and of the biological sciences generally, was in each case entirely disconnected from any reference to mathematics. The chief scientific excitement of the century was the Darwinian theory of evolution. Accordingly, mathematicians were in the background, so far as the general thought of that age was concerned. But this does not mean that mathematics was being neglected, or even that it was uninfluential. During the nineteenth century pure mathematics made almost as much progress as during all the preceding centuries from Pythagoras onwards. Of course progress was easier, because the technique had been perfected. But allowing for that, the change in mathematics between the years 1800 and 1900 is very remarkable. If we add in the previous hundred years, and take the two centuries preceding the present time, one is almost tempted to date the foundation of mathematics somewhere in the last quarter of the seventeenth century. The period of the discovery of the elements stretches from Pythagoras to Descartes, Newton, and Leibniz, and the developed science has been created during the last two hundred and fifty years. This is not a boast as to the superior genius of the modern world; for it is harder to discover the elements than to develop the science.

In the nineteenth century, the general influence of mathematics declined. The romantic movement in literature and the idealistic movement in philosophy were not the products of mathematical minds. Even in science, the growth of geology, zoology, and the biological sciences generally was entirely disconnected from mathematics. The century’s greatest scientific excitement was Darwin’s theory of evolution. So mathematicians remained in the background of the age’s general thought. But mathematics was not neglected, nor was it without influence. During the nineteenth century, pure mathematics made almost as much progress as it had in all the preceding centuries, beginning with Pythagoras. Of course, progress was easier because mathematical techniques had been perfected. But even allowing for that, the change in mathematics between 1800 and 1900 is remarkable. If we include the previous hundred years, we cover the two centuries before our own time. We are then almost tempted to date the founding of mathematics to the last quarter of the seventeenth century. The period when its basic elements were discovered stretches from Pythagoras to Descartes, Newton, and Leibniz. The fully developed science was created during the last two hundred and fifty years. This is not a boast about the greater genius of the modern world. It is harder to discover the basic elements than to develop the science.

Throughout the nineteenth century, the influence of the science was its influence on dynamics and physics, and thence derivatively on engineering and chemistry. It is difficult to overrate its indirect influence on human life through the medium of these sciences. But there was no direct influence of mathematics upon the general thought of the age.

Throughout the nineteenth century, mathematics influenced dynamics and physics, and through them, engineering and chemistry. It is hard to overstate the indirect influence mathematics had on human life through these sciences. But mathematics had no direct influence on the general thought of the age.

In reviewing this rapid sketch of the influence of mathematics throughout European history, we see that it had two great periods of direct influence upon general thought, both periods lasting for about two hundred years. The first period was that stretching from Pythagoras to Plato, when the possibility of the science, and its general character, first dawned upon the Grecian thinkers. The second period comprised the seventeenth and eighteenth centuries of our modern epoch. Both periods had certain common characteristics. In the earlier, as in the later period, the general categories of thought in many spheres of human interest, were in a state of disintegration. In the age of Pythagoras, the unconscious Paganism, with its traditional clothing of beautiful ritual and of magical rites, was passing into a new phase under two influences. There were waves of religious enthusiasm, seeking direct enlightenment into the secret depths of being; and at the opposite pole, there was the awakening of critical analytical thought, probing with cool dispassionateness into ultimate meanings. In both influences, so diverse in their outcome, there was one common element—an awakened curiosity, and a movement towards the reconstruction of traditional ways. The pagan mysteries may be compared to the Puritan reaction and to the Catholic reaction; critical scientific interest was alike in both epochs, though with minor differences of substantial importance.

This brief survey of mathematics in European history shows two great periods of direct influence on general thought. Each lasted about two hundred years. The first stretched from Pythagoras to Plato. During that time, Greek thinkers first began to recognize the possibility and general nature of mathematics. The second period covered the seventeenth and eighteenth centuries of the modern era. The two periods had certain features in common. In both, the basic categories of thought were breaking down in many areas of human interest. In the age of Pythagoras, an unreflective Paganism was entering a new phase under two influences. It had been clothed in traditional, beautiful rituals and magical rites. On one side were waves of religious enthusiasm, seeking direct insight into the hidden depths of being. On the other was the awakening of critical, analytical thought, examining ultimate meanings with cool detachment. Though these influences led to very different outcomes, they shared one element. Both showed an awakened curiosity and a movement to reconstruct traditional ways. The pagan mysteries may be compared to the Puritan reaction and the Catholic reaction. Critical scientific interest was similar in both periods, though it had minor differences of substantial importance.

In each age, the earlier stages were placed in periods of rising prosperity, and of new opportunities. In this respect, they differed from the period of gradual declension in the second and third centuries when Christianity was advancing to the conquest of the Roman world. It is only in a period, fortunate both in its opportunities for disengagement from the immediate pressure of circumstances, and in its eager curiosity, that the Age-Spirit can undertake any direct revision of those final abstractions which lie hidden in the more concrete concepts from which the serious thought of an age takes its start. In the rare periods when this task can be undertaken, mathematics becomes relevant to philosophy. For mathematics is the science of the most complete abstractions to which the human mind can attain.

In each age, the earlier stages occurred during periods of rising prosperity and new opportunities. In this respect, they differed from the gradual decline of the second and third centuries. During those centuries, Christianity was advancing toward the conquest of the Roman world. Only a fortunate age can directly revise the most fundamental abstractions hidden within its more concrete concepts. These concrete concepts are the starting points for the serious thought of an age. To undertake such a revision, an age must have opportunities to escape the immediate pressure of circumstances. It must also be eagerly curious. In the rare periods when this task can be undertaken, mathematics becomes relevant to philosophy. For mathematics is the science of the most complete abstractions the human mind can reach.

The parallel between the two epochs must not be pressed too far. The modern world is larger and more complex than the ancient civilization round the shores of the Mediterranean, or even than that of the Europe which sent Columbus and the Pilgrim Fathers across the ocean. We cannot now explain our age by some simple formula which becomes dominant and will then be laid to rest for a thousand years. Thus the temporary submergence of the mathematical mentality from the time of Rousseau onwards appears already to be at an end. We are entering upon an age of reconstruction, in religion, in science, and in political thought. Such ages, if they are to avoid mere ignorant oscillation between extremes, must seek truth in its ultimate depths. There can be no vision of this depth of truth apart from a philosophy which takes full account of those ultimate abstractions, whose interconnections it is the business of mathematics to explore.

The parallel between the two eras should not be taken too far. The modern world is larger and more complex than ancient civilization around the Mediterranean. It is larger and more complex even than the Europe that sent Columbus and the Pilgrim Fathers across the ocean. We can no longer explain our age with a simple formula that dominates, then lies dormant for a thousand years. Thus, the temporary decline of mathematical thinking that began with Rousseau already seems to be ending. We are entering an age of reconstruction in religion, science, and political thought. If such ages are to avoid merely swinging between extremes out of ignorance, they must seek truth at its deepest level. We cannot see truth at that depth without a philosophy that fully considers the most fundamental abstractions. Mathematics explores the connections among those abstractions.

In order to explain exactly how mathematics is gaining in general importance at the present time, let us start from a particular scientific perplexity and consider the notions to which we are naturally led by some attempt to unravel its difficulties. At present physics is troubled by the quantum theory. I need not now explain³ what this theory is, to those who are not already familiar with it. But the point is that one of the most hopeful lines of explanation is to assume that an electron does not continuously traverse its path in space. The alternative notion as to its mode of existence is that it appears at a series of discrete positions in space which it occupies for successive durations of time. It is as though an automobile moving at the average rate of thirty miles an hour along a road, did not traverse the road continuously; but appeared successively at the successive milestones, remaining for two minutes at each milestone.

To explain exactly how mathematics is becoming more important today, let us start with a particular scientific puzzle. We can then consider the ideas that arise naturally when we try to solve it. Physics is currently troubled by quantum theory. I need not explain³ the theory now to those who are not already familiar with it. But one of the most promising explanations assumes that an electron does not move continuously along its path through space. Instead, the electron appears at a series of separate positions in space. It occupies each position for a period of time, one after another. Imagine an automobile moving along a road at an average of thirty miles an hour. It would be as if the automobile did not travel continuously along the road. Instead, it would appear at each successive milestone and remain there for two minutes.

In the first place there is required the purely technical use of mathematics to determine whether this conception does in fact explain the many perplexing characteristics of the quantum theory. If the notion survives this test, undoubtedly physics will adopt it. So far the question is purely one for mathematics and physical science to settle between them, on the basis of mathematical calculations and physical observations.

First, mathematics must be used to determine whether this idea actually explains the many puzzling features of quantum theory. If the idea survives this test, physics will undoubtedly adopt it. So far, the question is purely for mathematics and physical science to settle between them, through mathematical calculations and physical observations.

But now a problem is handed over to the philosophers. This discontinuous existence in space, thus assigned to electrons, is very unlike the continuous existence of material entities which we habitually assume as obvious. The electron seems to be borrowing the character which some people have assigned to the Mahatmas of Tibet. These electrons, with the correlative protons, are now conceived as being the fundamental entities out of which the material bodies of ordinary experience are composed. Accordingly, if this explanation is allowed, we have to revise all our notions of the ultimate character of material existence. For when we penetrate to these final entities, this startling discontinuity of spatial existence discloses itself.

But now a problem is handed over to the philosophers. Electrons are said to exist discontinuously in space. This is very unlike the continuous existence we habitually assume material entities obviously have. The electron seems to be borrowing a trait that some people have assigned to the Mahatmas of Tibet. Electrons, together with their corresponding protons, are now thought to be the fundamental entities that make up ordinary material bodies. So if this explanation is accepted, we must revise all our ideas about the ultimate nature of material existence. For when we reach these most basic entities, we find this startling discontinuity in their existence in space.

There is no difficulty in explaining the paradox, if we consent to apply to the apparently steady undifferentiated endurance of matter the same principles as those now accepted for sound and light. A steadily sounding note is explained as the outcome of vibrations in the air: a steady colour is explained as the outcome of vibrations in ether. If we explain the steady endurance of matter on the same principle, we shall conceive each primordial element as a vibratory ebb and flow of an underlying energy, or activity. Suppose we keep to the physical idea of energy: then each primordial element will be an organized system of vibratory streaming of energy. Accordingly there will be a definite period associated with each element; and within that period the stream-system will sway from one stationary maximum to another stationary maximum,—or, taking a metaphor from the ocean tides, the system will sway from one high tide to another high tide. This system, forming the primordial element, is nothing at any instant. It requires its whole period in which to manifest itself. In an analogous way, a note of music is nothing at an instant, but it also requires its whole period in which to manifest itself.

The paradox is easy to explain if we apply the same principles to matter that we now accept for sound and light. Matter appears to endure steadily and without change. A steady musical note is explained as the result of vibrations in the air. A steady colour is explained as the result of vibrations in ether. If we explain the steady endurance of matter by the same principle, we can conceive of each primordial element as a vibrating ebb and flow of an underlying energy or activity. Suppose we keep to the physical idea of energy. Then each primordial element will be an organized system of vibrating streams of energy. Accordingly, each element will have a definite period. Within that period, the system of streams will sway from one stationary maximum to another. To borrow a metaphor from ocean tides, the system will sway from one high tide to another. The system that forms the primordial element is nothing at any instant. It needs its whole period to manifest itself. Similarly, a musical note is nothing at an instant. It too needs its whole period to manifest itself.

Accordingly, in asking where the primordial element is, we must settle on its average position at the centre of each period. If we divide time into smaller elements, the vibratory system as one electronic entity has no existence. The path in space of such a vibratory entity—where the entity is constituted by the vibrations—must be represented by a series of detached positions in space, analogously to the automobile which is found at successive milestones and at nowhere between.

So, when we ask where the primordial element is, we must use its average position at the center of each period. If we divide time into smaller elements, the vibrating system does not exist as a single electronic entity. The entity is constituted by its vibrations. So its path through space must be represented by a series of separate positions. This is like a car found at successive milestones and nowhere between them.

We first must ask whether there is any evidence to associate the quantum theory with vibration. This question is immediately answered in the affirmative. The whole theory centres round the radiant energy from an atom, and is intimately associated with the periods of the radiant wave-systems. It seems, therefore, that the hypothesis of essentially vibratory existence is the most hopeful way of explaining the paradox of the discontinuous orbit.

We must first ask whether there is any evidence linking quantum theory to vibration. The answer is immediately clear: yes. The whole theory centers on the radiant energy from an atom. It is closely linked to the periods of the radiant wave systems. So the hypothesis that existence is essentially vibratory seems the most promising way to explain the paradox of the discontinuous orbit.

In the second place, a new problem is now placed before philosophers and physicists, if we entertain the hypothesis that the ultimate elements of matter are in their essence vibratory. By this I mean that apart from being a periodic system, such an element would have no existence. With this hypothesis we have to ask, what are the ingredients which form the vibratory organism. We have already got rid of the matter with its appearance of undifferentiated endurance. Apart from some metaphysical compulsion, there is no reason to provide another more subtle stuff to take the place of the matter which has just been explained away. The field is now open for the introduction of some new doctrine of organism which may take the place of the materialism with which, since the seventeenth century, science has saddled philosophy. It must be remembered that the physicists’ energy is obviously an abstraction. The concrete fact, which is the organism, must be a complete expression of the character of a real occurrence. Such a displacement of scientific materialism, if it ever takes place, cannot fail to have important consequences in every field of thought.

Second, a new problem faces philosophers and physicists if we consider the hypothesis that the ultimate elements of matter are vibratory by nature. By this I mean that such an element would not exist except as a periodic system. Given this hypothesis, we must ask what ingredients make up the vibrating organism. We have already done away with matter and its appearance of enduring without change or differentiation. Unless some metaphysical principle forces us to do so, there is no reason to replace it with another, more subtle substance. The way is now open for a new theory of organism. It may replace the materialism with which science has burdened philosophy since the seventeenth century. We must remember that the physicists’ concept of energy is obviously an abstraction. The concrete fact, the organism, must fully express the character of a real occurrence. If scientific materialism is ever displaced in this way, the change is bound to have important consequences in every field of thought.

Finally, our last reflection must be, that we have in the end come back to a version of the doctrine of old Pythagoras, from whom mathematics, and mathematical physics, took their rise. He discovered the importance of dealing with abstractions; and in particular directed attention to number as characterizing the periodicities of notes of music. The importance of the abstract idea of periodicity was thus present at the very beginning both of mathematics and of European philosophy.

Finally, we must reflect that we have come back to a version of the doctrine of Pythagoras. Mathematics and mathematical physics began with him. He discovered the importance of working with abstractions. In particular, he drew attention to how numbers characterize the periodic patterns of musical notes. Thus, the abstract idea of periodicity was present at the very beginning of both mathematics and European philosophy.

In the seventeenth century, the birth of modern science required a new mathematics, more fully equipped for the purpose of analysing the characteristics of vibratory existence. And now in the twentieth century we find physicists largely engaged in analysing the periodicities of atoms. Truly, Pythagoras in founding European philosophy and European mathematics, endowed them with the luckiest of lucky guesses—or, was it a flash of divine genius, penetrating to the inmost nature of things?

In the seventeenth century, the birth of modern science required a new kind of mathematics. It needed to be better equipped to analyze the characteristics of vibratory existence. Now, in the twentieth century, physicists are largely engaged in analyzing the periodicities of atoms. Surely, when Pythagoras founded European philosophy and mathematics, he gave them the luckiest of lucky guesses. Or was it a flash of divine genius that penetrated to the innermost nature of things?