Chapter 52: MATHEMATICS

INTRODUCTION

It is necessary for us to observe the difference between problems in mathematics and the problem of the truth about mathematics. In the case of any science—in physics, logic, or metaphysics, as well as mathematics—it is one thing to examine the discourses or treatises of the scientists on the special subject matter of their field, and quite another to examine discussions of the science itself, its scope, branches, and unity, its objects, its methods, and its relation to other disciplines. The chapter on QUANTITY deals with the subject matter of arithmetic, geometry, and other branches of mathematics; here we are primarily concerned with the nature of mathematical science itself.

Sometimes reflections on the nature of a science are expressed by experts in its subject matter who comment on the scientific enterprise in which they are engaged in prefaces or interspersed remarks. Sometimes such reflections are the commentary on a particular science by those who may claim to speak with competence on the processes of the human mind, the nature of knowledge or of science in general, but who claim no special competence in the particular science under consideration. This is usually the commentary of philosophers who may assert their right to make all knowledge, as well as all reality, their province. The same man may, of course, be both a mathematician and a philosopher; as, for example, Plato, Descartes, Pascal.

In the case of mathematics, the disparity between discourse in and about the science could hardly escape notice. Even if no preliminary rule of caution were laid down, we should be struck by the contrast between the agreement mathematicians have been able to reach in the solution of their problems and the disagreement of the commentators on basic questions about mathematics. To this there may be one significant exception. Mathematics is honored for the precision of its concepts, the rigor of its demonstrations, the certitude of its truth. Even its detractors—like Swift or Berkeley—concede the exactitude and brilliance of mathematics while questioning its utility; or they admit its intellectual austerity while challenging some application of its method. Its “clearness and certainty of demonstration,” Berkeley writes, “is hardly anywhere else to be found.”

This general agreement about the quality of mathematical thought may explain why in all epochs mathematics has been looked upon as the type of certain and exact knowledge. Sometimes it is taken as more than a model for other sciences; it is regarded as the method of pure science itself or as the universal science. Sometimes its excellences are thought to be qualified by the limited or special character of its objects; or it is contrasted with other disciplines which, employing different methods, deal with more fundamental matters no less scientifically. But always the conclusions of mathematics serve to exemplify rational truth; always the method of mathematics represents the spirit of dispassionate thought; always mathematical knowledge symbolizes the power of the human mind to rise above sensible particulars and contingent events to universal and necessary relationships.

Mathematics means this not only to mathematicians and philosophers, but also to moralists and statesmen. “The objects of geometrical inquiry,” writes Alexander Hamilton, “are so entirely abstracted from those pursuits which stir and put in motion the unruly passions of the human heart, that mankind, without difficulty, adopt not only the more simple theorems of the science, but even those abstruse paradoxes which, however they may appear susceptible of demonstration, are at variance with the natural conceptions which the mind, without the aid of philosophy, would be led to entertain upon the subject…. But in the sciences of morals and politics, men are found far less tractable.” This, Hamilton points out, is not due merely to the passionate interest in their problems. “It cannot be pretended,” he says, “that the principles of moral and political knowledge have, in general, the same degree of certainty with those of mathematics.”

Admiration for mathematics often extends beyond enthusiasm for its exemplary virtues or delight in its intellectual beauty to the recognition of its influence on the whole history of thought. Yet here differences of opinion begin to appear.

In the ancient world Plato and Aristotle represent opposite estimates of the importance of mathematics for the rest of philosophy. For the Platonists, Aristotle says, “mathematics has come to be identical with philosophy, though they say that it should be studied for the sake of other things.” He complains of those students of science who “do not listen to a lecturer unless he speaks mathematically.” They make the error of supposing that “the minute accuracy of mathematics is … to be demanded in all cases,” whereas, according to Aristotle’s own view, “its method is not that of natural science.”

In the modern world, thinkers who are both mathematicians and philosophers, like Descartes and Whitehead, represent a return to the Platonic point of view; while Kant, even more than Aristotle, insists that the philosopher is grievously misled if he tries to follow the method of mathematics in his own inquiries. Whitehead charges Aristotle with having deposed mathematics from its high role “as a formative element in the development of philosophy”—a demotion which lasted until, with Descartes and others in the 17th century, mathematics recovered the importance it had for Plato.

Attempting to qualify his own enthusiasm, Whitehead admits that he would not “go so far as to say that to construct a history of thought without a 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.”

For Kant the madness lies not in the pursuit of mathematics itself, but in the delusion of the philosopher that he can proceed in the same way. “The science of mathematics,” Kant writes, “presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience. Such examples are always contagious, particularly when the faculty is the same, which naturally flatters itself that it will meet with the same success in other cases which it has had in one.” The expectation naturally arises that the method of mathematics “would have the same success outside the field of quantities.” But philosophers who understand their own task, Kant thinks, should not be infected by the “confidence … of those who are masters in the art of mathematics… as to their ability of achieving such success” by applying its method in other fields.

“The exactness of mathematics,” Kant holds, “depends on definitions, axioms, and demonstrations…. None of these can be achieved or imitated by the philosopher in the sense in which they are understood by the mathematician,” because, according to Kant, the validity of the mathematician’s definitions and demonstrations ultimately depends on the fact that he is able to construct the concepts he uses. The point is not that mathematics obtains its objects from reason rather than experience, but rather that it obtains them from reason by construction; as, for example, Euclid begins by constructing a triangle which corresponds with his definition of that figure.

Hence, Kant maintains, “we must not try in philosophy to imitate mathematics by beginning with definitions, except it be by the way of experiment…. In philosophy, in fact, the definition in its complete clearness ought to conclude rather than begin our work”; whereas in mathematics we cannot begin until we have constructed the objects corresponding to our definitions. “It follows from all this,” Kant concludes, “that it is not in accordance with the very nature of philosophy to boast of its dogmatical character, particularly in the field of pure reason, and to deck itself with the titles and ribands of mathematics.”

Differences of opinion about mathematics represent, for the most part, philosophical controversy concerning the nature of science or the objects of its knowledge. Mathematicians who engage in such controversy assume the role of philosophers in doing so, for mathematics itself is not concerned with questions of this sort. But there are some questions about mathematics which seem to call for a close study of the science itself and even for proficiency in its subject matter and operations. They are questions about the scope of mathematics and about the divisions of the science, in relation to one another and to its unity. On these issues, mathematicians disagree not only with philosophers, but among themselves and in their capacity as mathematicians.

These issues usually involve different interpretations of the history of mathematics. The problem is not one of the origin of mathematics.

The ancient opinion, found in Herodotus, Plato, and Aristotle, that the mathematical arts, especially geometry, were first developed by the Egyptians, is of interest because of the questions it raises about the circumstances of the origin of mathematics. Herodotus seems to suggest that geometry arose as an aid in the practice of surveying land. “From this practice,” he says, “geometry first came to be known in Egypt, whence it passed into Greece.” Aristotle, on the other hand, separating from the useful arts those which “do not aim at utility,” thinks the latter arose “first in the places where men first began to have leisure. That is why the mathematical arts were founded in Egypt, for there the priestly caste was allowed to be at leisure.”

The Greek development of mathematics very early distinguishes between the pure sciences of arithmetic and geometry and their useful applications in the arts of measurement. The Greeks conceived mathematics as essentially speculative rather than practical or productive. They also divorced it from empirical investigation of the sensible world. As arithmetic is concerned with numbers, not with numbered things, and geometry with figures, not with physical shapes, areas, or volumes, so Plato points out that music and astronomy belong to the mathematical sciences when they deal not with audible harmonies but with their numerical ratios, not with visible celestial motions but with their geometrical configurations.

Provoked by Glaucon’s interest in the usefulness of the mathematical arts, Socrates excludes their utility as being of no interest to the philosopher. He recommends arithmetic and its sister disciplines only so far as these sciences entirely ignore the world of sensible things. The reason why the philosopher “who has to rise out of the sea of change and lay hold of true being … must be an arithmetician,” he explains, is that arithmetic can have “a very great and elevating effect,” when it compels “the soul to reason about abstract number” and rebels “against the introduction of visible or tangible objects into the argument.” In the same way, only when it concerns itself with “knowledge of the eternal,” not with measuring earthly distances, will geometry “draw the soul towards truth, and create the spirit of philosophy.” The astronomer, like the geometer, “should employ problems, and let the heavens alone, if he would approach the subject in the right way”; and, like the astronomer, the student of harmony will work in vain, if he compares “the sounds and consonances which are heard only” and so fails to “reach the natural harmonics of number.”

About the non-empirical or non-experimental character of mathematics there has been little dispute. It is seldom suggested that the growth of mathematical knowledge depends upon improvement in methods of observation. But on the relation of mathematics to physics, which raises the whole problem of pure and applied mathematics, or of mathematical and experimental physics, there has been much controversy, especially in modern times.

Bacon, for example, adopts the ancient division of mathematics into pure and mixed, the former “wholly abstracted from matter and physical axioms.” Though he regards mathematics as a useful instrument in physics—“the investigation of nature” being “best conducted when mathematics are applied to physics”—he also insists upon the primacy of physics and upon its essentially experimental character. Physics has been corrupted, he says, by logic and by mathematics when these seek to dominate instead of to serve it. “It is a strange fatality that mathematics and logic, which ought to be but handmaids to physics, should boast their certainty before it, and even exercise dominion against it.”

The certainty and clarity which Hume is willing to attribute to mathematics cannot, in his opinion, be extended to mathematical physics. “The most perfect philosophy of the natural kind,” he thinks, “only staves off our ignorance a little longer…. Nor is geometry, when taken into the assistance of natural philosophy, ever able to remedy this defect, or lead us into the knowledge of ultimate causes, by all that accuracy of reasoning for which it is so justly celebrated. Every part of mixed mathematics,” Hume continues, “proceeds upon the assumption that certain laws are established by nature in her operations; and abstract reasonings are employed, either to assist experience in the discovery of these laws, or to determine their influence in particular instances, where it depends upon any precise degree of distance or quantity.” When mixed with physics, mathematics remains subordinate—at best an aid in the formulation and the discovery of the laws of nature.

A different view seems to be taken by the great mathematicians and physicists of the 17th century. Galileo, Descartes, and Newton tend to make mathematical analysis an integral part of physics. As the structure of the world is mathematical, so, too, must the science of nature be mathematical. Geometry, says Descartes, is “the science which furnishes a general knowledge of the measurement of all bodies.” If we retain the ancient distinction between geometry and mechanics, it can only be in terms of the assumption, “confirmed by the usage” of these names, that “geometry is precise and exact, while mechanics is not.”

In the preface to his Mathematical Principles of Natural Philosophy, Newton also says that “geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring.” What is called “rational mechanics” must not be confused with the manual arts of measurement which are imperfect and inexact; and it is therefore wrong to distinguish geometry from mechanics as that which is perfectly accurate from that which is less so. “But since the manual arts are chiefly employed in the moving of bodies, it happens that geometry is commonly referred to their magnitude, and mechanics to their motion.”

Newton himself does not abide by this distinction. His aim is to subject all the phenomena of nature “to the laws of mathematics” and to cultivate mathematics as far as it relates to natural philosophy. “I offer this work as the mathematical principles of philosophy, for the whole burden of philosophy consists in this—from the phenomena of motions to investigate the forces of nature and from these forces to demonstrate the other phenomena.” He regrets that he has not been able to deduce all the phenomena of nature “by the same kind of reasoning from mechanical principles.”

Fourier goes even further. “Mathematical analysis,” he says, is “as extensive as nature itself.” Mathematical analysis has “necessary relations with sensible phenomena.” In laying hold “of the laws of these phenomena,” mathematics “interprets them by the same language as if to attest the unity and simplicity of the plan of the universe, and to make still more evident that unchangeable order which presides over all natural causes.” This much had been said or implied by Descartes and Newton. But in addition to all this, Fourier, from his own experience in developing a mathematical theory of heat, comes to the conclusion that “profound study of nature is the most fertile source of mathematical discoveries.” Mathematics itself benefits from its alliance with physics; it increases in analytical power and in the generality of its formulations as physical inquiries extend the range of phenomena to be analyzed and formulated.

The relations of mathematics to physics are considered in the chapters on ASTRONOMY, MECHANICS, and PHYSICS. Mathematical physics must be examined in the light of the opinion that mathematics and physics are separate sciences, distinct in object and method. Furthermore, whereas some of the major contributions to mathematics appear in the great books of physics or natural philosophy (e.g., Archimedes, Kepler, Newton, Fourier), even more fundamental formulations of the science occur in great books devoted exclusively to mathematics: Euclid’s Elements (on geometry), Apollonius’ treatise On Conic Sections, Nicomachus’ Introduction to Arithmetic, Descartes’ Geometry, and Pascal’s mathematical papers. Others belonging to this latter group are listed in the Additional Readings. The great modern advances in mathematics are exemplified by the works of Gauss, Lobachevski, Hamilton, Riemann, Boole, Dedekind, Peano, Frege, Cantor, Hilbert.

It would be both natural and reasonable to inquire about the relation between the great works of mathematics included in this set and the equally great treatises or monographs, listed in the Additional Readings, which represent for the most part the contributions of the 19th century. But since the major question which immediately confronts us in such an inquiry concerns the relation of modern to ancient mathematics, we can examine the problem in terms of the works included in this set, for they represent both the continuity and the discontinuity in the tradition of mathematical science.

Galileo and Newton are disciples of Euclid and Archimedes; Fourier is a disciple of Newton and Descartes. But Descartes is the great innovator. He seems to be quite self-conscious of his radical departure from the ancients and from the state of mathematics as he found it in his own day. Yet the truth and power of his mathematical discoveries seem so evident to him that he cannot doubt the ancients must have had some inkling of it.

“I am quite ready to believe,” he writes, “that the greater minds of former ages had some knowledge of it, nature even conducting them to it. We have sufficient evidence that the ancient Geometricians made use of a certain analysis which they extended to the resolution of all problems, though they grudged the secret to posterity. At the present day also there flourishes a certain kind of Arithmetic, called Algebra, which designs to effect, when dealing with numbers, what the ancients achieved in the matter of figures. These two methods,” he claims, “are nothing else than the spontaneous fruit sprung from the inborn principles of the discipline here in question.”

Descartes does not regard his success as consisting in the advance of mathematical truth through discoveries based upon principles or conclusions already established. Nor would he even be satisfied to say that his use of algebra in developing analytical geometry created a new branch of mathematics. Rather, in his own view, it tended to unify all existing branches and to form a single universal method of analysis. In effect, it revolutionized the whole character of mathematics and laid the foundation for the characteristically modern development of that science since his day. “To speak freely,” he writes, “I am convinced that it is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency, as being the source of all others.”

One need not quite agree with Bertrand Russell that pure mathematics was not discovered until the 19th century, in order to perceive that the discoveries made in that century carry out the spirit of the Cartesian revolution. If one understands the difference between the universal mathematics of Descartes and the separate sciences of arithmetic and geometry as developed by the ancients; if one understands the difference between the theory of equations in Descartes and the theory of proportions in Euclid; if one understands how algebraic symbolism, replacing numbers by letters, frees both arithmetic and geometry from definite quantities, then the profound discontinuity between modern and ancient mathematics begins to be discernible.

There are other differences contributing to that discontinuity, such as the modern treatment of the infinite, the invention of the calculus, and the theory of functions. But what is of prime importance for the purpose of understanding the nature of mathematics, its objects, and its methods, is the perception of the discontinuity in any one or another of its manifestations. Here is a fundamental disagreement about the nature of mathematics which is not an issue between philosophers disputing the definition of the science, but rather an issue made by the actual work of mathematicians in ancient and modern times.

In his Battle of the Books—ancient and modern—Swift sees only the great poets and philosophers of the two epochs set against one another. The battle between the ancient and the modern books of mathematics might be as dramatically represented. In such affairs there is a natural tendency to prejudge the issue in favor of the modern contender. That prejudice has reason on its side in certain fields of knowledge where the perfection of new instruments and the discovery of new facts work to the advantage of the latecomer. But it is questionable whether in this dispute over the nature of mathematics the same advantage prevails.

When the issue is fairly explored by an examination of the differences between the great masterpieces of ancient and modern mathematics, it may be found impossible to say that truth lies more on one side than on the other, or that one conception of mathematics is more fruitful than another, because the two versions of the science may seem to be incommensurable in their aims, methods, and standards of accomplishment.

One example will illustrate this incommensurability. The ancient notion of number, as may be seen in Nicomachus’ Introduction to Arithmetic, limits the variety of numbers. A number always numbers a number of things, even though we can deal with the number itself apart from any set of numbered things. It is always a positive and integral quantity which, excepting unity itself, “the natural starting point of all numbers,” contains a multitude of discrete units.

Numbers are classified according to the way in which they are constituted of parts and according to the constitution of these parts. The primary division of numbers is into even and odd. “The even is that which can be divided into two equal parts without a unit intervening in the middle; and the odd is that which cannot be divided into two equal parts because of the aforesaid intervention of a unit.”

The even numbers are capable of subdivision into the even-times-even, the odd-times-even, and the even-times-odd; and the odd into the prime and incomposite, the secondary and composite, and the number which, in itself, is secondary and composite, but relatively is prime and incomposite. The peculiarities of these types of number are explained in the chapter on QUANTITY. There are still further classifications of numbers into superabundant, deficient, and perfect; and of the parts of numbers in relation to the numbers of which they are parts.

Finally, numbers are considered in terms of their geometrical properties, to be observed when their units are disposed discretely in spatial patterns, and in one, two, or three dimensions. There are linear, plane, and solid numbers, and among plane numbers, for example, there are triangular, square, pentagonal, hexagonal numbers, and so on.

The arithmetic operations of addition, subtraction, multiplication, and division are performed in the production of numbers or in the resolution of numbers into their parts. But though any two numbers can be added together or multiplied, the inverse operations cannot always be performed. A greater number cannot be subtracted from a less, for subtraction consists in taking a part from the whole, and leaving a positive remainder. Since division is the decomposition of a number into its parts, a number cannot be divided by one greater than itself, for the greater cannot be a part of the less.

In short, in Nicomachus’ theory of numbers what later came to be treated as negative numbers and fractions can have no place. Nicomachus will not carry out arithmetical operations in all possible directions without regard to the result obtained. He refuses to perform these operations when the results which would be obtained do not have for him the requisite mathematical reality. He does not find it repugnant to reason that subtraction and division, unlike addition and multiplication, are not possible for any two numbers; as, for example, subtracting a larger from a smaller number, or using a divisor which does not go into the dividend evenly, and so leaves a fractional remainder. On the contrary, Nicomachus finds it repugnant to reason to perform these operations in violation of their proper meaning, and to produce thereby results, such as negative quantities and fractions, which are for him not numbers, i.e., which cannot number any real thing.

Understanding the nature of square numbers, Nicomachus would be able to understand a square root, but he would not see why the operation of extracting the square root should be applied to numbers which are not square. Hence another kind of modern number, the irrational fraction which is generated by such operations as the extraction of the square of positive integers which are not perfect squares, would never appear in Nicomachus’ set of numbers; nor would the imaginary number, which is the result of applying the same operation to negative quantities.

When the arithmetical operations are performed algebraically, with unknowns as well as definite quantities, the solution of equations requires the employment of terms which Nicomachus would not admit to be numbers—negatives, fractions (both rational and irrational), imaginaries, and complex numbers, which are partly real and partly imaginary. Descartes finds nothing repugnant in these novel quantities. On the contrary, he would find it repugnant not to be able to perform the basic arithmetical operations without restriction. Algebra would be impossible, and with it the general method of analysis that proceeds in terms of the purely formal structure of equations from which all definite quantities have been excluded. It would also be impossible to do what Descartes thinks essential to the unity of mathematics, namely, to represent geometrical operations algebraically and to perform most algebraic operations geometrically.

Geometrical loci cannot be expressed by algebraic formulae or equations, unless there are as many numbers as there are points on a line. The number series for Nicomachus, without fractions and irrationals, is neither dense nor continuous. There are fewer numbers than there are points on a line. And without the use of zero, negative numbers, and fractions—none of which would be regarded as numbers by Nicomachus—it would be impossible for Descartes to construct a set of coordinates for the geometrical representation of equations, whereby all the points in a plane have their unique numerical equivalents.

The Cartesian synthesis of algebra and geometry, which in his view vastly increases the power of each, violates the ancient distinction between continuous and discontinuous quantities—magnitudes (like lines and planes) and multitudes (or numbers). Euclid, for example, treats the irrational or the incommensurable always as a relation of magnitudes, never of multitudes, or numbers; for him certain geometrical relationships cannot be expressed numerically. Arithmetic and geometry are not even coordinate, much less co-extensive sciences. Arithmetic is the simpler, the more elementary science, and is presupposed by geometry.

Other examples arising from the innovations of Descartes might be employed to show the chasm between the arithmetic and geometry of the ancients, and modern mathematics—such as the treatment of infinite magnitudes and numbers, the theory of functions, and the method of the calculus. But the multiplication of examples does not seem necessary to suggest that there may be no answer to the question, Is Descartes right, and Nicomachus and Euclid wrong? or to the question, Are the modern innovations improvements or corruptions of the mathematical arts and sciences?

These questions are not like questions concerning the truth or falsity of a proposition in mathematics or the validity of a proof. A given theorem in Euclid must, in the light of his definitions, axioms, and postulates, be either true or false; and accordingly Euclid’s demonstrations or constructions are either cogent or fallacious. The same rules apply to Descartes. But whether Euclid’s or Descartes’ conception of the whole mathematical enterprise is right seems to present a choice between disparate worlds, a choice to be made by reference to principles and purposes which are themselves not mathematical.

Modern mathematics may be much more useful in its physical applications, especially in the analysis and calculation of variable notions or quantities. It may have a special elegance and simplicity, as well as greater unity and even systematic rigor. But it may also purchase these qualities at the expense of the kind of intelligibility which seems to characterize ancient mathematics as a result of the insistence that its objects have an immediately recognizable reality. Ancient mathematics never occasioned such an extreme remark as that made by Bertrand Russell about modern mathematics—that it is “the science in which we never know what we are talking about, nor whether what we are saying is true.”

The question of the reality of the objects of mathematics is in part a problem for the mathematician and in part a question for the philosopher. The problem for the mathematician seems to be one of establishing the existence of the objects he defines. This can be illustrated by reference to Euclid’s Elements.

The basic principles, as Euclid expounds the science, seem to be threefold: definitions, postulates, and axioms or common notions. The axioms are called ‘common notions’ because they are truths common to other branches of mathematics as well as to geometry. The common notions are called “axioms” because their truth is supposed to be self-evident. In contrast, the postulates are peculiar to geometry, for they are written as rules of construction. They demand that certain operations be assumed possible, such as the drawing of a straight line or a circle, or the transposition of a figure from one portion of space to another without alteration of its form or quantity.

Euclid’s definitions include the definition of a straight line and a circle. His first two postulates, therefore, seem to ask us to assume that space is such that these defined geometrical objects exist in it as they are defined; or, in other words, that objects corresponding to the definitions have geometrical reality. But there are many definitions—of a triangle, of an equilateral triangle, of a parallelogram—for which Euclid states no postulate demanding that we assume the geometrical reality of the object defined. Hence before he undertakes to demonstrate the properties of these figures, he finds it necessary to prove that they can be constructed. Until they are constructed, and the construction demonstrated, the definitions state only possibilities to which no geometrical realities are known to correspond in the space determined by Euclid’s postulates.

In his first constructions, Euclid can employ only the definition of the figure itself, his axioms, and those postulates which permit him to use certain mechanical devices—the straight edge and the compass, which are the mechanical equivalents of his postulates that a straight line can be drawn between any two points and a circle described with any radius from any point upon a plane. When, for example, in the first proposition of Book I, Euclid thus demonstrates the construction of an equilateral triangle, he has proved the geometrical existence of that figure, or, in other words, its reality in the space of his postulates.

A number of questions can be asked about this and many other similar demonstrations. The postulates being assumptions, their truth can be questioned and an effort made to prove or disprove them. This type of questioning led to the development of the non-Euclidean geometries. After centuries of trying unsuccessfully to prove Euclid’s postulate about parallel lines, geometers like Lobachevski and Riemann postulated other conditions concerning parallels, with consequences for the properties of other geometrical figures.

The interior angles of a Euclidean triangle, for example, equal the sum of two right angles; in certain non-Euclidean triangles, they add up to more or less than two rights. One interpretation of this situation is that the truth of conclusions in geometry is entirely dependent on arbitrary assumptions. Another is that the several variants of the parallel postulate indicate the selection of different spaces in which to construct figures; and under each set of spatial conditions postulated, there is only one body of geometrical truths concerning the properties of the figures therein constructed.

Another type of question concerns the logical, as opposed to the geometrical, conditions of geometrical proof. In his essay On Geometrical Demonstration, Pascal declares the geometric method to be the most perfect available to men, for it “consists not in defining or in proving everything, nor in defining or proving nothing, but in maintaining itself in the middleground of not defining things which are clear to all men and in defining all others; and not proving everything known to men, but in proving all the other things.” This method, it seems, is not restricted to the subject matter of geometry; to Descartes and Spinoza, at least, it seems to be the method for demonstrating any theoretical truth. Descartes presents “arguments demonstrating the existence of God and the distinction between soul and body, drawn up in geometrical fashion”; and as its title page indicates, the whole of Spinoza’s Ethics is set forth “in geometrical order.”

It may be questioned whether the postulates which Descartes adds to his definitions and axioms, or those which Spinoza introduces beginning with Proposition 13 of Book II, function as postulates do in geometry, i.e., as rules of construction; it may similarly be questioned whether Spinoza is following the geometrical method in Book I where he proceeds without any postulates at all. But the more general question concerns the criteria for testing the consistency and the adequacy of the primitive propositions—the definitions, axioms, postulates—laid down as the foundation for all that is to be demonstrated. The investigation of this problem calls for an examination of the whole process of proof, from which has developed the modern theory of mathematical logic that challenges the universality and adequacy of the traditional logic of Aristotle, and asserts that mathematics and logic are continuous with one another—essentially the same discipline.

The issues raised by mathematical logic or the logic of mathematics are considered in the chapters on HYPOTHESIS, LOGIC, and REASONING. Here we must turn finally to one other question which is of interest principally to the philosopher rather than the mathematician. It concerns the objects of mathematics. It is a question about their reality or mode of existence which cannot be answered by the mathematical proof of a construction.

When, for example, Euclid constructs an equilateral triangle, the figure established cannot be the one imperfectly drawn upon paper. The postulated permission to use a ruler and compass does not remove the imperfection of these mechanical instruments or the inaccuracy in their physical use. The triangle whose properties the geometer tries to demonstrate must be perfect, as no actually drawn figure can be. The philosophical question, therefore, concerns the reality or existence of this ideal, perfect figure. The same question can be asked about pure numbers—numbers apart from all numbered things.

Are the objects of mathematics purely intelligible beings existing apart from the sensible world of material things? Or are they ideal entities—not in the sense of existing outside the mind, but in the sense of being ideas in the mind itself rather than perceptible particulars?

As indicated in the chapters on BEING, FORM, and IDEA, Plato and Aristotle seem to answer these questions differently. But there are further differences among those who regard mathematical objects as having being only in the mind.

Aristotle, Aquinas, Locke, and James, for example, think of the objects of mathematics as universals formed by abstraction from the particulars of sense and imagination. “The mathematicals,” such as numbers and figures, Aquinas writes, “do not subsist as separate beings.” Apart from numbered things and physical configurations, numbers and figures “have a separate existence only in the reason, in so far as they are abstracted from motion and matter.” Hobbes, Berkeley, and Hume, on the other hand, deny abstract ideas or universal concepts. “Let any man try to conceive a triangle in general,” Hume declares, “which is neither isosceles nor scalenum, nor has any particular length or proportion of sides; and he will soon perceive the absurdity of all the scholastic notions with regard to abstraction and general ideas.”

Despite these differences, there seems to be general agreement in the tradition of the great books that the truths of mathematics are rational rather than empirical; or, in the language of Kant and James, a priori rather than a posteriori. But the meaning of this agreement is not the same for those who think that truth in mathematics does not differ from truth in other sciences and those who think that mathematical truths stand alone precisely because they are not about matters of fact or real existence.

Plato, for whom all science is knowledge of purely intelligible objects, regards the mathematical sciences as inferior to dialectic in the knowledge of such objects “because they start from hypotheses and do not ascend to principles.” The students of such sciences, Plato writes, “assume the odd and the even and the figures and three kinds of angles and the like in their several branches of science; these are their hypotheses, which they and everybody are supposed to know, and therefore they do not deign to give any account of them either to themselves or others; but they begin with them, and go on until they arrive at last, and in a consistent manner, at their conclusion.”

For Aristotle, what differentiates mathematics from physics and metaphysics is the special character of its objects. Physics and metaphysics both deal with substances as they exist outside the mind, whereas the objects of mathematics are abstractions. Though figures and numbers “are inseparable in fact” from material substances, they are “separable from any particular kind of body by an effort of abstraction.” This does not deny, for example, that physical things have perceptible figures. It merely insists that the geometer does not treat figures as sensible, but as intelligible, that is, as abstracted from matter. Nevertheless, the truths of mathematics, no less than those of physics and metaphysics, apply to reality. All three sciences are further alike in demonstrating their conclusions rationally rather than by experiment. All three employ induction to obtain their principles, though metaphysics alone attains to the first principles of all science.

For Kant, “mathematical cognition is cognition by means of the construction of conceptions.” To explain this he cites the example of the construction of a triangle. “I construct a triangle, by the presentation of the object which corresponds to this conception, either by mere imagination (in pure intuition) or upon paper (in empirical intuition); in both cases completely a priori without borrowing the type of that figure from any experience…. We keep our eye merely on the act of the construction of the conception, and pay no attention to the various modes of determining it, for example, its size, the length of its sides, the size of its angles.” The a priori character of such intuitions, on which rests the a priori character of mathematical truths, does not mean that mathematics has no relevance to experience. Arithmetic and geometry are like physics, according to Kant; they are sciences of experience or nature but like pure (as opposed to empirical) physics, they are a priori sciences. Since Kant holds that experience itself is constituted by a priori forms of perception, he can ascribe the validity which mathematics has for all possible experience to the “a priori intuition of the pure forms of phenomena—space and time.”

Bertrand Russell rejects this “Kantian view which [asserts] that mathematical reasoning is not strictly formal, but always uses intuitions, i.e., the a priori knowledge of space and time. Thanks to the progress of Symbolic Logic… this part of the Kantian philosophy,” Russell holds, “is now capable of a final and irrevocable refutation.” Leibniz, before Kant, had advocated “the general doctrine that all mathematics is a deduction from logical principles,” but, according to Russell, he had failed to substantiate this insight, partly because of his “belief in the logical necessity of Euclidean geometry.” The same belief is, in Russell’s opinion, the cause of Kant’s error. “The actual propositions of Euclid… do not follow from the principles of logic alone; and the perception of this fact,” he thinks, “led Kant to his innovations in the theory of knowledge. But since the growth of non-Euclidean geometry, it has appeared that pure mathematics has no concern with the question whether the axioms and propositions of Euclid hold of actual space or not.”

Russell asserts that “by the help of ten principles of deduction and ten other premises of a general logical nature (e.g., ‘implication is a relation’), all mathematics can be strictly and formally deduced.” He regards “the fact that all Mathematics is Symbolic Logic” as “one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of Mathematics consists in the analysis of Symbolic Logic itself.” Though this view of mathematics may not be worked out in detail except in such treatises as Russell’s Principles of Mathematics and in the Principia Mathematica, on which he collaborated with Whitehead, the conception of mathematics as a purely formal science, analogous to (if not identical with) logic, does have some anticipations in the great books. For James, as for Locke and Hume, mathematics is strictly a science of the relations between ideas, not of real existences. “As regards mathematical judgments,” James writes, “they are all ‘rational propositions’… for they express results of comparison and nothing more. The mathematical sciences deal with similarities and equalities exclusively, and not with coexistences and sequences.” Both James and Locke, however, differ from Hume in thinking that there are sciences other than those of number and quantity which can demonstrate their conclusions with certitude.

The foregoing discussion indicates some of the differences among philosophers concerning the objects of mathematics, the conditions of its truth, and its relation to other sciences. These disagreements do not seem to take the form of an opposition between ancient and modern thought, like that between ancient and modern mathematicians concerning the nature of their science. The two oppositions do not run parallel to one another.

On the contrary, the objections which modern philosophers, especially Berkeley, Hume, and Kant, raise against the notion of infinite quantities seem to favor the ancient rather than the modern tenor of mathematical thought. Though the reasons they give do not derive from the same principles as those to which Plato and Aristotle appeal, they, like the ancients, appear to insist upon a certain type of intelligibility in the objects of mathematics, which seems to have been sacrificed in the mathematical development initiated by Descartes.

OUTLINE OF TOPICS

1. The science of mathematics: its branches or divisions; the origin and development of mathematics 1a. The distinction of mathematics from physics and metaphysics: its relation to logic 1b. The service of mathematics to dialectic and philosophy: its place in liberal education 1c. The certainty and exactitude of mathematical knowledge: the a priori foundations of arithmetic and geometry 1d. The ideal of a universal mathesis; the unification of arithmetic and geometry
2. The objects of mathematics: number, figure, extension, relation, order 2a. The apprehension of mathematical objects: by intuition, abstraction, imagination, construction; the forms of time and space 2b. The being of mathematical objects: their real, ideal, or mental existence 2c. Kinds of quantity: continuous and discrete quantities; the problem of the irrational
3. Method in mathematics: the model of mathematical thought 3a. The conditions and character of demonstration in mathematics: the use of definitions, postulates, axioms, hypotheses 3b. The role of construction: its bearing on proof, mathematical existence, and the scope of mathematical inquiry 3c. Analysis and synthesis: function and variable 3d. Symbols and formulae: the attainment of generality
4. Mathematical techniques 4a. The arithmetic and algebraic processes 4b. The operations of geometry 4c. The use of proportions and equations 4d. The method of exhaustion: the theory of limits and the calculus
5. The applications of mathematics to physical phenomena: the utility of mathematics 5a. The art of measurement 5b. Mathematical physics: the mathematical structure of nature

REFERENCES

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For additional information concerning the style of the references, see the Explanation of Reference Style; for general guidance in the use of The Great Ideas, consult the Preface.

1. The science of mathematics: its branches or divisions; the origin and development of mathematics

5 AESCHYLUS: Prometheus Bound [442-472] 44c-d esp [459-460] 44d 6 HERODOTUS: History, BK I, 70b-c 7 PLATO: Charmides, 7d-8a / Phaedrus, 138c-d / Gorgias, 254a-c / Republic, BK VII, 391b-397a / Timaeus, 451b-c / Statesman, 581a / Philebus, 633b-634b 8 ARISTOTLE: Posterior Analytics, BK I, CH 7 103c-d; CH 27 119b / Metaphysics, BK I, CH 1 [981b3-24] 500a; CH 9 [992a29-34] 510c; BK II, CH 3 513c-d 11 NICOMACHUS: Arithmetic, BK I, 812b-813d; BK II, 831d-832a 16 PTOLEMY: Almagest, BK I, 5a-6a 16 COPERNICUS: Revolutions of the Heavenly Spheres, BK I, 510a-b 17 PLOTINUS: Fifth Ennead, TR IX, CH 11, 250c-d 18 AUGUSTINE: Christian Doctrine, BK II, CH 38 654b-c 19 AQUINAS: Summa Theologica, PART I-II, Q 35, A 8, ANS 779c-780c 23 HOBBES: Leviathan, PART I, 72a-d; PART IV, 268c-d; 269b 30 BACON: Advancement of Learning, 46a-c 31 DESCARTES: Rules, IV, 5c-d; 6c-7c; XIV, 31c-32a / Discourse, PART I, 43b-c; PART II, 47b-d 31 SPINOZA: Ethics, PART I, APPENDIX, 370b-c 34 NEWTON: Principles, 1a-b 35 LOCKE: Human Understanding, BK IV, CH XII, SECT 7 360b-c; SECT 15 363a-b 35 HUME: Human Understanding, SECT IV, DIV 20 458b; SECT VII, DIV 48 470d-471c; SECT XII, DIV 131 508d-509a 41 GIBBON: Decline and Fall, 290a; 299b 42 KANT: Pure Reason, 5d-6b; 17d-18d; 46a-c; 211c-218d / Practical Reason, 295b-d; 330d-331a / Judgement, 553d [fn 1] 46 HEGEL: Philosophy of Right, additions, 40 122d-123b / Philosophy of History, PART I, 219a 53 JAMES: Psychology, 874a-878a

1a. The distinction of mathematics from physics and metaphysics: its relation to logic

7 PLATO: Euthydemus, 75b / Republic, BK VI, 386d-388a; BK VII, 391b-398c 8 ARISTOTLE: Posterior Analytics, BK I, CH 12 [77b27-34] 107a; [78a10-13] 107b-c / Physics, BK II, CH 2 [193b22-194a11] 270a-c; CH 9 [200a15-29] 277c-d / Heavens, BK I, CH 10 [279b32-280a11] 371b-c / Metaphysics, BK I, CH 8 [989b29-990a8] 508a; CH 9 [992a29-b9] 510c-d; BK II, CH 3 [995a15-20] 513d; BK III, CH 2 [996a21-36] 514d-515a; BK IV, CH 1 522a; BK VI, CH 1 547b,d-548c; BK XI, CH 1 [1059b15-21] 587c-d; CH 3 [1061a29-b12] 589c-d; CH 4 589d-590a; CH 7 592b-593a; BK XII, CH 8 [1073b1-7] 603d / Soul, BK I, CH 1 [403b10-16] 632d 9 ARISTOTLE: Ethics, BK VI, CH 8 [1142a12-19] 391b 16 PTOLEMY: Almagest, BK I, 5a-6a 19 AQUINAS: Summa Theologica, PART I, Q 1, A 1, REP 2 3b-4a; Q 7, A 3, ANS 32c-33c; Q 44, A 1, REP 3 238b-239a; Q 85, A 1, REP 2 451c-453c 20 AQUINAS: Summa Theologica, PART II-II, Q 9, A 2, REP 3 424b-425a; PART III, Q 7, A 12, REP 1 754c-755c 23 HOBBES: Leviathan, PART I, 56b-57a; 58a-c; 59b-c; 72a-d; PART IV, 267a-b; 268c-269a 28 GALILEO: Two New Sciences, SECOND DAY, 190b-c 30 BACON: Advancement of Learning, 46a-b 31 DESCARTES: Rules, II, 2d-3b; IV 5a-7d; VII, 12b-13a; XIV, 31c-32a / Discourse, PART II, 46c-47a / Meditations, I, 76b-c / Objections and Replies, 128d-129a; 169c-170a 33 PASCAL: Geometrical Demonstration, 445a-b 34 NEWTON: Principles, 1a-2a 35 LOCKE: Human Understanding, BK IV, CH III, SECT 29 322c-323a; CH XII, SECT 7-13 360b-362d passim 35 HUME: Human Understanding, SECT VII, DIV 48 470d-471c 39 SMITH: Wealth of Nations, BK V, 335b-337a 42 KANT: Pure Reason, 5a-9a; 15c-16c; 17d-19a; 211c-218d esp 215d-217a; 243c-248d passim, esp 244d-245a, 248c / Practical Reason, 295b-d; 311d-313d / Pref. Metaphysical Elements of Ethics, 376c-d. 43 MILL: Utilitarianism, 445b-c 53 JAMES: Psychology, 867a-870a esp 869a-870a

1b. The service of mathematics to dialectic and philosophy: its place in liberal education

7 PLATO: Republic, BK VI, 386d-388a; BK VII, 391b-398c; BK VIII, 403a-d / Timaeus, 448b-450a; 453b-454a; 458b-460b / Theaetetus, 515a-d / Philebus, 633a-635a / Laws, BK V, 691d-692a; 695c-697a; BK VII, 728b-730d 8 ARISTOTLE: Heavens, BK II, CH 1 [299a1-300a19] 390b-391c; CH 7-8 396d-399a,c / Generation and Corruption, BK II, CH 1 [329b5-24] 428d-429a / Metaphysics, BK I, CH 5-6 503d-506b; CH 8 [989b29-990a32] 508a-c; CH 9 [992a24-b9] 510c-d; BK XII, CH 8 [1073b3-7] 603d; BK XIV, CH 3 622d-623d; CH 5-6 624d-626d / Soul, BK I, CH 4 [408b33-409a9] 638d 9 ARISTOTLE: Ethics, BK VI, CH 8 [1142a12-19] 391b 11 NICOMACHUS: Arithmetic, BK I, 811a-813d 16 PTOLEMY: Almagest, BK I, 5b-6a 16 COPERNICUS: Revolutions of the Heavenly Spheres, BK I, 510a-b 16 KEPLER: Epitome, BK IV, 863b-872b passim 17 PLOTINUS: First Ennead, TR III, CH 3 10d-11a 18 AUGUSTINE: Confessions, BK IV, par 30 26b-c / Christian Doctrine, BK II, CH 16, 644d-645a; CH 38-39 654b-655b 23 HOBBES: Leviathan, PART I, 56b-57a; 59b-c; PART II, 164c; PART IV, 268c-d; 269b 24 RABELAIS: Gargantua and Pantagruel, BK I, 27d; BK IV, 278b 25 MONTAIGNE: Essays, 70a-d 28 GALILEO: Two New Sciences, SECOND DAY, 190b-c 30 BACON: Advancement of Learning, 16a-b; 46a-c 31 DESCARTES: Rules, I, 2a-3b; IV, 5a-7d; XIV, 29a-b / Discourse, PART I, 43b-c; PART II, 46c-48b / Geometry, BK I, 297a-b 31 SPINOZA: Ethics, PART I, APPENDIX, 370b-c 33 PASCAL: Pensées, 1-2 171a-172b / Geometrical Demonstration, 442a-446b 34 NEWTON: Principles, 1b-2a 36 SWIFT: Gulliver, PART III, 97a-b 42 KANT: Pure Reason, 15c-16c; 46a-c; 211c-218d esp 215a-c / Judgement, 551a-552c 51 TOLSTOY: War and Peace, BK I, 47b-48d 53 JAMES: Psychology, 882a-883a

1c. The certainty and exactitude of mathematical knowledge: the a priori foundations of arithmetic and geometry

7 PLATO: Philebus, 633b-634b 8 ARISTOTLE: Posterior Analytics, BK I, CH 27 119b / Metaphysics, BK I, CH 2 [982a25-28] 500c; BK II, CH 3 513c-d; BK XIII, CH 3 609a-610a 9 ARISTOTLE: Ethics, BK I, CH 3 [1094b12-28] 339d-340a 11 NICOMACHUS: Arithmetic, BK I, 811a-814b 16 PTOLEMY: Almagest, BK I, 5b-6a 23 HOBBES: Leviathan, PART I, 56b; 59b-c; PART IV, 268c-d; 269b 28 GALILEO: Two New Sciences, FOURTH DAY, 252a-b 30 BACON: Advancement of Learning, 61b-c; 65b / Novum Organum, BK I, APH 59 112b-c 31 DESCARTES: Rules, I, 2a-3b; IV, 5a-7d / Discourse, PART I, 43b-c; PART II, 47b-c; PART IV, 52d-53a; PART V, 54c / Meditations, I, 76b-c; V, 93a-95b / Objections and Replies, 128a-129a / Geometry, BK II, 304a-b 31 SPINOZA: Ethics, PART I, APPENDIX, 370b-c 33 PASCAL: Pensées, 1-2 171a-172b; 33 176b / Geometrical Demonstration, 430a-434a; 442a-446b 35 LOCKE: Human Understanding, BK IV, CH I, SECT 9-10 311b-c; CH III, SECT 18-20 317d-319c passim; SECT 29 322c-323a; CH IV, SECT 6-9 325a-326b; CH XII, SECT 1-8 358c-360c passim, esp SECT 7 360b-c; CH XIII, SECT 3 364a 35 BERKELEY: Human Knowledge, SECT 118 436b-c 35 HUME: Human Understanding, SECT IV, DIV 20 458a-b; SECT VII, DIV 48 470d-471c; SECT XII, DIV 131 508d-509a 36 SWIFT: Gulliver, PART III, 118b-119a 42 KANT: Pure Reason, 5a-8d; 15c-16c; 31b-d; 35b-36a; 46a-b; 68a-69c; 86b-c; 110a; 211c-218d / Practical Reason, 295b-d; 312c-d; 330d-331a / Pref. Metaphysical Elements of Ethics, 376c-d / Science of Right, 399a-b / Judgement, 551a-552a 43 FEDERALIST: NUMBER 31, 103c-104a; NUMBER 85, 257d-258d 43 MILL: Liberty, 283d-284b / Utilitarianism, 445b-c 45 LAVOISIER: Elements of Chemistry, PREF, 1a; 2b 45 FOURIER: Theory of Heat, 173a-b 53 JAMES: Psychology, 175a-176a; 874a-878a; 879b-882a

1d. The ideal of a universal mathesis: the unification of arithmetic and geometry

7 PLATO: Meno, 180d-182c / Theaetetus, 515a-c 8 ARISTOTLE: Categories, CH 6 [4b20-5b1] 9a-c / Posterior Analytics, BK I, CH 7 103c-d / Metaphysics, BK II, CH 3 513c-d; BK VI, CH 1 [1026a18-27] 548b; BK XI, CH 7 [1064b6-9] 592d-593a; BK XIII, CH 2 [1077a9-10] 608b; CH 3 [1077b17-23] 609a 11 EUCLID: Elements, BK III, 30a-40b; BK V 81a-98b; BK VII, DEFINITIONS, 16-19 127b 11 NICOMACHUS: Arithmetic, BK II, 831d-841c 16 KEPLER: Harmonies of the World, 1012b-1014b 23 HOBBES: Leviathan, PART I, 58a-c 31 DESCARTES: Rules 1a-40a,c esp II, 2d-3b, IV 5a-7d, VI 8a-10a, XIV-XXI 28a-40a,c / Discourse, PART II, 46c-48b; PART III, 50d / Geometry 295a-353b esp BK I, 295a-298b 33 PASCAL: Arithmetical Triangle, 447a-456a 36 SWIFT: Gulliver, PART III, 109b-111a 42 KANT: Pure Reason, 68a-69c 45 FOURIER: Theory of Heat, 172a-173b

2. The objects of mathematics: number, figure, extension, relation, order

7 PLATO: Charmides, 7a-8a / Meno, 176d-177a; 180b-183c / Republic, BK VI, 387b-c; BK VII, 393d / Philebus, 633d 8 ARISTOTLE: Categories, CH 6 9a-11a; CH 8 [10a11-16] 15a-b / Physics, BK IV, CH 11 [219b5-8] 299b / Metaphysics, BK III, CH 1 [996a13-15] 514c; CH 5 520c-521b passim; BK VII, CH 2 [1028b18-28] 551a-b; BK XI, CH 3 [1061a29-b4] 589c 9 ARISTOTLE: Rhetoric, BK I, CH 2 [1355b26-32] 595b 11 EUCLID: Elements, BK I, DEFINITIONS 1a-2a; BK VII, DEFINITIONS, 1-2 127a; BK XI, DEFINITIONS, 1 301a 11 NICOMACHUS: Arithmetic, BK I, 811d-812b 16 PTOLEMY: Almagest, BK I, 5a-b 18 AUGUSTINE: Christian Doctrine, BK II, CH 38 654b-c 19 AQUINAS: Summa Theologica, PART I, Q 7, A 3, ANS 32c-33c; A 4, ANS 33d-34c; Q 30, A 1, REP 4 167a-168a; Q 85, A 1, REP 2 451c-453c 23 HOBBES: Leviathan, PART I, 72a-d 30 BACON: Advancement of Learning, 46a-c 31 DESCARTES: Rules, II, 3a-b; IV, 7a-b; VI 8a-10a; XIV, 30d-33b / Discourse, PART II, 47b-d; PART IV, 52d-53a / Meditations, I, 76b-c / Objections and Replies, 217b-d / Geometry, BK II, 304a-306a; 316a-b 34 NEWTON: Principles, 1a-b 35 LOCKE: Human Understanding, BK II, CH XII, SECT 3-5 147d-148b 35 BERKELEY: Human Knowledge, SECT 118-132 436b-439c 35 HUME: Human Understanding, SECT XII, DIV 131 508d-509a 42 KANT: Pure Reason, 46a-c; 62a-d; 68a-69c; 211c-213a 53 JAMES: Psychology, 874a-878a esp 874a

2a. The apprehension of mathematical objects: by intuition, abstraction, imagination, construction; the forms of time and space

7 PLATO: Meno, 180b-183c / Republic, BK VI, 387b-d; BK VII, 393a-394a / Theaetetus, 535b-c 8 ARISTOTLE: Posterior Analytics, BK I, CH 18 111b-c / Metaphysics, BK IX, CH 9 [1051a23-33] 577b-c; BK XIII, CH 3 609a-610a / Memory and Reminiscence, CH 2 [452b17-23] 694b-d 9 ARISTOTLE: Ethics, BK III, CH 3 [1112b20-24] 358d; BK VI, CH 8 [1142a13-19] 391b 11 EUCLID: Elements, BK I, POSTULATES, 1-3 2a 19 AQUINAS: Summa Theologica, PART I, Q 1, A 1, REP 2 3b-4a; Q 85, A 1, REP 2 451c-453c 31 DESCARTES: Rules, XIV-XV 28a-33d / Discourse, PART II, 46d / Meditations, V, 93a-d; VI, 96b-d / Objections and Replies, 217b-d; 218c; 228c-d 33 PASCAL: Pensées, 1-2 171a-172b 35 LOCKE: Human Understanding, BK I, CH I, SECT 16 99b-c; BK II, CH XIII, SECT 5-6 149b-d; CH XVI 165c-167c passim 35 BERKELEY: Human Knowledge, INTRO, SECT 15-16 409a-d; SECT 12-13 415b-c; SECT 118-122 436b-437c 35 HUME: Human Understanding, SECT XII, DIV 122 505c-d; DIV 124-125 506a-507a esp DIV 125, 507b [fn 1] 42 KANT: Pure Reason, 16a-c; 17d-18d; 24d-25b; 31b-d; 35b-36a; 46a-c; 55c-56a; 62a-d; 68a-69c; 86b-c; 87b-c; 110a; 211c-212a / Practical Reason, 295b-d / Science of Right, 399a / Judgement, 551a-553c 52 DOSTOEVSKY: Brothers Karamazov, BK V, 121a-b 53 JAMES: Psychology, 302a-304b; 549a-552a esp 552b [fn 1]; 869a-870a; 874b-878a passim

2b. The being of mathematical objects: their real, ideal, or mental existence

7 PLATO: Phaedo, 228b-230c; 242c-243c / Republic, BK VI, 387b-c; BK VII, 391b-398c / Theaetetus, 535b-c; 541b-d / Sophist, 562c / Philebus, 636b-c / Seventh Letter, 809c-810b 8 ARISTOTLE: Prior Analytics, BK I, CH 41 [49b32-37] 68c / Posterior Analytics, BK I, CH 10 [76b39-77a3] 105d; CH 13 [79a6-10] 108c; CH 18 [81b40-b5] 111b-c / Topics, BK VI, CH 6 [143b11-33] 197b-c / Physics, BK II, CH 2 [193b23-194a11] 270a-c; BK III, CH 5 [204a8-34] 282a-b; BK IV, CH 1 [208b19-24] 287b-c; CH 11 [219b5-8] 299b; CH 14 [223b21-29] 303a / Heavens, BK III, CH 1 [299a1-300a19] 390b-391c; CH 7 [306a1-b2] 397b-d / Metaphysics, BK I, CH 5 [985b22-986a21] 503d-504b; CH 6 [987b10-34] 505c-506a; CH 8 [989b29-990a32] 508a-c; CH 9 [991b9-992a18] 509d-511a; BK III, CH 1 [995b13-18] 514a; [996a13-15] 514c; CH 2 [996a22-36] 514d-515a; [997b12-998a19] 516b-d; CH 5 [1001b26]-CH 6 [1002b25] 520c-521c; BK VII, CH 2 [1028b18-28] 551a-b; CH 10 [1035b32-1036a12] 559b-c; CH 11 [1036b32-1037a4] 560b-c; BK XI, CH 1 [1059b2-13] 587b-c; CH 2 [1060b36-b19] 588c-d; CH 3 [1061a29-b4] 589c; BK XII, CH 1 [1069a30-37] 598b; BK XIII, CH 10 [1075b25]-BK XIII, CH 3 [1078b5] 606c-610a; BK XIII, CH 6-9 611d-618c; BK XIV 619b,d-626d / Soul, BK III, CH 7 [431b13-19] 664b 9 ARISTOTLE: Ethics, BK I, CH 6 [1096a17-19] 341b; BK VI, CH 8 [1142a13-19] 391b 11 EUCLID: Elements, BK I, POSTULATES, 1-3 2a 11 NICOMACHUS: Arithmetic, BK I, 811a-812a; 813d-814b 18 AUGUSTINE: Confessions, BK X, par 19 76a-b / Christian Doctrine, BK II, CH 38 654b-c 19 AQUINAS: Summa Theologica, PART I, Q 5, A 3, REP 4 25a-d; Q 10, A 6, ANS 45c-46d; Q 11, A 1, REP 1 46d-47d; A 3, REP 2 49a-c; Q 30, A 1, REP 4 167a-168a; Q 44, A 1, REP 3 238b-239a; Q 85, A 1, REP 2 451c-453c 20 AQUINAS: Summa Theologica, PART III, Q 7, A 12, REP 1 754c-755c; PART III SUPPL, Q 83, A 2, ANS 976c-978c; A 3, REP 2 978c-980d 31 DESCARTES: Rules, XIV, 30b-32a / Discourse, PART IV, 52d-53a / Meditations, I, 76b-c; V, 93a-d; 96a / Objections and Replies, 169c-170a; 217b-d; 218c; 228c-229a 31 SPINOZA: Ethics, PART I, APPENDIX, 370b-c 33 PASCAL: Vacuum, 373a-b 35 LOCKE: Human Understanding, BK II, CH III, SECT 6 113c-d; BK III, CH III, SECT 19 259c-260a; BK IV, CH IV, SECT 5-8 324d-325c 35 BERKELEY: Human Knowledge, INTRO, SECT 12-16 408a-409d; SECT 12-13 415b-c; SECT 118-128 436b-438d passim, esp SECT 121-122 436d-437c, SECT 125-126 438a-c 35 HUME: Human Understanding, SECT IV, DIV 20 458a-b; SECT XII, DIV 122 505c-d; DIV 124-125 506a-507a esp DIV 125, 507b [fn 1] 42 KANT: Pure Reason, 16a-c; 24a-33d esp 31d-32c; 35b-36a; 46a-c; 62a-d; 87b-c; 91c-d; 94b-95a; 211c-213a esp 211c-212a / Practical Reason, 295b-d; 312c-d / Judgement, 551a-553c 45 FOURIER: Theory of Heat, 183a-b 53 JAMES: Psychology, 874a-878a esp 875a-876b; 880b-881a

2c. Kinds of quantity: continuous and discrete quantities; the problem of the irrational

7 PLATO: Meno, 180d-181c / Republic, BK VII, 403a-d / Parmenides, 499d-500c / Theaetetus, 515a-c 8 ARISTOTLE: Categories, CH 5 [3b32-4a9] 8a-b; CH 6 [4b23-5a14] 9a-c; CH 8 [10a26-11a14] 15d-16b / Physics, BK IV, CH 4 [211b29-b4] 290c; CH 5 [212a3-6] 291d; BK V, CH 3 [227a10-34] 307d-308b; BK VI, CH 1-2 312b,d-315d / Heavens, BK I, CH 1 [268a6-11] 359a; BK III, CH 1 [299a1-300a19] 390b-391c; CH 7 [306a26-b2] 397c-d; CH 8 [306b17-26] 398a / Generation and Corruption, BK I, CH 2 [315b25-317a17] 411b-413a; BK II, CH 10 [337b22-34] 439b-c / Metaphysics, BK I, CH 2 [983a11-20] 501b-c; BK III, CH 2 [994b23-26] 513a; BK V, CH 4 [1001b7-25] 520b-c; BK V, CH 6 [1015b35-1016a17] 536b-c; [1016b6-32] 537a-b; CH 13 [1020a9-14] 541b; BK VIII, CH 3 [1043b33-1044a14] 568b-d; BK X, CH 1 [1052b15-37] 578b,d; CH 3 [1054a20-29] 581a; BK XI, CH 12 [1068b26-1069a18] 597d-598a,c; BK XIV, CH 1 [1087b34-1088a14] 620a-b 11 EUCLID: Elements, BK V, DEFINITIONS, 4-5 81a; BK VII, DEFINITIONS, 2 127a; 20 127b; BK X 191a-300a esp PROP 1 191b-192a 11 ARCHIMEDES: Sphere and Cylinder, BK I, ASSUMPTIONS, 5 404b / Spirals, 484b / Quadrature of the Parabola, 527a-b 11 NICOMACHUS: Arithmetic, BK I, 811d-812a 17 PLOTINUS: Second Ennead, TR III, CH 7, 52c / Fourth Ennead, TR II, CH 1, 139d / Sixth Ennead, TR I, CH 4 253b-254b; TR III, CH 13, 287d-288a; CH 15 289a-c 19 AQUINAS: Summa Theologica, PART I, Q 3, A 1, REP 1 14b-15b; Q 7, A 3, REP 3-4 32c-33c; A 4, ANS 33d-34c; Q 11, A 2, REP 2 47d-48d; Q 30, A 3, ANS 169b-170c; Q 42, A 1, REP 1 224b-225d; Q 48. A 4, REP 3 262a-263a; Q 52, A 1 278d-279b 20 AQUINAS: Summa Theologica, PART II-II, Q 24, A 4, REP 1 491d-492b; PART III, Q 7, A 12, REP 1 754c-755c 28 GALILEO: Two New Sciences, FIRST DAY, 139c-153a passim; THIRD DAY, 201a-202a 31 DESCARTES: Discourse, PART IV, 52d-53a / Meditations, V, 93b 31 SPINOZA: Ethics, PART I, PROP 15, SCHOL 360b-361d 33 PASCAL: Geometrical Demonstration, 434a-439b 34 NEWTON: Principles, BK I, LEMMA I 25a; LEMMA II, SCHOL, 31a-32a 35 LOCKE: Human Understanding, BK II, CH XV, SECT 9 164b-d; CH XVI, SECT 3-4 165d-166b 35 BERKELEY: Human Knowledge, SECT 123-132 437c-439c 42 KANT: Pure Reason, 66d-72c; 124d-125b; 135a-137a,c; 152a-d; 161d-163a 51 TOLSTOY: War and Peace, BK XI, 469a-d

3. Method in mathematics: the model of mathematical thought

7 PLATO: Meno, 179d-183a / Republic, BK VI, 387b-d / Theaetetus, 515b-c 8 ARISTOTLE: Prior Analytics, BK I, CH 41 [49b32-50a4] 68c / Posterior Analytics, BK I, CH 10 104d-105d / Physics, BK II, CH 9 [200a15-29] 277c-d / Metaphysics, BK IV, CH 2 [996a22-36] 514d-515a; BK IX, CH 9 [1051a22-34] 577b-c; BK XI, CH 3 [1061a29-b4] 589c; BK XIII, CH 3 609a-610a 9 ARISTOTLE: Ethics, BK III, CH 3 [1112b20-24] 358d 11 ARCHIMEDES: Method 569a-592a esp 569b-570a 23 HOBBES: Leviathan, PART I, 56b; 58a-c; 59b-c 28 GALILEO: Two New Sciences, FOURTH DAY, 252a-b 30 BACON: Advancement of Learning, 65b / Novum Organum, BK I, APH 59 112b-c 31 DESCARTES: Rules 1a-40a,c esp II, 2a-3b, IV, 5c-7d, XIV-XXI 28a-40a,c / Discourse, PART II, 46c-48b / Meditations, 73a / Objections and Replies, 128a-129a / Geometry 295a-353b esp BK I, 295a-298b, BK II, 304a-306a, 316a-317a, BK III, 331b, 353a 31 SPINOZA: Ethics, PART I, APPENDIX, 370b-c 33 PASCAL: Pensées, 1-3 171a-172b / Geometrical Demonstration, 430a-434a; 442a-446b / Arithmetical Triangle, 451b-452a; 458b-459b; 464a-466a 34 NEWTON: Principles, BK II, LEMMA 2 and SCHOL 168a-170a 35 LOCKE: Human Understanding, BK IV, CH II, SECT 9-10 311b-c; CH III, SECT 18-20 317d-319c passim; SECT 29-30 322c-323c passim; CH IV, SECT 6-9 325a-326b; CH VII, SECT 11, 340c-341a; CH XII, SECT 1-8 358c-360c passim; SECT 14-15 362d-363b; CH XVII, SECT 11 378b 35 BERKELEY: Human Knowledge, INTRO, SECT 12 408a-b; SECT 19 410c; SECT 118-132 436b-439c passim 35 HUME: Human Understanding, SECT IV, DIV 20 458a-b; SECT VII, DIV 48 470d-471c; SECT XII, DIV 131 508d-509a 36 SWIFT: Gulliver, PART III, 118b-119a 42 KANT: Pure Reason, 5a-8d; 17d-18d; 46a-b; 211c-218d / Practical Reason, 302d-303b; 330d-331a / Science of Right, 399a-b 43 MILL: Liberty, 283d-284a 45 LAVOISIER: Elements of Chemistry, PREF, 2b 45 FOURIER: Theory of Heat, 172a-173b 51 TOLSTOY: War and Peace, BK XI, 469a-d; EPILOGUE II, 695b-c 53 JAMES: Psychology, 175a-176a; 874a-878a

3a. The conditions and character of demonstration in mathematics: the use of definitions, postulates, axioms, hypotheses

7 PLATO: Meno, 183b-c / Republic, BK VI, 386d-388a; BK VII, 397c-d 8 ARISTOTLE: Prior Analytics, BK I, CH 41 [49b32-37] 68c / Posterior Analytics, BK I, CH 1 [71a1-16] 97a-b; CH 2 [72a22-24] 98d; CH 5 101b-102b; CH 10 104d-105d; CH 12 106c-107c; CH 23 [84b6-9] 115b-c; CH 24 [85a20-31] 116b-c; [85b4-14] 116d-117a; [86a22-30] 117d-118a; CH 31 [87b33-39] 120a; BK II, CH 3 [91a1-5] 124b; CH 7 [92b12-25] 126c-d; CH 9 [93b21-25] 128a-b / Topics, BK I, CH 1 [101a5-17] 143c-d; BK V, CH 4 [132b32-34] 183b; BK VI, CH 4 [141b3-22]194d-195a; BK VII, CH 3 [153a6-11] 208a-b; BK VIII, CH 1 [157a1-3] 213a; CH 3 [158b29-159a1] 215c / Sophistical Refutations, CH 1 [171b12-18] 236b; [171b34-172a7] 236d / Physics, BK II, CH 9 [200a15-29] 277c-d / Heavens, BK III, CH 4 [302b27-31] 394a / Metaphysics, BK IV, CH 2 [996a21-36] 514d-515a; [996b18-21] 515b; CH 3 [998a25-27] 517a; BK V, CH 3 [1014a36-b1] 534c; BK VII, CH 10 [1036a2-13] 559b-c; BK XI, CH 3 [1061a29-b4] 589c; CH 4 [1061b17-25] 589d-590a; CH 7 [1063b36-1064a9] 592b; BK XIII, CH 2 [1077b1]-CH 3 [1078a32] 608d-609d; BK XIV, CH 2 [1089a22-26] 621c-d 9 ARISTOTLE: Generation of Animals, BK II, CH 6 [742b23-33] 283d-284a / Ethics, BK VII, CH 8 [1151a15-19] 402a 11 EUCLID: Elements 1a-396b esp BK I, DEFINITIONS-COMMON NOTIONS 1a-2a 11 ARCHIMEDES: Sphere and Cylinder, BK I, DEFINITIONS-ASSUMPTIONS 404a-b / Spirals, 484b / Quadrature of the Parabola, 527a-b / Method, 569b-570a; PROP 1 571a-572b esp 572b 19 AQUINAS: Summa Theologica, PART I, Q 85, A 8, REP 2 460b-461b 23 HOBBES: Leviathan, PART I, 56b; 58a-c; 59b-c 28 GILBERT: Loadstone, PREF, 1b-c 28 GALILEO: Two New Sciences, FOURTH DAY, 252a-b 31 DESCARTES: Rules, II, 3a; XIV-XXI 28a-40a,c / Discourse, PART I, 43b-c; PART II, 46d; 47b-c; PART IV, 52d-53a / Meditations, I, 76b-c; V, 93a-d / Objections and Replies, 128a-d / Geometry, BK II, 304a-b; 316a-b; BK III, 353a 33 PASCAL: Pensées, 1-5 171a-173a / Vacuum, 365b-366a; 373a-b / Geometrical Demonstration, 430a-434a; 442a-446b / Arithmetical Triangle, 451b-452a; 458b-459b; 464a-466a 34 NEWTON: Principles, 1a-b 34 HUYGENS: Light, PREF, 551b-552a 35 LOCKE: Human Understanding, BK III, CH XVI, SECT 4 166a-b; BK IV, CH I, SECT 9 308c-309b; CH II, SECT 9-10 311b-c; CH III, SECT 29, 322c; CH VII, SECT 11, 340c-341a; CH XII, SECT 1-8 358c-360c passim; SECT 14-15 362d-363b 35 BERKELEY: Human Knowledge, INTRO, SECT 12 408a-b; SECT 15-16 409a-d; SECT 118 436b-c 35 HUME: Human Understanding, SECT VII, DIV 48 470d-471c; SECT XII, DIV 131 508d-509a 42 KANT: Pure Reason, 17d-18d; 46a-b; 68a-69c; 91c-d; 110a; 110d-111a; 211c-218d / Practical Reason, 302d-303b; 330d-331a / Pref. Metaphysical Elements of Ethics, 376c-d 43 FEDERALIST: NUMBER 31, 103c 43 MILL: Utilitarianism, 445b-c 45 LAVOISIER: Elements of Chemistry, PREF, 2b 53 JAMES: Psychology, 869a-870a; 874a-878a

3b. The role of construction: its bearing on proof, mathematical existence, and the scope of mathematical inquiry

7 PLATO: Meno, 180b-182c / Republic, BK VI, 387b-c 8 ARISTOTLE: Heavens, BK I, CH 10 [279b32-280a12] 371b-c / Metaphysics, BK IX, CH 9 [1051a22-34] 577b-c 11 EUCLID: Elements, BK I, POSTULATES, 1-3 2a; PROP 1 2b-3a 23 HOBBES: Leviathan, PART IV, 267a-b 31 DESCARTES: Rules, XIV-XVII 28a-39d / Geometry, BK II, 304a-306a; 316a-b; BK III, 331b-332b 33 PASCAL: Vacuum, 373a-b 34 NEWTON: Principles, 1a-b 42 KANT: Pure Reason, 31b-d; 68a-69c; 86b-c; 91c-d; 94b-95a; 211c-215a; 217c-d / Science of Right, 399a-b / Judgement, 551a-553c 53 JAMES: Psychology, 673b

3c. Analysis and synthesis: function and variable

9 ARISTOTLE: Ethics, BK III, CH 3 [1112b20-24] 358d 11 ARCHIMEDES: Sphere and Cylinder, BK II, PROP 1 434b-435b; PROP 3-7 437a-443b / Method, 569b-570a; PROP 1, 572b 11 APOLLONIUS: Conics, BK II, PROP 44-47 710b-713a; PROP 49-51 714b-726a; BK III, PROP 54-56 793b-798b 31 DESCARTES: Rules, IV, 5c-7a; VI 8a-10a; XII, 20d-23b; 24a-b; XIII 25b-27d; XIV, 31a-33b; XVI-XXI 33d-40a,c / Objections and Replies, 128a-d / Geometry 295a-353b esp BK I, 296b-297b, 298b-304a, BK II, 308a-314b 33 PASCAL: Geometrical Demonstration, 430a 34 NEWTON: Principles, BK I, LEMMA 19 57b-58b / Optics, BK I, 543a-b 35 LOCKE: Human Understanding, BK IV, CH XII, SECT 15 363a-b; CH XVII, SECT 11 378b 45 LAVOISIER: Elements of Chemistry, PREF, 1a; 2b 45 FOURIER: Theory of Heat, 172a-173b; 196a-251b

3d. Symbols and formulae: the attainment of generality

8 ARISTOTLE: Prior Analytics, BK I, CH 41 [49b32-37] 68c / Posterior Analytics, BK I, CH 10 [76b39-77a2] 105d 11 NICOMACHUS: Arithmetic, BK II, 832a 23 HOBBES: Leviathan, PART I, 56a 31 DESCARTES: Rules, XIV-XXI 28a-40a,c / Discourse, PART II, 46d; 47b-d / Geometry 295a-353b esp BK I, 295a-298b, BK II, 314a-b, BK III, 353a 35 LOCKE: Human Understanding, BK IV, CH III, SECT 19 318b-319a 35 BERKELEY: Human Knowledge, INTRO, SECT 16 409b-d; SECT 121-122 436d-437c 42 KANT: Pure Reason, 68a-69c; 211d-212a; 212d-213a 45 FOURIER: Theory of Heat, 172b-173b; 181a-b; 233b-240b

4. Mathematical techniques

4a. The arithmetic and algebraic processes

8 ARISTOTLE: Topics, BK VIII, CH 14 [163b23-28] 222b 11 EUCLID: Elements, BK VII-IX 127a-190b 11 ARCHIMEDES: Measurement of a Circle, PROP 3 448b-451b / Conoids and Spheroids, LEMMA 455b-456a; LEMMA to PROP 2 456b-457b / Spirals, PROP 10-11 488a-489b / Sand-Reckoner 520a-526b 11 NICOMACHUS: Arithmetic, BK I, 814b-821d 16 KEPLER: Epitome, BK V, 990a-b 23 HOBBES: Leviathan, PART I, 56a; 58a-b 28 GALILEO: Two New Sciences, SECOND DAY, 193b-c 31 DESCARTES: Rules, XVI 33d-35c; XVIII 36b-39d / Discourse, PART II, 46d; 47b-d / Geometry, BK I, 295a-296a 33 PASCAL: Arithmetical Triangle 447a-473b esp 451b-452a, 458b-459b, 464a-466a / Correspondence with Fermat 474a-487a passim 35 LOCKE: Human Understanding, BK IV, CH XVII, SECT 11 378b 35 BERKELEY: Human Knowledge, INTRO, SECT 19 410c; SECT 121-122 436d-437c 42 KANT: Pure Reason, 17d-18d; 68a-69c; 212d-213a; 217c-d 53 JAMES: Psychology, 874b-876b

4b. The operations of geometry

7 PLATO: Meno, 180b-183c / Republic, BK VI, 387b-c 8 ARISTOTLE: Categories, CH 7 [7a29-33] 12c-d 11 EUCLID: Elements, BK I-IV 1a-80b esp BK I, POSTULATES 2a, PROP 1-4 2b-4b, PROP 9-12 7a-9a, PROP 22-23 13b-15a, PROP 31 19b, PROP 42 24b-25a, PROP 44-46 25b-28a, BK II, PROP 14 40a-b; BK VI 99a-126a esp PROP 9-13 106b-109a, PROP 18 112a-b, PROP 25 119a-b, PROP 28-29 121a-123a; BK XI-XIII 301a-396b 11 ARCHIMEDES: Sphere and Cylinder, BK I-II 403a-446a / Measurement of a Circle 447a-451b / Conoids and Spheroids 452a-481b / Spirals 482a-501a / Quadrature of the Parabola 527a-537b / Book of Lemmas 561a-568b 11 APOLLONIUS: Conics 603a-804b esp BK I, PROP 11-13 615a-620a 28 GALILEO: Two New Sciences, FIRST DAY, 149d-150d 31 DESCARTES: Rules, XVI 33d-35c; XVIII 36b-39d / Discourse, PART IV, 52d-53a / Geometry 295a-353b esp BK I, 295a-296a, 297b-298b, BK II, 304a-308a 34 NEWTON: Principles, 1a-b; BK I, LEMMA 15-PROP 29, SCHOL 50a-75b esp LEMMA 19 57b-58b 42 KANT: Judgement, 551a-552a 53 JAMES: Psychology, 550b [fn 1]; 673b; 876b-878a

4c. The use of proportions and equations

8 ARISTOTLE: Posterior Analytics, BK II, CH 17 [99a8-10] 135a / Topics, BK VIII, CH 3 [158b29-35] 215c 9 ARISTOTLE: Ethics, BK II, CH 6 [1106a26-36] 351d-352a; BK V, CH 3 [1131a30-b15] 378d-379b 11 EUCLID: Elements, BK V-VI 81a-126a esp BK V, DEFINITIONS, 5 81a, 7 81a-b, BK VI, PROP 23 117a-b; BK VII, DEFINITIONS, 20 127b; PROP 11-14 134b-136a; PROP 17-22 137a-140a; PROP 33 145a-146a; BK VIII, PROP 1-13 150a-161b; PROP 18-27 163b-170a; BK IX, PROP 8-13 174a-180a; PROP 15-19 181a-183b; PROP 35-36 188b-190b; BK X, PROP 5-9 195a-198b; PROP 11 199b; PROP 14 201a-202a; BK XI, PROP 25 320b-321b; PROP 32-34 327b-332b; PROP 37 335b-336a; BK XII 338a-368b 11 ARCHIMEDES: Conoids and Spheroids, PROP 4 459b-460b 11 APOLLONIUS: Conics 603a-804b passim 11 NICOMACHUS: Arithmetic, BK I-II, 821d-831d; BK II, 841c-848d 16 KEPLER: Harmonies of the World, 1012b-1014b 31 DESCARTES: Rules, VI 8a-10a; XIV, 28b-29a; XVIII-XXI 36b-40a,c / Discourse, PART II, 47b-d / Geometry 295a-353b esp BK III, 332b-341b 51 TOLSTOY: War and Peace, BK XIV, 590a-c

4d. The method of exhaustion: the theory of limits and the calculus

11 EUCLID: Elements, BK X, PROP 1 191b-192a; BK XII, PROP 2 339a-340b; PROP 10-12 351b-359a; PROP 18 367a-368b 11 ARCHIMEDES: Sphere and Cylinder, BK I, ASSUMPTIONS, 5 404b; PROP 13-14 411a-414a; PROP 33-34 424b-427a; PROP 42 431b-432b; PROP 44 433a-b / Measurement of a Circle, PROP 1 447a-448a / Conoids and Spheroids, PROP 4 459a-460b; PROP 21-22 470a-471b; PROP 25-30 473a-479b / Spirals, PROP 18-20 492b-495b; PROP 24-27 496b-500a / Equilibrium of Planes, BK I, PROP 6-7 503b-504b / Quadrature of the Parabola 527a-537b esp PROP 16 533b-534a, PROP 24 537a-b / Method 569a-592a passim 16 KEPLER: Epitome, BK V, 973a-975a; 979b-983b 28 GALILEO: Two New Sciences, SECOND DAY, 193b-194d; THIRD DAY, 205b-d; 224b-c 31 DESCARTES: Geometry, BK II, 316b-317a 33 PASCAL: Equilibrium of Liquids, 395a-b 34 NEWTON: Principles, BK I, LEMMA 1-11 and SCHOL 25a-32a esp LEMMA II, SCHOL, 31a-b; BK II, LEMMA 2 and SCHOL 168a-170a 45 FOURIER: Theory of Heat, 172b; 177a; 181a-b; 221a-248b 51 TOLSTOY: War and Peace, BK XI, 469a-d; EPILOGUE II, 695b

5. The applications of mathematics to physical phenomena: the utility of mathematics

7 PLATO: Republic, BK VII, 391b-398c; BK VIII, 403b-d; BK IX, 424d-425b / Timaeus, 448b-450a; 453b-454a / Statesman, 585a-b / Philebus, 633c-d / Laws, BK V, 691d-692a; 695c-697a; BK VII, 728b-730d 8 ARISTOTLE: Posterior Analytics, BK I, CH 9 [76a3-25] 104b-d; CH 13 [78b31-79a16] 108b-c; CH 27 [87a32-33] 119b / Physics, BK II, CH 2 [194a7-11] 270b-c; BK VII, CH 5 333a-d / Heavens, BK II, CH 14 [297a4-7] 388c / Meteorology, BK III, CH 3 477a-478a; CH 5 480a-481c / Metaphysics, BK III, CH 2 [997b12-998a19] 516b-d 9 ARISTOTLE: Gait of Animals, CH 9 247a-248a / Politics, BK V, CH 12 [1316a1-17] 518d-519a 11 ARCHIMEDES: Equilibrium of Planes 502a-519b / Floating Bodies 538a-560b 11 NICOMACHUS: Arithmetic, BK I, 812d-813a 14 PLUTARCH: Marcellus, 252a-255a 16 COPERNICUS: Revolutions of the Heavenly Spheres, BK I, 510b 16 KEPLER: Epitome, BK V, 964b-965a 18 AUGUSTINE: Confessions, BK V, par 3-6 27c-28c / Christian Doctrine, BK II, CH 29 650d-651c 19 AQUINAS: Summa Theologica, PART I-II, Q 35, A 8, ANS 779c-780c 20 AQUINAS: Summa Theologica, PART II-II, Q 9, A 2, REP 3 424b-425a 23 HOBBES: Leviathan, PART I, 73b; PART IV, 267a-b; 268c-d 24 RABELAIS: Gargantua and Pantagruel, BK IV, 278b 28 GALILEO: Two New Sciences, FIRST DAY, 131d-132a; SECOND DAY–FOURTH DAY 178a-260a,c 30 BACON: Advancement of Learning, 46b-c 31 DESCARTES: Rules, IV, 7a-b; XIV, 31b-d / Discourse, PART I, 43b-c / Objections and Replies, 169c-170a / Geometry, BK II, 322b-331a 34 NEWTON: Principles, 1a-2a; DEF VIII 7b-8a; COROL II 15a-16b; BK I, PROP 69, SCHOL, 131a; BK III 269a-372a 34 HUYGENS: Light 551a-619b 35 BERKELEY: Human Knowledge, SECT 120 436d; SECT 131 439b-c 35 HUME: Human Understanding, SECT IV, DIV 27 460c-d 36 SWIFT: Gulliver, PART I, 78b; PART III, 94b-103a 40 GIBBON: Decline and Fall, 661c-662c 42 KANT: Pure Reason, 68a-69c / Practical Reason, 300d [fn 1]; 330d-331a / Judgement, 551a-552a 45 FOURIER: Theory of Heat, 169a-173b; 175b; 177a-251b esp 183a-184a 45 FARADAY: Researches in Electricity, 831b-c 51 TOLSTOY: War and Peace, BK XI, 469a-d; BK XIV, 589c-590c; 609b; EPILOGUE II, 694d-695c 53 JAMES: Psychology, 348a-359a; 882a-884b

5a. The art of measurement

5 ARISTOPHANES: Birds [992-1020] 555a-b 6 HERODOTUS: History, BK I, 70b-c; BK IV, 139b-c 7 PLATO: Republic, BK X, 431b-d / Statesman, 594a-595a / Philebus, 633a-634d / Laws, BK V, 691d-692a; 695c-697a 8 ARISTOTLE: Physics, BK IV, CH 12 [220b15-31] 300c-d; CH 14 [223b12-224a1] 303c-d; BK VII, CH 4 330d-333a / Heavens, BK II, CH 14 [298a15-20] 389d / Metaphysics, BK III, CH 2 [997b27-33] 516c; BK V, CH 6 [1016b18-31] 537b; BK X, CH 1 [1052b15-1053b31] 579a-580a 11 EUCLID: Elements, BK V 81a-98b esp DEFINITIONS, 5 81a, 7 81b; BK VII, DEFINITIONS, 20 127b 11 ARCHIMEDES: Sand-Reckoner 520a-526b 13 VIRGIL: Aeneid, BK VI [847-853] 233b-234a 16 PTOLEMY: Almagest, BK I, 14a-28b; BK II, 38b-39b; BK V, 143a-144a; 165a-176a; BK VII-VIII, 233a-258b 16 COPERNICUS: Revolutions of the Heavenly Spheres, BK I, 532b-556b; BK II, 558b-559b; 567b; 586b-621b; BK IV, 705a-725a 19 AQUINAS: Summa Theologica, PART I, Q 3, A 5, REP 2 17c-18b; Q 10, A 6, ANS and REP 4 45c-46d; Q 66, A 4, REP 3 348d-349d 28 GILBERT: Loadstone, BK IV, 85c-89c; BK V, 92a-95a 28 GALILEO: Two New Sciences, FIRST DAY, 136d-137c; 148d-149c; 164a-166c; THIRD DAY, 207d-208c 28 HARVEY: Motion of the Heart, 286c-288c 30 BACON: Novum Organum, BK II, APH 44-48 175d-188b 31 DESCARTES: Rules, XIV, 31b-33b / Geometry, BK I, 296a 33 PASCAL: Great Experiment 382a-389b 34 NEWTON: Principles, 1a-b; LAWS OF MOTION, SCHOL, 20a-22a; BK II, GENERAL SCHOL 211b-219a; PROP 40, SCHOL 239a-246b 34 HUYGENS: Light, CH I, 554b-557b 35 LOCKE: Human Understanding, BK II, CH XVI, SECT 8 167c 39 SMITH: Wealth of Nations, BK I, 14b 42 KANT: Judgement, 497a-498d esp 498b-d 45 LAVOISIER: Elements of Chemistry, PART I, 14a-c; 33b-36a; 41a-44d; PART III, 87d-90a; 91a-95a; 96b-103b 45 FOURIER: Theory of Heat, 184b-185b; 249a-251b 50 MARX: Capital, 20b

5b. Mathematical physics: the mathematical structure of nature

7 PLATO: Timaeus, 448b-450a; 453b-454a; 458a-460b 8 ARISTOTLE: Posterior Analytics, BK I, CH 9 104b-d; CH 13 [78b31-79a16] 108b-c / Physics, BK II, CH 2 [194a7-11] 270b-c / Heavens, BK III, CH 1 [299a1-300a19] 390b-391c / Metaphysics, BK I, CH 9 [992a29-b9] 510c-d; BK II, CH 3 [995a15-17] 513d; BK III, CH 2 [997b12-998a19] 516b-d; BK XII, CH 8 [1073b1-17] 603d-604a; BK XIII, CH 3 609a-610a 11 NICOMACHUS: Arithmetic, BK I, 812b-d; 813d-814a 16 PTOLEMY: Almagest, BK I, 5b-6a 16 KEPLER: Epitome, BK IV, 863b-872b / Harmonies of the World, 1023b-1080b passim 19 AQUINAS: Summa Theologica, PART I, Q 1, A 1, REP 2 3b-4a; Q 32, A 1, REP 2 175d-178a; PART I-II, Q 35, A 8, ANS 779c-780c 20 AQUINAS: Summa Theologica, PART II-II, Q 9, A 2, REP 3 424b-425a 23 HOBBES: Leviathan, PART I, 72a-d; PART IV, 268c-d 28 GALILEO: Two New Sciences, FIRST DAY, 131b-132a; 133b; FOURTH DAY, 245b-d; 252a-b 30 BACON: Advancement of Learning, 46b-c / Novum Organum, BK II, APH 8 140b 31 DESCARTES: Rules, IV, 7a-c / Discourse, PART I, 43b-c; PART III, 50d / Objections and Replies, 169c-170a 33 PASCAL: Pensées, 119 195a 34 NEWTON: Principles, 1a-2a; DEF VIII 7b-8a; BK I, 111b; PROP 69, SCHOL, 131a; BK III, 269a; GENERAL SCHOL, 371b-372a 34 HUYGENS: Light, PREF, 551b-552a 35 BERKELEY: Human Knowledge, SECT 119 436c 35 HUME: Human Understanding, SECT IV, DIV 27 460c-d 42 KANT: Judgement, 551a-552a 45 FOURIER: Theory of Heat, 169a-b; 172a-173b; 175b; 177a; 183a-184a 45 FARADAY: Researches in Electricity, 831b-c 52 DOSTOEVSKY: Brothers Karamazov, BK V, 121a-b 53 JAMES: Psychology, 126a-b; 348a-359a; 675b; 876a-b; 882a-884b

CROSS-REFERENCES

For:

• The relation of mathematics to other arts and sciences, see ASTRONOMY 2c; MECHANICS 3; METAPHYSICS 3b; PHILOSOPHY 1b; PHYSICS 1b, 3; SCIENCE 5c.
• The quality of necessity in mathematical truth, and for the theory of the a priori foundations of arithmetic and geometry in the transcendental forms of space and time, see NECESSITY AND CONTINGENCY 4d; SENSE 1c; SPACE 4a; TIME 6c.
• The controversy over the character and existence of the objects of mathematics, see BEING 7d(3); QUANTITY 1; SPACE 5.
• The discussion of the mental processes by which mathematical objects are apprehended, see IDEA 1a, 2f-2g; KNOWLEDGE 6a(3), 6c(4); MEMORY AND IMAGINATION 1a, 6c(2)-6d; SENSE 5a; UNIVERSAL AND PARTICULAR 2b.
• The consideration of the specific objects of mathematical inquiry, such as numbers and figures, ratios and proportions, continuous and discontinuous quantities, finite and infinite quantities, see INFINITY 3a-3c; QUANTITY 1b-4c, 7; RELATION 1d, 5a(3); SPACE 3a-3c.
• The general theory of mathematical method or logic, see DEFINITION 6a; HYPOTHESIS 3; JUDGMENT 8b-8c; LOGIC 4a; REASONING 6b; TRUTH 4c; and for the particular techniques of arithmetic, geometry, algebra, and calculus, see MECHANICS 3b-3d; QUANTITY 1b, 6b; RELATION 5a(3); SPACE 3d.
• Other discussions of applied mathematics or mathematical physics, and of the role of measurement, see ASTRONOMY 2c; MECHANICS 3; PHYSICS 3, 4d; QUANTITY 3d(1), 6-6a, 6c; SCIENCE 5c.

ADDITIONAL READINGS

Listed below are works not included in Great Books of the Western World, but relevant to the idea and topics with which this chapter deals. These works are divided into two groups: I. Works by authors represented in this collection. II. Works by authors not represented in this collection. For the date, place, and other facts concerning the publication of the works cited, consult the Bibliography of Additional Readings which follows the last chapter of The Great Ideas.

I.

• AQUINAS. On the Trinity of Boethius, QQ 5-6
• HOBBES. Six Lessons to the Savilian Professors of Mathematics
• NEWTON. The Method of Fluxions and Infinite Series
• ———. Universal Arithmetic
• BERKELEY. A Defence of Free Thinking in Mathematics
• KANT. Prolegomena to Any Future Metaphysic, par 6-13
• HEGEL. The Phenomenology of Mind, INTRO
• J. S. MILL. A System of Logic, BK II, CH 5
• ———. An Examination of Sir William Hamilton’s Philosophy, CH 27

II.

• R. BACON. Opus Majus, PART IV
• ORESME. Treatise on the Breadth of Forms
• SUAREZ. Disputationes Metaphysicae, IV (9), X (3), XL (3, 5-6), XLI (4), XLVI (13)
• BARROW. Lectiones Mathematicae
• ———. Thirteen Geometrical Lectures
• LEIBNIZ. Early Mathematical Manuscripts
• SACCHERI. Euclides Vindicatus (Vindication of Euclid)
• VOLTAIRE. “Geometry,” in A Philosophical Dictionary
• EULER. Elements of Algebra
• CARNOT. Réflexions sur la métaphysique du calcul infinitésimal
• GAUSS. Untersuchungen über Höhere Arithmetik
• SCHOPENHAUER. The World as Will and Idea, VOL II, SUP, CH 13
• DE MORGAN. On the Study and Difficulties of Mathematics
• COMTE. The Philosophy of Mathematics
• ———. The Positive Philosophy, BK I
• LOBACHEVSKI. Geometrical Researches on the Theory of Parallels
• WHEWELL. The Philosophy of the Inductive Sciences, VOL I, BK II, CH 11-12, 14
• G. PEACOCK. A Treatise on Algebra
• B. PEIRCE. An Elementary Treatise on Curves, Functions, and Forces
• W. R. HAMILTON. Lectures on Quaternions
• RIEMANN. Über die Hypothesen welche der Geometrie zu Grunde liegen (The Hypotheses of Geometry)
• BOOLE. A Treatise on Differential Equations
• ———. A Treatise on the Calculus of Finite Differences
• DEDEKIND. Essays on the Theory of Numbers
• CLIFFORD. Preliminary Sketch of Biquaternions
• ———. On the Canonical Form and Dissection of a Riemann’s Surface
• JEVONS. On Geometrical Reasoning
• LEWIS CARROLL. Euclid and His Modern Rivals
• GIBBS. Collected Works
• C. S. PEIRCE. Collected Papers, VOL III, par 553-562, 609-645
• FREGE. Grundgesetze der Arithmetik
• BURNSIDE. Theory of Groups of Finite Order
• CANTOR. Contributions to the Founding of the Theory of Transfinite Numbers
• HILBERT. The Foundations of Geometry
• PEANO. Arithmetices Principia
• ———. Formulaire de mathématique
• ———. Arithmetica generale e algebra elementare
• BONOLA. Non-Euclidean Geometry
• E. W. HOBSON. The Theory of Functions of a Real Variable and the Theory of Fourier’s Series
• O. VEBLEN and LENNES. Introduction to Infinitesimal Analysis
• KLEIN. Famous Problems of Elementary Geometry
• ———. Elementary Mathematics from an Advanced Standpoint
• POINCARÉ. Science and Hypothesis, PART I
• ———. The Value of Science, PART I, CH 1; PART II, CH 3
• ———. Science and Method, BK I, CH 2-3; BK II, CH 3
• CASSIRER. Substance and Function, PART I, CH 2-3; SUP VI
• E. V. HUNTINGTON. The Continuum, and Other Types of Serial Order
• ———. The Fundamental Propositions of Algebra
• J. W. YOUNG. Lectures on Fundamental Concepts of Algebra and Geometry
• JOURDAIN. The Nature of Mathematics
• O. VEBLEN and YOUNG. Projective Geometry
• B. RUSSELL. Principles of Mathematics, CH I
• ———. Philosophical Essays, CH 13
• ———. Mysticism and Logic, CH 4-5
• ———. Introduction to Mathematical Philosophy, CH 18
• N. R. CAMPBELL. What Is Science?, CH 6-9
• MARITAIN. An Introduction to Philosophy, CH 11 (3)
• ———. Theonas, Conversations of a Sage, VI
• NICOD. Foundations of Geometry and Induction
• WHITEHEAD. A Treatise on Universal Algebra
• ———. An Introduction to Mathematics
• ———. Science and the Modern World, CH 2
• G. N. LEWIS. The Anatomy of Science, ESSAY I-II
• HILBERT and ACKERMANN. Grundzüge der theoretischen Logik
• BUCHANAN. Poetry and Mathematics
• M. R. COHEN. Reason and Nature, BK II, CH 1
• A. E. TAYLOR. Philosophical Studies, CH II
• GILSON. The Unity of Philosophical Experience, CH 8
• DEWEY. Logic, the Theory of Inquiry, CH 20
• CARNAP. Foundations of Logic and Mathematics
• BELL. The Development of Mathematics
• G. H. HARDY. A Course of Pure Mathematics
• ———. A Mathematician’s Apology
• KASNER and NEWMAN. Mathematics and the Imagination
• COURANT and ROBBINS. What Is Mathematics?
• WEYL. The Philosophy of Mathematics and Natural Science