International Catholic University
We have been discussing the philosophy and history of the liberal arts, and we have come to the point where we can discuss the mathematical arts of the Trivium. As we have seen in an earlier discussion of the traditional liberal arts, there are two groups of arts: the linguistic arts of the Trivium, grammar, logic and rhetoric, and the four arts of the Quadrivium, the mathematical arts, which were in the tradition where arithmetic, geometry, "music" in a special sense, and astronomy. Now we have seen what reason there is, after making an analysis of the sign, to identify the three arts of the Trivium, with three arts necessary for using the linguistic sign. Now in this lecture I propose to do two things: first, to consider how the Quadrivial arts were distinguished from one another in the tradition; second, given the tremendous developments in mathematics in modern times, to propose a reformulation of the Quadrivium to fit the new situation.
The arts of the Quadrivium show their age more than those of the Trivium. They are more conditioned by time and have progressed more both in theory and practice -- although logic is an exception here, as we will see. Although great advances have been made in linguistics and communication theory, the same is not true of the products they have produced, that is the works of literary art. The epics of Homer, Virgil, Dante, Milton and Goethe are certainly as good and as great as anything produced since in imaginative literature. Yet the practice as well as the theory of the mathematical arts have far surpassed that of their predecessors. As mathematicians are proud of claiming, we have been living through a golden age of mathematics.
The arts of the Quadrivium, their identification, enumeration, and distinction was the work of the ancient Pythagoreans. Nicomachus noted three distinctions in his Introduction to Arithmetic that he wrote about 100 A.D. He claimed that these three distinctions with respect to quantity served to identify four arts.
The first distinction is that which distinguishes the discreet from the continuous. The natural or positive whole numbers are discreet, 1, 2, 3, etc. They consist of many separate and distinct individuals. On the other hand a line is continuous, no breaks, no separations. It can be cut or divided but each of the sections so cut constitutes a whole without parts. This distinction between the discreet and continuous marks the difference between multitude and magnitude.
Each of these, multitude and magnitude, can be considered in two distinct ways. Number as discreet can be considered either as it is in itself or as it is related to other numbers. The study of number in itself, that is of the whole numbers 1, 2, 3, etc. gives rise to the art of arithmetic. Consideration of numbers in relation to one another as double, half, one-fifth, etc. yields a mathematical art that was known as "music" in the ancient world. The name "music" rests on the discovery of the Pythagoreans, that musical intervals depend on the relation of whole numbers. Thus a length of wire or tubes of the same substance and tension were vibrated. The harmonic octave will sound when the wire lengths are as two to one. The fifth as in one to two. The fourth as four to two. Hence the ancient art of music was the study of ratios of numbers and could apply to the music of words, that is to verse, as well as to that of musical instruments, as it was by Saint Augustine in his treatise, De Musica, On Music.
Spatial magnitude can likewise be studied in either of two ways. As it is in itself while at rest, which gives us the art of geometry. Or magnitudes may be considered as in motion, as moving circles, for example, and then we have astronomy. These arts have all obtained their great books. In fact one of the most famous books of all time is the Elements of Euclid since it provided the basic introduction to the study of mathematics really down to the 20th century -- the basis for the study of elementary mathematics.
Of the thirteen books of the Euclid's elements, three of them, book seven, eight, and nine, concern the arithmetic of whole numbers. The rest of the books deal with plane and solid geometry. In other words, his Elements includes arithmetic as well as geometry. It is significant that Euclid represents whole numbers by length of line. The Introduction to Arithmetic of Nicomacheios, which also achieved the status of a great book, represents numbers graphically by plane figures. The square of two for example appears as a square with four elements.
Music or harmony fails to inspire a book on its art to equal, at least in fame, those of the other arts. The astronomy of the great books was provided by Ptolemy in his work entitled, The Mathematical Composition which later came to be known by its Arabic title, the Almagest which means the greatest. And that too down to the time of Kepler and even after was the basic book for the study of astronomy.
The Quadrivium possesses no account of the battle among its arts corresponding to that which we found among the arts of the Trivium. Yet in effect such a battle occurred and a much more grievous one in that it resulted in the destruction of the old arts, at least in their old form, and led to logic deserting its former colleagues, leaving the Trivium to join the Quadrivium. This story can be told briefly by considering the following incidents. In arithmetic the introduction of the Arabic notation for numbers including zero. In geometry the invention of analytical geometry and then of non-Euclidean geometries. In astronomy the Heliocentric theory of our planetary system, replacing that Geocentric system of Ptolemy. In the mathematics the invention of the calculus. And in logic, the mathematization of logic in our own time.
As a result of these developments the ancient Quadrivium has been destroyed. These incidents occur in the story of the arithmetization of mathematics and the overthrow of geometry as the first of the mathematical arts. That this has been a recent development is evident from the fact that Descartes presented his great discovery of analytical geometry as a work in geometry and entitled it so. Newton published his, Principles of Natural Philosophy in the axiomatic form of geometry, although he performed his calculations by the calculus that he had invented.
The first essential step in arithmetization came with the invention of zero and the adoption of the Arabic notation for numbers that we still use, 1, 2, 3, 4 and up. The Greek and Roman practice of using letters to represent numbers was highly cumbersome and made calculation difficult without the help of mechanical aid. Just the expression of very large numbers, for example, was extremely difficult, as shown by a little treatise written by the great physicist and mathematician Archimedes entitled The Sand Reckoner. In this treatise he claims to discover the number of all the grains of the sand in the world. It is not only the problem that seems incredible but the fact that he can express it in his cumbersome notation. So he's as proud of being able to express a very, very large number as much as making the claim that he can count all the grains of sand in the world. Modern translations of The Sand Reckoner obscure this fact by expressing the solution in Arabic numerals. It's no problem then. In our own day only Arthur Eddington, the astronomer and physicist, has attempted a similar feat by claiming that he can determine the cosmological number of the total number of particles in the Einsteinian universe.
With the adoption of a more satisfactory notation future progress depended on increasing the possibilities of calculation or computability. This is achieved by inventing new number systems to overcome obstacles in the way. Ancient arithmetic limited itself to natural positive whole numbers, 1, 2, 3, 4, etc. Hence in some cases certain operations became impossible. Subtraction is the simplest case, since it cannot be done unless the subtrahend is smaller than the minuend. Three cannot be subtracted from two, for example, if one is limited to positive whole numbers. The invention of negative numbers overcame this obstacle, and two minus three is equal to minus one. You've got a negative number to express the result.
Division faces a similar difficulty since here the dividend must be greater than the divisor. Now the invention of fractions, or rational numbers, rational because they establish a ratio, solves the problem because then you can't divide two into one but you can express one half. So we've overcome the limit set up on subtraction by the invention of negative numbers and of division by the invention of fractions. But not all results of division yield whole numbers. The diagonal of a square for example, if the sides of the square are 1 then 1 squared plus 1 squared equals 2 and by the Pythagorean theorem the result is the length of the diagonal is the square root of 2. It cannot be expressed as the ratio of two whole numbers. In fact, it's equal to, an unending decimal so we get a new kind of number, the irrational numbers, so called because they can't be expressed as a ratio of two whole numbers.
Yet this doesn't complete the needs we have for arithmetical operations because there still remains an operation that cannot be performed, namely the extraction of the square root of negative numbers. Hence to get the square root of a negative number a new number was invented called "i," imaginary number which is the square root of negative one. With this all our arithmetical operations of addition, multiplication, are real, and together with imaginary numbers obtain the complex numbers, and our number system is complete. There is no arithmetical operation that we cannot now perform and get an arithmetical or numerical result. The range of our ability to calculate and compute has been enormously extended.
However arithmetic has not yet reached its full development. It still has to overcome the difference separating the discrete from the continuous which the ancients saw as a radical break between arithmetic and geometry. The victory was slow in coming, but come it did. That the attack was fully underway is evident in the opening sentence of Descartes' Geometry. His very first sentence advances the claim that any problem in geometry can easily be reduced to such terms that a knowledge of certain straight lines is sufficient for its construction. When dealing with many lines he takes one line as unity in order to relate it as closely as possible to numbers. He then proceeds to show how this claim can be justified and in doing so he invents analytical or coordinate geometry. To take an example of how this is accomplished, let's consider the equation "ax + by = c" and represent it graphically. Draw two lines at right angles. Let the horizontal line be the x axis. Divide it at the middle between positive numbers and negative numbers. Let the y axis be the vertical one. Divide it at the middle above into +y and below to -y. Now let us make one x, a measure along the x axis to 1 and along the y axis to 1 and make a point, and along the -x axis to 1, and -y axis 1 and make a point. Now since we know that ax + by = c we've got 1x + -1y, that is 1x - 1y = 0. And we establish those points accordingly and draw a line, and we've got a straight line. And the straight line is defined by the equation ax + by = c where ax, by and c are all numbers. So the geometrical line is now defined in terms of the relation of numbers.
Since all the variable letters in our equation represent numbers the geometric line is no longer anything that need be done geometrically; it can be done with numbers. So lines, curves, figures, solids and their relations can all be determined by equations which ultimately are but variable expressions for numbers. Thus Euclid's Pythagorean Theorem proven as a relation between the sides of a triangle, and squares erected on them can now be stated, as we saw, much more simply as A2 + B2 = C2, where at issue is a matter of numbers even though they may also be taken as the lengths of the sides of the triangle. The arithmetical algebraic expression is admittedly more abstract than the corresponding geometric expressions, but it is simpler and easier to work with. Geometry has been arithmetized.
Geometry, at least as understood by Euclid and his followers, suffered still another great shock with the discovery of non-Euclidean geometries. This discovery resulted from doubts about Euclid's fifth postulate maintaining that parallel lines never meet. By relying on this postulate it can be proven that the interior angles of a triangle equal to two right angles. So you see the parallel postulate is equivalent to defining the interior angles of the triangle as equal or 180 degrees, to right angles.
Euclid himself may have had some doubts about the postulate since he postponed appealing to or using that postulate as long as possible in his first book. The converse of the postulate is stated and proven in the twenty-eighth theorem of the first book, and efforts were continuously made after Euclid to prove the postulate and so convert it into a theorem. But all such efforts resulted in failure. Finally attempts were made to abandon the postulate and were now gone in the nineteenth century so that the sum of the interior angles of the triangle might either be greater or less than 180 degrees. It then proved possible to develop consistent geometries on either supposition, and non-Euclidean geometry was born, due primarily to the work of Lobachevsky and Riemann.
Euclidean geometry might hold for flat or plane space but it did not hold for all space. For in a curved space the non-Euclidean geometries and not the Euclidean geometry are true. Such a situation might have remained no more than an abstract mathematical speculation, but that situation changed with the coming of Einstein and his Theory of Relativity. For this called for a non-Euclidean geometry to explain the relation of events occurring in a space that is curved.
While such incidents and developments were shaking the position of arithmetic and geometry as they were understood in the ancient Quadrivium, even more serious upsets had happened to astronomy. Whereas geometry as presented by Euclid still remained true as far as it went, the principle basis for ancient astronomy of Ptolemy is replaced. The work of Copernicus, Kepler, Galileo and Newton, based on the heliocentric theory of planetary motions around the sun, replaced the geocentric theory of Ptolemy with its circular orbits of planets traveling around the earth. At the same time the mathematical and theoretical methods of analysis were tremendously more powerful. Newton in effect achieved a marriage of celestial and terrestrial dynamics by producing an explanation of movements in either the heavens or the earth. We are still experiencing its effect upon the traditional understanding of the mathematical arts.
The mathematization of logic threatened to breech the gaps separating Trivium from Quadrivium. Here, if any one great book can be singled out for this attempt, it would be the Principia Mathematica written by Bertrand Russell and Alfred North Whitehead. This huge work, ten years in writing, was published in three large volumes in 1910 to 1913. It contains literally hundreds of propositions. By 1900, when the work was begun, mathematical or symbolic logic was already well developed. A notation had been constructed consisting almost entirely of letters and special signs or symbols for the operations to be performed with them, so that the use of English or any other natural language was reduced to a minimum. Efforts were made to reduce to the smallest possible number the undefined notions. Rules of entrance were stated, definitions identifying the objects and operations providing the subject matter to be investigated.
Putting the system to work resulted in producing and proving a string of propositions logically following from one another ultimately depending upon the primitive presuppositions. What made it look still more like a piece of mathematics was that it was presented as an axiomatic system, just as Euclid's elements were. And up until 1900 the principle examples of the axiomatic method were Euclid's Elements, Newton's Principia Mathematica, and Spinoza's Ethics.
The first part of the Russell-Whitehead Principia is entitled Mathematical Logic, and it opens with an account of the theory of deduction. This is the theory of how one proposition can be inferred from another, which depends upon the fact that the two propositions are so related that one is a consequence of the other. The development of this theory is needed to accomplish the purpose of the work as a whole. For as the first sentence of this theory's explication declares, this is nothing less than the deduction of pure mathematics from its logical foundations. In other words, the aim of the Principia Mathematica was to reduce mathematics to logic. For this reason it is the position known in the philosophy of mathematics as logicism. However the effort that is readily to be known as the arithmetizing of logic, and the great German logician Gottlieb Frege held that logic is a branch of arithmetic.
In our discussion of the mathematization logic and Russell and Whitehead's Principia Mathematica, we saw that its aim was to show and prove that all of mathematics could be reduced to logic. Or to put it another way, as the German logician Frege did that logic itself is a branch of arithmetic. This great and ambitious project ultimately turns out to have been a failure, as pointed out in the best account of the development of logic, a book of that title written by William and Martha Kneal. Although Whitehead and Russell continue to maintain the thesis of Frege, the expedients to which they are driven reveal the peculiarity of their usage of the word logic. Perhaps the most glaring case of this is their introduction of what they call the axiom of infinity. Arithmetic needs an infinity of natural numbers to make possible some of its operations, notably that of mathematical induction, to assure that any number has a successor. The need is met by postulating the existence of an infinity of such numbers, and that is what Russell and Whitehead do -- a need that had not been thought by logic from the time of Aristotle until 1900, with the advent of mathematical reductionists like Russell and Whitehead. Hence, as Kneal claims, it is better to retain the old understanding of logic as the study of the principles that assure the validity of inference, and that its laws are those of the laws of the other sciences. Logic is thus the science of sciences, as Aristotle called it, or the art of arts, as Saint Thomas called it.
The mathematical form that the study of logic has achieved in modern times, recent times, possesses great advantage for the analysis of argument and for making explicit by identifying and distinguishing the different elements that enter into it. It is especially adapted to the clarification of highly technical reasonings as those of mathematics. However this of itself is no reason for denying that there is a gulf separating mathematics from logic. What then is the result of the attack of modern mathematics upon the ancient Quadrivium of arithmetic, geometry, music and astronomy? The distinctions on which it was based have been overcome, as we have seen. Considered as the elementary mathematics necessary for continuing on to a higher science, including those of mathematics, the Quadrivium has to be reformulated, and has been in effect for the past three centuries. If the choice now were to be limited to only four preparatory studies in mathematics, the four today would consist of arithmetic, geometry, algebra, and the calculus, algebra and the calculus replacing the "music" and astronomy of the old Quadrivium. For with grounding in these four of basic arithmetic and geometry and algebra and the calculus, one would possess the mathematical language that is the international learned tongue of the sciences today.
Thus it would appear that nothing at all remains of the traditional Quadrivium. Arithmetic and geometry still remain basic but are no longer understood in the same way. The distinction between the discreet and the continuous has been overcome and is no longer regarded as fundamental to mathematics. And yet the distinction between the discreet and the continuous has not disappeared entirely. It has turned up in the quantum theory of the atom. As the name indicates, energy is the basic building block of the physical world, and energy itself has to be conceived as occurring in quanta, that's in discreet amounts. Or it can also be conceived as a continuous wave. In fact the atomic scientist Nils Bohr maintains that both the discreet and the continuous are needed to provide an adequate account of physical phenomena, since atomic phenomena in some ways are better accounted for if they are considered to be discreet entities. On the other hand there are other phenomena that are better described as considering the atomic particles being a wave which is continuous.
Mathematics since the time of its development by the ancient Greeks has always been prized for the power and beauty of its reasoning, and indeed for its ability to form and train the faculty of reasoning itself. As long as Euclid was studied as the basic introductory work to mathematics, Euclid's geometry provided the basic training for the logic of argument. It provided the basic understanding of what a proof is and the means of constructing and establishing a proof. In fact it's no exaggeration to claim that mathematics has provided the clearest and most explicit instance of reason, of reason itself at work, of reason reasoning, in its development of the axiomatic method. The earliest and most extensive use of this method is to be found in the thirteen books of Euclid's Elements.
I want to consider for a while the axiomatic method, how it's constructed and what its value is. Now Euclid himself was not the inventor of the method. Plato in his dialog, the Republic, gave an outline of it in his analysis of the kinds of knowledge. Aristotle made an extensive analysis of the axiomatic method; he also was the first to actually use it. He presented his Theory of Syllogisms in logic in axiomatic form by showing how all the valid moods of the syllogism can be derived from a few taken as primary or axiomatic. His analysis of the method in the Posterior Analytics, his theory of methodology, is a good introduction to the discussion of Euclid's use of it.
Both Aristotle and Euclid were members of Plato's academy but Euclid was younger by perhaps a generation since Euclid flourished about 300 B.C. whereas Aristotle died in 322. Hence there is a possibility that Euclid may have known about Aristotle's theory of the axiomatic metod. According to the account given in the Posterior Analytics by Aristotle, every demonstrative science is concerned with three things: the subject matter, or that with which the science is concerned; the things proven about it, its attributes; and that from which the demonstration begins. The last, that from which the demonstration begins, are indemonstrable principles or starting points -- indemonstrable since not everything can be proven nor need be proven. Now these indemonstrables are two kinds. Some are common to many sciences, an example of which is the axiom that if equals are taken from equals, equals remain. Others are special or peculiar to a given science, and of these Aristotle distinguishes three again. There is the subject matter of which both the meaning and existence is assumed, such as points and lines in geometry. Secondly, the attributes of the subject of which only the meaning is assumed, whereas their existence is demonstrated by means of the axioms and previously established conclusions taken as premises. And third, postulates, the existence of which is assumed and one from which something can be assumed regarding the subject.
When we turn to Euclid's first book of the Elements dealing with plane geometry, we find much of what Aristotle had taught us to expect, but much more fully developed, since we are now embarked upon the course of a demonstrative science. The book begins by distinguishing three different kinds of principles, starting points: definitions, postulates, and common notions, the last of which corresponds to what Aristotle called axioms. As seen from the fact that Euclid's third theorem is the same as the example, the third common notion is the same as the example cited by Aristotle. Equals taken from equals are equal.
The three different principles clearly differ with regard to their relations to meaning and existence. That the definitions are intended to establish meaning only is evident once definition four is compared with postulate one. Postulate one declares that a straight line can be drawn from any point to any point, whereas the definition says only that a straight line is a line that lies evenly with the points on itself. Now this definition may not seem very illuminating. It may be appealing to our visual experience of sighting along an even row of objects, for example. If so the postulate would also imply that only one straight line can join two points. At any rate the appeal to sense experience can at most only serve as a crutch to help us grasp the meaning of the definition. Since a line is defined as a breadthless length, a length with no breadth, a geometrical line is not a thing that can be observed by sight.
Still a different relation between meaning and existence appears in definition twenty of an equilateral triangle as compared, for example, with the proof of its construction that an equilateral triangle can be constructed. For in this case it is shown that the postulated possibility of drawing a straight line and describing a circle enable us to prove that an equilateral triangle is possible by actually constructing one by drawing straight lines and circles -- although again the diagram that is drawn is but a crutch for the intellect since the lines, the circles drawn are three dimensional objects, not plane figures. That the definitions establish the subject matter of book one is evident merely from the two just considered.
Yet it is worth reflecting further on just what kind of an object is given in these lines and circles. It's obviously not a visual object or one that can be sensed, as is clear from the definitions of point, line, surface. Perhaps these objects can be imagined even though they cannot be seen. But when we reach the definition of parallel straight lines we have also gone beyond the imaginable, for these are lines that are produced indefinitely without end and never meet. In short the subject matter of geometry or mathematics in general is conceptual and not either sensible or imaginary. It is by means of and with the help of words and diagrams, which are things that can be sensed, that we can think about and discover the properties of these purely conceptual objects.
It is also clear that Euclid is beginning from what is logically or conceptually prior and moving from there to the later and more complex, since a point or a line as he defines them is certainly not the sort of thing that we first encounter. Indeed among objects of sense experience closest to his geometrical ones the ones that he treats last are those with which we have the most immediate experience, namely solids of three dimensions, which Euclid considers in the last three of his thirteen books.
The postulates are, if anything, even more important than the definitions for establishing the objects of geometry. For it is there in those postulates that so constitute their existence that properties or attributes about them can be inferred, or deduced. In effect Euclid's postulates establish a unique kind of space. The first three postulates concerning the straight line and circle lay down the only kind of lines to be dealt with. And in postulating that they can be of any magnitude, they imply that his space is continuous and not discreet, for if it is possible to describe the circle of any size it can be indefinitely small and so leave no room for any gaps in it. However of the five postulates that Euclid has it is the fourth and the fifth that make Euclid's geometry uniquely Euclidian.
Postulate four asserts that the right angle is a determinate magnitude and so can serve as a standard or criterion by which to measure other angles. These are as less (that is an acute angle), or greater (that is an obtuse angle). Since all right angles are equal it in effect says, that the space in which they exist is homogenous or all of the same kind -- and accordingly makes it possible to test figures by coincidence or application, placing one upon another. This underlies the fourth axiom that things which coincide with one another are equal. The fourth postulate is also necessary for the statement of the great fifth postulate, the uniquely Euclidian postulate. The parallel postulate it is. It has been written about it that this postulate must ever be regarded as among the most epic-making achievements in the domain of geometry.
Now to appreciate this claim you must see how it is used and why it is necessary. But for that we must first understand what it says. Parallel lines have been defined in definition twenty-three as straight lines which never meet however far they are extended. The postulate now provides a criterion by which to determine when two straight lines are not parallel but will intersect if extended sufficiently far enough. The test consists in letting a single straight line fall across the two lines that are suspected of being parallel. Let a straight line fall across the two parallel lines and then determine whether the interior angles within these two lines is less than two right angles. For if the interior angles falling within the two angles is less than two angles the lines are postulated to meet if they are produced far enough on the side in which the two angles are less than two right angles. So if, drawing this line across two lines being tested for being parallel, the interior angles between those two in this line crossing it are less than two right angles, then those two lines are going to meet if indefinitely extended. So they are not parallel. So there is a test for parallelism you see which is provided in this fifth postulate.
Readers of Euclid from a very early date took exception to this postulate about parallels from the belief that it should be proven rather than assumed. They had at least two reasons for this belief. One, that Euclid himself proves the converse of the problem, the other side of it, the reverse of it, in proposition 128, namely that if a transverse line crosses two lines so as to make the interior angles equal to two right angles the two lines are parallel. It is by no means necessary that if lines converge they must ultimately meet, for there are lines that continually get ever closer without ever meeting, so called asymptotes. Hence from antiquity until the 19th century myriads of attempts were made to prove and not assume the proposition asserted in the fifth postulate. However in the nineteenth century it was definitely proven that it could not be demonstrated using Euclid's methods and his genius came to be better appreciated for having recognized that it had to be assumed as a postulate for his geometry.
Why that must be and something of what it involves can be gathered from seeing how Euclid uses the postulate. Euclid seems to have wanted to postpone as long as possible the treatment of parallels. For he does not begin it until the twenty-seventh proposition of the first book. He is then more than halfway through his first book. Up to this point he had been investigating properties of the triangle. He then devotes five propositions to the consideration of parallels, the first two of which make no appeal to the fifth postulate, which is invoked for the first time in the twenty-ninth theorem. The need for the postulate becomes apparent when he returns to the triangle in the thirty-second theorem, the very important proposition that the three interior angles of a triangle are equal to two right angles.
There is thus an intimate, indeed an essential connection, an interdependence between the Euclidian triangle, the theory of parallels, and the meeting of lines asserted in the fifth postulate, which is why it has come to be known as the parallel postulate. Thus the proposition about triangles in 132 is in effect the equivalent of the fifth postulate that the interior angles of a triangle are equal to two right angles. If the sum of the interior angles is greater than two right angles, as it is in the case of a triangle on the surface of the sphere, one obtains the non-Euclidian geometry discovered by Riemann. If the interior angles of a triangle are less than two right angles you get the non-Euclidian geometry of Lobachevsky, the so-called space of negative curvature. Hence by means of his fifth postulate Euclid in effect specifies that his is a geometry of a space that is flat, or of zero curvature. Enough of the wonders of the parallel postulate.
Of the third kind of a principle, that of a common notions or axioms, no such lengthy consideration is needed. In calling them common Euclid implies, as already expressly noted by Aristotle, that they are used in more than one science. This is obviously the case of all except possibly the fourth. For the first common notion, two things equal to same thing are equal to each other, is as true of number as it is of size. And that the whole is greater than the part. It is a typical example of the self-evident truths, since it is evident once the meaning of whole and part are understood. The fourth common notion concerns coincidence, and that one seems to be peculiar to geometry. It certifies the superimposition of one figure on another as it attests to the equality of the figures. It would not seem to be common to arithmetic, for example. One does not prove the equal numerosity of two groups of things by placing the items in one group on those of another, unless coincidence is taken to be a matching of one thing with one of the other, as we know that the number of chairs in a room equals the number of people in it if every chair is occupied by one person and no one is without a chair. If you call that coincidence, then there is an arithmetical example of it.
Aristotle claimed that axioms are indemonstrable and require no proof or postulation, since they are self evident. And there seems to be no doubt that they have more evidence in themselves than the postulates do, the common notions that is. It is sometimes objected that Euclid fails to state explicitly in his preliminary matter all the principles that he uses when he comes to prove his propositions. Geometers have frequently added to the number of postulates and axioms, which accounts for the different numbers found in different editions of Euclid. But Euclid of course had not himself been aware of such a lack and assumed principles without knowing it. Yet one should not rule out the possibility that Euclid was aware and decided himself not to make explicit some of the principles that he uses. Aristotle noted, as we have seen, that there is no need to make everything explicit that is obvious and well known.
Hence it is the postulates that are special and peculiar to geometry, the axioms being common to many sciences. Why, it might be asked, should Euclid be said to be using axiomatic rather than a postulational method? Wouldn't it be more accurate to use the latter name? Euclid might agree, but for many modern mathematicians, logicians, and philosophers axioms are no less postulational than the postulates. In denying that there aren't any self-evident axioms these people are denying that there are any axioms in the old sense as principles distinct from postulates, with the result that the two words have come to be used interchangeably. In any case, an axiomatic system is one that begins from certain indemonstrable principles from which certain other propositions can be deduced as conclusions. And that certainly describes what Euclid does for all the books after this first one. The postulates and axioms given here apply to all the thirteen books. With this we have concluded our discussion of the destruction and reformulation of the Quadrivium.