This is chapter 11 of the "The
Phenomenon of Science" by Valentin F.
Turchin
Contents:
DURING THE TIME of Pythagoras and the early Pythagoreans, the concept of number occupied the dominant place in Greek mathematics. The Pythagoreans believed that God had made numbers the basis of the world order. God is unity and the world is plurality. The divine harmony in the organization of the cosmos is seen in the form of numerical relationships. A substantial part in this conviction was played by the Pythagoreans discovery of the fact that combinations of sounds which are pleasant to hear are created in the cases where a string is shortened by the ratios formed by whole numbers such as 1:2 (octave), 2:3 (fifth), 3:4 (fourth), and so on. The numerical mysticism of the Pythagoreans reflected their belief in the fact that, in the last analysis, all the uniformities of natural phenomena derive from the properties of whole numbers.
We see here an instance of the human inclination to overestimate new discoveries. The physicists of the late nineteenth century, like the Pythagoreans, believed that they had a universal key to all the phenomena of nature and with proper effort would be able to use this key to reveal the secret of any phenomenon. This key was the notion that space was filled by particles and fields governed by the equations of Newton and Maxwell. With the discovery of radioactivity and the diffraction of electrons, however, the physicists' arrogant posture crumbled.
In the case of the Pythagoreans the same function was performed by discovery of the existence of incommensurable line segments, that is, segments such that the ratio of their lengths is not expressed by any ratio of whole numbers (rational number). The side of a square, and its diagonal are incommensurable, for example. It is easy to prove this statement using the Pythagorean theorem. In fact, let us suppose the opposite, namely that the diagonal of a square stands in some ratio m:n to its side. If the numbers m and n have common factors they can be reduced, so we shall consider that m and n do not have common factors. This means that in measuring length by some unitary segment, the length of the side is n and the length of the diagonal is m. It follows from the Pythagorean theorem that the equality m2= 2n2 must occur. Therefore, m2 must be divisible by 2, an consequently 2 must be among the factors of m, that is, m = 2m1. Making this substitution we obtain 4m12 = 2n2 , that is, 2m12 = n2 . This means that n also must be divisible by 2, which contradicts the assumption that m and n do not have common factors. Aristotle often refers to this proof. It is believed that the proof had already been discovered by the Pythagoreans.
If there are quantities which for a given scale are not expressed by numbers then the number can no longer be considered the foundation of foundations; it is removed from its pedestal. Mathematicians then must use the more general concept of geometric quantity and study the relations among quantities that may (although only occasionally) be expressed in a ratio of whole numbers. This approach lies at the foundation of all Greek mathematics beginning with the classical period. The relations we know as algebraic equalities were known to the Greeks in geometric formulation as relations among lengths, areas, and volumes of figures constructed in a definite manner.
FIGURE 11.1 shows the well-known geometric interpretation of the relationship
(a +b)2 = a2+2ab+b2.
Figure 11.1. Geometric interpretation of the identity (a +b)2 = a2+2ab+b2.
The equality (a+b)(a-b) = a2 - b2, which is equally commonplace from an algebraic point of view, requires more complex geometric consideration. The following theorem from the second book of Euclid's Elements corresponds to it.
Figure 11.2. Geometric interpretation of the identity (a-b)(a+b) = a2 - b2
"If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of the section is equal to the square on the half.''[1]
The theorem is proved as follows. Rectangle ABFE is equal to rectangle BDHF. Rectangle BCGF is equal to rectangle GHKJ. If square FGJI is added to these two rectangles (which together form rectangle ACGE which is ''contained by the unequal segments of the whole'') what we end up with is precisely rectangle BDKI, which is constructed ''on the half.'' Thus we have the equality (a+b)(a--b) + b2= a2 which is equivalent to the equality above but does not contain the difficult-to-interpret subtraction of areas.
Clearly, if these very simple algebraic relations require great effort to understand the formulation of the theorem--as well as inventiveness in constructing the proof--when they are expressed geometrically, then it is impossible to go far down this path. The Greeks proved themselves great masters in everything concerning geometry proper, but the line of mathematical development that began with algebra and later gave rise to the infinitesimal analysis and to modern axiomatic theories (that is to say, the line of development involving the use of the language of symbols rather than the language of figures) was completely inaccessible to them. Greek mathematics remained limited, confined to the narrow framework; of concepts having graphic geometric.
DURING THE ALEXANDRIAN EPOCH (330 200 B.C.) two great learned men lived in whose work Greek mathematics reached its highest point. They were Archimedes (287-212 B.C.) and Apollonius (265?-170? B.C.)
In his works on geometry Archimedes goes far beyond the limits of the figures formed by straight lines and circles. He elaborates the theory of conic sections and studies spirals. Archimedes's main achievement in geometry is his many theories on the areas. volumes, and centers of gravity of figures and bodies formed by other than just straight lines and plane surfaces. He uses the "method of exhaustion.'' To illustrate the range of problems solved by Archimedes we shall list the problems included in his treatise entitled The Method whose purpose, as can be seen from the title, is not a full summary of results but rather an explanation of the method of his work. The Method contains solutions to the following 13 problems: area of a parabolic segment, volume of a spher, volume of a spheroid (ellipsoid of rotation), volume of a segment of a paraboloid of rotation, center of gravity of a segment of a paraboloid of rotation, center of gravity of a hemisphere, volume of a segment of a sphere, volume of a segment of a spheroid, center of gravity of a segment of a sphere, center of gravity of a segment of a spheroid, center of gravity of a segment of a hyperboloid of rotation, volume of a segment of a cylinder, and volume of the intersection of two cylinders (the last problem is without proof).
Archimedes's investigations in the field of mechanics were just as important as his work on geometry. He discovered his famous ''law'' and studied the laws of equilibrium of bodies. He was extraordinarily skillful in making different mechanical devices and attachments. It was thanks to machines built under his direction that the inhabitants of Syracuse, his native city, repulsed the Romans' first attack on their city. Archimedes often used mechanical arguments as support in deriving geometric theorems. It would be a mistake to suppose, however, that Archimedes deviated at all from the traditional Greek way of thinking. He considered a problem solved only when he had found a logically flawless geometric proof. He viewed his mechanical inventions as amusements or as practical concerns of no scientific importance whatsoever. ''Although these inventions,'' Plutarch writes, ''made his superhuman wisdom famous, he nonetheless wrote nothing on these matters because he felt that the construction of all machines and all devices for practical use in general was a low and ignoble business. He himself strove only to remove himself, by his handsomeness and perfection, far from the kingdom of necessity."
Of all his achievements Archimedes himself was proudest of his proof that the volume of a sphere inscribed in a cylinder is two thirds of the volume of the cylinder. In his will he asked that a cylinder with an inscribed sphere be shown on his gravestone. After the Romans took Syracuse and one of his soldiers (against orders, it is said) killed Archimedes, the Roman general Marcellus authorized Archimedes' relatives to carry out the wish of the deceased.
Apollonius was primarily famous for his work on the theory of conic sections. His work is in fact a consistent algebraic investigation of second-order curves expressed in geometric language. In our day any college student can easily repeat Appolonius' results by employing the methods of analytic geometry. But Apollonius needed to show miraculous mathematical intuition and inventiveness to do the same thing within a purely geometric approach.
''AFTER APOLLONIUS,'' writes B. van der Waerden, ''Greek mathematics comes to a dead stop. It is true that there were some epigones, such as Diocles and Zenodorus, who, now and then, solved some small problem, which Archimedes and Apollonius had left for them, crumbs from the board of the great. It is also true that compendia were written, such as that of Pappus of Alexandria (300 A.D.); and ,geometry was applied to practical and to astronomical problems, which led to the development of plane and spherical trigonometry. But apart from trigonometry, nothing great nothing new appeared. The ,geometry of the conics remained in the form Apollonius gave it, until Descartes. Indeed the works of Apollonius were but little read and were even partly lost. The Method of Archimedes was lost sight of and the problem of integration remained where it was, until it was attacked new in the seventeenth century. . . "[2]
The decline of Greek mathematics was in part caused by external factors--the political storms that engulfed Mediterranean civilization. Nonetheless, internal factors were decisive. In astronomy, van der Waerden notes, development continued steadily along an ascending line; there were short and long periods of stagnation, but after them the work was taken up again from the place where it had stopped. In geometry, however, regression plainly occurred. The reason is found, of course, in the lack of an algebraic language.
"Equations of the first and second degree," we read in van der Waerden, ''can be expressed clearly in the language of geometric algebra and, if necessary, also those of the third degree. But to get beyond this point, one has to have recourse to the bothersome tool of proportions .
"Hippocrates, for instance, reduced the cubic equation x3 = V to the proportion
a :x =x :y = y:b.
and Archimedes wrote the cubic
x2 (a-x) = bc2
in the form
(a-x) :b = c2: x2."
"In this manner one can get to equations of the fourth degree; examples can be found in Apollonius.... But one cannot get any further; besides, one has to be a mathematician of genius, thoroughly versed in transforming proportions with the aid of geometric, to obtain results by this extremely cumbersome method. Anyone can use our algebraic notation, but only a gifted mathematician can deal with the Greek theory of proportions and with geometric algebra.
Something has to be added, that is, the difficulty of the written tradition.
Reading a proof in Apollonius requires extended and concentrated study. Instead of a concise algebraic formula, one finds a long sentence, in which each line segment is indicated by two letters which have to be located in the figure. To understand the line of thought, one is compelled to transcribe these sentences in modern concise formulas . . .
An oral explanation makes it possible to indicate the line segments with the fingers; one can emphasize essentials and point out how the proof was found. All of this disappears in the written formulation of the strictly classical style. The proofs are logically sound, but they are not suggestive. One feels caught as in a logical mouse trap, but one fails to see the guiding line of thought.
As long as there was no interruption. as long as each generation could hand over its method to the next, everything went well and the science flourished. But as soon as some external cause brought about an interruption in the oral tradition. and only books remained. it became extremely difficult to assimilate the work of the great precursors and next to impossible to pass beyond it.[3]
But why, despite their high mathematical sophistication and abundance of talented mathematicians, did the Greeks fail to create an algebraic language? The usual answer is that their high mathematical sophistication was in fact what hindered them, more specifically their extremely rigorous requirements for logical strictness in a theory, for the Greeks could not consider irrational numbers, in which the values of geometric quantities are ordinarily expressed, as numbers: if line segments were incommensurate it was considered that a numerical relationship for them simply did not exist. Although this explanation is true in general, still it must be recognized as imprecise and superficial. Striving toward logical strictness cannot by itself be a negative factor in the development of mathematics. If it acts as a negative factor this will evidently be only in combination with certain other factors and the decisive role in this combination should certainly not be ascribed to the striving for strictness. Perfect logical strictness in his final formulations and proofs did not prevent Archimedes from using guiding considerations which were not strict. Then why did it obstruct the creation of an algebraic language? Of course, this is not simply a matter of a high standard of logical strictness, it concerns the whole way of thinking, the philosophy of mathematics. In creating the modern algebraic language Descartes went beyond the Greek canon, but this in no way means that he sinned against the laws of logic or that he neglected proof. He considered irrational numbers to be ''precise" also, not mere substitutions for their approximate values. Some problems with logic arose after the time of Descartes, during the age of swift development of the infinitesimal analysis. At that time mathematicians were so carried away by the rush of discoveries that they simply were not interested in logical subtleties. In the nineteenth century came time to pause and think. and then a solid logical basis was established for the analysis.
We shall grasp the causes of the limitations of Greek mathematics after we review the substance of the revolution in mathematics made by Descartes.
ADVANCES IN GEOMETRY forced the art of solving equations into the background. But this art continued to develop and gave rise to arithmetie algebra. The emergence of algebra from arithmetic was a typical metasystem transition. When an equation must be solved, whether it is formulated in everyday conversational language or in a specialized language, this is an arithmetic problem. And when the general method of solution is pointed out--by example, as is done in elementary school. or even in the form of a formula--we still do not go beyond arithmetic. Algebra begins when the equations themselves become the object of activity, when the properties of equations and rules for converting them are studied. Probably everyone who remembers his first acquaintance with algebra in school (if this was at the level of understanding, of course, not blind memorization) also remembers the happy feeling of surprise experienced when it turns out that various types of arithmetic problems whose solutions had seemed completely unrelated to one another are solved by the same conversions of equations according to a few simple and understandable rules. All the methods known previously fall into place in a harmonious system, new methods open up, new equations and whole classes of equations come under consideration (the law of branching growth of the penultimate level), and new concepts appear which have absolutely no meaning within arithmetic proper: negative, irrational, and imaginary numbers.
In principle the creation of a specialized language is not essential for the development of algebra. In fact, however, only with the creation of a specialized language does the metasystem transition in people's minds conclude. The specialized language makes it possible to see with one's eyes that we are dealing with some new reality, in this case with equations, which can be viewed as an object of computations just as the objects (numbers) of the preceding level were. People typically do not notice the air they breathe and the language they speak. But a newly created specialized language goes outside the sphere of natural language and is in part nonlinguistic activity. This facilitates the metasystem transition. Of course, the practical advantages of using the specialized language also play an enormous part here; among them are making expressions visible, reducing time spent recopying, and so on.
The Arab scholar Muhammed ibn Musa al-Khwarizmi (780-850) wrote several treatises on mathematics which were translated into Latin in the twelfth century and served as the most important textbooks in Europe for four centuries. One of them, the Arithmetic, gave Europeans the decimal system of numbers and the rules (algorithms--the name is based on al-Khwarizmi) for performing the four arithmetic operations on numbers written in this system. Another work was entitled Book of Al Jabr Wa'l Muqabala. The purpose of the book was to teach the art of solving equations, an art which is essential, as the author writes, ''in cases of inheritance, division of property, trade, in all business relationships, as well as when measuring land, laying canals, making geometric computations, and in other cases....'' Al Jabr and al Mugabala are two methods al-Khwarizmi uses to solve equations. He did not think up these methods himself; they were described and used in the Arithmetica of the Greek mathematician Diophantus (third century A.D.), who was famous for his methods of solving whole-number (''diophantine'') equations. In the same Arithmetica of Diophantus we find the rudiments of letter symbolism. Therefore, if anyone is to be considered the progenitor of arithmetic algebra it should obviously be Diophantus. But Europeans first heard of algebraic methods from al-Khwarizmi while the works of Diophantus became known much later. There is no special algebraic symbolism in al-Khwarizmi, not even in rudimentary form. The equations are written in natural language. But for brevity's sake, we shall describe these methods and give our examples using modern symbolic notation.
Al Jabr involves moving elements being subtracted from one part of the equation to the other; al Muqabala involves subtracting the same element from both parts of the equation. Al-Khwarizmi considers these procedures different because he does not have the concept of a negative number. For example let us take the equation
7x - 11 = 5x - 3.
Applying the al Jabr method twice, for the 11 and 3, which are to be subtracted, we receive:
7x + 3 = 5x + 11.
Now we use the al Muqabala method twice, for 3 and 5x. We receive
2x = 8.
From this we see that
x = 4.
So although al-Khwarizmi does not use a special algebraic language, his book contains the first outlines of the algebraic approach. Europeans recognized the merits of this approach and developed it further. The very word algebra comes from the name of the first of al-Khawarizmi's methods.
IN THE FIRST HALF of the sixteenth century the efforts of Italian mathematicians led to major changes in algebra which were associated with very dramatic events. Scipione del Ferro (1465-1526). a professor at the University of Bologna, found a general solution to the cubic equation x3 +px = q where p and q are positive. But del Ferro kept it secret, because it was very valuable in the problem-solving competitions which were held in Italy at that time. Before his death he revealed his secret to his student Fiore. In 1535 Fiore challenged the brilliant mathematician Niccolo Tartaglia (1499-1557) to a contest. Tartaglia knew that Fiore possessed a method of solving the cubic equation, so he made an all-out effort and found the solution himself. Tartaglia won the contest, but he also kept his discovery secret. Finally, Girolamo Cardano (1501-1576) tried in vain to find the algorithm for solving the cubic equation. In 1539 he finally appealed to Tartaglia to tell him the secret. Having received a "sacred oath'' of silence from Cardano, Tartaglia unveiled the secret, but only partially and in a rather unintelligible form. Cardano was not satisfied and made efforts to familiarize himself with the manuscript of the late del Ferro. In this he was successful, and in 1545 he published a book in which he reported his algorithm, which reduces the solution of a cubic equation to radicals (the "Cardano formula''). This same book contained one more discovery made by Cardano's student Luigi (Lodovico) Ferrari (1522-1565): the solution of a quartic equation in radicals. Tartaglia accused Cardano of violating his oath and began a bitter and lengthy polemic. It was under such conditions that modern mathematics made its first significant advances.
Using a tool suggests ways to improve it. While striving toward a uniform solution to equations, mathematicians found that it was extremely useful in achieving this goal to introduce certain new objects and treat them as if these were numbers. And in fact they were called numbers although it was understood that they differed from ''real'' numbers: this was seen in the fact that they were given such epithets as "false'' "fictitious,'' ''incomprehensible,'' and "imaginary." What they correspond to in reality remained somewhat or entirely unclear. Whether their use was correct also remained debatable. Nonetheless, they began to be used increasingly widely, because with them it was possible to obtain finite results containing only "real" numbers which could not be obtained otherwise. A person consistently following the teachings of Plato could not use ''unreal'' numbers. But the Indian, Arabic, and Italian mathematicians were by no means consistent Platonists. For them a healthy curiosity and pragmatic considerations outweighed theoretical prohibitions. In this, however, they did make reservations and appeared to be apologizing for their ''incorrect'' behavior.
All "unreal'' numbers are products of the reverse movement in the arithmetic model: formally they are solutions to equations that cannot have solutions in the area of "real'' numbers. First of' all we must mention negative numbers. They are found in quite developed form in the Indian mathematician all Bahascara (twelfth century), who performed all four arithmetic operations on such numbers. The interpretation of the negative number as a debt was known to the Indians as early as the seventh century. In formulating the rules of operations on negative numbers. Bhascara calls them ''debts,'' and calls positive numbers ''property.'' He does not choose to declare the negative number an abstract concept like the positive number. "People do not approve of abstract negative numbers,'' Bhascara writes. The attitude toward negative numbers in Europe in the fifteenth and sixteenth centuries was similar. In geometric interpretation negative roots are called ''false'' as distinguished from the 'true'' positive roots. The modern interpretation of negative numbers as points lying to the left of the zero point did not appear until Descartes' Géométrie (1637). Following tradition, Descartes called negative roots false.
Formal operations on roots of numbers that cannot be extracted exactly go back to deep antiquity, when the concept of incommensurability of line segments had not yet appeared. In the fifteenth and sixteenth centuries people handled them cavalierly: they were helped here, of course, by the simple geometric interpretation. An understanding of the theoretical difficulty which arises from the incommensurability of line segments can be .seen by the fact that the numbers were called "irrational."
The square of any number is positive: therefore the square root of a negative number does not exist among positive, negative, rational, or irrational numbers. But Cardano was daring enough to use (not w without reservations) the roots of negative numbers. ' Imaginary'' numbers thus appeared. The logic of using algebraic language drew mathematicians inexorably down an unknown path. It seemed wrong and mysterious. but intuition suggested that all these impossible numbers were profoundly meaningful and that the new path would prove useful. And it certainly did.
THE RUDIMENTS of algebraic letter symbolism are first encountered, as mentioned above, in Diophantus. Diophantus used a character resembling the Latins to designate an unknown. It is hypothesized that this designation originates from the last letter of the Greek word for number: a[rho][iota][theta]u[omicron][sigma] (arithmos). He also had abbreviated notations for the square cube, and other degrees of the unknown quantity. He did not have an addition sign: quantities being added were written in a series, something like an upside-down Greek letter [psi] was used as the subtraction sign, while the first letter of the Greek word [iota]d[omicron]d for ''equal" was used as the equal sign, everything else was expressed in words. Known quantities were always written in concrete numerical form while there were no designations for known, but arbitrary numbers.
Diophantus' Arithmetica became known in Europe in 1463. In the late fifteenth and early sixteenth centuries European mathematicians first Italians and then others began to use abbreviated notations. These abbreviations gradually wandered from arithmetic algebra to geometric, and unknown geometric quantities also began to be designated by letters. In the late sixteenth century the Frenchman François Vieta (1540-1603) took the next important step. He introduced letter designations for known quantities and was thus able to write equations in general form. Vieta also introduced the term "coefficient.'' In external appearance Vieta's symbols are still rather far from modern ones. For example, Vieta writes
instead of our notation
D(2B3 - D3)
By the beginning of the seventeenth century the situation in European mathematics was as follows. There were two algebras. The first was arithmetic based on symbols created by the Europeans themselves and representing a substantial advance in comparison with the arithmetic of the ancients. The second algebra. geometric algebra. was part of geometry. It was taken. as was the whole of geometry, from the Greeks. The fundamentals were from Euclid's Elements and the further development came primarily from the works of Pappus of Alexandria and Apollonius, who had been thoroughly studied by that time. Nothing fundamentally new had been done in this field. We cannot say that there was no relationship at all between these two algebra: equation of degrees higher than the first could only receive geometric interpretation, for where else could squares, cubes, and higher degrees of an unknown number occur but in computing areas, volumes and manipulations of line segments related to complex systems of proportions? The very names of the second and third degrees, the square and the cube, illustrate this very eloquently. Nonetheless, the gap between the concepts of quantity (or magnitude) and number remained and in full conformity with the Greek canon only geometric proofs w ere considered real. When geometric objects--lengths, areas, and volumes--appeared in equations they operated either as geometric quantities or as concrete numbers. Geometric quantities were thought of as necessarily something spatial and, because of incommensurability not reducible to a number.
This was the situation that René Descartes (1596-1650), one of the greatest thinkers who has ever lived, encountered.
DESCARTES' ROLE as a philosopher is generally recognized. But when Descartes as a mathematician is discussed it is usually indicated that he "refined algebraic notations and created analytical geometry.'' Sometimes it is added that at approximately the same time the basic postulates of analytic geometry were proposed, independently of Descartes by his countryman Pierre de Fermat (1601-1665), while Vieta had already made full use of algebraic symbols. It comes out, thus, that there is no special cause to praise Descartes the mathematician, and in fact many authors writing about the history of mathematics do not give him his due. However, Descartes carried out a revolution in mathematics. He created something incomparably greater than analytic geometry (understood as the theory of curves on a plane). What he created was a new approach to describing the phenomena of reality: the modern mathematical language.
It is sometimes said that Descartes ''reduced geometry to algebra" which means, of course numerical algebra, arithmetic algebra. This is a flagrant mistake. It is true that Descartes overcame the gap between quantity and number, between geometry and arithmetic. He did not achieve this by reducing one language to the other, however: he created a new language, the language of algebra. Not arithmetic algebra, not geometric algebra, simply algebra. In syntax the new language coincided with arithmetic algebra. but by semantics it coincided with geometric. In Descartes' language the symbols do not designate number or quantities, but relations of quantities. This is the essence of the revolution called out by Descartes.
The modern reader will perhaps shrug his shoulders and think ''So what"? Could this logical nuance really have been very important?'' As it turns out, it was. It was precisely this ''nuance'' that had prevented the Greeks from taking the next step in their mathematics.
We have become so accustomed to placing irrational numbers together
with rational ones that we are no longer aware of the profound difference
which exists between them. We write 
as
we write 4/5 and we call number
and, when necessary, substitute an approximate value for it. And there
is no way we can understand why the ancient Greeks responded with such
pain to the incommensurability of line segments. But if we think a little,
we cannot help agreeing with the Greeks that 
is
not a number. It can be represented as an infinite process which generates
the sequential characters of expansion of in
decimal fractions. It can also be pictured in the form of a boundary line
in the field of rational numbers--one that divides rational numbers into
two classes: those which are less than and
those which are greater than .
In this case the rule is very simple: the rational number a belongs
to the first class if a2 < 2 and to the second where
this is not true. Finally, can
be pictured in the form of a relation between two line segments, between
the diagonal of a square and its side in the particular case. These representations
are equivalent to one another but they are not at all equivalent to the
representation of the whole or fractional number.
This by no means implies that we are making a mistake or not being sufficiently
strict when we deal with as
a number. The goal of mathematics is to create linguistic models of reality,
and all means which lead to this goal are good. Why shouldn't our language
contain characters of the type in
addition to ones such as 4/5? It is my language and I will do what I want
to with it.'' The only important thing is that we be able to interpret
these characters and perform linguistic conversions on them. But we are
able to interpret .
In practical computations the first of the three representations in the
preceding paragraph may serve a. the basis of interpretation. while in
geometry the third can be used. We can also carry out other computations
with them. All that remains now is to refine the terminology. Let us stipulate
that we shall use the term rational numbers for what were formerly
called numbers, name the new objects irrational numbers, and use
the term numbers for both (real numbers according to modern mathematical
terminology). Thus, in the last analysis, there is no difference in principle
between and
4/5 and we have proved wiser than the Greeks. This wisdom was brought in
as contraband by all those who operated with the symbol as
a number, while recognizing that it was "irrational." It was
Descartes who substantiated this wisdom and established it as law.
THE GREEKS' failure to create algebra is profoundly rooted in their philosophy. They did not even have arithmetic algebra. Arithmetic equations held little interest for them: after all, even quadratic equations do not, generally speaking, have exact numerical solutions. And approximate calculations and everything bound up with practical problems were uninteresting to them. On the other hand, the solution could have been found by geometric construction! But even if we assume that the Greek mathematicians of the Platonic school were familiar with arithmetic letter symbols it is difficult to imagine that they would have performed Descartes' scientific feat. To the Greeks relations were not ideas and therefore did not have real existence. Who would ever think of using a letter to designate something that does not exist? The Platonic idea is a generalized image. a form a characteristic: it can be pictured in the imagination as a more or less generalized object. All this is primary and has independent existence an existence even more real than that of things perceived by the senses. But what is a relation of line segments? Try to picture it and you will immediately see that what you are picturing is precisely two line segments, not any kind of relation. The concept of the relation of quantities reflects the process of measuring one by means of the other. But the process is not an idea in the Platonic sense: it is something secondary that does not really exist. Ideas are eternal and invariable, and by this alone have nothing in common with processes.
Interestingly. the concept of the relation of quantities, which reflects characteristics of the measurement process, was introduced in strict mathematical form as early as Eudoxus and was included in the fifth book of Euclid's Elements. This was exactly the concept Descartes used. But the relation as an object is not found in Eudoxus or in later Greek mathematicians: after being introduced it slowly gave way to the proportion which it is easy to picture as a characteristic of four line segments formed by two parallel lines intersecting the sides of an angle.
The concept of the relation of quantities is a linguistic construct, and quite a complex one. But Platonism did not permit the introduction of constructs in mathematics: it limited the basic concepts of mathematics to precisely representable static spatial images. Even fractions were considered somehow irregular by the Platonic school from the point of view of real mathematics. In The Republic we read: "If you want to divide a unit. learned mathematicians will laugh at you and will not permit you to do it: if you change a unit for small pieces of money they believe it has been turned into a set and are careful to avoid viewing the unit as consisting of parts rather than as a whole.'' With such an attitude toward rational numbers, why even talk about irrational ones!
We can briefly summarize the influence of Platonic idealism on Greek mathematics as follows. By recognizing mathematical statements as objects to work with. the Greeks made a metasystem transition of enormous importance but then they immediately objectivized the basic elements of mathematical statements and began to view them as part of a nonlinguistic reality, "the world of ideas." In this way they closed off the path to further escalation of critical thinking to becoming aware of the basic elements (concepts) of mathematics as phenomema of language and to creating increasingly more complex mathematical constructs. The development of mathematics in Europe was a continuous liberation from the fetters of Platonism.
IT IS VERY INSTRUCTIVE to compare the mathematical world of Descartes and Fermat. As a mathematician Fermat was as gifted as Descartes, perhaps even more so. This can be seen from his remarkable works on number theory. But he was an ardent disciple of the Greeks and continued the traditions. Fermat set forth his discoveries on number theory in remarks in the margins of Diophantus' Arithmetica His works on geometry originated as the result of efforts to prove certain statements referred to by Pappus as belonging to Apollonius, but presented without proofs. Reflecting on these problems, Fermat began to systematically represent the position of a point on a plane by the lengths of two line segments: the abscissa and the cardinal and represent the curve as an equation relating these segments. This idea was not at all new from a geometric point of view: it was a pivotal idea not only in Apollonius but even as far back as Archimedes, and it originates with even more ancient writers. Archimedes describes conic sections by their ''symptoms" that is, the proportions which connect the abscissas and ordinates of the points. As an example, let us take an ellipsis with the longer (major) axis AB.
Figure 11.3.
Perpendicular line PQ which is dropped from a certain point of the ellipsis P to axis AB is called the ordinate, and segments AQ and QB are the abscissas of this point (both terms are Latin translations of Archimedes' Greek terms). The ratio of the area of a square constructed on the ordinate to the area of the rectangle constructed on the two abscissas is the same for all points P lying on the ellipsis. This is the "symptom'' of the ellipsis, that is, in essence, its equation. It can be written as Y2: X1 X2 = const. Analogous symptoms are established for the hyperbola and parabola. How is this not a system of coordinates ?
Unlike the ancients, Fermat formulates the symptoms as equations in Vieta's language, not in the form of proportions described by words. This makes conversions easier: specifically, it can be seen immediately that it is more convenient to leave one abscissa than two. But the approach continues to be purely geometric, spatial.
Fermat set forth his ideas in the treatise "Ad locos planos et solidos isagoge" (Introduction to Plane and Solid Loci). This work was published posthumously in 1679, but it had been known to French mathematicians as early as the 1630s, somewhat earlier than Descartes' mathematical works.
Descartes famous Géométrie came out in 1637. Descartes was not of course, at all influenced by Fermat (it is unknown whether he even read Fermat's treatise); Descartes' method took shape in the 1620's, long before the Géométrie was published. Nonetheless, the properly geometrical ideas of Descartes and Fermat are practically identical. But Descartes created a new algebra based on the concept of the relation of geometric quantities. In Vieta only similar quantities can be added and subtracted and coefficients must include an indication of their geometric nature. For example, the equation which we would write as
A3 + BA = D.
Vieta wrote as follows:
A cubus + B planum in A aequatur D solido.
This means the cube with edge A added to area B multiplied by A is equal to the volume D. Vieta and Fermat are intellectual prisoners of the Greek geometric algebra. Descartes breaks with it decisively. The relations Descartes' algebra deals with are not geometric, spatial objects, but theoretical concepts, "numbers." Descartes is not restricted by the requirement for uniformity of things being added or the general requirement of a spatial interpretation; he understands raising to a power as repeated multiplication and indicates the number of factors by a small digit above and to the right. Descartes' symbolism virtually coincides with our modern system.
FERMAT WAS ONLY a mathematician; Descartes was above all a philosopher. His reflections went far beyond mathematics and dealt with the problems of the essence of being and knowledge. Descartes was the founder of the philosophy of rationalism which affirms the human being's unlimited ability to understand the world on the basis of a small number of intuitively clear truths and proceeding forward step by step using definite rules or methods. These two words are key words for all Descartes' philosophy. The name of his first philosophical composition was Regulae ad directionem ingenii (Rules for the Direction of the Mind), and his second was Discours de la méthode (Discourse on the Method). The Discours de la méthode was published in 1637 in a single volume with three physico-mathematical treatises: "La Dioptrique" (the Dioptric), "les Meteores" (Meteors) and "la Gémétrie'' (Geometry). The Discours preceded them as a presentation of the philosophical principles on which the following parts were based. In this Discours Descartes proposes the following four principles of investigation:
The first of these was to accept nothing as true which I did not clearly recognize to be so: that is to say, carefully to avoid precipitation and prejudice in judgements and to accept in them nothing more than what was presented to my mind so clearly and distinctly that I could have no occasion to doubt it.
The second was to divide up each of the difficulties which I examined into as many parts as possible, and as seemed requisite in order that it might be resolved in the best manner possible.
The third was to carry on my reflections into order, commencing with objects that were the more simple and easy to understand, in order to rise little by little, or by degrees, to knowledge of the most complex assuming order, even it a fictitious one, among those which do not follow a natural sequence relatively to one another.
The last was in all cases to make enumerations so complete and reviews so general that I should be certain of having omitted nothing.[4]
Descartes arrived at his mathematical ideas guided by these principles. Here is how he himself describes his path in Discours de la méthode:
And I have not much trouble in discovering which objects it was necessary to begin with, for I already knew that it was with the most simple and those most easy to apprehend. Considering also that of all those who have hitherto sought for the truth in the Sciences, it has been the mathematicians alone who have been able to succeed in making any demonstrations, that is to say, producing reasons which are evident and certain. I did not doubt that it had been by means of a similar kind that they carried on their investigations.... But for all that I had no intention of trying to master all those particular sciences that receive in common the name of Mathematics; but observing that, although the objects are different, they do not fail to agree in this, that they take nothing under consideration but the various relationships or proportions which are present in these objects. I thought that it would be better if I only examined those proportions in their general aspect, and without viewing them otherwise than in the objects which would serve most to facilitate a knowledge of them. Not that I should in any way restrict them to these objects. for I might later on all the more easily apply them to all other objects to which they were applicable. Then, having carefully noted that in order to comprehend the proportions I should sometimes require to consider each one in particular. and sometimes merely keep them in min(l. or take them in groups. I thought that in order the better to consider them in detail, I should picture them in the form of lines, because I could find no method more simple nor mole capable of being distinctly represented to my imagination and senses. I considered, however, that in order to keep them in my memory or to embrace several at once, it would be essential that I should explain them by means of certain formulas, the shorter the better. And for this purpose it was requisite that I should borrow all that is best in Geometrical Analysis and Algebra, and correct the errors of the one by the other.[5]
We can see from this extremely interesting testimony that Descartes was clearly aware of the semantic novelty of his language based on the abstract concept of the relation and applicable to all the phenomena of reality. Lines serve only to illustrate the concept of the relation, just as a collection of little sticks serves to illustrate the concept of number. In their mathematical works Descartes and subsequent mathematicians have followed tradition and used the term "quantity" for that which is designated by letters, but semantically these are not the spatial geometric quantities of the Greeks but rather their relations. In Descartes the concept of quantity is just as abstract as the concept of number. But of course, it cannot be reduced to the concept of number in the exact meaning of the word, that is, the rational number. Explaining his notations in the Géométrie, Descartes points out that they are similar (but not identical) to the notations of arithmetic algebra:
Just as arithmetic consists of only four or five operations, namely, addition, subtraction, multiplication, division, and the extraction of roots, which may be considered a kind of division, so in geometry, to find required lines it is merely necessary to add or subtract other lines: or else, taking one line which I shall call unit) in order to relate it as closely) as possible to numbers, and which can in general be chosen arbitrarily and having given two other lines, to find a fourth line which shall be to one of the given lines as the other is to unity which is the same as multiplication: or, again, to find a fourth line which is to one of the given lines as unity is to the other (which is equivalent to division): or, finally, to find one, two, or several mean proportionals between units and some other line (which is the same as extracting the square root. cube root, etc., of the given line). And I shall not hesitate to introduce these arithmetical terms into geometry, for the sake of greater clearness."[6]
The semantics of Descartes' algebraic language are much more complex than the semantics of the arithmetic and geometric languages which rely on graphic images. The use of such a language changes one's view of the relation between language and reality. It is discovered that the letters of mathematical language may signify not only numbers and figures. but also something much more abstract (to be more precise ''constructed''). This is where the invention of new mathematical languages and dialects and the introduction of new constructs began. The precedent was set by Descartes. Descartes in fact laid the foundation for describing the phenomena of reality by means of formalized symbolic languages.
The immediate importance of Descartes' reform was that it untied the hands of mathematicians to create. in abstract symbolic form. the infinitesimal analysis whose basic ideas in geometric form were already known to the ancients. If we go just half a century from the publication date of the Géométrie we find ourselves in the age of Leibnitz and Newton, and 50 more years brings us to the age of Euler.
The history of science shows that the greatest glory usually doesnot go to those who lay the foundations and, of course, not to those who work on the small finishing touches: rather it goes to those who are the first in a new line of thought to obtain major results which strike the imagination of their contemporaries or immediate descendants. In European physico-mathematical science this role was played by Newton. But as Newton said, ''If I have seen further than Descartes, it is by standing on the shoulders of giants.'' This is, of course, evidence of the modesty of a brilliant scientist but it is also a recognition of the debt of the first great successes to the pioneers who showed the way. The apple which made Newton famous grew on a tree planted by Descartes.
[1] The Thirteen Books of Euclid's Elements, translated and annotated by T. L. Heath, (Cambridge: Cambridge University Press, 1908), vol. 1. p. 382.
[2] B. van der Waerden Science Awakening (New York: Oxford University Press, 1969), p. 264.
[3] B. van der Waerden Science Awakening, p. 266.
[4] Descartes, Spinoza, Great Books of the Western World, Encyclopaedia Britanica Inc., Vol 31, 1952, p. 47.
[5] Descartes, ibid., p. 47.
[6] Descartes, ibid., p. 295