This is chapter 10 of the "The
Phenomenon of Science" by Valentin F.
Turchin
Contents:
NEITHER IN Egyptian nor in Babylonian texts do we find anything even remotely resembling mathematical proof. This concept was introduced by the Greeks, and is their greatest contribution. It is obvious that some kind of guiding considerations were employed earlier in obtaining new formulas. We have even cited an example of a grossly incorrect formula (for the area of irregular quadrangles among the Egyptians) which was plainly obtained from externally plausible ''general considerations.'' But only the Greeks began to give these guiding considerations the serious attention they deserved. The Greeks began to analyze them from the point of view of how convincing they were, and they introduced the principle according to which every proposition concerning mathematical formulas, with the exception of just a small number of "completely obvious'' basic truths, must be proved--derived from these ''perfectly obvious" truths in a convincing manner admitting of no doubt. It is not surprising that the Greeks, with their democratic social order, created the doctrine of mathematical proof. Disputes and proofs played an important part in the life of the citizens of the Greek city-state (polis). The concept of proof already existed; it was a socially significant reality. All that remained was to transfer it to the field of mathematics, which was done as soon as the Greeks became acquainted with the achievements of the ancient Eastern civilizations. It must be assumed that a certain part here was also played by the role of the Greeks as young, curious students in relation to the Egyptians and Babylonians, their old teachers who did not always agree with one another. In fact, the Babylonians determined the area of a circle according to the formula 3r2, while the Egyptians used the formula (8/9 2r)2. Where was the truth? This was something to think about and debate.
The creators of Egyptian and Babylonian mathematics have remained anonymous. The Greeks preserved the names of their wise men. The first, Thales of Miletus, is also the first name included in the history of science. Thales lived in the sixth century B.C. in the city of Miletus on the Asia Minor coast of the Aegean Sea. One date in his life has been firmly established: in 585 B.C. he predicted a solar eclipse--unquestionable evidence of Thales's familiarity with the culture of the ancient civilizations, because the experience of tens and hundreds of years is required to establish the periodicity of eclipses. Thales had no Greek predecessors, and could therefore only have taken his knowledge of astronomy from the scientists of the East. Thales, the Greeks assert, gave the world the first mathematical proofs. Among the propositions (theorems) proved by him they mention the following:
1 The diameter divides a circle into two equal parts.
2 The base angles of an isosceles triangle are equal.
3 Two triangles which have an identical side and identical angles adjacent to it are equal.
In addition, Thales was the first to construct a circle circumscribed about a right triangle (and it is said that he sacrificed an ox in honor of this discovery).
The very simple nature of these three theorems and their intuitive obviousness shows that Thales was entirely aware of the importance of proof as such. Plainly, these theorems were proved not because there was doubt about their truth but in order to make a beginning at systematically finding proof and developing a technique for proof. With such a purpose it is natural to begin by proving the simplest propositions.
Suppose triangle ABC is isosceles, which is to say side AB is equal to side BC.
Figure 10.1. Isosceles triangle.
Let us divide angle ABC into two equal parts by line BD. Let us mentally fold our drawing along line BD. Because angle ABD is equal to angle CBD, line BA will lie on line BC, and because the length of the segments AB and BC is equal, point A will lie on point C. Because point D remains in place, angles BCD and BAD must be equal. Whereas formerly it only seemed to us that angles BCD and BAD were equal (Thales probably spoke this way to his fellow citizens), we have now proved that these angles necessarily and with absolute precision must be equal (the Greeks said "similar'') to one another: that is, they match when one is placed on the other.
The problem of construction is more complex and here the result is not at all obvious beforehand. Let us draw a right triangle.
Figure 10.2. Construction of a circle described around a right triangle
May a circle be drawn such that all three vertices of the triangle appear on it? And if so, how'? It is not clear. But suppose that intuition suggests a solution to us. We divide the hypotenuse BC into two equal segments at point D. We connect it with point A. If segment AD is equal in magnitude to segment DC (and therefore also to BD) we can easily draw the required circle by putting the point of a compass at point D and taking segment DC as the radius. But is it true that AD =DC, that is to say triangle ADC is an isosceles triangle? It is not clear. It seems probable, but in any case it is far from obvious. Now we shall take the crucial step. We shall add point E to our triangle, making rectangle ABEC and draw in a second diagonal AE. Suddenly it becomes obvious that triangle ADC is isosceles. Indeed, from the overall symmetry of the drawing it is clear that the diagonals are equal and intersect at the point which divides them in half--at point D. We have not yet arrived at proof, but we already are at that level of clarity where formal completion of the proof presents no difficulty. For example, relying on the equality of the opposite sides of the rectangle (which can be derived from even more obvious propositions if we wish), we complete the proof by the following reasoning: triangles ABC and AEC are equal because they have side AC in common, sides AB and EC are equal, and angles BAC and ECA are right angles; therefore angle EAC is equal to angle BCA. That is, triangle ADC is an isosceles triangle, which is what had to be proved.
SO, FROM a few additional points and lines on a drawing, a chain of logical reasoning, and simple and obvious truths we receive truths which are by no means simple and by no means obvious, but whose correctness no one can doubt for a minute. This is worth sacrificing an ox to the gods for! One can imagine the delight the Greeks experienced upon making such a discovery. They had struck a vein of gold and they diligently began working it. In the time of Pythagoras (550 B.C.) the study of mathematics was already very widespread among people who had leisure time and was considered a noble, honorable, and even sacred matter. Advances and discoveries, each more marvelous than the one before, poured from the horn of plenty.
The appearance of proof was a metasystem transition within language. The formula was no longer the apex of linguistic activity. A new class of linguistic objects appeared, proof, and there was a new type of linguistic activity directed to the study and production of formulas. This was a new stage in the control hierarchy and its appearance called forth enormous growth in the number of formulas (the law of branching of the penultimate level).
The metasystem transition always means a qualitative leap forward--a flight to a new step, swift, explosive development. The mathematics of the countries of the Ancient East remained almost unchanged for up to two millennia, and a person of our day reads about it with the condescension of an adult toward a child. But in just one or two centuries the Greeks created all of the geometry our high school students sweat over today. Even more, for the present-day geometry curriculum covers only a part of the achievements of the initial, ''classical,'' period of development of Greek mathematics and culture (to 330 B.C.). Here is a short chronicle of the mathematics of the classical period.
585 B.C. Thales of Miletus. The first geometric theorems.
550 B.C. Pythagoras and his followers. Theory of numbers. Doctrine of harmony. Construction of regular polyhedrons. Pythagorean theorem. Discovery of incommensurable line segments. Geometric algebra. Geometric construction equivalent to solving quadratic equations.
500 B.C. Hippasas, Pythagorean who was forced to break with his comrades because he shared his knowledge and discoveries with outsiders (this was forbidden among the Pythagoreans). Specifically, he gave away the construction of a sphere circumscribed about a dodecahedron.
430 B.C. Hippocrates of Chios (not to be confused with the famous doctor Hippocrates of Kos). He was considered the most famous geometer of his day. He studied squaring the circle, making complex geometric constructions. He knew the relationship between inscribed angles and arcs, the construction of a regular hexagon, and a generalization of the Pythagorean theorem for obtuse- and acute-angled triangles. Evidently, he considered all these things elementary truths. He could square any polygon, that is, construct a square of equal area for it.
427-348 B.C. Plato. Although Plato himself did not obtain new mathematical results, he knew mathematics and it sometimes played an important part in his philosophy--just as his philosophy played an important part in mathematics. The major mathematicians of his time, such as Archytas, Theaetetus, Eudoxus, were Plato's friends; they were his students in the field of philosophy and his teachers in the field of mathematics.
390 B.C. Archytas of Tarentum. Stereometric solution to the problem of doubling the cube--that is, constructing a cube with a volume equal to twice the volume of a given cube.
370 B.C. Eudoxus of Cnidus. Elegant, logically irreproachable theory of proportions closely approaching the modern theory of the real number. The ''exhaustion method,'' which forms the basis of the modern concept of the integral.
384-322 B.C. Aristotle. He marked the beginning of logic and physics. Aristotle's works reveal a complete mastery of the mathematical method and a knowledge of mathematics, although he, like his teacher Plato, made no mathematical discoveries. Aristotle the philosopher is inconceivable without Aristotle the mathematician.
300 B.C. Euclid. Euclid lived in a new and different age, the Alexandrian Epoch. In his famous Elements Euclid collected and systematized all the most important works on mathematics which existed at the end of the fourth century B.C. and presented them in the spirit of the Platonic school. For more than 2,000 years school courses in geometry have followed Euclid's Elements to some extent.
WHAT IS MATHEMATICS? What does this science deal with ? These questions were raised by the Greeks after they had begun to construct the edifice of mathematics on the basis of proofs, for the aura of absolute validity, of virtual sanctity, which mathematical knowledge acquired thanks to the existence of the proofs immediately made it stand out against the background of other everyday knowledge. The answer was given by the Platonic theory of ideas. This theory formed the basis of all Greek philosophy, defined the style and way of thinking of educated Greeks, and exerted an enormous influence on the subsequent development of philosophy and science in the Greco-Roman-European culture.
It is not difficult to establish the logic which led Plato to his theory. What does mathematics talk about'? About points, lines, right triangles, and the like. But are there in nature points which do not have dimensions? Or absolutely straight and infinitely fine lines? Or exactly equal line segments, angles, or areas? It is plain that there are not. So mathematics studies nonexistent, imagined things; it is a science about nothing. But this is completely unacceptable. In the first place, mathematics has unquestionably produced practical benefits. Of course, Plato and his followers despised practical affairs, but this was a logical result of philosophy, not a premise. In the second place, any person who studies mathematics senses very clearly that he is dealing with reality, not with fiction, and this sensation cannot be rooted out by any logical arguments. Therefore, the objects of mathematics really exist but not as material objects, rather as images or ideas, because in Greek the word "idea" in fact meant "image'' or "form.''[1] Ideas exist outside the world of material things and independent of it. Material things perceived by the senses are only incomplete and temporary copies (or shadows) of perfect and eternal ideas. The assertion of the real, objective existence of a world of ideas is the essence of Plato's teaching (''Platonism'').
For many centuries hopelessly irresolvable disputes arose among the Platonists over attempts to in some way give concrete form to the notion of the world of ideas and its interaction with the material world. Plato himself wisely remained invulnerable, avoiding specific, concrete terms and using a metaphorical and poetic language. But he did have to enter a polemic with his student Eudoxus, who not only proved mathematical theorems but also defended trading in olive oil. Such a position of course restricted the influx of new problems and ideas and fostered a canonization and regimentation of scientific thought, thus retarding its development. But beyond this, Platonism also had a more concrete negative effect on mathematics. It prevented the Greeks from creating algebraic language. This could be done only by the less educated and more practical Europeans. Later on we shall consider in more detail the history of the creation of modern algebraic language and the inhibiting role of Platonism, but first we shall discuss the answers given by modern science to the questions posed in Platonic times and how the answers given by Plato look in historical retrospect.
FOR US MATHEMATICS is above all a language that makes it possible to create a certain kind of models of reality: mathematical models. As in any other language (or branch of language) the linguistic objects of mathematics, mathematical objects, are material objects that fix definite functional units, mathematical concepts. When we say that the objects ''fix functional units'' we take this to mean that a person, using the discriminating capabilities of his brain, performs certain linguistic actions on these objects or in relation to them. It is plain that it is not the concrete form (shape, weight, smell) of the mathematical object which is important in mathematics; it is the linguistic activity related to it. Therefore the terms ''mathematical object'' and ''mathematical concept'' are often used as synonyms. Linguistic activity in mathematics naturally breaks into two parts: the establishment of a relationship between mathematical objects and nonlinguistic reality (this activity defines the meanings of mathematical concepts), and the formulations of conversions within the language, mathematical calculations and proofs. Often only the second part is what we call ''mathematics'' while we consider the first as the ''application of mathematics.''
Points, lines, right triangles, and the like are all mathematical objects. They make up our geometric drawings or stereometric models: spots of color, balls of modeling clay, wires, pieces of cardboard, and the like. The meanings of these objects are known. The point, for example, is an object whose dimensions and shape may be neglected. Thus the ''point'' is simply an abstract concept which characterizes the relation of an object to its surroundings. In some cases we view our planet as a point. But when we construct a geometric model we usually make a small spot of color on the paper and say, ''Let point A be given.'' This spot of color is in fact linguistic object Li, and the planet Earth may be the corresponding object (referent) Rj. There are no other true or ideal'' points, that is, without dimensions. It is often said that there are no ''true'' points in nature, but that they exist only in our imagination. This commonplace statement is either absolutely meaningless or false, depending on how it is interpreted. In any case it is harmful, because it obscures the essence of the matter. There are no "true'' points in our imagination and there cannot be any. When we say that we are picturing a point we are simply picturing a very small object. Only that which can be made up of the data of sensory experience can be imagined, and by no means all of that. The number 1,000 for example, cannot be imagined, large numbers, ideal points, and lines exist not in our imagination, but in our language, as linguistic objects we handle in a certain way. The rules for handling them reveal the essence of mathematical concepts, specifically the ''ideality of the point.'' The dimensions of points on a drawing do not influence the development of the proof, and if two points must be set so close that they merge into one, we can increase the scale.
But aren't the assertions of mathematics characterized by absolute prehave an entirely different status. By itself this language is, of course, discrete also, but empirical assertions reflect semantic conversions L1->S1 leading us into the area of nonlinguistic activity which is neither discrete nor deterministic. When we say that two rods have equal length this means that every time we measure them the result will be the same. Experience, however, teaches us that if we can increase the precision of measurement without restriction, sooner or later we shall certainly obtain different values for the length, because an empirical assertion of absolutely exact equality is completely senseless. Other assertions of empirical language which have meaning and can be expressed in the language of predicate calculus, for example ''rod no. 1 is smaller than rod no. 2," possess the same ''absolute precision'' (which is a trivial consequence of the discrete nature of the language) as mathematical assertions of the equality of segments. This assertion is either ''exactly'' true or "exactly'' false. Because of variations in the measuring process, however, neither is absolutely reliable.
NOW LET US DISCUSS the reliability of mathematical assertions. Plato deduced it from the ideal nature of the object of mathematics, from the fact that mathematics does not rely on the illusory and changing data of sensory experience. According to the mathematician, drawings and symbols are nothing but a subsidiary means for mathematics; the real objects Plato deals with are contained in his imagination and represent the result of perception of the world of ideas through reason, just as sensory experience is the result of perception of the material world through the sense organs. Imagination obviously plays a crucial part in the work of the mathematician (as it does, we might note, in all other areas of creative activity). But it is not entirely correct to say that mathematical objects are contained in the imagination: basically they are still contained in drawings and texts, and the imagination takes them up only in small parts. Rather than holding mathematical objects in our imagination we pass them through and the characteristics of our imagination determine the functioning of mathematical language. As for the source which determines the content of our imagination, here we disagree fundamentally with Plato. The source is the same sensory experience used in the empirical sciences. Therefore, even though it uses the mediation of imagination, mathematics creates models of the very same. unique (as far as we know) world we live in.
However, although they constructed a stunningly beautiful edifice of logically strict proofs, the Greek mathematicians nonetheless left a number of gaps in the structure; and these gaps, as we have already noticed, lie on the lowest stories of the edifice--in the area of definitions and the most elementary properties of the geometric figures. And this is evidence of a veiled reference to the sensory experience so despised by the Platonists. The mathematics of Plato's times provides even clearer material than does present-day mathematics to refute the thesis that mathematics is independent of experience.
The first statement proved in Euclid's first book contains a method of constructing an equilateral triangle according to a given side. The method is as follows.
Figure 10.3. Construction of an equilateral triangle.
Suppose AB is the given side of the triangle. Taking point A as the center we describe circle [pi]A with radius AB. We describe a similar circle ([pi][Beta]) from point B We use C to designate either of the points of intersection of these circles . Triangle ABC is equilateral, for AC = CB = AB.
There is a logical hole in this reasoning: how does it follow that the circles constructed by us will intersect at all'? This is a question fraught with complications, for the fact that point of intersection C exists cannot be related either to the attributes of a circle or even to the attributes of a pair of circles (for they by no means always intersect). We are dealing here with a more specific characteristic of the given situation. Euclid probably sensed the existence of a hole here, but he could not find anything to plug it with.
But how are we certain that circles [pi]A and [pi]B intersect? In the last analysis, needless to say, we know from experience. From experience in contemplating and drawing straight lines, circles, and lines in general, from unsuccessful attempts to draw circles [pi]A and [pi]B so that they do not intersect.
So Plato's view that the mathematics of his day was entirely independent of experience cannot be considered sound. But the question of the nature of mathematical reliability requires further investigation, for to simply make reference to experience and thus equate mathematical reliability with empirical reliability would mean to rush to the opposite extreme from Platonism. Certainly, we feel clearly that mathematical reliability is somehow different from empirical reliability, but how'?
The assertion that circles of radius AB with centers of A and B intersect (for brevity we shall designate this assertion E1 ) seems to us almost if not completely reliable; we simply cannot imagine that they would not intersect. We cannot imagine.... This is how mathematical reliability differs from the empirical! When we are talking about the sun rising tomorrow, we can imagine that the sun will not rise and it is only on the basis of experience that we believe that it probably will rise. Here there are two possibilities and the prediction as to which one will happen is probabilistic. But when we say that two times two is four and that circles constructed as indicated above inter
sect we cannot imagine that it could be otherwise. We see no other possibility, and therefore these assertions are perceived as absolutely reliable and independent of concrete facts we have observed.
IT IS VERY INSTRUCTIVE for an understanding of the nature of mathematical reliability to carry our analysis of the assertion E1 through to the end. Because we still have certain doubts that the circles in figure 10.3 necessarily intersect, let us attempt to picture a situation where they do not. If this attempt fails completely it will mean that assertion E1 is mathematically reliable and cannot be broken down into simpler assertions: then it should be adopted as an axiom. But if through greater or lesser effort of imagination we are able to picture a situation in which [pi]A and [pi]B do not intersect, it must be expected that this situation contradicts some simpler and deeper assertions which do possess mathematical reliability. Then we shall adopt them as axioms and the existence of the contradiction will serve as proof of E1. This is the usual way to establish axioms in mathematics.
First let us draw circle [pi]A. Then we shall put the point of the compass at point B and the writing element at point A and begin to draw circle [pi]B. We shall move from the center of circle [pi]A toward its periphery and at a certain moment (this is how we picture it in our imagination) we must either intersect circle [pi]A or somehow skip over it, thus breaking circle [pi]B.
Figure 10.4.
But we imagine circle pB as a continuous line and it becomes clear to us that the attributes of continuousness, which are more fundamental and general than the other features of this problem, lie at the basis of our confidence that circles [pi]A and [pi][Beta] will intersect. Therefore we set as our goal proving assertion E1 beginning with the attributes of continuousness of the circle. For this we shall need certain considerations related to the order of placement of points on a straight line. We include the concepts of continuousness and order among the basic. undefined concepts of geometry, like the concepts of the point, the straight line, or distance.
Here is one possible way to our goal. We introduce the concept of ''inside'' (applicable to a circle) by means of the following definition:
D1: It is said that point A lies inside circle [pi] if it does not lie on [pi] and any straight line passing through point A intersects [pi] at two points in such a way that point A lies between the points of intersection. If the point is neither on nor inside the circle it is said that it lies outside the circle.
The concept of ''between'' characterizes the order of placement of three points on a straight line. It may be adopted as basic and expressed, through the more ,general concept of ''order,'' by the following definition:
D2: It is said that point A is located between points B1 and B2 if these three points are set on one straight line and during movement along this line they are encountered in the order B1, A, and B2 or B2, A, and B1.
We shall adopt the following propositions as axioms:
A1: The center of a circle lies inside it.
A2: The arc of a circle connecting any two points of the circle is continuous.
A3: If point A lies inside circle [pi] and point B is outside it, and these two points are joined by a continuous line, then there is a point where this line intersects the circle.
Relying on these axioms, let us begin with the proof. According to the statement of the problem, circle [pi]B passes through center A of circle [pi]A. If we have confidence that there is at least one point of circle [pi][Beta] that does not lie inside [pi]A we shall prove E1. Indeed, if it lies on [pi]A then E1 has been proved. If it lies outside [pi]A then the arc of circle [pi]B connects it with the center, that is, with an inside point of circle [pi]A. Therefore, according to axioms A2 and A3 there is a point of intersection of [pi]B and [pi]A.
But can we be confident that there is a point on circle [pi]B which is outside [pi]A? Let us try to imagine the opposite case. It is shown in figure 10.5.
This is the second attempt to imagine a situation which contradicts the assertion being proved. Whereas the first attempt immediately came into explicit contradiction with the continuousness of a circle, the second is more successful. Indeed, stretching things a bit we can picture this case. We take a compass, put its point at point B and the pencil at point A. We begin to draw the circle without taking the pencil from the paper and when the pencil returns to the starting point of the line we remove it and see that we have figure 10.5. And why not?
To prove that this is impossible we must prove that in this case the center of circle [pi]B is necessarily outside it. We shall be helped in this by the following theorem:
T1: If circle [pi]1, lies entirely inside circle [pi]2 then every inside point of circle [pi]1 is also an inside point of circle [pi]2.
To prove this we shall take an arbitrary inside point A of circle [pi]1, which is shown in figure 10.6.
We draw a straight line through it. According to definition D1 it intersects [pi]1, at two points: B1 and B2 Because B1 (just as B2) lies inside [pi]2 this straight line also intersects [pi]2 at two points: C1 and C2. We have received five points on a straight line and they are connected by the following relationships of order: A lies between B1 and B2; B1 and B2 lie between C1 and C2. That point A proves to be between points C1 and C2 in this situation seems so obvious to us that we shall boldly formulate it as still another axiom.
A3: If points B1 and B2 on a straight line both lie between C1andC2, then any point A lying between B1 and B2 also lies between C1 andC2.
Because we can take any point inside [pi]1 as A and we can draw any straight line through it, theorem T1 is proven.
Now it is easy to complete the proof of E1. If circle [pi]B lies entirely inside [pi]A then according to theorem T1 its center B must also lie inside [pi]A. But according to the statement of the problem point B is located on [pi]A. Therefore [pi]B contains at least one point which is not inside in relation to [pi]A
So to prove one assertion E1 we needed four assertions (axioms A1-A4), but then these assertions express very fundamental and general models of reality related to the concepts of continuousness and order and we cannot even imagine that they are false. The only question that can be raised refers to axiom A1 which links the concept of center. which is metrical (that is, including the concept of measurement) in nature, with the concept of ''inside," which relies exclusively on the concepts of continuousness and order. It may be desired that this connection be made using simpler geometric objects, under conditions which are easier for the functioning of imagination. This desire is easily met. For axiom A1 let us substitute the following axiom:
A1': if on a straight line point A and a certain distance (segment) R are given, then there are exactly two points on the straight line which are set at distance R from point A, and point A lies between these two points.
Relying on this axiom we shall prove assertion A1 as a theorem. We shall draw an arbitrary straight line through the center of the circle. According to axiom A1' there will be two points on it which are set at distance R (radius of the circle) from the center. Because a circle is defined as the set of all points which are located at distance R from the center, these points belong to the circle. According to axiom A1' the center point lies between them and therefore, according to definition D1, it is an inside point. In this way axiom A1 has been reduced to axiom A1'. Now try to imagine a point on a straight line which does not have two points set on different sides from it at the given distance!
THE PRIMARY PROPOSITIONS of arithmetic in principle possess the same nature as the primary propositions of geometry, but they are perhaps even simpler and more obvious and denial of them is even more inconceivable than denial of geometric axioms. As an example let us take the axiom which says that for any number
a + 0 =a
The number O depicts an empty set. Can you imagine that the number of elements in some certain set would change if it were united with an empty set? Here is another arithmetic axiom: for any numbers a and b
a+ (b+ 1) = (a+b) + 1,
that is, if we increase the number b by one and add the result to a, we shall obtain the same number as if we were to add a and b first and then increase the result by one. If we analyze why we are unable to imagine a situation that contradicts this assertion, we shall see that it is a matter of the same considerations of continuousness that also manifest themselves in geometric axioms. In the process of counting, it is as if we draw continuous lines connecting the objects being counted with the elements of a standard set and, of course, lines in time (let us recall the origin of the concept ''object'') whose continuousness ensures that the number is identical to itself.
Natural auditory language transferred to paper gives rise to linear language, that is, a system whose subsystems are all linear sequences of signs. Signs are objects concerning which it is assumed only that we are able to distinguish identical ones from different ones. The linearity of natural languages is a result of the fact that auditory language unfolds in time and the relation of following in time can be modeled easily by the relation of order of placement on a timeline. The specialization of natural language led to the creation of the linear, symbolic mathematical language which now forms the basis of mathematics.
Operating within the framework of linear symbolic languages we are constantly taking advantage of certain other attributes which seem so obvious and self-evident that we don't even want to formulate them in the form of axioms. As an example let us take this assertion: if symbol a is written to the left of symbol b and symbol c is written on the right the same word (sequence of characters) will be received as when b is written to the right of a and followed by c. This assertion and others like it possess mathematical reliability for we cannot imagine that it would be otherwise. One of the fields of modern mathematics, the theory of semi-groups, studies the properties of linear symbolic systems from an axiomatic point of view and declares the simplest of these properties to be axioms.
All three kinds of axioms, geometric, arithmetic, and linear-symbolic, possess the same nature and in actuality rely on the same fundamental concepts. concepts such as identity, motion, continuousness, and order. There is no difference in principle among these groups of axioms. And if one term were to be selected for them they should be called geometric or geometric-kinematic because they all reflect the attributes of our space-time experience and space-time imagination. The only more or less significant difference which can be found is in the group of "properly geometric'' axioms; some of the axioms concerning straight lines and planes reflect more specific experience related to the existence of solid bodies. The same thing evidently applies to metric concepts. But this difference too is quite arbitrary. Can we say anything serious about those concepts which we would have if there were no solid bodies in the world?
Thus far we have been discussing the absolute reliability of axioms. But where do we get our confidence in the reliability of assertions obtained by logical deduction from axioms? From the same source, our imagination refuses to permit a situation in which by logical deduction we obtain incorrect results from correct premises. Logical deduction consists of successive steps. At each step, relying on the preceding proposition. we obtain a new one. From a review of formal logical deduction (chapter 11) it will be seen that our confidence that at every step we can only receive a true proposition from higher true propositions is based on logical axioms [2] which seem to us just as reliable as the mathematical axioms considered above. And this is for the same reason, that the opposite situation is absolutely inconceivable. Having this confidence we acquire confidence that no matter how many steps a logical deduction may contain it will still possess this attribute. Here we are using the following very important axiom:
The axiom of induction: Let us suppose that function f (x) leaves attribute P (x) unchanged, that is
(
x){P (x) =>P
[f (x)]}
We will use f n(x) to signify the result of sequential n-time application of function f (x), that is
f 1(x) =f (x), f n(x) = f [fn-1 (x)].
Then f n(x) will also leave attribute P (x) unchanged for any n, that is
(n)(
x){P (x) => P [f n(x)]}
By their origin and nature logical axioms and the axiom of induction (which is classed with arithmetic because it includes the concept of number) do not differ in any way from the other axioms; they are all mathematical axioms. The only difference is in how they are used. When mathematical axioms are applied to mathematical assertions they become elements of a metasystem within the framework of a system of mathematically reliable assertions and we call them logical axioms. Thanks to this, the system of mathematically reliable assertions becomes capable of development. The great discovery of the Greeks was that it is possible to add one certainty to another certainty and thus obtain a new certainty.
THE DESCRIPTION of mathematical axioms as models of reality which are true not only in the sphere of real experience but also in the sphere of imagination relies on their subjective perception. Can it be given a more objective characterization?
Imagination emerges in a certain stage of development of the nervous system as arbitrary associating of representations. The preceding stage was the stage of nonarbitrary associating (the level of the dog). It is natural to assume that the transition from nonarbitrary to arbitrary associating did not produce a fundamental change in the material at the disposal of the associating system, that is, in the representations which form the associations. This follows from the hierarchical principle of the organization and development of the nervous system in which the superstructure of the top layers has a weak influence on the lower ones And it follows from the same principle that m the process of the preceding transition, from fixed concepts to nonarbitrary associating, the lowest levels of the system of concepts remained unchanged and conditioned those universal, deep-seated properties of representations that were present before associating and that associating could not change. Imagination cannot change them either. These properties are invariant in relation to the transformations made by imagination. And they are what mathematical axioms rely on.
If we picture the activity of the imagination as shuffling and fixing certain elements. ''pieces'' of sensory perception. then axioms are models which are true for any piece and. therefore, for any combination of them. The ability of the imagination to break sensory experience up into pieces is not unlimited; emerging at a certain stage of development it takes the already existing system of concepts as its background, as a foundation not subject to modification. Such profound concepts as motion, identity, and continuousness were part of this background and therefore the models which rely on these concepts are universally true not only for real experience but also for any construction the imagination is capable of creating.
Mathematics forms the frame of the edifice of natural sciences. Its axioms are the support piles that drive deep into the neuronal concepts, below the level where imagination begins to rule. This is the reason for the stability of foundation which distinguishes mathematics from empirical knowledge. Mathematics ignores the superficial associations which make up our everyday experience, preferring to continue constructing the skeleton of the system of concepts which was begun by nature and set at the lowest levels of the hierarchy. And this is the skeleton on which the "noncompulsory'' models we class with the natural sciences will form, just as the ''noncompulsory'' associations of representations which make up the content of everyday experience form on the basis of inborn and "compulsory" concepts of the lowest level. The requirements dictated by mathematics are compulsory: when we are constructing models of reality we cannot bypass them even if we want to. Therefore we always refer the possible falsehood of a theory beyond the sphere of mathematics. If a discrepancy is found between the theory and the experiment it is the external, "noncompulsory" part of the theory that is changed but no one would ever think of expressing the assumption that, in such a case, the equality 2 + 2 = 4 has proved untrue.
The ''compulsory" character of classical mathematical models does not contradict the appearance of mathematical and physical theories which at first glance conflict with our space-time intuition (for example, non-Euclidian geometry or quantum mechanics). These theories are linguistic models of reality whose usefulness is seen not in the sphere of everyday experience but in highly specialized situations. They do not destroy and replace the classical models; they continue them. Quantum mechanics, for example, relies on classical mechanics. And what theory can get along without arithmetic? The paradoxes and contradictions arise when we forget that the concept constructs which are included in a new theory are new concepts, even when they are given old names. We speak of a ''straight line'' in non-Euclidian geometry and call an electron a ''particle'' although the linguistic activity related to these words (proof of theorems and quantum mechanics computations) is not at all identical to that for the former theories from which the terms were borrowed. If two times two is not four then either two is not two, times is not times, or four is not four.
The special role of mathematics in the process of cognition can be expressed in the form of an assertion, that mathematical concepts and axioms are not the result of cognition of reality, rather they are a condition and form of cognition. This idea was elaborated by Kant and we may agree with it if we consider the human being to be entirely given and do not ask why these conditions and forms of cognition are characteristic of the human being. But when we have asked this question we must reach the conclusion that they themselves are models of reality developed in the process of evolution (which, in one of its important aspects, is simply the process of cognition of the world by living structures). From the point of view of the laws of nature there is no fundamental difference between mathematical and empirical models; this distinction reflects only the existence in or~anization of the human mind of a certain border line which separates inborn models from acquired ones. The position of this line, one must suppose, contains an element of historical accident. If it had originated at another level, perhaps we would not be able to imagine that the sun may tail to rise or that human beings could soar above the earth in defiance or gravity.
PLATO'S IDEALISM was the result of a sort of projection of the elements of language onto reality. Plato's ''ideas'' have the same origin as the spirits in primitive thinking; they are the imagined of really existing names. In the first stages of the development of critical thinking the nature of abstraction in the interrelationship of linguistic objects and non-linguistic activity is not yet correctly understood. The primitive name-meaning unit is still pressing on people an idea of a one-to-one correspondence between names and their meanings. For words that refer to concrete objects the one-to-one correspondence seems to occur because we picture the object as some one thing. But what will happen with general concepts (universals)? In the sphere of the concrete there is no place at all for their meanings; everything has been taken up by unique' concepts, for a label with a name can be attached to each object. The empty place that form is filled by the "idea." Let us emphasize that Plato's idealism is far from including an assertion of the primacy of the spiritual over the material, which is to say it is not spiritualism (this term, which is widely used in Western literature, is little used in our country and is often replaced by the term "idealism,'' which leads to inaccuracy). According to Plato spiritual experience is just as empirical as sensory experience and it has no relation to the world of ideas. Plato's ''ideas'' are pure specters, and they are specters born of sensory, not spiritual, experience.
From a modern cybernetic point of view only a strictly defined, unique situation can be considered a unique concept. This requires an indication of the state of all receptors that form the input of the nervous system. It is obvious that subjectively we are totally unaware of concepts that are unique in this sense. Situations that are merely similar become indistinguishable somewhere in the very early stages of information processing and the representations with which our consciousness is dealing are generalized states, that is to say, general, or abstract, concepts (sets of situations). The concepts of definite objects which traditional logic naively takes for the primary elements of sensory experience and calls ''unique" concepts are in reality, as was shown above, very complex constructions which require analysis of the moving picture of situations and which rely on more elementary abstract concepts such as continuousness, shape, color, or spatial relations. And the more ''specific'' a concept is from the logical point of view, the more complex it will be from the cybernetic point of view. Thus, a specific cat differs from the abstract cat in that a longer moving picture of situations is required to give meaning to the first concept than to the second. Strictly speaking the film may even be endless, for when we have a specific cat in mind we have in mind not only its ''personal file'' which has been kept since its birth, but also its entire genealogy. There is no fundamental difference in the nature of concrete and abstract concepts; they both reflect characteristics of the real world. If there is a difference, it is the opposite of what traditional logic discerns: abstract, general concepts of sensory and spiritual experience (which should not be confused with mathematical constructs) are simpler and closer to nature than concrete concepts which refer to the definite objects. Logicians were confused by the fact that concrete concepts appeared in language earlier than abstract ones did. But this is evidence of their relatively higher position in the hierarchy of neuronal concepts, thanks to which they emerged at the point of connection with linguistic concepts.
The Platonic theory of ideas, postulating a contrived, ideal existence of generalized objects, puts one-place predicates (attributes) in a position separate from multiplace predicates (relations). This theory assigned attributes the status of true existence but denied it to relations, which became perfectly evident in Aristotle's loci. The concrete, visual orientation and static quality in thinking which were so characteristic of the Greeks in the classical period came from this. In the next chapter we shall see how this way of thinking was reflected in the development of mathematics.
[1] The resemblance in sound between the Greek idea and the Russian vid is not accidental; they come from a common Indo-European root. (Compare also Latin "vidi" - past tense of "to see.")
[2] For those who are familiar with mathematimal logic let us note that this is in the broad sense. including the rules of inference.