This is chapter 12 of the "The Phenomenon of Science" by Valentin F. Turchin
"THE NEXT STOP IS APRELEVKA STATION", a hoarse voice announces through the loudspeaker. ''I repeat Aprelevka Station. The train does not have a stop at Pobeda Station.''
You are riding a commuter train on the Kiev Railroad and because you have forgotten to bring a book and there is nothing for you to do you begin reflecting on how carelessly we still treat our native language. Really, what an absurd expression ''does not have a stop.'' Wouldn't it be simpler to say ''does not stop"? Of these bureaucratic governmental expressions, people write about it all the time, but it hasn't done any good yet.
If you do not get off at Aprelevka, however, and you have time for further reflection you will see that this is by no means a matter of a careless attitude toward our native language; in fact ''does not have a stop'' does not mean quite the same thing as ''does not stop.'' The concept of the stop in railroad talk is not the same as the concept of ceasing movement. The following definition, not too elegant but accurate enough, can be given: a stop is a deliberate cessation of the train's movement accompanied by the activities necessary to ensure that passengers get on and off the train. This is a very important concept for railroad workers and it is linked to the noun ''stop'' not to the verb ''to stop.'' Thus if the engineer stopped the train but did not open the pneumatic doors, the train ''stopped" but it did not ''have a stop."
The railroad worker who made the announcement did not, of course, perform such a linguistic analysis. He simply used the ordinary professional term, which enabled him to express his thought exactly, even if it seemed somewhat clumsy to a nonprofessional. This is an instance of a very common phenomenon: when language is used for comparatively narrow professional purposes there is a tendency to limit the number of terms used and to give them more precise and constant meanings. We say the language is formalized. If this process is carried through to its logical conclusion the language will be completely formalized.
The concept of a formalized language can be defined as follows. Let us refer to our diagram of the use of linguistic models of reality (see figure 9.5) and put the question: how is the conversion L_{1} -> L_{2} performed, on what information does it depend? We can picture two possibilities:
1 . The conversion L_{1} -> L_{2} is determined exclusively by linguis tic objects L_{1} which participate in it and do not depend on those nonlinguistic representations S_{1} which correspond to them according to the semantics of the language. In other words, the linguistic activity depends only on the ''form'' of the language objects not on their ''content'' (meaning).
2. The result of the conversion of linguistic object L_{1} depends less on the type of object L_{1} itself than on representation S_{1} it generates in the person's mind, on the associations in which it is included, and therefore on the person's personal experience of life. In the first case we call the language formalized, while in the second case it is unformalized. We should emphasize that complete formalization of a language does not necessarily mean complete algorithmization of it, the situation where all linguistic activity amounts to fulfilling precise and unambiguous prescriptions as a result of which each linguistic object L_{1} is converted into a completely definite object L_{2}. The rules of conversion L_{1} -> L_{2} can be formalized as more or less rigid constraints and leave a certain freedom of action; the only important thing is that these constraints depend on the type of object L_{1} and potential objects L_{2 }by themselves alone and not on the meanings of the linguistic objects.
The definition we have given of a formalized language applies to the case where language is used to create models of reality. When a language serves as a means of conveying control information (the language of orders) there is a completely analogous division into two possible types of responses:
1. The person responds in a strictly formal manner to the order, that is, his actions depend only on the information contained in the text of the order, which is viewed as an isolated material system.
2. The person's actions depend on those representations and associations the order evokes in him. Thus, he actually uses much more information than that contained in the text of the order.
There is no difference in principle between the language of orders and the language of models. The order "Hide!'' can be interpreted as the model "If you don't hide your life is in danger.'' The difference between the order and the model is a matter of details of information use. In both cases the formalized character of the language leads to a definite division of syntax and semantics, a split between the material linguistic objects and the representations related to them; the linguistic objects acquire the characteristics of an independent system.
Depending on the type of language which is used we may speak of informal and formal thinking. In informal thinking, linguistic objects are primarily important to the extent that they evoke definite sets of representations in us. The words here are strings by which we extract from our memory particles of our experience of life; we relive them, compare them, sort through them, and so on. The result of this internal work is the conversion of representations S_{1} -> S_{2}, which models the changes R_{1} -> R_{2} in the environment. But this does not mean that informal thinking is identical to nonlinguistic thinking. In the first place, by itself the dismembering of the stream of perceptions depends on a system of concepts fixed in language. In the second place, in the process of the conversion S_{1} -> S_{2} the "natural form'' of the linguistic object, the word, plays a considerable part. Very often we use associations among words, not among representations. Theretore the formula for nonformal thinking can be represented as follows: (S_{1},L_{1}) -> (S_{2}, L_{2}).
In ormal thinking we operate with linguistic objects as if they were certain independent and self-sufficient essences, temporarily forgetting their meanings and recalling them only when it is necessary to interpret the result received or refine the initial premises. The formula for formal thinking is as follows: S_{1} ->L_{1}-> L_{2} ->S_{2}.
In order for formal thinking to yield correct results, the semantic system of the language must possess certain characteristics we describe by such terms as ''precision,'' ''definiteness,'' and ''lack of ambiguity.'' If the semantic system does not possess these characteristics, we shall not be able to introduce such formal conversions L_{1} -> L_{2} in order that, by using them, we may always receive a correct answer. Of course, it is possible to establish the formal rules of conversions somehow and thus obtain a formalized language, but this will be a language that sometimes leads to false conclusions. Here is an example of a deduction which leads to a false result because of ambiguity in the semantic system:
Vanya is a gypsy.
The gypsies came to Europe from India.
In practice, thus, semantic precision and syntactical formalization are inseparable, and a language that satisfies both criteria is called formalized. But the leading criterion is the syntactical one, for the very concept of a precise semantic system can be defined strictly only through syntax. And indeed, the semantic system is precise if it is possible to establish formalized syntax which yields only true models of reality.
BECAUSE the syntactical conversions L_{1} -> L_{2} within the framework of a formalized language are determined entirely by the physical type of objects L_{i}, the formalized language is in essence a machine that produces different changes of symbols. For a completely algorithmized language, such as the language of arithmetic, this thesis is perfectly obvious and is illustrated by the existence of machines in the ordinary, narrow sense of the word (calculators and electronic computers) that carry out arithmetic algorithms. If the rules of conversion are constraints only, it is possible to construct an algorithm that determines whether the conversion L_{1} -> L_{2} is proper for given L_{1} and L_{2}. It is also possible to construct an algorithm (a ''stupid'' one) which for a given L_{1} begins to issue all proper results for L_{2} and continues this process to infinity if the number of possible L_{2 }is unlimited. In both cases we are dealing with a certain language machine, that can work without human intervention.
The formalization of a language has two direct consequences. In the first place, the process of using linguistic models is simplified because precise rules for converting L_{1} -> L_{2} appear. In the extreme case of complete algorithmization, this conversion can generally be carried out automatically. In the second place, the linguistic model becomes independent of the human brain which created it, and becomes an objective model of reality. Its semantic system reflects, of course, concepts that have emerged in the process of the development of the culture of human society, but in terms of syntax it is a language machine that could continue to work and preserve its value as a model of reality even if the entire human race were to suddenly disappear. By studying this model an intelligent being with a certain knowledge of the object of modeling would probably be able to reproduce the semantic system of the language by comparing the model to his own knowledge. Let us suppose that people have built a mechanical model of the Solar System in which the planets are represented by spheres of appropriate diameters revolving on pivots around a central sphere, representing the sun, in appropriate orbits with appropriate periods. Then let us suppose that this model has fallen into the hands (perhaps the tentacles?) of the inhabitants of a neighboring stellar system, who know some things about our Solar System--for example, the distances of some planets from the sun or the times of their revolutions. They will be able to understand what they have in front of them, and they will receive additional information on the Solar System. The same thing is true of scientific theories, which are models of reality in its different aspects, built with the material of formalized symbolic language. Like a mechanical model of the Solar System, each scientific theory can in principle be deciphered and used by any intelligent beings.
Language can be characterized not only by the degree of its formalization but also by the degree of its abstraction, which is measured by the abundance and complexity of the linguistic constructs it uses. As we noted in chapter 7, it would be more correct to speak of the ''construct quality'' of a language rather than of its abstractness, but the former term [the Russian ''konstruktnost''] has not yet been accepted. Therefore we shall use the term ''abstractness.'' We shall call a language which does not use constructs or uses only those of the very lowest level ''concrete,'' and we shall call a language which does use complex constructs ''abstract.'' Although this is a conditional and relative distinction, its meaning is nonetheless perfectly clear. And it does not depend on dividing languages into formalized and unformalized, which are different aspects of language. By combining all these aspects we obtain four types of languages used in the four most important spheres of linguistic activity. They can be arranged according to the table below:
Concrete Language | Abstract Language | |
Unformalized Language | Art | Philosophy |
Formalized Language | Descriptive Sciences | Theoretical sciences (mathematics) |
Neither the vertical nor the horizontal division is strict and unambiguous; the differences are more of a quantitative nature. There are transitional types on the boundaries between these "pure'' types of language.
Art is characterized by unformalized and concrete language. Words are important only as symbols which evoke definite complexes of representations and emotions. The emotional aspect is ordinarily decisive, but the cognitive aspect is also very fundamental. In the most significant works of art these aspects are inseparable. The principal expressive means is the image, which may be synthetic but always remains concrete.
Moving leftward across the table, we come next to philosophy, which is characterized by abstract-act, informal thinking. The combination of an extremely high degree of constructs among the concepts used and an insignificant degree of formalization requires great effort by the intuition and makes philosophical language unquestionably the most difficult of the tour types of language. When art raises abstract ideas it comes close to philosophy. On the other hand, philosophy will use the artistic image now and again to stimulate the intuition, and here it borders on art.
On the bottom right half of our table we find the theoretical sciences, characterized by an abstract and formalized language. Science in general is characterized by formalized language; the difference between the descriptive and theoretical sciences lies in a different degree of use of concept-constructs. The language of descriptive science must be concrete and precise; formalization of syntax by itself does not play a large part, but rather acts as a criterion of the precision of the semantic system (logical consistency of definitions, completeness of classifications, and so on).
The models of the world given by the descriptive sciences [bottom left of the table] are expressed in terms of ordinary neuronal concepts or concepts with a low degree of construct usage and, properly speaking, as models they are banal and monotypic: if some particular thing is done (for example, a trip to Australia or cutting open the abdominal cavity of a frog) it will be possible to see some other particular thing. On the other hand, the whole essence of the theoretical sciences is that they give fundamentally new models of reality: scientific theories based on concept-constructs not present at the neuronal levels. Here the formalization of syntax plays the decisive part. The most extreme of the theoretical sciences is mathematics, which contains the most complex constructs and uses a completely formalized language. Properly speaking mathematics is the formalized language used by the theoretical sciences.
Moving back up from the descriptive sciences we are again in the sphere of art. Somewhere on the border between the descriptive sciences and art lies the activity of the journalist or naturalist-writer.
ALTHOUGH THE LANGUAGE of science is formalized, scientists cannot restrict themselves to purely formal thinking. The use of a complete and finished theory does indeed demand formal operations that do not go outside the framework of a definite language, but the creation of a new theory always involves going beyond the formal system; it is always a metasystem transition of greater or lesser degree.
Of course, we certainly cannot say that everyone who does not break down old formalisms is working on banal and uncreative things. This applies only to those who operate in accordance with already available algorithms, essentially performing the functions of a language machine. But fairly complex formal systems cannot be algorithmized and they offer a broad area for creative activity. Actions within the framework of such a system can be compared to playing chess. In order to play chess well one must study for a long time, memorize different variations and combinations, and acquire a specific chess intuition. In the same way the scientist who is dealing with a complex formalized language (that is to say, with mathematics either pure or applied) develops in himself, through long study and training, an intuition for his language, often a very narrow one, and obtains new theoretical results. This is, of course, activity which is both noble and creative.
All the same, going beyond the old formalism is an even more serious creative step. If the scientists we were discussing above could be called chess-player-scientists, then the scientists who create new formalized languages and theories can be called philosopher-scientists. We saw an example of these two types of scientist in our discussion of Fermat and Descartes in chapter 11. The concepts of new theories do not emerge in precise and formalized form from a vacuum. They become crystallized gradually, during a process of abstract but not formalized thinking--i.e., philosophical thinking. And whereas here too intuition is required, it is of a different type-- philosophical. ''The sciences,'' Descartes wrote in his Discours de la méthode ''borrow their principles from philosophy.''
The creation of fundamental scientific theories lies in the borderline area between philosophy and science. As long as a scientist operates with conventional concepts within the framework of conventional formalized language he does not need philosophy. He is like the chess player who pictures the same pieces on the same board, but solves different problems. And he does obtain new results, relying on his intuition for chess. But in this he will never, in his game of chess, go beyond the limits inherent in his language. To improve language itself, to formalize what has not yet been formalized means to go into philosophy. If a new theory does not contain this element it is only a consequence of old theories. It can be said that the amount of what is new in any theory corresponds exactly to the amount of philosophy in it.
From the above discussion the importance of philosophy for the activity of the scientist is clear. In the Dialectic of Nature, F. Engels wrote:
Naturalists imagine that they are free of philosophy when they ignore or downgrade it. But because they cannot take a step without thinking, and thinking demands logical categories and they borrow these categories uncritically either from the everyday, general consciousness of so-called educated people among whom the remnants of long-dead philosophical systems reign, from crumbs picked up in required university courses in philosophy (which are not only fragmentary views, but also a hodgepodge of the views of people affiliated with the most diverse and usually the most despicable schools), or from uncritical and unsystematic reading of every kind of philosophical works--in the end they are still subordinate to philosophy but, unfortunately, it is usually the most despicable philosophy and those who curse philosophy most of all are slaves to the worst vulgarized remnants of the worst philosophical systems.[1]
That sounds amazingly modern!
THE CONVERSION of language, occurring as a result of formalization, into a reality independent of the human mind which creates it has far-reaching consequences. The just-created language machine (theory), as a part of the human environment, becomes an object of study and description by means of the new language. In this way a metasystem transition takes place. In relation to the described language the new language is a metalanguage and the theories formulated in this language and concerned with theories in the language-object are metatheories. If the metalanguage is formalized, it may in turn become an object of study by means of the language of the next level and this metasystem transition can be repeated without restriction.
In this way, the formalization of a language gives rise to the stairway effect (see chapter 5). Just as mastering the general principles of making tools to influence objects gives rise to multiple repetitions of the metasystem transition and the creation of the hierarchical system of industrial production, so mastering the general principle of describing (modeling) reality by means of a formalized language gives rise to creation of the hierarchical system of formalized languages on which the modern exact sciences are based. Both hierarchies have great height. It is impossible to build a jet airplane with bare hands. The same thing is true of the tools needed to build an airplane. One must begin with the simplest implements and go through the whole hierarchy of complexity of instruments before reaching the airplane. In exactly the same way, in order to teach the savage quantum mechanics, one must begin with arithmetic.
THE ESSENCE of what occurred in mathematics in the seventeenth century was that the general principle of using formalized language was mastered. This marked the beginning of movement up the stairway; it led to grandiose achievements and continues to the present day. It is true that this principle was not formulated so clearly then as now, and the term ''formalized language" did not appear until the twentieth century. But such a language was in fact used. As we saw. Descartes' reform was the first step along this path. The works of Descartes, in particular the quotations given above, show that this step was far from accidental: rather it followed from his method of learning the laws of nature which, if we put it in modern terms, is the method of creating models using formalized language. Descartes was aware of the universality of his method and its mathematical character. In the Regulae ad directionem ingenii he expresses his confidence that there must be ''some general science which explains everything related to order and measure without going into investigation of any particular objects.'' This science," he writes, should be called "universal mathematics."
Another great mathematician-philosopher of the seventeenth century, G. Leibnitz (1646-1716), understood fully the importance of the formalization of language and thinking. Throughout his life Leibnitz worked to develop a symbolic calculus to which he ,gave the Latin name characteristica universalis. Its goal was to express all clear human thoughts and reduce logical deduction to purely mechanical operations. In one of his early works Leibnitz states, ''The true method should be our Ariadne's thread, that is, a certain palpable and rough means which would guide the reason like lines in geometry and the forms of operations prescribed for students of arithmetic. Without this our reason could not make the long journey without getting off the road.'' This essentially points out the role of formalized language as the material fixer of concept-constructs--i.e., its main role. In his historical essay on the foundations of mathematics[2] N. Bourbaki writes:
The many places in the works of Leibnitz where he mentions his grandiose project and the progress which would follow upon its realization show how clearly he understood formalized language as a pure combination of characters in which only their coupling is important, so that a machine will be able to derive all theorems and it will be possible to resolve all incomplete or mistaken understanding by simple calculation. Although such hopes might seem excessive, it must be admitted that it was in fact under the constant influence of them that Leibnitz created a significant share of his mathematical writings, above all his works on the symbolism of infinitesimal calculus. He himself was very well aware of this and openly linked his ideas of introducing indexes and determinants and his draft of the "geometric calculus" to his "charateristica.'' But he felt that his most significant work would be symbolic logic.... And although he was not able to create such calculus, at least he started work to carry out his intention three times.
Leibnitz's ideas on the characteristica universalis. were not elaborated in his day. The work of formalizing logic did not get underway until the second half of the nineteenth century. But Leibnitz's ideas are testimony to the fact that the principle of describing reality by means of formalized logic is an inborn characteristic of European mathematics, and has always been the source of its development, even though different authors have been aware of this to different degrees.
It is not our purpose to set forth the history of modern mathematics or to give a detailed description of the concepts on which it is based; a separate book would be required for that. We shall have to be satisfied with a brief sketch that only touches that aspect of mathematics which is most interesting to us in this book--specifically, the system aspect.
The leitmotif in the development of mathematics during the last three centuries has been the gradually deepening awareness of mathematics as a formalized language and the resulting growth of multiple levels in it, occurring through metasystem transitions of varying scale.
We shall now review the most important manifestations of this process; they can be called variations on a basic theme, performed on different instruments and with different accompaniment. Simultaneously with upward growth in the edifice of mathematics there was an expansion of all its levels, including the lowest one--the level of applications .
WE HAVE ALREADY spoken of ''impossible'' numbers--irrational, negative, and imaginary numbers. From the point of view of Platonism the use of such numbers is absolutely inadmissible and the corresponding symbols are meaningless. But Indian and Arabic mathematicians began to use them in a minor way, and then in European mathematics they finally and irreversibly took root and received reinforcement in the form of new ''nonexistent" objects, such as an infinitely remote point of a plane. This did not happen all at once, though. For a long time the possibility of obtaining correct results by working with ''nonexistent'' objects seemed amazing and mysterious. In 1612 the mathematician Clavius, discussing the rule that ''a minus times a minus yields a plus'' wrote: ''Here is manifested the weakness of human reason which is unable to understand how this can be true.'' In 1674, discussing a certain relation between complex numbers, Huygens remarked: ''There is something incomprehensible to us concealed here.'' A favorite expression of the early eighteenth century was the ''incomprehensible riddles of mathematics.'' Even Cauchy in 1821 had very dim notions of operations on complex quantities.[3]
The last doubts and uncertainties related to uninterpreted objects were cleared up only with the introduction of the axiomatic approach to mathematical theories and final awareness of the ''linguistic nature'' of mathematics. We now feel that there is no more reason to be surprised at or opposed to the presence of such objects in mathematics than to be surprised at or opposed to the presence of parts in a car in addition to the four wheels, which are in direct contact with the ground and set the car in motion. Complex numbers and objects like them are the internal "wheels'' of mathematical models; they are connected with other ''wheels,'' but not directly with the ''ground,'' that is, the elements of nonlinguistic reality. Therefore one may go right on and operate with them as formal objects (that is, characters written on paper) in accordance with their properties as defined by axioms. And there is no reason to grieve because you cannot go to the pastry shop and buy square root(-15) rolls.
AWARENESS OF THE PRINCIPLE of describing reality by means of formalized language gives rise, as we have seen, to the stairway effect. Here is an example of a stairway consisting of three steps. Arithmetic is a theory we apply directly to such objects of nonlinguistic reality as apples, sheep, rubles, and kilograms of goods. In relation to it school algebra is a metatheory that knows only one reality--numbers and numerical equalities--while its letter language is a metalanguage in relation to the language of the numerals of arithmetic. Modern axiomatic algebra is a metatheory in relation to school algebra. It deals with certain objects (whose nature is not specified) and certain operations on these objects (the nature of the operations is also not specified). All conclusions are drawn from the characteristics of the operations. In the applications of axiomatic algebra to problems formulated in the language of school algebra, objects are interpreted as variables and operations are arithmetic operations. But modern algebra is applied with equal success to other branches of mathematics, for example to analysis and geometry.
A thorough study of mathematical theory generates new mathematical theories which consider the initial theory in its different aspects. Therefore, each of these theories is in a certain sense simpler than the initial theory, just as the initial theory is simpler than reality, which it always considers in some certain aspect. The models are dismembered and a set of simpler models is isolated from the complex one. Formally speaking, new theories are just as universal as the initial theory: they can be applied to any objects, regardless of their nature, if they satisfy the axioms. With the axiomatic approach different mathematical theories form what is, strictly speaking, a hierarchy of complexity, not of control. When we consider the models that in fact express laws of nature (the ones used in applications of mathematics), however, we see that mathematical theories are very clearly divided into levels according to the nature of the objects to which they are actually applied. Arithmetic and elementary geometry are in direct contact with nonlinguistic reality, but a certain theory of groups is used to create new physical theories from which results expressed in the language of algebra and analysis are extracted and then "put in numbers": only after this are they matched with experimental results. This distribution of theories by levels corresponds overall to the order in which they arose historically, because they arose through successive metasystem transitions. The situation here is essentially the same as in the hierarchy of implements of production. It is possible to dig up the ground with a screwdriver, but that tool was not invented for this purpose and really is needed only by someone working with screws and bolts. Group theory can be illustrated by simple examples from everyday life or elementary mathematics, but it is really used only by mathematicians and theoretical physicists. A clerk in a store or an engineer in the field has no more use for group theory than the primitive has for a screwdriver.
ACCORDING TO THE ancient Greeks, the objects of mathematics had real existence in the "world of ideas.'' Some of the properties of these objects seemed in the mind to be absolutely indisputable; they were declared axioms. Others, which were not so obvious, had to be proved using the axioms. With such an approach there was no great need to precisely formulate and to completely list all the axioms: if some ''indisputable'' attribute of objects is used in a proof, it is not that important to know whether it has been included in a list of axioms or not: the truth of the property being proved does not suffer. Although Euclid did give a list of definitions and axioms (including postulates) in his Elements as we saw in chapter 10, now and again he used assumptions which are completely obvious intuitively but not included in the list of axioms. As for his definitions, there are more of them than there are objects defined, and they are completely unsuitable for use in the proof process. The list of definitions in the first book of the Elements begins as follows:
There are a total of 34 definitions. The Swiss geometer G. Lambert (1728-1777) noted in this regard: ''What Euclid offers in this abundance of definitions is something like a nomenclature. He really proceeds like, for example, a watchmaker or other artisan who is beginning to familiarize his apprentices with the names of the tools of his trade."
The trend toward formalization of mathematics generated a trend toward refinement of definitions and axioms. Leibnitz called attention to the fact that Euclid's construction of an equilateral triangle relies on an assumption that does not follow from the definitions and axioms (we reviewed this construction in chapter 10). But it was only the creation of non-Euclidean geometry by N. 1. Lobachevsky (1792- 1856), J. Bolyai (1802-1860), and K. Gauss (1777- 1855) which brought universal recognition of the axiomatic approach to mathematical theories as the fundamental method of mathematics. At first Lobachevsky's "imaginary'' (conceptual) geometry, like all "imaginary'' phenomena in mathematics, encountered distrust and hostility. Soon the irrefutable fact of the existence of this geometry began to change the point of view of mathematicians concerning the relation between mathematical theory and reality. The mathematician could not refuse Lobachevsky's geometry the right to exist, because this geometry was proved to be noncontradictory. It is true that Lobachevsky's geometry contradicted our geometrical intuition, but with a sufficiently small parameter of spatial curvature it was indistinguishable from Euclidean geometry in small spatial volumes. As for the cosmic scale, it is not at all obvious that we can trust our intuition there, because our intuition forms under the influence of experience limited to small volumes. Thus we face two competing geometries and the question arises: which of them is ''true''?
When we ponder this question it becomes clear that the word ''true'' is not placed in quotation marks without reason. Strictly speaking, the experiment cannot answer the question of the truth or falsehood of geometry: it can only answer the question of its usefulness or lack of usefulness, or more precisely its degree of usefulness, for there are perhaps no theories which are completely useless. The experiment deals with physical, not geometric, concepts. When we turn to the experiment we are forced to give some kind of interpretation to geometric objects, for example to consider that straight lines are realized by light beams. If we discover that the sum of the angles of a triangle formed by light beams is less than 180 degrees, this in no way means that Euclidean geometry is "false.'' Possibly it is "true," but the light is propagated not along straight lines but along arcs of circumference or some other curved lines. To speak more precisely, this experiment will demonstrate that light beams cannot be considered as Euclidean straight lines. Euclidean geometry itself will not be refuted by this. The same thing applies, of course, to non-Euclidean geometry also. The experiment can answer the question of whether the light beam is an embodiment of the Euclidean straight line or the Lobachevsky straight line, and this of course is an important argument in choosing one geometry or the other as the basis for physical theories. But it does not take away the right to existence of the geometry which ''loses out.'' It may perhaps do better next time and prove very convenient for describing some other aspect of reality.
Such considerations led to a reevaluation of the relative importance of the nature of mathematical objects and their properties (including relations as properties of pairs, groups of three, and other such objects). Whereas formerly objects seemed to have independent, real existence while their properties appeared to be something secondary and derived from their nature, now it was the properties of the objects, fixed in axioms, which became the basis by which to define the specific nature of the given mathematical theory while the objects lost all specific characteristics and, in general, lost their ''nature,'' which is to say, the intuitive representations necessarily bound up with them. In axiomatic theory the object is something which satisfies the axioms. The axiomatic approach finally took root at the turn of the twentieth century. Of course, intuition continued to be important as the basic (and perhaps only) tool of mathematical creativity, but it came to be considered that the final result of creative work was the completely formalized axiomatic theory which could be interpreted to apply to other mathematical theories or to nonlinguistic reality.
THE FORMALIZATION of logic was begun (if we do not count Leibnitz's first attempts) in the mid-nineteenth century in the works of G. Boole (1815-1864) and was completed by the beginning of the twentieth century, primarily thanks to the work of Schroeder, C. S. Peirce, Frege, and Peano. The fundamental work of Russell and Whitehead, the Principia Mathematica, which came out in 1910, uses a formalized language which, disregarding insignificant variations, is still the generally accepted one today. We described this language in chapter 6, and now we shall give a short outline of the formalization of logical deduction.
There are several formal systems of logical deduction which are equivalent to one another. We shall discuss the most compact one. It uses just one logical connective, implication É, and one quantifier, the universal quantifier ". But then it includes a logical constant which is represented by the symbol 0 and denotes an identically false statement. Using this constant it is possible to write the negation of statement p as p É 0, and from negation and implication it is easy to construct the other logical connectives. The quantifier of existence is expressed through negation and the quantifier of generality, so our compressed language is equivalent to the full language considered in chapter 6. The formal system (language machine) contains five axioms and two rules of inference. The axioms are the following:
In this p, q, and r are any propositions; in A4 and A5 the entry q(r) means that one of the free variables on which proposition q depends has been isolated: the entry q(t) means that some term t has been substituted for this variable: finally, in A4 it is assumed that variable r does not enter p as a free variable.
It is easy to ascertain that these axioms correspond to our intuition. Axioms A1-A3 involve only propositional calculus and their truth can be tested by the truth tables of logical connectives. It turns out that they are always true, regardless of the truth values assumed by propositions p, q, and r. A4 says that if q(r) follows for any r from proposition p which does not depend on r, the truth of q(r) for any r follows from p. A5 is in fact a definition of the universal quantifier: if q(r) is true for all r, then it is also true for any t.
The rules of inference may be written concisely in the following way:
In this notation the premises are above the line and the conclusion is below. The first rule (which traditionally bears the Latin name modus ponens) says that if there are two premises, proposition p and a proposition which affirms that q follows from p, then we deduce proposition p as the conclusion. The second rule, the rule of generalization, is based on the idea that if it has been possible to prove a certain proposition p(x), which contains free variable r, it may be concluded that the proposition will be true for any value of this variable.
The finite sequence of formulas D = (d_{1}, d_{2}, . . . , d_{n}) such that d_{n }coincides with q and each formula d_{n} is either a formula from a set of premises X, a logical axiom, or a conclusion obtained according to the rules of inference from the preceding formulas d_{j }is called the logical deduction of formula q from the set of formulas (premises) X. When we consider axiomatic theory, the aggregate of all axioms of the given theory figures as the set X and the logical deduction of a certain formula is its proof.
Thus, the formula's proof itself became a formal object, a definite type of formula (sequence of logical statements) and as a result the possibility of purely syntactical investigation of proofs as characteristics of a certain language machine. This possibility was pointed out by the greatest mathematician of the twentieth century, David Hilbert (1862-1943), who with his students laid the foundations of the new school. Hilbert introduced the concept of the metalanguage and called the new school metamathematics. The term metasystem which we introduced at the start of the book (and which is now generally accepted) arose as a result of generalizing Hilbert's terminology. Indeed, the transition to investigating mathematical proofs by mathematical means is a brilliant example of a large-scale metasystem transition.
The basic goal pursued by the program outlined by Hilbert was to prove that different systems of axioms were consistent (noncontradictory). A system of axioms is called contradictory if it is possible to deduce from it a certain formula q and its negation --q. It is easy to show that if there is at least one such formula, that is to say if the theory is contradictory, then any formula can be deduced from it. For an axiomatic theory, therefore, the question of the consistency of the system of axioms on which it is based is extremely important. This question admits a purely syntactical treatment: is it possible from the given formulas (strings of characters), following the given formal rules, to obtain a given formal result? This is the formulation of the question from which Hilbert began: it then turned out that there are also other important characteristics of theories which can be investigated by syntactical methods. Many very interesting and important results, primarily of a negative nature, were obtained in this way.
THE CONCEPT of the aggregate or set is one of the most fundamental concepts given to us by nature: it precedes the concept of number. In its primary form it is not differentiated into the concepts of the finite and infinite sets, but this differentiation appears very early: in any case, in very ancient written documents we can already find the concept of infinity and the infinite set. This concept was used in mathematics from ancient times on, remaining purely intuitive, taken as self-explanatory and not subject to special consideration, until Geory Cantor (1845- 1918) developed his theory of sets in the 1870s. It soon became the basis of all mathematics. In Cantor the concept of the set (finite or infinite) continues to be intuitive. He defines it as follows: ''By a set we mean the joining into a single whole of objects which are clearly distinguishable by our intuition or thought.'' Of course, this "definition" is no more mathematical than Euclid's ''definition'' that "The point is that which does not have parts.'' But despite such imprecise starting points, Cantor (once again, like the Greek geometers) created a harmonious and logically consistent theory with which he was able to put the basic concepts and proofs of mathematical analysis into remarkable order. (''It is simply amazing,'' writes Bourbaki, ''what clarity is gradually acquired in his writing by concepts which, it seemed, were hopelessly confused in the classical conception of the "continuum.")[4] In set theory mathematicians received a uniform method of creating new concept-constructs and obtaining proofs of their properties. For example, the real number is the set of all sequences of rational numbers which have a common limit: the line segment is a set of real numbers: the function is the set of pairs (x, f) where x and f are real numbers.
By the end of the nineteenth century Cantor's set theory had become recognized and was naturally combined with the axiomatic method. But then the famous ''crisis of the foundations'' of mathematics burst forth and continued for three decades. ''Paradoxes,'' which is to say constructions leading to contradiction, were found in set theory. The first paradox was discovered by Burali-Forti in 1897 and several others appeared later. As an example we will give Russell's paradox (1905), which can be presented using only the primary concepts of set theory and at the same time not violating the requirements of mathematical strictness. This is the paradox. Let us define M as the set of all those sets which do not contain themselves as an element. It would seem that this is an entirely proper definition because the formation of sets from sets is one of the bases of Cantor's theory. However, it leads to a contradiction. In order to make this clearer we shall use P(x) to signify the property of set X of being an element of itself. In symbolic form this will be
P(x) <=> x x (1)
Then, according to the definition of set M, all its elements X gave the property which is the opposite of P(x):
x M <=> -P(x) (2)
Then we put the question: is set M itself an element, that is, is P(M) true? If P(M) is true, then M M according to definition (1). But in this case, substituting M for X in proposition (2) we receive -P(x) , for if M is included in set M, then according to the definition of the latter it should not have property P. On the other hand, if P(M) is false, then -P(M) occurs; then according to (2) M should be included in M, that is, P(M) is true. Thus, P(M) cannot be either true or false. From the point of view of formal logic we have proved two implications:
P(M) => -P(M)
-P(M) É =>P(M)
If the implication is expressed through negation and disjunction and we use the property of disjunction AVA = A, the first statement will become -P(M) while the second will become P(M). Therefore, a formal contradiction takes place and therefore anything you like may be deduced from set theory!
The paradoxes threatened set theory and the mathematical analysis based on it. Several philosophical-mathematical schools emerged which proposed different ways out of this blind alley. The most radical school was headed by Brouwer and came to be called intuitionism; this school demanded not only a complete rejection of Cantor's set theory, but also a radical revision of logic. Intuitionist mathematics proved quite complex and difficult to develop, and because it threw classical analysis onto the scrap heap most mathematicians found this position unacceptable. ''No one can drive us from the heaven which Cantor created for us,'' Hilbert announced, and he found a solution which kept the basic content of set theory and at the same time eliminated the paradoxes and contradictions. With his followers Hilbert formulated the main channel along which the current of mathematical thought flowed.
Hilbert's solution corresponds entirely to the spirit of development of European mathematics. Whereas Cantor viewed his theory from a profoundly Platonist standpoint, as an investigation of the attributes of really existing and actually infinite sets, according to Hilbert the sets must be viewed as simply certain objects that satisfy axioms, while the axioms must be formulated so that definitions leading to paradoxes become impossible. The first system of set theory axioms which did not give rise to contradictions was proposed in 1908 by Zermelo and later modified. Other systems were also proposed, but the attitude toward set theory remained unchanged. In modern mathematics set theory plays the role of the frame, the skeleton which joins all its parts into a single whole but cannot be seen from the outside and does not come in direct contact with the external world. This situation can be truly understood and the formal and contentual aspects of mathematics combined only from the "linguistic" point of view regarding mathematics. This point of view, which we have followed persistently throughout this book, leads to the following conception. There are no actually infinite sets in reality or in our imagination. The only thing we can find in our imagination is the notion of potential infinity--that is, the possibility of repeating a certain act without limitation. Here we must agree fully with the intuitionist criticism of Cantor's set theory and give due credit to its insight and profundity. To use set theory in the way it is used by modern mathematics, however, it is not at all necessary to force one's imagination and try to picture actual infinity. The "sets'' which are used in mathematics are simply symbols, linguistic objects used to construct models of reality. The postulated attributes of these objects correspond partially to intuitive concepts of aggregateness and potential infinity; therefore intuition helps to some extent in the development of set theory, but sometimes it also deceives. Each new mathematical (linguistic) object is defined as a ''set'' constructed in some particular way. This definition has no significance for relating the object to the external world, that is for interpreting it: it is needed only to coordinate it with the frame of mathematics, to mesh the internal wheels of mathematical models. So the language of set theory is in fact a metalanguage in relation to the language of contentual mathematics, and in this respect it is similar to the language of logic. If logic is the theory of proving mathematical statements, then set theory is the theory of constructing mathematical linguistic objects.
Precisely why did the intuitive concept of the set form the basis of mathematical construction? To define a newly introduced mathematical object means to point out its semantic ties with objects introduced before. With the exception of the trivial case where we are talking about redesignation, replacing a sign with a sign, there are always many such ties, and many objects introduced earlier can participate in them. And so, instead of saying that the new object is related in such-and-such ways to such-and-such old objects, it is said that the new object is a set constructed of the old objects in such-and-such a manner. For example, a rational number is the result of dividing two natural numbers: the numerator by the denominator. The number 5/7 is object X such that the value of the function "numerator" (X) is 5 and the value of the function ''denominator'' (X) is 7. In mathematics, however, the rational number is defined simply as a pair of natural numbers. In exactly the same way it would be necessary to speak only of the realization of a real number by different sequences of rational numbers, understanding this to mean a definite semantic relation between the new and old linguistic objects. Instead of this, it is said that the real number is a set of sequences of rational numbers. At the present time the terminology should be considered a vestige of Platonic views according to which what is important is not the linguistic objects but the elements of ''ideal reality" concealed behind them, and therefore an object must be defined as a "real'' set to acquire the right to exist. The idea of the set was promoted to "executive work" in mathematics as one of the aspects of the relation of name and meaning (specifically, that the meaning is usually a construction which includes a number of elements), and it is hardly necessary to prove that the relation of name and meaning always has been and always will be the basis of linguistic construction.
AT THE CONCLUSION of this chapter we cannot help saying a few words about Bourbaki's multivolume treatise entitled Eléments de mathematique. Nicholas Bourbaki is a collective pseudonym used by a group of prominent mathematicians, primarily French, who joined together in the 1930s. Eléments de mathematique started publication in 1939.
Specialists from different fields of mathematics joined together in the Bourbaki group on the basis of a conception of mathematics as a formalized language. The goal of the treatise was to present all the most important achievements of mathematics from this point of view and to represent mathematics as one formalized language. And although Bourbaki's treatise has been criticized by some mathematicians for various reasons, it is unquestionably an important milestone in the development of mathematics along the path of self-awareness.
Bourbaki's conception was set forth in layman's terms in the article ''The Architecture of Mathematics.'' At the start of the article the author asks: is mathematics turning into a tower of Babel, into an accumulation of isolated disciplines? Are we dealing with one mathematics or with several? The answer given to this question is as follows. Modern axiomatic mathematics is one formalized language that expresses abstract mathematical structures that are not distinct, independent objects but rather form a hierarchical system. By a ''structure'' Bourbaki means a certain number of relations among objects which possess definite properties. Leaving the objects completely undefined and formulating the properties of relations in the form of axioms and then extracting the consequences from them according to the rules of logical inference, we obtain an axiomatic theory of the given structure. Translated into our language, a structure is the semantic aspect of a mathematical model. Several types of fundamental generating structures may be identified. Among them are algebraic structures (which reflect the properties of the composition of objects), structures of order, and topological structures (properties related to the concepts of contiguity, limit, and continuity). In addition to the most general structure of the given type--that is, the structure with the smallest number of axioms--we find in each type of generating structure structures obtained by including additional axioms. Thus, group theory includes the theory of finite groups, the theory of abelian groups, and the theory of finite abelian groups. Combining generating structures produces complex structures such as, for example, topological algebra. In this way a hierarchy of structures emerges.
How is the axiomatic method employed in creative mathematics? This is where, Bourbaki writes, the axiomatic method is closest to the experimental method. Following Descartes, it "divides difficulties in order to resolve them better.'' In proofs of a complex theory it tries to break down the main groups of arguments involved and, taking them separately, deduce consequences from them (the dismemberment of models or structures, which we discussed above). Then, returning to the initial theory, it again combines the structures which have been identified beforehand and studies how they interact with one another. We conclude with this citation:
From the axiomatic point of view, mathematics appears thus as a storehouse of abstract forms--the mathematical structures: and it so happens--without our knowing why--that certain aspects of empirical reality fit themselves into these forms, as if through a kind of preadaptation. Of course, it cannot be denied that most of these forms had originally a very definite intuitive content; but it is exactly by deliberately throwing out this content that it has been possible to give these forms all the power which they were capable of displaying and to prepare them for new interpretations and for the development of their full power.[5]
[1] Engels, F. Dialektika prirody (The Dialectic of Nature). Gospolitizdat Publishing House, 1955, p. 165.
[2] Bourbaki, N. Elements d'histoire des mathematiques. Paris: Hermann. The quote is from the first essay, in the session "Formalization of Logic."
[3] This opinion and the quotations cited above were taken from H. Weyl's book The Philo.sophy of Mathematics (Russian edition O filosofi matematiki. Moscow-Leningrad, 1934).
[4] Bourbaki, fisrt essay, section "Set Theory."
[5] Bourbaki, "The Architecture of Mathematiques."