Ergebnis für URL: http://pespmc1.vub.ac.be/POS/Turchap12.html#Heading8 This is chapter 12 of the [1]"The Phenomenon of Science" by [2]Valentin F.
Turchin
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Contents:
* [3]FORMALIZED LANGUAGE
* [4]THE LANGUAGE MACHINE
* [5]FOUR TYPES OF LINGUISTIC ACTIVITY
* [6]SCIENCE AND PHILOSOPHY
* [7]FORMALIZATION AND THE METASYSTEM TRANSITION
* [8]THE LEITMOTIF OF THE NEW MATHEMATICS
* [9]"NONEXISTENT" OBJECTS
* [10]THE HIERARCHY OF THEORIES
* [11]THE AXIOMATIC METHOD
* [12]METAMATHEMATICS
* [13]THE FORMALIZATION OF SET THEORY
* [14]BOURBAKI'S TREATISE
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CHAPTER TWELVE.
FROM DESCARTES TO BOURBAKI
FORMALIZED LANGUAGE
"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[15] 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.
__________________________________
Therefore, Vanya 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.
THE LANGUAGE MACHINE
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.
FOUR TYPES OF LINGUISTIC ACTIVITY
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.
SCIENCE AND PHILOSOPHY
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.[16][1]
That sounds amazingly modern!
FORMALIZATION AND THE METASYSTEM TRANSITION
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 LEITMOTIF OF THE NEW MATHEMATICS
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[17][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 .
"NONEXISTENT" OBJECTS
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.[18][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.
THE HIERARCHY OF THEORIES
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.
THE AXIOMATIC METHOD
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:
1. The point is that which does not have parts.
2. The line is a length without width.
3. The ends of lines are points.
4. A straight line is a line which lies the same relative to all its points.
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.
METAMATHEMATICS
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:
* A1. p É=> (q ÉÉ=> p)
* A2. [p É=>É (q ÉÉ=> r)] ÉÉ=> [(p ÉÉ=> q ) ÉÉ=> (p ÉÉ=> r)]
* A3. [(p É=>É 0) É=>É 0] É=>É p
* A4. ("x)[ p É=>Éq (x)] É=>É [p É=>É ("x)q (x)]
* A5. ("x)q (x) ÉÉ=> q (t)]
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:
[IMG.FIG12.0.GIF]
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 FORMALIZATION OF SET THEORY
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".)[19][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 [element.gif] 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 [element.gif] 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 [element.gif] 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.
BOURBAKI'S TREATISE
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.[20][5]
_________________________________________________________________________________
[21][1] Engels, F. Dialektika prirody (The Dialectic of Nature). Gospolitizdat
Publishing House, 1955, p. 165.
[22][2] Bourbaki, N. Elements d'histoire des mathematiques. Paris: Hermann. The
quote is from the first essay, in the session "Formalization of Logic".
[23][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).
[24][4] Bourbaki, fisrt essay, section "Set Theory".
[25][5] Bourbaki, "The Architecture of Mathematiques".
____________________________________________________________________________
References
1. http://pespmc1.vub.ac.be/POS/default.html
2. http://pespmc1.vub.ac.be/turchin.html
3. http://pespmc1.vub.ac.be/POS/Turchap12.html#Heading2
4. http://pespmc1.vub.ac.be/POS/Turchap12.html#Heading3
5. http://pespmc1.vub.ac.be/POS/Turchap12.html#Heading4
6. http://pespmc1.vub.ac.be/POS/Turchap12.html#Heading5
7. http://pespmc1.vub.ac.be/POS/Turchap12.html#Heading6
8. http://pespmc1.vub.ac.be/POS/Turchap12.html#Heading7
9. http://pespmc1.vub.ac.be/POS/Turchap12.html#Heading8
10. http://pespmc1.vub.ac.be/POS/Turchap12.html#Heading9
11. http://pespmc1.vub.ac.be/POS/Turchap12.html#Heading10
12. http://pespmc1.vub.ac.be/POS/Turchap12.html#Heading11
13. http://pespmc1.vub.ac.be/POS/Turchap12.html#Heading12
14. http://pespmc1.vub.ac.be/POS/Turchap12.html#Heading13
15. http://pespmc1.vub.ac.be/POS/turchap9.html#IMG.FIG9.5.GIF
16. http://pespmc1.vub.ac.be/POS/Turchap12.html#fn0
17. http://pespmc1.vub.ac.be/POS/Turchap12.html#fn1
18. http://pespmc1.vub.ac.be/POS/Turchap12.html#fn2
19. http://pespmc1.vub.ac.be/POS/Turchap12.html#fn3
20. http://pespmc1.vub.ac.be/POS/Turchap12.html#fn4
21. http://pespmc1.vub.ac.be/POS/Turchap12.html#fnB0
22. http://pespmc1.vub.ac.be/POS/Turchap12.html#fnB1
23. http://pespmc1.vub.ac.be/POS/Turchap12.html#fnB2
24. http://pespmc1.vub.ac.be/POS/Turchap12.html#fnB3
25. http://pespmc1.vub.ac.be/POS/Turchap12.html#fnB4
Usage: http://www.kk-software.de/kklynxview/get/URL
e.g. http://www.kk-software.de/kklynxview/get/http://www.kk-software.de
Errormessages are in German, sorry ;-)