Ergebnis für URL: http://pespmc1.vub.ac.be/POS/Turchap10.html#Heading10 This is chapter 10 of the [1]"The Phenomenon of Science" by [2]Valentin F.
Turchin
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Contents:
* [3]PROOF
* [4]THE CLASSICAL PERIOD
* [5]PLATO'S PHILOSOPHY
* [6]WHAT IS MATHEMATICS?
* [7]PRECISION IN COMPARING QUANTITIES
* [8]THE RELIABILITY OF MATHEMATICAL ASSERTIONS
* [9]IN SEARCH OF AXIOMS
* [10]CONCERNING THE AXIOMS OF ARITHMETIC AND LOGIC
* [11]PLATONISM IN RETROSPECT
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CHAPTER TEN.
From Thales to Euclid
PROOF
NEITHER IN Egyptian nor in Babylonian texts do we find anything even remotely
resembling mathematical proof. This concept was introduced by the Greeks, and is
their greatest contribution. It is obvious that some kind of guiding
considerations were employed earlier in obtaining new formulas. We have even
cited an example of a grossly incorrect formula (for the area of irregular
quadrangles among the Egyptians) which was plainly obtained from externally
plausible ''general considerations.'' But only the Greeks began to give these
guiding considerations the serious attention they deserved. The Greeks began to
analyze them from the point of view of how convincing they were, and they
introduced the principle according to which every proposition concerning
mathematical formulas, with the exception of just a small number of "completely
obvious'' basic truths, must be proved--derived from these ''perfectly obvious"
truths in a convincing manner admitting of no doubt. It is not surprising that
the Greeks, with their democratic social order, created the doctrine of
mathematical proof. Disputes and proofs played an important part in the life of
the citizens of the Greek city-state (polis). The concept of proof already
existed; it was a socially significant reality. All that remained was to transfer
it to the field of mathematics, which was done as soon as the Greeks became
acquainted with the achievements of the ancient Eastern civilizations. It must be
assumed that a certain part here was also played by the role of the Greeks as
young, curious students in relation to the Egyptians and Babylonians, their old
teachers who did not always agree with one another. In fact, the Babylonians
determined the area of a circle according to the formula 3r^2, while the
Egyptians used the formula (8/9 2r)^2. Where was the truth? This was something to
think about and debate.
The creators of Egyptian and Babylonian mathematics have remained anonymous. The
Greeks preserved the names of their wise men. The first, Thales of Miletus, is
also the first name included in the history of science. Thales lived in the sixth
century B.C. in the city of Miletus on the Asia Minor coast of the Aegean Sea.
One date in his life has been firmly established: in 585 B.C. he predicted a
solar eclipse--unquestionable evidence of Thales's familiarity with the culture
of the ancient civilizations, because the experience of tens and hundreds of
years is required to establish the periodicity of eclipses. Thales had no Greek
predecessors, and could therefore only have taken his knowledge of astronomy from
the scientists of the East. Thales, the Greeks assert, gave the world the first
mathematical proofs. Among the propositions (theorems) proved by him they mention
the following:
1 The diameter divides a circle into two equal parts.
2 The base angles of an isosceles triangle are equal.
3 Two triangles which have an identical side and identical angles adjacent to it
are equal.
In addition, Thales was the first to construct a circle circumscribed about a
right triangle (and it is said that he sacrificed an ox in honor of this
discovery).
The very simple nature of these three theorems and their intuitive obviousness
shows that Thales was entirely aware of the importance of proof as such. Plainly,
these theorems were proved not because there was doubt about their truth but in
order to make a beginning at systematically finding proof and developing a
technique for proof. With such a purpose it is natural to begin by proving the
simplest propositions.
Suppose triangle ABC is isosceles, which is to say side AB is equal to side BC.
[IMG.FIG10.1.GIF]
Figure 10.1. Isosceles triangle.
Let us divide angle ABC into two equal parts by line BD. Let us mentally fold our
drawing along line BD. Because angle ABD is equal to angle CBD, line BA will lie
on line BC, and because the length of the segments AB and BC is equal, point A
will lie on point C. Because point D remains in place, angles BCD and BAD must be
equal. Whereas formerly it only seemed to us that angles BCD and BAD were equal
(Thales probably spoke this way to his fellow citizens), we have now proved that
these angles necessarily and with absolute precision must be equal (the Greeks
said "similar'') to one another: that is, they match when one is placed on the
other.
The problem of construction is more complex and here the result is not at all
obvious beforehand. Let us draw a right triangle.
[IMG.FIG10.2.GIF]
Figure 10.2. Construction of a circle described around a right triangle
May a circle be drawn such that all three vertices of the triangle appear on it?
And if so, how'? It is not clear. But suppose that intuition suggests a solution
to us. We divide the hypotenuse BC into two equal segments at point D. We connect
it with point A. If segment AD is equal in magnitude to segment DC (and therefore
also to BD) we can easily draw the required circle by putting the point of a
compass at point D and taking segment DC as the radius. But is it true that AD
=DC, that is to say triangle ADC is an isosceles triangle? It is not clear. It
seems probable, but in any case it is far from obvious. Now we shall take the
crucial step. We shall add point E to our triangle, making rectangle ABEC and
draw in a second diagonal AE. Suddenly it becomes obvious that triangle ADC is
isosceles. Indeed, from the overall symmetry of the drawing it is clear that the
diagonals are equal and intersect at the point which divides them in half--at
point D. We have not yet arrived at proof, but we already are at that level of
clarity where formal completion of the proof presents no difficulty. For example,
relying on the equality of the opposite sides of the rectangle (which can be
derived from even more obvious propositions if we wish), we complete the proof by
the following reasoning: triangles ABC and AEC are equal because they have side
AC in common, sides AB and EC are equal, and angles BAC and ECA are right angles;
therefore angle EAC is equal to angle BCA. That is, triangle ADC is an isosceles
triangle, which is what had to be proved.
THE CLASSICAL PERIOD
SO, FROM a few additional points and lines on a drawing, a chain of logical
reasoning, and simple and obvious truths we receive truths which are by no means
simple and by no means obvious, but whose correctness no one can doubt for a
minute. This is worth sacrificing an ox to the gods for! One can imagine the
delight the Greeks experienced upon making such a discovery. They had struck a
vein of gold and they diligently began working it. In the time of Pythagoras (550
B.C.) the study of mathematics was already very widespread among people who had
leisure time and was considered a noble, honorable, and even sacred matter.
Advances and discoveries, each more marvelous than the one before, poured from
the horn of plenty.
The appearance of proof was a metasystem transition within language. The formula
was no longer the apex of linguistic activity. A new class of linguistic objects
appeared, proof, and there was a new type of linguistic activity directed to the
study and production of formulas. This was a new stage in the control hierarchy
and its appearance called forth enormous growth in the number of formulas (the
law of branching of the penultimate level).
The metasystem transition always means a qualitative leap forward--a flight to a
new step, swift, explosive development. The mathematics of the countries of the
Ancient East remained almost unchanged for up to two millennia, and a person of
our day reads about it with the condescension of an adult toward a child. But in
just one or two centuries the Greeks created all of the geometry our high school
students sweat over today. Even more, for the present-day geometry curriculum
covers only a part of the achievements of the initial, ''classical,'' period of
development of Greek mathematics and culture (to 330 B.C.). Here is a short
chronicle of the mathematics of the classical period.
585 B.C. Thales of Miletus. The first geometric theorems.
550 B.C. Pythagoras and his followers. Theory of numbers. Doctrine of harmony.
Construction of regular polyhedrons. Pythagorean theorem. Discovery of
incommensurable line segments. Geometric algebra. Geometric construction
equivalent to solving quadratic equations.
500 B.C. Hippasas, Pythagorean who was forced to break with his comrades because
he shared his knowledge and discoveries with outsiders (this was forbidden among
the Pythagoreans). Specifically, he gave away the construction of a sphere
circumscribed about a dodecahedron.
430 B.C. Hippocrates of Chios (not to be confused with the famous doctor
Hippocrates of Kos). He was considered the most famous geometer of his day. He
studied squaring the circle, making complex geometric constructions. He knew the
relationship between inscribed angles and arcs, the construction of a regular
hexagon, and a generalization of the Pythagorean theorem for obtuse- and
acute-angled triangles. Evidently, he considered all these things elementary
truths. He could square any polygon, that is, construct a square of equal area
for it.
427-348 B.C. Plato. Although Plato himself did not obtain new mathematical
results, he knew mathematics and it sometimes played an important part in his
philosophy--just as his philosophy played an important part in mathematics. The
major mathematicians of his time, such as Archytas, Theaetetus, Eudoxus, were
Plato's friends; they were his students in the field of philosophy and his
teachers in the field of mathematics.
390 B.C. Archytas of Tarentum. Stereometric solution to the problem of doubling
the cube--that is, constructing a cube with a volume equal to twice the volume of
a given cube.
370 B.C. Eudoxus of Cnidus. Elegant, logically irreproachable theory of
proportions closely approaching the modern theory of the real number. The
''exhaustion method,'' which forms the basis of the modern concept of the
integral.
384-322 B.C. Aristotle. He marked the beginning of logic and physics. Aristotle's
works reveal a complete mastery of the mathematical method and a knowledge of
mathematics, although he, like his teacher Plato, made no mathematical
discoveries. Aristotle the philosopher is inconceivable without Aristotle the
mathematician.
300 B.C. Euclid. Euclid lived in a new and different age, the Alexandrian Epoch.
In his famous Elements Euclid collected and systematized all the most important
works on mathematics which existed at the end of the fourth century B.C. and
presented them in the spirit of the Platonic school. For more than 2,000 years
school courses in geometry have followed Euclid's Elements to some extent.
PLATO'S PHILOSOPHY
WHAT IS MATHEMATICS? What does this science deal with ? These questions were
raised by the Greeks after they had begun to construct the edifice of mathematics
on the basis of proofs, for the aura of absolute validity, of virtual sanctity,
which mathematical knowledge acquired thanks to the existence of the proofs
immediately made it stand out against the background of other everyday knowledge.
The answer was given by the Platonic theory of ideas. This theory formed the
basis of all Greek philosophy, defined the style and way of thinking of educated
Greeks, and exerted an enormous influence on the subsequent development of
philosophy and science in the Greco-Roman-European culture.
It is not difficult to establish the logic which led Plato to his theory. What
does mathematics talk about'? About points, lines, right triangles, and the like.
But are there in nature points which do not have dimensions? Or absolutely
straight and infinitely fine lines? Or exactly equal line segments, angles, or
areas? It is plain that there are not. So mathematics studies nonexistent,
imagined things; it is a science about nothing. But this is completely
unacceptable. In the first place, mathematics has unquestionably produced
practical benefits. Of course, Plato and his followers despised practical
affairs, but this was a logical result of philosophy, not a premise. In the
second place, any person who studies mathematics senses very clearly that he is
dealing with reality, not with fiction, and this sensation cannot be rooted out
by any logical arguments. Therefore, the objects of mathematics really exist but
not as material objects, rather as images or ideas, because in Greek the word
"idea" in fact meant "image'' or "form.''[12][1] Ideas exist outside the world of
material things and independent of it. Material things perceived by the senses
are only incomplete and temporary copies (or shadows) of perfect and eternal
ideas. The assertion of the real, objective existence of a world of ideas is the
essence of Plato's teaching (''Platonism'').
For many centuries hopelessly irresolvable disputes arose among the Platonists
over attempts to in some way give concrete form to the notion of the world of
ideas and its interaction with the material world. Plato himself wisely remained
invulnerable, avoiding specific, concrete terms and using a metaphorical and
poetic language. But he did have to enter a polemic with his student Eudoxus, who
not only proved mathematical theorems but also defended trading in olive oil.
Such a position of course restricted the influx of new problems and ideas and
fostered a canonization and regimentation of scientific thought, thus retarding
its development. But beyond this, Platonism also had a more concrete negative
effect on mathematics. It prevented the Greeks from creating algebraic language.
This could be done only by the less educated and more practical Europeans. Later
on we shall consider in more detail the history of the creation of modern
algebraic language and the inhibiting role of Platonism, but first we shall
discuss the answers given by modern science to the questions posed in Platonic
times and how the answers given by Plato look in historical retrospect.
WHAT IS MATHEMATICS?
FOR US MATHEMATICS is above all a language that makes it possible to create a
certain kind of models of reality: mathematical models. As in any other language
(or branch of language) the linguistic objects of mathematics, mathematical
objects, are material objects that fix definite functional units, mathematical
concepts. When we say that the objects ''fix functional units'' we take this to
mean that a person, using the discriminating capabilities of his brain, performs
certain linguistic actions on these objects or in relation to them. It is plain
that it is not the concrete form (shape, weight, smell) of the mathematical
object which is important in mathematics; it is the linguistic activity related
to it. Therefore the terms ''mathematical object'' and ''mathematical concept''
are often used as synonyms. Linguistic activity in mathematics naturally breaks
into two parts: the establishment of a relationship between mathematical objects
and nonlinguistic reality (this activity defines the meanings of mathematical
concepts), and the formulations of conversions within the language, mathematical
calculations and proofs. Often only the second part is what we call
''mathematics'' while we consider the first as the ''application of
mathematics.''
Points, lines, right triangles, and the like are all mathematical objects. They
make up our geometric drawings or stereometric models: spots of color, balls of
modeling clay, wires, pieces of cardboard, and the like. The meanings of these
objects are known. The point, for example, is an object whose dimensions and
shape may be neglected. Thus the ''point'' is simply an abstract concept which
characterizes the relation of an object to its surroundings. In some cases we
view our planet as a point. But when we construct a geometric model we usually
make a small spot of color on the paper and say, ''Let point A be given.'' This
spot of color is in fact linguistic object L[i], and the planet Earth may be the
corresponding object (referent) R[j]. There are no other true or ideal'' points,
that is, without dimensions. It is often said that there are no ''true'' points
in nature, but that they exist only in our imagination. This commonplace
statement is either absolutely meaningless or false, depending on how it is
interpreted. In any case it is harmful, because it obscures the essence of the
matter. There are no "true'' points in our imagination and there cannot be any.
When we say that we are picturing a point we are simply picturing a very small
object. Only that which can be made up of the data of sensory experience can be
imagined, and by no means all of that. The number 1,000 for example, cannot be
imagined, large numbers, ideal points, and lines exist not in our imagination,
but in our language, as linguistic objects we handle in a certain way. The rules
for handling them reveal the essence of mathematical concepts, specifically the
''ideality of the point.'' The dimensions of points on a drawing do not influence
the development of the proof, and if two points must be set so close that they
merge into one, we can increase the scale.
But aren't the assertions of mathematics characterized by absolute prehave an
entirely different status. By itself this language is, of course, discrete also,
but empirical assertions reflect semantic conversions L[1]->S[1] leading us into
the area of nonlinguistic activity which is neither discrete nor deterministic.
When we say that two rods have equal length this means that every time we measure
them the result will be the same. Experience, however, teaches us that if we can
increase the precision of measurement without restriction, sooner or later we
shall certainly obtain different values for the length, because an empirical
assertion of absolutely exact equality is completely senseless. Other assertions
of empirical language which have meaning and can be expressed in the language of
predicate calculus, for example ''rod no. 1 is smaller than rod no. 2," possess
the same ''absolute precision'' (which is a trivial consequence of the discrete
nature of the language) as mathematical assertions of the equality of segments.
This assertion is either ''exactly'' true or "exactly'' false. Because of
variations in the measuring process, however, neither is absolutely reliable.
PRECISION IN COMPARING QUANTITIES
NOW LET US DISCUSS the reliability of mathematical assertions. Plato deduced it
from the ideal nature of the object of mathematics, from the fact that
mathematics does not rely on the illusory and changing data of sensory
experience. According to the mathematician, drawings and symbols are nothing but
a subsidiary means for mathematics; the real objects Plato deals with are
contained in his imagination and represent the result of perception of the world
of ideas through reason, just as sensory experience is the result of perception
of the material world through the sense organs. Imagination obviously plays a
crucial part in the work of the mathematician (as it does, we might note, in all
other areas of creative activity). But it is not entirely correct to say that
mathematical objects are contained in the imagination: basically they are still
contained in drawings and texts, and the imagination takes them up only in small
parts. Rather than holding mathematical objects in our imagination we pass them
through and the characteristics of our imagination determine the functioning of
mathematical language. As for the source which determines the content of our
imagination, here we disagree fundamentally with Plato. The source is the same
sensory experience used in the empirical sciences. Therefore, even though it uses
the mediation of imagination, mathematics creates models of the very same. unique
(as far as we know) world we live in.
However, although they constructed a stunningly beautiful edifice of logically
strict proofs, the Greek mathematicians nonetheless left a number of gaps in the
structure; and these gaps, as we have already noticed, lie on the lowest stories
of the edifice--in the area of definitions and the most elementary properties of
the geometric figures. And this is evidence of a veiled reference to the sensory
experience so despised by the Platonists. The mathematics of Plato's times
provides even clearer material than does present-day mathematics to refute the
thesis that mathematics is independent of experience.
The first statement proved in Euclid's first book contains a method of
constructing an equilateral triangle according to a given side. The method is as
follows.
[IMG.FIG10.3.GIF]
Figure 10.3. Construction of an equilateral triangle.
Suppose AB is the given side of the triangle. Taking point A as the center we
describe circle [pi][A] with radius AB. We describe a similar circle
([pi][[Beta]]) from point B We use C to designate either of the points of
intersection of these circles . Triangle ABC is equilateral, for AC = CB = AB.
There is a logical hole in this reasoning: how does it follow that the circles
constructed by us will intersect at all'? This is a question fraught with
complications, for the fact that point of intersection C exists cannot be related
either to the attributes of a circle or even to the attributes of a pair of
circles (for they by no means always intersect). We are dealing here with a more
specific characteristic of the given situation. Euclid probably sensed the
existence of a hole here, but he could not find anything to plug it with.
But how are we certain that circles [pi][A] and [pi][B] intersect? In the last
analysis, needless to say, we know from experience. From experience in
contemplating and drawing straight lines, circles, and lines in general, from
unsuccessful attempts to draw circles [pi][A] and [pi][B ]so that they do not
intersect.
So Plato's view that the mathematics of his day was entirely independent of
experience cannot be considered sound. But the question of the nature of
mathematical reliability requires further investigation, for to simply make
reference to experience and thus equate mathematical reliability with empirical
reliability would mean to rush to the opposite extreme from Platonism. Certainly,
we feel clearly that mathematical reliability is somehow different from empirical
reliability, but how'?
The assertion that circles of radius AB with centers of A and B intersect (for
brevity we shall designate this assertion E[1] ) seems to us almost if not
completely reliable; we simply cannot imagine that they would not intersect. We
cannot imagine.... This is how mathematical reliability differs from the
empirical! When we are talking about the sun rising tomorrow, we can imagine that
the sun will not rise and it is only on the basis of experience that we believe
that it probably will rise. Here there are two possibilities and the prediction
as to which one will happen is probabilistic. But when we say that two times two
is four and that circles constructed as indicated above inter
sect we cannot imagine that it could be otherwise. We see no other possibility,
and therefore these assertions are perceived as absolutely reliable and
independent of concrete facts we have observed.
THE RELIABILITY OF MATHEMATICAL ASSERTIONS
IT IS VERY INSTRUCTIVE for an understanding of the nature of mathematical
reliability to carry our analysis of the assertion E[1] through to the end.
Because we still have certain doubts that the circles in figure 10.3 necessarily
intersect, let us attempt to picture a situation where they do not. If this
attempt fails completely it will mean that assertion E[1] is mathematically
reliable and cannot be broken down into simpler assertions: then it should be
adopted as an axiom. But if through greater or lesser effort of imagination we
are able to picture a situation in which [pi][A] and [pi][B ]do not intersect, it
must be expected that this situation contradicts some simpler and deeper
assertions which do possess mathematical reliability. Then we shall adopt them as
axioms and the existence of the contradiction will serve as proof of E[1]. This
is the usual way to establish axioms in mathematics.
First let us draw circle [pi][A]. Then we shall put the point of the compass at
point B and the writing element at point A and begin to draw circle [pi][B]. We
shall move from the center of circle [pi][A] toward its periphery and at a
certain moment (this is how we picture it in our imagination) we must either
intersect circle [pi][A] or somehow skip over it, thus breaking circle [pi][B].
[IMG.FIG10.4.GIF]
Figure 10.4.
But we imagine circle p[B] as a continuous line and it becomes clear to us that
the attributes of continuousness, which are more fundamental and general than the
other features of this problem, lie at the basis of our confidence that circles
[pi][A] and [pi][[Beta]] will intersect. Therefore we set as our goal proving
assertion E[1] beginning with the attributes of continuousness of the circle. For
this we shall need certain considerations related to the order of placement of
points on a straight line. We include the concepts of continuousness and order
among the basic. undefined concepts of geometry, like the concepts of the point,
the straight line, or distance.
Here is one possible way to our goal. We introduce the concept of ''inside''
(applicable to a circle) by means of the following definition:
D[1]: It is said that point A lies inside circle [pi] if it does not lie on [pi]
and any straight line passing through point A intersects [pi] at two points in
such a way that point A lies between the points of intersection. If the point is
neither on nor inside the circle it is said that it lies outside the circle.
The concept of ''between'' characterizes the order of placement of three points
on a straight line. It may be adopted as basic and expressed, through the more
,general concept of ''order,'' by the following definition:
D[2]: It is said that point A is located between points B[1] and B[2] if these
three points are set on one straight line and during movement along this line
they are encountered in the order B[1], A, and B[2] or B[2], A, and B[1].
We shall adopt the following propositions as axioms:
A[1]: The center of a circle lies inside it.
A[2]: The arc of a circle connecting any two points of the circle is continuous.
A[3]: If point A lies inside circle [pi] and point B is outside it, and these two
points are joined by a continuous line, then there is a point where this line
intersects the circle.
Relying on these axioms, let us begin with the proof. According to the statement
of the problem, circle [pi][B] passes through center A of circle [pi][A]. If we
have confidence that there is at least one point of circle [pi][[Beta]] that does
not lie inside [pi][A] we shall prove E[1]. Indeed, if it lies on [pi][A] then
E[1] has been proved. If it lies outside [pi][A] then the arc of circle [pi][B]
connects it with the center, that is, with an inside point of circle [pi][A].
Therefore, according to axioms A[2] and A[3] there is a point of intersection of
[pi][B ]and [pi][A].
But can we be confident that there is a point on circle [pi][B] which is outside
[pi][A]? Let us try to imagine the opposite case. It is shown in figure 10.5.
[IMG.FIG10.5.GIF]
This is the second attempt to imagine a situation which contradicts the assertion
being proved. Whereas the first attempt immediately came into explicit
contradiction with the continuousness of a circle, the second is more successful.
Indeed, stretching things a bit we can picture this case. We take a compass, put
its point at point B and the pencil at point A. We begin to draw the circle
without taking the pencil from the paper and when the pencil returns to the
starting point of the line we remove it and see that we have figure 10.5. And why
not?
To prove that this is impossible we must prove that in this case the center of
circle [pi][B] is necessarily outside it. We shall be helped in this by the
following theorem:
T[1]: If circle [pi][1], lies entirely inside circle [pi][2] then every inside
point of circle [pi][1 ]is also an inside point of circle [pi][2].
To prove this we shall take an arbitrary inside point A of circle [pi][1, ]which
is shown in figure 10.6.
[IMG.FIG10.6.GIF]
We draw a straight line through it. According to definition D[1] it intersects
[pi][1], at two points: B[1] and B[2] Because B[1 ](just as B[2]) lies inside
[pi][2 ]this straight line also intersects [pi][2] at two points: C[1 ]and C[2].
We have received five points on a straight line and they are connected by the
following relationships of order: A lies between B[1 ]and B[2]; B[1 ]and B[2 ]lie
between C[1 ]and C[2]. That point A proves to be between points C[1 ]and C[2] in
this situation seems so obvious to us that we shall boldly formulate it as still
another axiom.
A[3]: If points B[1] and B[2 ]on a straight line both lie between C[1]andC[2],
then any point A lying between B[1] and B[2 ]also lies between C[1 ]andC[2].
Because we can take any point inside [pi][1] as A and we can draw any straight
line through it, theorem T[1] is proven.
Now it is easy to complete the proof of E[1]. If circle [pi][B] lies entirely
inside [pi][A] then according to theorem T[1] its center B must also lie inside
[pi][A]. But according to the statement of the problem point B is located on
[pi][A]. Therefore [pi][B ]contains at least one point which is not inside in
relation to [pi][A]
So to prove one assertion E[1] we needed four assertions (axioms A[1]-A[4]), but
then these assertions express very fundamental and general models of reality
related to the concepts of continuousness and order and we cannot even imagine
that they are false. The only question that can be raised refers to axiom A[1]
which links the concept of center. which is metrical (that is, including the
concept of measurement) in nature, with the concept of ''inside," which relies
exclusively on the concepts of continuousness and order. It may be desired that
this connection be made using simpler geometric objects, under conditions which
are easier for the functioning of imagination. This desire is easily met. For
axiom A[1] let us substitute the following axiom:
A[1]': if on a straight line point A and a certain distance (segment) R are
given, then there are exactly two points on the straight line which are set at
distance R from point A, and point A lies between these two points.
Relying on this axiom we shall prove assertion A[1] as a theorem. We shall draw
an arbitrary straight line through the center of the circle. According to axiom
A[1]' there will be two points on it which are set at distance R (radius of the
circle) from the center. Because a circle is defined as the set of all points
which are located at distance R from the center, these points belong to the
circle. According to axiom A[1]' the center point lies between them and
therefore, according to definition D[1], it is an inside point. In this way axiom
A[1] has been reduced to axiom A[1]'. Now try to imagine a point on a straight
line which does not have two points set on different sides from it at the given
distance!
IN SEARCH OF AXIOMS
THE PRIMARY PROPOSITIONS of arithmetic in principle possess the same nature as
the primary propositions of geometry, but they are perhaps even simpler and more
obvious and denial of them is even more inconceivable than denial of geometric
axioms. As an example let us take the axiom which says that for any number
a + 0 =a
The number O depicts an empty set. Can you imagine that the number of elements in
some certain set would change if it were united with an empty set? Here is
another arithmetic axiom: for any numbers a and b
a+ (b+ 1) = (a+b) + 1,
that is, if we increase the number b by one and add the result to a, we shall
obtain the same number as if we were to add a and b first and then increase the
result by one. If we analyze why we are unable to imagine a situation that
contradicts this assertion, we shall see that it is a matter of the same
considerations of continuousness that also manifest themselves in geometric
axioms. In the process of counting, it is as if we draw continuous lines
connecting the objects being counted with the elements of a standard set and, of
course, lines in time (let us recall the origin of the concept ''object'') whose
continuousness ensures that the number is identical to itself.
Natural auditory language transferred to paper gives rise to linear language,
that is, a system whose subsystems are all linear sequences of signs. Signs are
objects concerning which it is assumed only that we are able to distinguish
identical ones from different ones. The linearity of natural languages is a
result of the fact that auditory language unfolds in time and the relation of
following in time can be modeled easily by the relation of order of placement on
a timeline. The specialization of natural language led to the creation of the
linear, symbolic mathematical language which now forms the basis of mathematics.
Operating within the framework of linear symbolic languages we are constantly
taking advantage of certain other attributes which seem so obvious and
self-evident that we don't even want to formulate them in the form of axioms. As
an example let us take this assertion: if symbol a is written to the left of
symbol b and symbol c is written on the right the same word (sequence of
characters) will be received as when b is written to the right of a and followed
by c. This assertion and others like it possess mathematical reliability for we
cannot imagine that it would be otherwise. One of the fields of modern
mathematics, the theory of semi-groups, studies the properties of linear symbolic
systems from an axiomatic point of view and declares the simplest of these
properties to be axioms.
All three kinds of axioms, geometric, arithmetic, and linear-symbolic, possess
the same nature and in actuality rely on the same fundamental concepts. concepts
such as identity, motion, continuousness, and order. There is no difference in
principle among these groups of axioms. And if one term were to be selected for
them they should be called geometric or geometric-kinematic because they all
reflect the attributes of our space-time experience and space-time imagination.
The only more or less significant difference which can be found is in the group
of "properly geometric'' axioms; some of the axioms concerning straight lines and
planes reflect more specific experience related to the existence of solid bodies.
The same thing evidently applies to metric concepts. But this difference too is
quite arbitrary. Can we say anything serious about those concepts which we would
have if there were no solid bodies in the world?
Thus far we have been discussing the absolute reliability of axioms. But where do
we get our confidence in the reliability of assertions obtained by logical
deduction from axioms? From the same source, our imagination refuses to permit a
situation in which by logical deduction we obtain incorrect results from correct
premises. Logical deduction consists of successive steps. At each step, relying
on the preceding proposition. we obtain a new one. From a review of formal
logical deduction (chapter 11) it will be seen that our confidence that at every
step we can only receive a true proposition from higher true propositions is
based on logical axioms [13][2] which seem to us just as reliable as the
mathematical axioms considered above. And this is for the same reason, that the
opposite situation is absolutely inconceivable. Having this confidence we acquire
confidence that no matter how many steps a logical deduction may contain it will
still possess this attribute. Here we are using the following very important
axiom:
The axiom of induction: Let us suppose that function f (x) leaves attribute P (x)
unchanged, that is
( [forall.GIF] x){P (x) =>P [f (x)]}
We will use f ^n(x) to signify the result of sequential n-time application of
function f (x), that is
f ^1(x) =f (x), f ^n(x) = f [f^n-1 (x)].
Then f ^n(x) will also leave attribute P (x) unchanged for any n, that is
( [forall.GIF] n)( [forall.GIF] x){P (x) => P [f ^n(x)]}
By their origin and nature logical axioms and the axiom of induction (which is
classed with arithmetic because it includes the concept of number) do not differ
in any way from the other axioms; they are all mathematical axioms. The only
difference is in how they are used. When mathematical axioms are applied to
mathematical assertions they become elements of a metasystem within the framework
of a system of mathematically reliable assertions and we call them logical
axioms. Thanks to this, the system of mathematically reliable assertions becomes
capable of development. The great discovery of the Greeks was that it is possible
to add one certainty to another certainty and thus obtain a new certainty.
DEEP-SEATED PILINGS
THE DESCRIPTION of mathematical axioms as models of reality which are true not
only in the sphere of real experience but also in the sphere of imagination
relies on their subjective perception. Can it be given a more objective
characterization?
Imagination emerges in a certain stage of development of the nervous system as
arbitrary associating of representations. The preceding stage was the stage of
nonarbitrary associating (the level of the dog). It is natural to assume that the
transition from nonarbitrary to arbitrary associating did not produce a
fundamental change in the material at the disposal of the associating system,
that is, in the representations which form the associations. This follows from
the hierarchical principle of the organization and development of the nervous
system in which the superstructure of the top layers has a weak influence on the
lower ones And it follows from the same principle that m the process of the
preceding transition, from fixed concepts to nonarbitrary associating, the lowest
levels of the system of concepts remained unchanged and conditioned those
universal, deep-seated properties of representations that were present before
associating and that associating could not change. Imagination cannot change them
either. These properties are invariant in relation to the transformations made by
imagination. And they are what mathematical axioms rely on.
If we picture the activity of the imagination as shuffling and fixing certain
elements. ''pieces'' of sensory perception. then axioms are models which are true
for any piece and. therefore, for any combination of them. The ability of the
imagination to break sensory experience up into pieces is not unlimited; emerging
at a certain stage of development it takes the already existing system of
concepts as its background, as a foundation not subject to modification. Such
profound concepts as motion, identity, and continuousness were part of this
background and therefore the models which rely on these concepts are universally
true not only for real experience but also for any construction the imagination
is capable of creating.
Mathematics forms the frame of the edifice of natural sciences. Its axioms are
the support piles that drive deep into the neuronal concepts, below the level
where imagination begins to rule. This is the reason for the stability of
foundation which distinguishes mathematics from empirical knowledge. Mathematics
ignores the superficial associations which make up our everyday experience,
preferring to continue constructing the skeleton of the system of concepts which
was begun by nature and set at the lowest levels of the hierarchy. And this is
the skeleton on which the "noncompulsory'' models we class with the natural
sciences will form, just as the ''noncompulsory'' associations of representations
which make up the content of everyday experience form on the basis of inborn and
"compulsory" concepts of the lowest level. The requirements dictated by
mathematics are compulsory: when we are constructing models of reality we cannot
bypass them even if we want to. Therefore we always refer the possible falsehood
of a theory beyond the sphere of mathematics. If a discrepancy is found between
the theory and the experiment it is the external, "noncompulsory" part of the
theory that is changed but no one would ever think of expressing the assumption
that, in such a case, the equality 2 + 2 = 4 has proved untrue.
The ''compulsory" character of classical mathematical models does not contradict
the appearance of mathematical and physical theories which at first glance
conflict with our space-time intuition (for example, non-Euclidian geometry or
quantum mechanics). These theories are linguistic models of reality whose
usefulness is seen not in the sphere of everyday experience but in highly
specialized situations. They do not destroy and replace the classical models;
they continue them. Quantum mechanics, for example, relies on classical
mechanics. And what theory can get along without arithmetic? The paradoxes and
contradictions arise when we forget that the concept constructs which are
included in a new theory are new concepts, even when they are given old names. We
speak of a ''straight line'' in non-Euclidian geometry and call an electron a
''particle'' although the linguistic activity related to these words (proof of
theorems and quantum mechanics computations) is not at all identical to that for
the former theories from which the terms were borrowed. If two times two is not
four then either two is not two, times is not times, or four is not four.
The special role of mathematics in the process of cognition can be expressed in
the form of an assertion, that mathematical concepts and axioms are not the
result of cognition of reality, rather they are a condition and form of
cognition. This idea was elaborated by Kant and we may agree with it if we
consider the human being to be entirely given and do not ask why these conditions
and forms of cognition are characteristic of the human being. But when we have
asked this question we must reach the conclusion that they themselves are models
of reality developed in the process of evolution (which, in one of its important
aspects, is simply the process of cognition of the world by living structures).
From the point of view of the laws of nature there is no fundamental difference
between mathematical and empirical models; this distinction reflects only the
existence in or~anization of the human mind of a certain border line which
separates inborn models from acquired ones. The position of this line, one must
suppose, contains an element of historical accident. If it had originated at
another level, perhaps we would not be able to imagine that the sun may tail to
rise or that human beings could soar above the earth in defiance or gravity.
CONCERNING THE AXIOMS OF ARITHMETIC AND LOGIC
PLATO'S IDEALISM was the result of a sort of projection of the elements of
language onto reality. Plato's ''ideas'' have the same origin as the spirits in
primitive thinking; they are the imagined of really existing names. In the first
stages of the development of critical thinking the nature of abstraction in the
interrelationship of linguistic objects and non-linguistic activity is not yet
correctly understood. The primitive name-meaning unit is still pressing on people
an idea of a one-to-one correspondence between names and their meanings. For
words that refer to concrete objects the one-to-one correspondence seems to occur
because we picture the object as some one thing. But what will happen with
general concepts (universals)? In the sphere of the concrete there is no place at
all for their meanings; everything has been taken up by unique' concepts, for a
label with a name can be attached to each object. The empty place that form is
filled by the "idea". Let us emphasize that Plato's idealism is far from
including an assertion of the primacy of the spiritual over the material, which
is to say it is not spiritualism (this term, which is widely used in Western
literature, is little used in our country and is often replaced by the term
"idealism,'' which leads to inaccuracy). According to Plato spiritual experience
is just as empirical as sensory experience and it has no relation to the world of
ideas. Plato's ''ideas'' are pure specters, and they are specters born of
sensory, not spiritual, experience.
From a modern cybernetic point of view only a strictly defined, unique situation
can be considered a unique concept. This requires an indication of the state of
all receptors that form the input of the nervous system. It is obvious that
subjectively we are totally unaware of concepts that are unique in this sense.
Situations that are merely similar become indistinguishable somewhere in the very
early stages of information processing and the representations with which our
consciousness is dealing are generalized states, that is to say, general, or
abstract, concepts (sets of situations). The concepts of definite objects which
traditional logic naively takes for the primary elements of sensory experience
and calls ''unique" concepts are in reality, as was shown above, very complex
constructions which require analysis of the moving picture of situations and
which rely on more elementary abstract concepts such as continuousness, shape,
color, or spatial relations. And the more ''specific'' a concept is from the
logical point of view, the more complex it will be from the cybernetic point of
view. Thus, a specific cat differs from the abstract cat in that a longer moving
picture of situations is required to give meaning to the first concept than to
the second. Strictly speaking the film may even be endless, for when we have a
specific cat in mind we have in mind not only its ''personal file'' which has
been kept since its birth, but also its entire genealogy. There is no fundamental
difference in the nature of concrete and abstract concepts; they both reflect
characteristics of the real world. If there is a difference, it is the opposite
of what traditional logic discerns: abstract, general concepts of sensory and
spiritual experience (which should not be confused with mathematical constructs)
are simpler and closer to nature than concrete concepts which refer to the
definite objects. Logicians were confused by the fact that concrete concepts
appeared in language earlier than abstract ones did. But this is evidence of
their relatively higher position in the hierarchy of neuronal concepts, thanks to
which they emerged at the point of connection with linguistic concepts.
The Platonic theory of ideas, postulating a contrived, ideal existence of
generalized objects, puts one-place predicates (attributes) in a position
separate from multiplace predicates (relations). This theory assigned attributes
the status of true existence but denied it to relations, which became perfectly
evident in Aristotle's loci. The concrete, visual orientation and static quality
in thinking which were so characteristic of the Greeks in the classical period
came from this. In the next chapter we shall see how this way of thinking was
reflected in the development of mathematics.
_________________________________________________________________________________
[14][1] The resemblance in sound between the Greek idea and the Russian vid is
not accidental; they come from a common Indo-European root. (Compare also Latin
"vidi" - past tense of "to see".)
[15][2] For those who are familiar with mathematimal logic let us note that this
is in the broad sense. including the rules of inference.
____________________________________________________________________________
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/Turchap10.html#Heading3
4. http://pespmc1.vub.ac.be/POS/Turchap10.html#Heading4
5. http://pespmc1.vub.ac.be/POS/Turchap10.html#Heading5
6. http://pespmc1.vub.ac.be/POS/Turchap10.html#Heading6
7. http://pespmc1.vub.ac.be/POS/Turchap10.html#Heading7
8. http://pespmc1.vub.ac.be/POS/Turchap10.html#Heading8
9. http://pespmc1.vub.ac.be/POS/Turchap10.html#Heading9
10. http://pespmc1.vub.ac.be/POS/Turchap10.html#Heading10
11. http://pespmc1.vub.ac.be/POS/Turchap10.html#Heading11
12. http://pespmc1.vub.ac.be/POS/Turchap10.html#fn0
13. http://pespmc1.vub.ac.be/POS/Turchap10.html#fn1
14. http://pespmc1.vub.ac.be/POS/Turchap10.html#fnB0
15. http://pespmc1.vub.ac.be/POS/Turchap10.html#fnB1
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 ;-)