From Thales to Euclid
The Phenomenon of Science
Valentin F. Turchin
Web: http://pespmc1.vub.ac.be/turchin.html
VTurchin [at] BellAtlantic.net
Professor Emeritus of Computer Sciences.
The City College,
The City University
of New York
Contents:
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 3r2, while the Egyptians
used the formula (8/9 2r)2. Where was the truth?
This was something to think about and debate.
The creators of Egyptian and Babylonian mathematics have remained
anonymous. The Greeks preserved the names of their wise men. The first,
Thales of Miletus, is also the first name included in the history
of science. Thales lived in the sixth century B.C. in the city of
Miletus on the Asia Minor coast of the Aegean Sea. One date in his
life has been firmly established: in 585 B.C. he predicted a solar
eclipse--unquestionable evidence of Thales's familiarity with the
culture of the ancient civilizations, because the experience of tens
and hundreds of years is required to establish the periodicity of
eclipses. Thales had no Greek predecessors, and could therefore only
have taken his knowledge of astronomy from the scientists of the East.
Thales, the Greeks assert, gave the world the first mathematical proofs.
Among the propositions (theorems) proved by him they mention the following:
1 The diameter divides a circle into two equal parts.
2 The base angles of an isosceles triangle are equal.
3 Two triangles which have an identical side and
identical angles adjacent to it are equal.
In addition, Thales was the first to construct a circle circumscribed
about a right triangle (and it is said that he sacrificed an ox in
honor of this discovery).
The very simple nature of these three theorems and their intuitive
obviousness shows that Thales was entirely aware of the importance
of proof as such. Plainly, these theorems were proved not because
there was doubt about their truth but in order to make a beginning
at systematically finding proof and developing a technique for proof.
With such a purpose it is natural to begin by proving the simplest
propositions.
Suppose triangle ABC is isosceles, which is to say side AB
is equal to side BC.
Figure 10.1. Isosceles triangle.
Let us divide angle ABC into two equal parts by line BD.
Let us mentally fold our drawing along line BD. Because angle
ABD is equal to angle CBD, line BA will lie on
line BC, and because the length of the segments AB and
BC is equal, point A will lie on point C. Because
point D remains in place, angles BCD and BAD
must be equal. Whereas formerly it only seemed to us that angles
BCD and BAD were equal (Thales probably spoke this way
to his fellow citizens), we have now proved that these angles
necessarily and with absolute precision must be equal (the Greeks
said "similar'') to one another: that is, they match when one is placed
on the other.
The problem of construction is more complex and here the result is
not at all obvious beforehand. Let us draw a right triangle.
Figure 10.2. Construction of a circle
described around a right triangle
May a circle be drawn such that all three vertices of the triangle
appear on it? And if so, how'? It is not clear. But suppose that intuition
suggests a solution to us. We divide the hypotenuse BC into
two equal segments at point D. We connect it with point A.
If segment AD is equal in magnitude to segment DC (and
therefore also to BD) we can easily draw the required circle
by putting the point of a compass at point D and taking segment
DC as the radius. But is it true that AD =DC, that is
to say triangle ADC is an isosceles triangle? It is not clear.
It seems probable, but in any case it is far from obvious. Now we
shall take the crucial step. We shall add point E to our triangle,
making rectangle ABEC and draw in a second diagonal AE.
Suddenly it becomes obvious that triangle ADC is isosceles.
Indeed, from the overall symmetry of the drawing it is clear that
the diagonals are equal and intersect at the point which divides them
in half--at point D. We have not yet arrived at proof, but
we already are at that level of clarity where formal completion of
the proof presents no difficulty. For example, relying on the equality
of the opposite sides of the rectangle (which can be derived from
even more obvious propositions if we wish), we complete the proof
by the following reasoning: triangles ABC and AEC are
equal because they have side AC in common, sides AB and
EC are equal, and angles BAC and ECA are right angles;
therefore angle EAC is equal to angle BCA. That is,
triangle ADC is an isosceles triangle, which is what had to
be proved.
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.''[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
Li, and the planet Earth may be the corresponding
object (referent) Rj. There are no other true or
ideal'' points, that is, without dimensions. It is often said that
there are no ''true'' points in nature, but that they exist only in
our imagination. This commonplace statement is either absolutely meaningless
or false, depending on how it is interpreted. In any case it is harmful,
because it obscures the essence of the matter. There are no "true''
points in our imagination and there cannot be any. When we say that
we are picturing a point we are simply picturing a very small object.
Only that which can be made up of the data of sensory experience can
be imagined, and by no means all of that. The number 1,000 for example,
cannot be imagined, large numbers, ideal points, and lines exist not
in our imagination, but in our language, as linguistic objects we
handle in a certain way. The rules for handling them reveal the essence
of mathematical concepts, specifically the ''ideality of the point.''
The dimensions of points on a drawing do not influence the development
of the proof, and if two points must be set so close that they merge
into one, we can increase the scale.
But aren't the assertions of mathematics characterized by absolute
prehave an entirely different status. By itself this language is,
of course, discrete also, but empirical assertions reflect semantic
conversions L1->S1 leading us into
the area of nonlinguistic activity which is neither discrete nor deterministic.
When we say that two rods have equal length this means that every
time we measure them the result will be the same. Experience, however,
teaches us that if we can increase the precision of measurement without
restriction, sooner or later we shall certainly obtain different values
for the length, because an empirical assertion of absolutely exact
equality is completely senseless. Other assertions of empirical language
which have meaning and can be expressed in the language of predicate
calculus, for example ''rod no. 1 is smaller than rod no. 2," possess
the same ''absolute precision'' (which is a trivial consequence of
the discrete nature of the language) as mathematical assertions of
the equality of segments. This assertion is either ''exactly'' true
or "exactly'' false. Because of variations in the measuring process,
however, neither is absolutely reliable.
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.
Figure 10.3. Construction of an equilateral
triangle.
Suppose AB is the given side of the triangle. Taking point
A as the center we describe circle [pi]A with radius
AB. We describe a similar circle ([pi][Beta]) from
point B We use C to designate either of the points of
intersection of these circles . Triangle ABC is equilateral,
for AC = CB = AB.
There is a logical hole in this reasoning: how does it follow that
the circles constructed by us will intersect at all'? This is a question
fraught with complications, for the fact that point of intersection
C exists cannot be related either to the attributes of a circle or
even to the attributes of a pair of circles (for they by no means
always intersect). We are dealing here with a more specific characteristic
of the given situation. Euclid probably sensed the existence of a
hole here, but he could not find anything to plug it with.
But how are we certain that circles [pi]A and [pi]B
intersect? In the last analysis, needless to say, we know from experience.
From experience in contemplating and drawing straight lines, circles,
and lines in general, from unsuccessful attempts to draw circles [pi]A
and [pi]B so that they do not intersect.
So Plato's view that the mathematics of his day was entirely independent
of experience cannot be considered sound. But the question of the
nature of mathematical reliability requires further investigation,
for to simply make reference to experience and thus equate mathematical
reliability with empirical reliability would mean to rush to the opposite
extreme from Platonism. Certainly, we feel clearly that mathematical
reliability is somehow different from empirical reliability, but how'?
The assertion that circles of radius AB with centers of A
and B intersect (for brevity we shall designate this assertion
E1 ) seems to us almost if not completely reliable;
we simply cannot imagine that they would not intersect. We cannot
imagine.... This is how mathematical reliability differs from
the empirical! When we are talking about the sun rising tomorrow,
we can imagine that the sun will not rise and it is only on the basis
of experience that we believe that it probably will rise. Here there
are two possibilities and the prediction as to which one will happen
is probabilistic. But when we say that two times two is four and that
circles constructed as indicated above inter
sect we cannot imagine that it could be otherwise. We see no other
possibility, and therefore these assertions are perceived as absolutely
reliable and independent of concrete facts we have observed.
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 E1
through to the end. Because we still have certain doubts that the
circles in figure 10.3 necessarily intersect, let us attempt to picture
a situation where they do not. If this attempt fails completely it
will mean that assertion E1 is mathematically reliable
and cannot be broken down into simpler assertions: then it should
be adopted as an axiom. But if through greater or lesser effort of
imagination we are able to picture a situation in which [pi]A
and [pi]B do not intersect, it must be expected that this
situation contradicts some simpler and deeper assertions which do
possess mathematical reliability. Then we shall adopt them as axioms
and the existence of the contradiction will serve as proof of E1.
This is the usual way to establish axioms in mathematics.
First let us draw circle [pi]A. Then we shall put the
point of the compass at point B and the writing element at
point A and begin to draw circle [pi]B. We shall
move from the center of circle [pi]A toward its periphery
and at a certain moment (this is how we picture it in our imagination)
we must either intersect circle [pi]A or somehow skip over
it, thus breaking circle [pi]B.
Figure 10.4.
But we imagine circle pB as
a continuous line and it becomes clear to us that the attributes of
continuousness, which are more fundamental and general than the other
features of this problem, lie at the basis of our confidence that
circles [pi]A and [pi][Beta] will intersect.
Therefore we set as our goal proving assertion E1
beginning with the attributes of continuousness of the circle. For
this we shall need certain considerations related to the order of
placement of points on a straight line. We include the concepts of
continuousness and order among the basic. undefined concepts of geometry,
like the concepts of the point, the straight line, or distance.
Here is one possible way to our goal. We introduce the concept of
''inside'' (applicable to a circle) by means of the following definition:
D1: It is said that point A lies inside
circle [pi] if it does not lie on [pi] and any straight line passing
through point A intersects [pi] at two points in such a way
that point A lies between the points of intersection. If the
point is neither on nor inside the circle it is said that it lies
outside the circle.
The concept of ''between'' characterizes the order of placement of
three points on a straight line. It may be adopted as basic and expressed,
through the more ,general concept of ''order,'' by the following definition:
D2: It is said that point A is located between
points B1 and B2 if these three
points are set on one straight line and during movement along this
line they are encountered in the order B1, A,
and B2 or B2, A, and B1.
We shall adopt the following propositions as axioms:
A1: The center of a circle lies inside it.
A2: The arc of a circle connecting any two points
of the circle is continuous.
A3: If point A lies inside circle [pi] and
point B is outside it, and these two points are joined by a
continuous line, then there is a point where this line intersects
the circle.
Relying on these axioms, let us begin with the proof. According to
the statement of the problem, circle [pi]B passes through
center A of circle [pi]A. If we have confidence
that there is at least one point of circle [pi][Beta] that
does not lie inside [pi]A we shall prove E1.
Indeed, if it lies on [pi]A then E1 has
been proved. If it lies outside [pi]A then the arc of circle
[pi]B connects it with the center, that is, with an inside
point of circle [pi]A. Therefore, according to axioms A2
and A3 there is a point of intersection of [pi]B and
[pi]A.
But can we be confident that there is a point on circle [pi]B
which is outside [pi]A? Let us try to imagine the opposite
case. It is shown in figure 10.5.
This is the second attempt to imagine a situation which contradicts
the assertion being proved. Whereas the first attempt immediately
came into explicit contradiction with the continuousness of a circle,
the second is more successful. Indeed, stretching things a bit we
can picture this case. We take a compass, put its point at point B
and the pencil at point A. We begin to draw the circle without
taking the pencil from the paper and when the pencil returns to the
starting point of the line we remove it and see that we have figure
10.5. And why not?
To prove that this is impossible we must prove that in this case
the center of circle [pi]B is necessarily outside it. We
shall be helped in this by the following theorem:
T1: If circle [pi]1, lies entirely inside
circle [pi]2 then every inside point of circle [pi]1
is also an inside point of circle [pi]2.
To prove this we shall take an arbitrary inside point A of
circle [pi]1, which is shown in figure 10.6.
We draw a straight line through it. According to definition D1
it intersects [pi]1, at two points: B1
and B2 Because B1 (just as B2)
lies inside [pi]2 this straight line also intersects [pi]2
at two points: C1 and C2. We have
received five points on a straight line and they are connected by
the following relationships of order: A lies between B1
and B2; B1 and B2
lie between C1 and C2.
That point A proves to be between points C1 and
C2 in this situation seems so obvious to us that
we shall boldly formulate it as still another axiom.
A3: If points B1 and B2
on a straight line both lie between C1andC2,
then any point A lying between B1 and B2
also lies between C1 andC2.
Because we can take any point inside [pi]1 as A
and we can draw any straight line through it, theorem T1
is proven.
Now it is easy to complete the proof of E1. If
circle [pi]B lies entirely inside [pi]A then
according to theorem T1 its center B must
also lie inside [pi]A. But according to the statement of
the problem point B is located on [pi]A. Therefore
[pi]B contains at least one point which is not inside in
relation to [pi]A
So to prove one assertion E1 we needed four assertions
(axioms A1-A4), but then these
assertions express very fundamental and general models of reality
related to the concepts of continuousness and order and we cannot
even imagine that they are false. The only question that can be raised
refers to axiom A1 which links the concept of center.
which is metrical (that is, including the concept of measurement)
in nature, with the concept of ''inside," which relies exclusively
on the concepts of continuousness and order. It may be desired that
this connection be made using simpler geometric objects, under conditions
which are easier for the functioning of imagination. This desire is
easily met. For axiom A1 let us substitute the following
axiom:
A1': if on a straight line point A and a
certain distance (segment) R are given, then there are exactly
two points on the straight line which are set at distance R
from point A, and point A lies between these two points.
Relying on this axiom we shall prove assertion A1
as a theorem. We shall draw an arbitrary straight line through the
center of the circle. According to axiom A1' there
will be two points on it which are set at distance R (radius
of the circle) from the center. Because a circle is defined as the
set of all points which are located at distance R from the
center, these points belong to the circle. According to axiom A1'
the center point lies between them and therefore, according to definition
D1, it is an inside point. In this way axiom A1
has been reduced to axiom A1'. Now try to imagine
a point on a straight line which does not have two points set on different
sides from it at the given distance!
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 [2]
which seem to us just as reliable as the mathematical axioms considered
above. And this is for the same reason, that the opposite situation
is absolutely inconceivable. Having this confidence we acquire confidence
that no matter how many steps a logical deduction may contain it will
still possess this attribute. Here we are using the following very
important axiom:
The axiom of induction: Let us suppose that function f
(x) leaves attribute P (x) unchanged, that is
(
x){P (x) =>P
[f (x)]}
We will use f n(x) to signify the result of sequential
n-time application of function f (x), that is
f 1(x) =f (x), f n(x)
= f [fn-1 (x)].
Then f n(x) will also leave attribute P (x) unchanged
for any n, that is
(n)(
x){P (x) => P [f n(x)]}
By their origin and nature logical axioms and the axiom of induction
(which is classed with arithmetic because it includes the concept
of number) do not differ in any way from the other axioms; they are
all mathematical axioms. The only difference is in how they are used.
When mathematical axioms are applied to mathematical assertions they
become elements of a metasystem within the framework of a system of
mathematically reliable assertions and we call them logical axioms.
Thanks to this, the system of mathematically reliable assertions becomes
capable of development. The great discovery of the Greeks was that
it is possible to add one certainty to another certainty and thus
obtain a new certainty.
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.
[1]
The resemblance in sound between the Greek idea and the Russian
vid is not accidental; they come from a common Indo-European
root. (Compare also Latin "vidi" - past tense of "to see.")
[2]
For those who are familiar with mathematimal logic let us note that
this is in the broad sense. including the rules of inference.