-
Euclidean geometry: foundations and paradoxes 2
EUCLIDEAN GEOMETRY: FOUNDATIONS AND PARADOXES
George Mpantes www.mpantes.gr
Philosophical foundations: axiomatic method
Conceptual foundations: the Elements
The paradoxes
The criticism
The paradox of the parallel axiom
The evolution
Introduction .
The foundations of geometry are conceptual and philosophical .
The first
are outlined in the famous book of Euclid 'the Elements ' and
the latter, which
are deeper, in another famous book of antiquity "the Analytica
posterioria " ( in
the middle of the fourth century B.C ) of Aristotle , in which
he develops his
theory of scientific knowledge. It is a text from Aristotle's
Organon that
deals with demonstration, definition, and scientific
knowledge.
Aristotle was not a mathematician he but was working at the time
of an
active mathematical practice and his writings reflected as well
influenced that
practice. In mathematics he distinguished the models of logical
reasoning,
whence he derived the principles of axiomatic method as accepted
in his time.
So by the turn of the century the stage was set for Euclids
epoch-
making application of Aristotles new ideas of Knowledge :
Knowledge of the fact
differs from knowledge of the reasoned fact ,.Analytica
posterioria)
-
3
Philosophical foundations - axiomatic method .
The philosophical foundations of Euclidean geometry is the
axiomatic
method which is the greatest contribution of the Greeks in
Western science .
Without it, there is no any science . Mathematics of course
existed before
Euclid, but mathematics after Euclid was a science, that is the
mathematical
conclusions are assured by logical rather than empirical
demonstrations.
The axiomatic method is based on deductive reasoning and a
classical
example of it, is the following
Premises 1. All men are mortal
2. Socrates is a man
Follows the
Conlusion 3. Socrates is mortal
If we adopt the premises and the system
Aristotles logic that is employed, then the
conclusion is incontestable and the reasoning is
valid. The concern of an expert in the use of
deduction is not the truth of the conclusion but
in the validity of the reasoning, it is a theme of
logic, and he wants to be able to assert that his
conclusions applied by the premises.
So the deductive reasoning is an
algorithm of logical demonstration, and axiomatic method starts
since the
Greeks discovered this deductive reasoning. This method finally
led to the top
-
Euclidean geometry: foundations and paradoxes 4
of the creation, that is the mathematical proof1 , and all these
(deductive
reasonig, axiomatic method and mathematical proof) were a new
form of
perception and thought, transforming the empirical calculation
of the
Babylonians and Egyptians , in what is known today as
mathematical science .
But what is it and how did appear axiomatic method ? A good
image that
appears to work , gives us H. Eves: as deductive reasoning in
geometry of the
Pythagoreans were increasing , and the logical chain lengthens
and many
intertwined , born the terrible idea , the whole geometry to
make a unique chain
considerations "( foundations of mathematics ) . This unique
chain would start
somewhere. So one should accept without proof some proposals and
all other
recommendations of the system to produce the original , with the
only help of
the principles of logic ( deductive reasoning ),in the belief
that the axiomatic
method organizes and promotes logical reasoning producing "new
and necessary
knowledge2." Euclid applied it for the first time in the entire
geometry ( 300
BC).
So axiomatic method means of constructing a scientific theory,
in which
this theory has as its basis certain points of departure
premises, (hypotheses)
axioms or postulates, from which all the remaining assertions of
this discipline
(theorems) must be derived through a purely logical method by
means of proofs.
In Posterioria Analytica, Aristotle attempted to show how his
logical
theory could apply to scientific knowledge. He argues that a
science must be
based on axioms-postulates (self-evident truths), from which one
can draw
1 The mathematical proof is the culmination of mathematical
creation , it did not arise by
some sort of experience, it is not interpreted mechanically by
the method of trial and error , or
by coincidence. Its intellectual process is unspecified as in
music or poetry . It is a flash that
illuminates the minds of creators , and belongs to another
unknown world ! It's that strange joy we
felt in school when we were proving an exercise in geometry. We
all knew - we experienced the
aura of mathematical proof and no need to say anything else.
This aura is the answer to every foundational program of
philosophy in mathematics
2 New, because you learn something that you did not know before,
and necessary because the
conclusion is inescapable (A. Doxiadis, Logicomix)
-
5
definitions and hypotheses. The axioms, said Aristotle , are
known to be true by
our infallible intuition. Moreover we must have such truths on
which to base our
reasoning. If instead , reasoning were to use some facts not
known to be truths,
further reasoning would be needed to establish these facts and
this process
would have to be repeated endlessly. There would then be an
infinite regress.
The theoretical foundations of these systems are in the
Aristotelian
account of first principles, where are the bases of every
science as we read:
scientific knowledge through reasoning is impossible unless
a
man knows the first immediate principles. In every systematic
inquiry
(methodos) where there are first principles, or causes, or
elements,
knowledge and science result from acquiring knowledge of these;
for
we think we know something just in case we acquire knowledge of
the
primary causes, the primary first principles, all the way to
the
elements. It is clear, then, that in the science of nature
as
elsewhere, we should try first to determine questions about the
first
principles. Aristotle Phys. (184a1021) (
) (Phys. 184a1021)
.... the first basis from which a thing is known" (Met.
1013a1415).
A first principle is one that cannot be deduced from any
other
.By the first principles of a subject I mean those the truth
of which is not possible to prove. What is denoted by the first
terms
and those derived from them is assumed; but , as regards
their
existence, must be assumed for the principles but proved for
the
rest.. Thus what a unit is, what a straight line is , or what a
triangle
is, must be assumed and the existence of the unit and of
magnitude
must also be assumed but the existence of the rest must be
proved..
Aristotle
How are these first principles to be established? At the end
of
Analytica posterioria ii, Aristotle says that they are arrived
at by the
-
Euclidean geometry: foundations and paradoxes 6
repeated visual sensations, which leave their marks in the
memory. We then
reflect on these memories and arrive by a process of intuition
() at the first
principles.
The first principles should revolve around three things:
every demonstrative science has to do with three things:
(1) the things which are assumed to exist , namely the subject
-
matter in each case , the essential properties of which the
science
investigates, (2) the so-called common axioms3 , which are
the
primary source of demonstration and (3) the properties ,
with
regard to which all that is assumed is the meaning of the
respective
terms used..Aristotle , Analytica posterioria.
1. the definitions of the genus of science , which merely
explain the meaning of the terms involved in the project ( e.g.
the
definition of the 'Elements ' an acute angle is an angle less
than a
right angle ) Definitions are not hypotheses , for they do not
assert
the existence or non-existence of anything , only require to
be
understood ;
a definition is therefore not a hypotheses, a hypothesis is
that from the truth of which , if assumed , a conclusion can
be
established.
2. the common principles, or axioms , which are general
principles
that apply to any field of study in any science and are
considered
self-evident
( eg If equals are added to equals, then the wholes are
equal
)
3. the postulates for which science assumes what they mean
and
linked to a specific science , the properties , with regard to
which all
3 Today are the common notions , and the axioms identical with
postulates. For Aristotle
an axiom is common to all sciences , whereas a postulate is
related to a particular science; an axiom
is self-evident whereas a postulate is not; an axiom is assumed
with the ready asset of the learner
, whereas a postulate is assumed without perhaps , the assent of
the learner.
-
7
that is assumed is the meaning of the respective terms
used..Aristotle , Analytica posterioria
From these
considerations it follows that
there will be no scientific
knowledge of the first
principles, and since except
intuition nothing can be truer
than scientific knowledge, it
will be intuition that
apprehends the first
principles-a result which also
follows from the fact that
demonstration cannot be the
originative source of
reasoning, nor, consequently,
scientific knowledge of
scientific knowledge. If,
therefore, it is the only
other kind of true thinking except scientific knowing, intuition
will be the
originative source of scientific knowledge. And the originative
source of science
grasps the original basic principes, while science as a whole is
similarly related as
originative source to the whole body of fact. 4
Here we should beware :
There are distinctions between an hypothesis and a postulate
in
Aristotle:
4 posterioria analytica, internet classic archive, translated
by
B.D.B Muse
-
Euclidean geometry: foundations and paradoxes 8
anything that the teacher assumes , though it is matter of
proof, without proving it himself, is a hypothesis if the
thing
assumed is believed by the learner, and it is moreover a
hypothesis,
not absolutely, but relatively to the particular pupil; but if
the same
thing is assumed when the learner either has no opinion on
the
subject or is of contrary opinion , it is a postulate. This is
the
difference between a hypothesis and a postulate; for a postulate
is
that which is rather contrary than otherwise to the opinion of
the
learner , or whatever is assumed and used without being
proved
although matter of demonstration. Posterior Analytics
Historically , the ancient Greeks conceived of postulates as
being self-
evident truths, unproven claims or recognized as truth , that
are accepted
without proof which are defined as such by the unerring
intuition ( Aristotle) .
But as we see above, he refined and extended this concept of
postulates in a
way that made it much stronger : A postulate may not appeal to a
persons sense
of what is right , but it has been adopted as basic in order
that the work may
proceed.
" .... The postulate is an assumption not necessarily obvious,
nor
necessarily accepted by the student ." That is, we postulate
true even though
this is not proved logically nor easily apparent .
.This is a philosophical approach that was not understood , and
which
was destined to play an important historical role in the
development of the
axiomatic method and the whole of western science . The previous
looser view of
self-evident truths was retained by Euclid (or at least his
followers) in his
systematic organization of geometry as an axiom-based set of
deductive proofs.
This was the reason for the fruitless investigations on the
theory of parallels
for centuries , as we will see below. Is intuition an essential
element of the
structure of deductive reasoning ?
Today the postulates and axioms are identical. An axiom is not
proved
but it is chosen, is the spiritual stamp of the creator of the
theory. For
example, in classical mechanics, Aristotles axioms are the laws
of Newton. It is
-
9
neither logical nor perfectly obvious that, a body on which no
forces are exerted
is moving indefinitely . Alike for the axiom of Einstein on the
strange and
incomprehensible motion of light , similarly with the "many
stories " of
Feynman on quantum particles.
Finally we say that a proof in an axiom system L, is an ordered
list of
proposals p1, p2,,, pn such that every proposal of the list,
either it is a postulate
or has been obtained from previous proposals of the list, in
accordance with
the rules of system . A theorem is just a sentence of L, for
which there is a
logical chain p1, p2,,, pn = of proposals , which concludes the
. Thus the
organization of knowledge in an axiomatic system places the
burden of truth in
the axioms of the system, rather than in a distribution of truth
to the whole
body of knowledge.
Conceptual foundations of Euclids geometry .
Geometry is the first historical example of the development
of
perceptual abilities of human beings, to pass from the
experience and intuition
(the space around us and of the space relations of objects
inside it), in a
science of pure forms. But pure forms here are the concepts ,
which are the
basic entities of our perceptual space . The lines and shapes
are for Euclid, the
ideal excess of experience and intuition .
Geometry , as is known , is dealing with space, after make clear
what is
space . Space for the geometry is a set of points and lines . So
if space refers
to the surface of a sphere , the points of space are the points
of the surface of
the sphere and the lines (straight) of our space is the great
circles of the
sphere.
The flat two-dimensional space , i.e., our familiar plane, is
fully described
by Euclids geometry, with points and straight lines our familiar
shapes. These
shapes behave in a certain way , as described by Euclid in
"Elements " which are
-
Euclidean geometry: foundations and paradoxes 10
the transference of the first principles of Aristotle in
geometry , ie the
conceptual foundations of Euclidean geometry .
The classic example is that of Euclids Elements; its hundreds
of
propositions can be deduced from a set of definitions,
postulates, and common
notions or axioms: all three types constitute the first
Aristotles principles of
geometry .
The first principles of the Elements, contain 23 definitions, 9
common
notions or axioms and 5 postulates, which postulates are nothing
other despite
affairs for the behaviour of points and straight lines of plane.
If therefore we
say that Euclids postulates are in effect in space, it amounts
with we ask if the
space is Euclidean.
Book 1 of Euclid's Elements opens with a set of unproved
assumptions:
definitions (), postulates, and common notions ( ).
The definitions, are merely explanations of the meaning of the
terms.
Definition 1.
A point is that which has no part.
Definition 2.
A line is breadthless length.
Definition 3.
The ends of a line are points.
Definition 4.
A straight line is a line which lies evenly with the points on
itself.
Definition 5.
A surface is that which has length and breadth only.
Definition 6.
The edges of a surface are lines.
Definition 7.
A plane surface is a surface which lies evenly with the straight
lines on
itself.
Definition 8.
-
11
A plane angle is the inclination to one another of two lines in
a plane
which meet one another and do not lie in a straight line.
Definition 9.
And when the lines containing the angle are straight, the angle
is called
rectilinear.
Definition 10.
When a straight line standing on a straight line makes the
adjacent
angles equal to one another, each of the equal angles is right,
and the straight
line standing on the other is called a perpendicular to that on
which it stands.
Definition 11.
An obtuse angle is an angle greater than a right angle.
Definition 12.
An acute angle is an angle less than a right angle.
Definition 13.
A boundary is that which is an extremity of anything.
Definition 14.
A figure is that which is contained by any boundary or
boundaries.
Definition 15.
A circle is a plane figure contained by one line such that all
the straight
lines falling upon it from one point among those lying within
the figure equal one
another.
Definition 16.
And the point is called the center of the circle.
Definition 17.
A diameter of the circle is any straight line drawn through the
center
and terminated in both directions by the circumference of the
circle, and such a
straight line also bisects the circle.
Definition 18.
A semicircle is the figure contained by the diameter and the
circumference cut off by it. And the center of the semicircle is
the same as
that of the circle.
Definition 19.
-
Euclidean geometry: foundations and paradoxes 12
Rectilinear figures are those which are contained by straight
lines,
trilateral figures being those contained by three, quadrilateral
those contained
by four, and multilateral those contained by more than four
straight lines.
Definition 20.
Of trilateral figures, an equilateral triangle is that which has
its three
sides equal, an isosceles triangle that which has two of its
sides alone equal, and
a scalene triangle that which has its three sides unequal.
Definition 21.
Further, of trilateral figures, a right-angled triangle is that
which has a
right angle, an obtuse-angled triangle that which has an obtuse
angle, and an
acute-angled triangle that which has its three angles acute.
Definition 22.
Of quadrilateral figures, a square is that which is both
equilateral and
right-angled; an oblong that which is right-angled but not
equilateral; a rhombus
that which is equilateral but not right-angled; and a rhomboid
that which has its
opposite sides and angles equal to one another but is neither
equilateral nor
right-angled. And let quadrilaterals other than these be called
trapezia.
Definition 23
Parallel straight lines are straight lines which, being in the
same plane
and being produced indefinitely in both directions, do not meet
one another in
either direction.
For example definition 10 , tells what a right angle is and how
an angle
may be identified as a right angle , but says nothing about the
existence of
right angles , nor does it state what is assumed about such
angles. These later
functions are left to the postulates and to deduced
propositions. Thus postulate
4 informs us that all right angles are equal and Proposition 11
proves that right
angle exists
The common notions
Common notion 1.
Things which equal the same thing also equal one another.
-
13
Common notion 2.
If equals are added to equals, then the wholes are equal.
Common notion 3.
If equals are subtracted from equals, then the remainders are
equal.
Common notion 4.
Things which coincide with one another equal one another.
Common notion 5.
The whole is greater than the part.
The postulates are called both in the manuscripts of the
Elements and in the ancient exegetic tradition.
The postulates (axioms) are the following:
1. A straight line can be drawn from any point to any point.
2. A finite straight line can be produced continuously in a
straight line.
3. A circle may be described with any center and radius.
4. All right angles are equal to one another
5. (fig.1 )If a straight line falling on two straight lines
makes the interior
angles on the same side together less than two right angles ,
the two straight
lines, if produced infinitely , meet on that side on which
the angles are together less than two right angles
(+
-
Euclidean geometry: foundations and paradoxes 14
famous statement in mathematical history.
We can observe that the first principles of Euclids Elements fit
quite
well the Aristotelian account of definitions, postulates and
axioms as given in
Analytica posterioria , we have seen before.
Exactly Euclid accepts that every deductive system requires
assumptions from which the deduction may proceed. Therefore
as
initial premises , Euclid puts down five postulates or
assumed
statements about his subject matter, in addition he lists five
common
notions , that he needs for the proofs. These notions are not
peculiar
to his subject matter but are general principles valid in any
field of
study. Now in the postulates a number of terms occur, such as
point,
straight line, tight angle, and circle , of which it is not
certain that
the reader has a precise notion. Hence some definitions are
also
given.Howard Eves
All these are an exact construction o Aristotles views!
The combinations of these first principles , will produce
through
deductive reasoning the probative science of geometry (
theorems) .
The part of the proposals of geometry based on the 5th postulate
is the
pure Euclidean geometry , while the set of proposals that are
not based on the
fifth postulate , are the absolute geometry .
Examples of proposals of pure Euclidean geometry are:
The sum of the angles of a triangle are two right .
The sum of the exterior angles of polygon is 4 right angles.
The Pythagorean theorem and its extensions .
The length of the circumference is 2r etc.
Proposals of absolute geometry are the first 28 proposals of "
Elements
" ( constructions ) is e.g. it is possible to construct an
equilateral triangle with
a given side.
But simultaneously a question is born, that is not answered .
How do we
know that the axioms we have taken are the right axioms ? What
does the
expression right axioms mean? For example , are they free of
contradictions
-
15
? Each theorem of geometry is proved with these axioms or do we
need more ,
that Euclid overlooked ? What relationship should be between the
axioms ?
All these will join the investigation after two thousand years!
They are
the secrets of axiomatic bases, whose discovery in mathematical
practice will
start randomly with the terrible idea of Lobatchewski . Until
then there was no
contradiction, (even though critical investigations have
revealed a number of
defects in its logical structure) and is well known that
Euclidean geometry has
been the bible of science for many centuries 5.
The paradoxes of Euclidean geometry .
The paradoxes of Euclidean geometry are of some special kind:
the
fallacies here lay not in assuming something contrary to our
first principles but
in assuming something that is not implied by them. Sometimes
unconsciously (e.g
the infinitude of a straight line), others intuitively (the
proofs by
superposition) or tacitly (the intersection of the circles in
the proposition 1).
So we have no logical contradictions but rather logical defects
on its structure.
But the first mans transition from intuitive perception to the
deductive
study of abstract forms (axiomatic method), and in such an early
and extensive
application as Euclids , could not be perfect and final. The
remnants of empirical
perception, are often insisting into the deductive reasoning.
The transition
always leaves unresolved items , flaws , ambiguities , in the
beginning of every
branch of mathematics.
But when the subject matter of the axiomatic method excised
completely
from the empirical basis of intuition (non Euclidean geometry)
then the logic
purity and only this, would be the only driver of the process.
The material
axiomatics of Greeks became formal axiomatics , the modern
axiomatic method
.Then a need was felt for a truly satisfactory logical treatment
of Euclidean
geometry . Such an organization of Euclidean geometry was first
accomplished in 5 The 13th century Campanus translated the
"Elements" in Latin, and in 1482 we had the first
printed edition of Euclid in Europe.
-
Euclidean geometry: foundations and paradoxes 16
1882 by the German mathematician Moritz Pasch and later by
Hilbert, Birkhoff,
and Tarski.
The criticism (the definitions) .
The first point of criticism in Euclid was the issue of
definitions. Euclid
following the Greek plan of material axiomatic method , attempts
to define or
at least explain all the terms of his method. What is a point ?
Something that
has not parts or size. What is it ? This resembles the
definition of "nothing" . In
fact we mean point like something a very small , very specific
blot and if we
are pushed to explain what we mean by the very small, very
specific blot, will say
: well we mean point. The same happens with line: length without
breath. So
they are easily saw to be circular and therefore from a logical
point of view,
inadequate.
In fact, we cant define explicitly all terms, one through the
other , this
can not happen without avoid circularity , and there will always
be some
overarching terms that are defined implicitly , in the sense
that these are
things that are explained by the axioms , axioms are ultimately
definitions for
the prime terms . Here is the recipe for the modern axiomatic
method. But
Geometry for Greeks was not an abstract study but an
idealization of physical
space around us. And how we define the point? It took millennia
to be answered :
we simply overlook a definition. Hilbert stated that " for every
pair of points
there is a straight line that contains them. The proposal does
not require us to
know what is the point , but when we have two of them , there is
another thing
called straight, that contains them . The primitive terms in
Hilberts treatment
of plane Euclidean geometry are point, (straight) line, on,
between, and
congruent.
A paradox on definitions: every triangle is isosceles.
-
17
Given an arbitrary triangle ABC , draw the
angle bisector of A and the perpendicular bisector
of segment BC at D as n figure 1. if they are
parallel then ABC is isosceles. If not, they
intersect at a point P, and we draw the
perpendiculars PE, PF . The triangles labelled are
equal. Therefore PE=PF. Also the triangles labeled are equal
right triangles so
PB=PC. From this follows that the triangles are similar and
equal so we have
BE+EA=CF+FA so the triangle ABC is isosceles.
But if we attempt to construct accurately the points and lines
of the
figure we will discover that the actual
configuration doesnt look like the figure
1.the point P falls outside the triangle. But
again if we assume that the points E and F
also fall outside the triangle, we still conclude
that the triangle is isosceles. This too is a
incorrect configuration.
The actual configuration is of the
figure 2.
Now we see that even though AE=AF and BE=FC it doesnt follow
that
AB=AC, because while F is between A and C E is not between A and
B . this
illustrates the importance of betweeness as a concept in
geometry (axioms of
order in modern axiomatics, M.Pasch))
Paradoxes on propositions .
Bertrand Russell wrote an article The Teaching of Euclid in
which he was
highly critical of the Euclid's axiomatic approach. Although
this article is very
interesting, it seems extremely harsh to criticise Euclid in the
way that Russell
does. As someone once said, Euclid's main fault in Russell's
eyes is that he
-
Euclidean geometry: foundations and paradoxes 18
hadn't read the work of Russell. The article appeared in The
Mathematical
Gazette in 1902. Its full reference is B Russell, The Teaching
of Euclid, The
Mathematical Gazette 2 (33) (1902), 165-167. We give below some
items of
Russell's article6.
Proposition 1.
To construct an equilateral triangle on a given finite straight
line.
Euclid : the intersection of the circles (A,AB) and (B, BA) is
the point C.so AB=BG=GA
BUT
Russell : Here Euclid assumes that the circles used in the
construction intersect - an assumption not noticed by Euclid
because of the dangerous habit of using a figure. We require as
a
lemma, before the construction can be known to succeed,
the following:
If A and B be any two given points, there is at least one point
C whose distances from A and B are both equal to AB.
This lemma may be derived from an axiom of continuity. The fact
that in elliptic
space it is not always possible to construct an equilateral
triangle on a given base, shows also that Euclid has assumed the
straight line to be not a closed
curve - an assumption which certainly is not made explicit. When
these facts are
taken account of, it will be found that the first proposition
has a rather long
proof, and presupposes the fourth.
Postulate 2.
It is an implicit assumption of Euclid is that straight has
infinite extent.
While in postulate 2 states that the line can be produced
indefinitely , it is
strictly logically imply that a straight line is infinite in
extent, but that is
unlimited. The arc of a maximum circle joining two points on the
sphere can be
6 www.mathpages com.
-
19
produced indefinitely but does not imply that it has infinite
extent , is simply
unlimited . Need , says Russell, an axiom that " every straight
line there is at
least one point whose distance from a point on the straight or
outside exceeds a
given distance ."
Proposition 4.
Another point of criticism of Russell is the fourth proposition
that is the
proofs by superposition
If two triangles have two sides equal to two sides
respectively,
and have the angles contained by the equal straight lines equal,
then they
also have the base equal to the base, the triangle equals the
triangle, and
the remaining angles equal the remaining angles respectively,
namely
those opposite the equal sides
Russell says :
.The fourth proposition is a tissue of nonsense. Superposition
is a
logically worthless device; for if our triangles are spatial,
not material, there is a
logical contradiction in the notion of moving them, while if
they are material,
they cannot be perfectly rigid, and when superposed they are
certain to be
slightly deformed from the shape they had before. What is
presupposed, if
anything analogous to Euclid's proof is to be retained, is the
following very
complicated axiom:
Given a triangle ABC and a straight line DE, there are two
triangles, one on either side of DE, having their vertices at D,
and one side along DE, and equal in all respects to the triangle
ABC.
Another point on Russells critic is about the sixth
proposition
Proposition 6.
If in a triangle two angles equal one another, then the
sides
opposite the equal angles also equal one another.
This proposition requires an axiom which may be stated as
follows:
-
Euclidean geometry: foundations and paradoxes 20
If OAA', OBB', OCC' be three lines in a plane,
meeting two transversals in A, B, C, A', B', C'
respectively; and if O be not between A and A', nor B
and B', nor C and C', or be between in all three cases;
then, if B be between A and C, B' is between A' and
C'.
Proposition 8.
If two triangles have the two sides equal to two sides
respectively,
and also have the base equal to the base, then they also have
the
angles equal which are contained by the equal straight
lines.
the same fallacy as I.4, and requires the same axiom as to the
existence of congruent triangles in different places.
In the following propositions, we require the equality of all
right angles, which is
not a true axiom, since it is demonstrable. [Cf. Hilbert,
Grundlagen der
Geometris, Leipzig, 1899, p. 16.]
Proposition 12.
To draw a straight line perpendicular to a given infinite
straight
line from a given point not on it.
involves the assumption that a circle meets a line in two points
or in none, which has not been in any way demonstrated. Its
demonstration requires an axiom of
continuity, by the help of which the circle can be dispensed
with as an
independent figure.
Proposition 16.
In any triangle, if one of the sides is produced, then the
exterior
angle is greater than either of the interior and opposite
angles.
is false in elliptic space, although Euclid does not explicitly
employ any assumption which fails for that space. Implicitly, he
uses the following:
If ABC be a triangle, and E the middle point of AC; and if BE be
produced to F
so that BE = EF, then CF is between CA and BC produced.
Many more general criticisms might be passed on Euclid's
methods, and on his
conception of Geometry; but the above definite fallacies seem
sufficient to
-
21
show that the value of his work as a masterpiece of logic has
been very grossly
exaggerated. (Russell)
So much logic from Russell , and yet the logical gaps in
Euclid's
presentation did not produce ambiguities or doubts concerning
the accepted
rules of the calculus . There was a little more intuition rather
meticulous
adherence to logic . Besides so happened to all branches of
mathematics in early
investigations. The mathematicians of all times were
communicating and
discussing the Euclidean proofs, but with the discovery of
geometry
Lobatchewsky , the logical problems should be addressed.
The paradox of the parallel axiom .
this concern over Euclids fifth postulate furnished the
stimulus for the development of a great deal of modern
mathematics and also led to deep and revealing inquiries into
the
logical and philosophical foundations of the subject Howard
Eves
But the greatest paradox of Euclidean geometry , one that marked
the
history of geometry until the 19th century is the fifth
postulate , the famous
axiom of parallels
What exactly was happening ?
Surely the fifth postulate lacks the terseness and the
simple
comprehensibility possessed by the other four , after entering
in the
description for the behavior of the lines, the magic infinite .
It was not clear
and acceptable to talk about the intersection of two lines ...
to infinity. This
proposal did not appear outset immediately apparent to geometers
(
Papafloratos ) , but Aristotle had warned : " .. A postulate may
not appeal to a
persons sense of what is right , nor necessarily accepted by the
student .. ' .
The actual origin of the controversy seems to be geometric ,
arising from
the system itself . The searching of twenty centuries opened by
Proclus , who
was under the illusion that he possessed a proof of the
postulate, raised the
issue: He notes that two sentences of the first Book of Elements
are converse
-
Euclidean geometry: foundations and paradoxes 22
and moreover Euclid himself proved the second as a theorem:
1 . Postulate 5.
That, if a straight line falling on two straight lines makes
the
interior angles on the same side less than two right angles, the
two
straight lines, if produced indefinitely, meet on that side on
which are
the angles less than the two right angles. (fig.1)
Proposition 17.
In any triangle the sum of any two angles is less than two
right
angles.
For the proof of 17 , is not used the 5th postulate.
Therefore Proclus considers that it is not possible for two
converse
propositions, one to be proved while the other can not be proven
for true or
false. However, if a proposition can be demonstrated , then it
is not "legal" to
put in a postulate, and here he was right .
He continues :
when the two right angles are reduced ( + < 180 , Figure
1 ) is true and the fact that the straight lines e and e
converge is
true and necessary . But the statement that they will meet
sometime since they converge more and more as they are
produced,
is plausible but not necessary in the absence of some
argument
showing that it is true. It is a known fact that some lines
exist which
approach each other indefinitely, but yet remain
nonintersecting7.
May not the same thing which happens in the case of the
lines
referred to be possible in the case of the straight lines? ..and
thus
a proof of the fifth postulate is necessary
Proclus conclusion may be condensed in the phrase : "
There were many attempts to prove the parallel postulate and
many
substitutes devised for its replacement. Of the various
substitutions or
alternatives for the parallel postulate that have been either
proposed or tacitly
7 Our known as asymptotic lines
-
23
assumed are these by Proclus , Ptolemy , Neptunium , Geminus ,
Wallis,
Saccheri, Carnot, Laplace, Lambert, Clairaut, Legendre,
W.Bolyai, Gauss, and all
these efforts were based on an assumption equivalent to original
postulate of
Euclid . some of them are:
Playfair: In a plane, given a line and a point not on it, at
most one line parallel to
the given line can be drawn through the point.
Gauss : "there is no upper limit to the area of a triangle
."
Legendre : there exists at least one triangle having the sum of
its three angles
equal to two right angles.
Lambert and Clairaut: if in a quadrilateral three angles are
right angles , then
the fourth is also a right angle..
Saccheri8: if in a quadrilateral a pair of opposite sides are
equal and if the
angles adjacent to a third side are right angles , then the
other two angles, if the
assumption of the 5th postulate is not to be employed, might
both be right angles, obtuse
angles or acute angles9.
The efforts to find an acceptable understanding of the status
for the
Euclidean axiom were so numerous and so futile that in 1759 D
Alembert called
the problem of the parallel axiom the scandal of the elements of
geometry.
It is interesting to show the equivalence of all alternative
axioms of
Euclid, with it . To do this we must show that the alternative
is a theorem for
the Euclidean system, and conversely that the Euclidean fifth
postulate follows
8 We must mention here that the man who made the first really
scientific
assault on the problem of Euclids parallel postulate was
Saccheri
attempt to prove the fifth postulate he states three
assumptions: the acute angle
(hyperbolic geometry), the obtuse (elliptic geometry) and right
geometry (Euclidean). The
theorems produced with the assumption that the sum of the angles
of a triangle is less than 180
degrees form a kind of geometry as logic as Euclidean. However
the Saccheri did not realize it.
(elementary geometry for high school).
9 The work of Saccheri (first part) has been translated into
English and can be easy read
by any student of elementary plane geometry.
-
Euclidean geometry: foundations and paradoxes 24
as a theorem from the Euclidean system in which we replace the
5th postulate,
with this alternative .
The evolution .
But the causes of endless efforts of research proving the fifth
axiom,
are deeper. They are mainly in the philosophical foundations
rather than
conceptual.
It is the very structure of deductive reasoning, i.e. this
corner stone of
scientific geometry.
We must remember that if we change the 'mortal' with 'immortal',
in the
classical example of productive reasoning, then the conclusion
"therefore
Socrates is immortal" is valid! The premises are true or false,
but the reasoning
is only valid or invalid!
The persistence for the fifth postulate arose in that, in
classical
axiomatic method when confirming or denying an axiom (ie a
premise) spoke
about something true or false, this was given for the premises.
It was so obvious
for the classical axiomatic that seemed completely inconceivable
that such a
claim on the truth or falsity of a premise, could be
meaningless. Mathematicians
did not grasp at least the tighter definition of the concept of
the axiom
(postulate) by Aristotle we saw above, that the axiom should not
be unanimity,
but the main point were valid considerations after the
axioms!
The removal of mathematics from the world of direct experience
(and
here lies the ubiquitous infinity but will not analyze), brought
this development.
The empirical origin of Euclids geometrical axioms and
postulates was lost sight
of , indeed was never even realized. The intersection of two
lines at infinity is
neither true nor false. To suppose that there are not parallel
lines (premise-
axiom, do the mortal immortal) and to infer from there that the
sum of the
angles of a triangle is greater than two right angles, is a
matter of valid
reasoning and nothing else! ..
-
25
These secrets of the axiomatic bases were discovered randomly,
when it
became clear that the fifth postulate is impossible to prove, as
its refusal from
Lobatchewsky did not arise a logical contradiction. To make a
long story short, it
was found that by varying one of Euclids fundamental assumpions
(5th postulate)
, it was possible to construct two other geometrical doctrines ,
perfectly
consistent in every respect , though differing widely from
Euclidean geometry .
These are known as non-Euclidean geometries of Lobatchewski and
of Riemann.
Lovatchewski denied the 5th postulate and assumed that an
indefinite number of
non-intersecting straight lines could be drawn as parallels
(Playfair) and
Riemann assumed that none could be drawn. This was the big idea
of the new era.
The mathematical freedom came after replacing the 5th post, that
changed the
knowledge of centuries on the axiomatic system. What were
ultimately the
axioms? How could an axiom that determines the nature of the
whole geometry
and forms the basis for most theorems, not to be proved ... or
be obvious and
self-evident as the others? Yet this happened! The phenomenon at
infinity
leaves open the possibility that the straight line could be
defined and
otherwise, beyond the empirical description of Euclid, which was
one of the
many. But it was slow to grasp, and when done, the material
axiomatic of Greeks
evolved into formal axiomatic.. The truth of the axioms were not
assured of
anything.
And Euclid? Did he know meta-mathematics? of course not, but
rather
the intuitive conception of the phenomenon was so strong, that
led him to this
attitude of silence, leaving open the question of independence
for the next.
The story of the 5th postulate will end the 19th century with
the
independent work of Bolyai (son) and Lobatchewski. Until then,
the axiomatic
foundations of geometry were the five postulates of Euclid.
George Mpantes www.mpantes.gr
Sources:
-
Euclidean geometry: foundations and paradoxes 26
(X) ,
Foundations and fundamentals concepts of Mthematics Howard
Eves
(Dover)
www.mpantes.gr
www.mathpages.com
The teaching of Euclic (Bertrand Russel internet)
www.mathifone.gr
Mathematics, the loss of certainty (Morris Kline , Oxford
University
press)
: Steward Shapiro,
George Mpantes mpantes on scribd