euclidean geometry, foundations and the logical paradoxes
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Philosophical foundations: axiomatic methodConceptual foundations: the “Elements”The paradoxesThe criticismThe paradox of the parallel axiomThe evolutionTRANSCRIPT
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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)
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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
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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)
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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
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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.
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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
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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
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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
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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.
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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.
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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.
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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
(+
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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
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? 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.
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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.
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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
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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.
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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:
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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
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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
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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
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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.
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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! ..
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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:
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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