euclidean geometry, foundations and the logical paradoxes

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


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


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


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


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.


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

(+


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.


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


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:


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


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.


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:


(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