the rise of axiomatic method

7/22/2019 The Rise of Axiomatic Method

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The Rise of

AXIOMATIC METHODProperties of Axiomatic Method

Euclidean Axiomatic Geometry

Finite Geometries

University of Southeastern Philippines

Tagum Mabini Campus


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Properties of xiomatic MethodThe axiomatic method is based on a system of

deductive reasoning.

In deductive system, statements used in an argument

must be derived, or based upon, prior statements used

in the argument. These prior statement must themselves

be derived from even earlier statements and so on.

UNIVERSITY OF SOUTHEASTERN PHILIPPINES


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Properties of xiomatic MethodLogical Traps

The trap of producing a never -ending stream of prior statements.

The trap of circular reasoning.



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Properties of xiomatic MethodFour components of

deductive system

Undefined terms

Axioms

Defined Terms Theorems



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Properties of xiomatic MethodFour components of

deductive system

Consistency

Independence

Completeness



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Properties of xiomatic MethodAn axiomatic system is

consistent if no two

statements (these could

be two axioms, an axiom

and theorem or two

theorems) contradict

each other.


CONSISTENCY

INDEPENDENCE

COMPLETENESS


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Properties of xiomatic MethodAn individual axiom in

an axiomatic system iscalled independentif it

cannot be proved from

the other axiom.


INDEPENDENCE

CONSISTENCY

COMPLETENESS


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Properties of xiomatic MethodAn axiomatic system is

called complete if it is

impossible to add a newconsistent and independent

axiom to the system. The

new axiom can use onlydefined and undefined terms

of the original system.


COMPLETENESS

CONSISTENCY

INDEPENDENCE


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Highlights of the dvancement ofModern Geometry


The need for statements about

the order of points on a line.

The need for a statement about

the concept of betweenness.

The need for a statement

guaranteeing the uniqueness of a

line joining two distinct

points.The need for a list of undefined

termsLO

GICAL

PRO

BLEMS


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The need for a definite

statement about the continuity

of lines and circles.

The need for statement about

the infinite extent of a

straight line.

The need to state that if the

straight line enters a triangleat a vertex then it must

intersect the opposite side.LO

GICAL

PRO

BLEMS


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The need for more logical

approach that does not depend

on the concept of

superposition.

LO

GICAL

PRO

BLEMS


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Moritz Pasch (1882) devised a modernset of axioms of Euclidean Geometry.

David Birkhoff (1902) used the six undefinedterms: point, line, plane, between, congruentand on.

Moritz Pasch (1882) devised a modernset of axioms of Euclidean Geometry.


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Finite Geometries


AXIOMS

1. There exist exactly three distinct

points in the geometry.

2. Each two distinct points are onexactly one line.

3. Not all the points of the

geometry are on the same line.

4. Each two distinct lines are on atleast one point.


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Finite Geometries


Theorem1.1

Each two distinct lines are on

exactly one point.

Proof:By axiom 4, two distinct lines are on at

least one point. Assume two lines lie on

more than one point. If lines l and m lieon the points P and Q, then axiom 2 is

contradicted, since P and Q would be on

two lines, l and m.


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Finite Geometries


Theorem1.2The three point geometry has exactly three

lines.

Proof:From axiom 2, each pair of points is one exactly one

line. Each possible pair of points is on a distinct line, so

the geometry has at least three lines. Suppose there is a

fourth line must have a point in common with each of

the other three lines, by theorem 1.1, so that it must alsobe on two of the three points, which contradicts axiom

2. Therefore, there can be no more than three lines in

the geometry.


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Finite Geometries


AXIOMS

1. There exist exactly four

lines.2. Any two distinct lines have

exactly one point on both of

them.3. Each point is on exactly two

lines.


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Finite Geometries


Theorem1.3The four line geometry has exactly six

points.

Proof:From Axiom 1, there are six pairs of lines. The

number six is obtained as the combination of

four things taken two at a time. By axiom 2,

each pair of lines has exactly one point on bothof them. Suppose two of these six points are

not distinct. That would be contradiction of

axiom 3. Why?


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Finite Geometries


Theorem1.3The four line geometry has

exactly six points.

Proof:Because this point would be on more than two

line. Also, by Axiom 3, no point could exist in

the geometry other than those six on the pairs

of lines.


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Finite Geometries


Theorem1.4Each line of the four line geometry

has exactly three points in it.

Proof:By axiom 2 each line of the geometry has a

point in common with each of the other three

lines and all three of these distinct points are

on the given line. Suppose there were a fourthpoint on one line. Then by Axiom 3, it must

also be on one of the other lines. But this is

impossible. Why?


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Finite Geometries


Theorem1.4Each line of the four line geometry

has exactly three points in it.

Proof:Because the other lines already determine

exactly one point with the given line, and, by

Axiom 2, they can only determine one point.

Thus, each line of the geometry has exactlythree points on it.


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Finite Geometries


AXIOMS

1. The total number of points in

this geometry is four.2. For any two distinct points P1

and P2, there is precisely one

line that contains both P1and P2.

3. Each line is on exactly twopoints.


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Finite Geometries


Theorem1.5

The four point geometry has exactly

six lines.

Proof?

Theorem1.6

Each point of the four point geometry

has exactly three lines in it.

Proof?


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Finite Geometries


AXIOMS

1. There exist at least one line.

2. Every line of the geometry has exactly

three points in it.3. Not all points of the geometry are on

the same line.

4. For each two distinct points, there

exists exactly one line on both ofthem.

5. Each two lines have at least one point

on both of them.


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Finite Geometries


Theorem1.7Each two distinct lines have exactly one

point in common.

Proof:By Axiom 5, two lines have at least one point

on both of them. In a manner similar to the

proof of Theorem 1.3, suppose that one pair of

lines has two points in common. Theassumption that they have two distinct points

would have two lines containing both of them.


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Finite Geometries


A different finite geometry can be obtained

from Fanos geometry by a modification of

the last axiom. The new geometry, called

Youngsgeometry, has four axioms of Fanos

geometry along with the following substituteaxiom for axiom 5.

5. If a point does not lie on a given line, then

there exists exactly one line on that point thatdoes not intersect the given line


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Finite Geometries


Can you think of a representation

of Youngsgeometry?

How many points and lines doesYoungs geometry include? Can

you provide a proof?

the rise of axiomatic method

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