dinsdag 26 februari 2013

definitions, postulates, axioms and propositions.

Euclid (300bc)
definitions, postulates, axioms and propositions.

Definitions

A definition describes what is being defined, and gives it a name.
To use it  we must agree to use the name in precisely that way.  
A definition however does not assert that what has that name exists –
and we may not assume it does.
For the statements we do not prove must be as few as possible,
 so must be the assumptions that things exist.
We may assume, for example, that points exist, but very little else.
A definition is required only to be understood.

Some of the 23 Definitions.
· A point is that which has no part.
· A line is breadthless length.
· The extremities of a line are points.
· A straight line is a line which lies evenly with the points on itself.
· A surface is that which has length and breadth only.
· The extremities of a surface are lines.
· A plane surface is a surface which lies evenly with the straight lines on itself.
· A boundary is that which is an extremity of anything.
· A figure is that which is contained by any boundary or boundaries.
· 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.

Postulates.

Postulates are the fundamental propositions used to prove other statements known as theorems.
Once a theorem has been proven it is may be used in the proof of other theorems. In this way, an entire branch of mathematics can be built up from a few postulates.

We require that the figures of geometry -- the triangles, squares, circles -- be more than ideas.  We must be able to draw them.
 The fact that we can draw a figure is what permits us to say that it exists.  
For we may not assume that what we have called a "triangle" or a "circle" actually exists.
The first three Postulates narrowly set down what we are permitted to draw.  
Everything else we must prove.  Each of those Postulates is therefore a "problem" -- a construction -- that we are asked to consider solved:  "Grant the following."

Let the following be postulated:
1.      To draw a straight line from any point to any point.
2.      To produce a finite straight line continuously in a straight line.
3.      To describe a circle with any centre and distance.
4.      That all right angles are equal to one another.
5.      That, if a straight line falling on two straight lines make 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.

Axioms or Common Notions

The distinction between a postulate and an axiom is that a postulate is about the specific subject at hand, in this case, geometry; while an axiom is a statement we acknowledge to be more generally true; it is in fact a common notion. 
Yet each has the same logical function, which is to authorize statements in the proofs that follow.
1.      Things which are equal to the same thing are also equal to one another.
2.      If equals be added to equals, the wholes are equal.
3.      If equals be subtracted from equals, the remainders are equal.
4.      Things which coincide with one another are equal to one another.
5.      The whole is greater than the part. 

Propositions

The theory is  followed by 48 propositions.
There are two kinds of propositions: theorems and problems.
 A theorem proposes a statement to prove.
A problem proposes a task to accomplish.

Some of the 48 Propositions;
· To place a straight line equal to a given straight line with one end at a given point.
·  To cut off from the greater of two given unequal straight lines a straight line equal to the less. 
· To bisect a given finite straight line. 
· To draw a straight line at right angles to a given straight line from a given point on it. 
· If a straight line stands on a straight line, then it makes either two right angles or angles whose sum equals two right angles. 
· If with any straight line, and at a point on it, two straight lines not lying on the same side make the sum of the adjacent angles equal to two right angles, then the two straight lines are in a straight line with one another. 
· If a straight line falling on two straight lines makes the alternate angles equal to one another, then the straight lines are parallel to one another.
· Straight lines parallel to the same straight line are also parallel to one another.


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