One of the foremost contribution of ancient ___ mathematicians to the foundations of mathematics was the ____

___ deals with the measurement and computation of lengths of line segments, areas of figures, and sizes of angles.

The word geometry comes from the word ___ which means ___ earth and ____ which means

____is originally land surveying.

Original name of geometry

Father of Geometry

Euclid, famously called the Father of Geometry, lived in the city of ____ around ___ (date)

He invented the form of mathematical proof that is still used today.

Name of book. Euclid gathered up all of the known mathematics of his time, as well as a lot of his own, and then he subjected it all to logical, mathematic proofs.

Euclid's 5 Geometric Postulates:

is a basic statement assumed to be true and requiring no proof of its truthfulness.

are statements that are considered true without proof or validation.

are statements proved to be true using postulates, definitions, other established theorems, and logic.

is a list of undefined terms together with a list of statements called axioms that are presupposed to be true.

is any statement that can be proven using logical deduction from the axioms.

An axiomatic system consists of:

concepts accepted without definition/explanation.

In the setting of basic geometry these migh include line or point.

forms the basis of the necessary technical vocabulary

logical statements regarding the undefined terms which are accepted without proof.

A set of unproved initial assumptions.

For example: There exists a line joining any two given points.

concepts defined in terms of 1 and 2.

For instance a triangle could be defined in terms of three non-collinear points.

logical statements deduced from 1-3.

expressing properties of the undefined objects, which are derived from the axioms by the laws of logic

Find the undefined terms: Axiom 1: Every robot has at least two paths Axiom 2: Every path has at least two robots Axiom 3: A minimum of one robot exists

Find the undefined terms: Axiom 1: Each committee is a set of three members. AxIom Z: cach member is on exactiv two committees AXIOM 3: NO two members may be together on more than one committee. Axiom 4: There is at least one committee.

is an argument demonstrating the truth of a theorem within an axiomatic system.

is an interpretation of the undefined terms such that all the axioms/postulates are true.

Properties of Axiomatic Systems

has internal logic that is not self-contradictory

If there is a model for an axiomatic system, then the system is called___

if it cannot be proven from the other axioms.

if every true statement can be proven from the axioms.

___set out a list of 23 unsolved mathematical problems to focus the direction of research in the 20th Century.

——developed a more modern axiomatic system, and his key idea was to detach "points", "lines" and "planes" from real-world objects.

who reduced geometry to a series of axioms and contributed substantially to the establishment of the formalistic foundations of mathematics.

Basic terms of hilbert:

Basic relations of hilbert:

Hilbert grouped his axioms into the following groups and we will follow his approach:

As mentioned above, David Hilbert (1862 - 1943) proposed an axiomatic system dealing with geometrical objects like "points", "straight lines", and "planes", detaching them from their specific meaning. Hilbert wanted to avoid any attempt to define exactly what a point, a straight line, or a plane is. He rather focused on the relationships between these objects. The following definitions and axioms demonstrate this approach.