Thread: Hilbert's Axioms

Hello,

Can anyone please explain me what is Hilbert's Axioms?

Originally Posted by shounakbhatta

Can anyone please explain me what is Hilbert's Axioms?

There are entire textbooks devoted to that question.
Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg is one of the best organized and is quite readable.

Originally Posted by Plato

There are entire textbooks devoted to that question.
Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg is one of the best organized and is quite readable.

Well,

I am going through "Foundations of Geometry" by David Hilbert. What I feel the concept of point, line and plane can be substituted by tables, chair etc. It unified both plane geometry and solid geometry. What I mean to ask you is that the axioms of Hilbert introduced, did it try to evade Euclid's axioms or it has been substituted?

Essentially Hilbert's axioms (or postulates) clarified Euclid's axioms with a couple of additions- adding, for example, "continuity" of lines.

Originally Posted by HallsofIvy

Essentially Hilbert's axioms (or postulates) clarified Euclid's axioms with a couple of additions- adding, for example, "continuity" of lines.

Hello,

There are certain things which I would like to know on Hilbert's axioms:

(1) Axioms of Connection
(2) Axioms of Order
(3) Axioms of parallel
(4) Axioms of congruence
(5) Axioms of continuity.

Apart from 3 and 5 which are Euclid and Archimedes, the other 3 are they new axioms invented by Hilbert? Or are they the same 5 axioms of Euclid, which has been proven in a different manner?

Did Hilbert tried to prove something like: Geometry can be proven WITHOUT AXIOM?

Can anyone please explain the following lines:

"Axioms are not taken as self-evident truths. Geometry may treat things, about which we have powerful intuitions, but it is not necessary to assign any explicit meaning to the undefined concepts. The elements, such as point, line, plane, and others, could be substituted, as Hilbert says, by tables, chairs, glasses of beer and other such objects. It is their defined relationships that are discussed." (Source David Hilbert - Wikipedia, the free encyclopedia)

Originally Posted by shounakbhatta

Can anyone please explain the following lines:

"Axioms are not taken as self-evident truths. Geometry may treat things, about which we have powerful intuitions, but it is not necessary to assign any explicit meaning to the undefined concepts. The elements, such as point, line, plane, and others, could be substituted, as Hilbert says, by tables, chairs, glasses of beer and other such objects. It is their defined relationships that are discussed." (Source David Hilbert - Wikipedia, the free encyclopedia)

Hilbert is the father of Formalism, see the Wikipedia article. All that the above says is that when doing geometry Hilbert is wearing his Formalist hat.

CB