Dear Mohamed I. Riffi ,

The axiomatic system includes primitive statements. That is postulated as facts and called axioms. Obviously, all axioms in the same system should be independent and consistent.

To talk about proving axioms has no meaning. Unless we assume that axiom under consideration is not independent of the other axioms and one can try to show his claim.

A famous example is the well known geometric axioms of Euclid where we have five axioms:

Quoting:

1. A line can be drawn from a point to any other point.

2. A finite line can be extended indefinitely.

3. A circle can be drawn, given a center and a radius.

4. All right angles are ninety degrees.

5. If a line intersects two other lines such that the sum of the interior angles on one side of the intersecting line is less than the sum of two right angles, then the lines meet on that side and not on the other side. (also known as the Parallel Postulate).

In fact, the last axiom is equivalent to say the sum of the interior angles of any triangle is π (180 degrees).

Hundred of mathematicians claimed that the fifth axiom is not independent and can be deduced from the first four axioms.

But all failed to prove that claim.

Those attempts were the base stone to create new geometries such as the ( elliptic geometry), (hyperbolic geometry),etc, and hundreds of Non- Euclidean geometries where the sum of the angles of the triangle in some of such geometries is greater than π and in other geometries, it is less than π.

Nowadays facts and observations about the surrounding universe proved that our universe is Non-Euclidean one. So trying to prove axioms may lead to new creative ideas that change the whole axiomatic system into a new more efficient model. Why not?

Best regards

Hi, in a mathematical theory an axiom is, by definition, taken to be true and is used as a starting point for further reasoning. Thus, an axiom is assumed to be true and don't have to be proved.

Hi, why is an axiom true?

One interesting question is where to start from. How do you prove the first theorem, if you don’t know anything yet? You can’t prove something using nothing. You need a few building blocks to start with, and these are called Axioms.

Dear Mohamed I. Riffi ,

The axiomatic system includes primitive statements. That is postulated as facts and called axioms. Obviously, all axioms in the same system should be independent and consistent.

To talk about proving axioms has no meaning. Unless we assume that axiom under consideration is not independent of the other axioms and one can try to show his claim.

A famous example is the well known geometric axioms of Euclid where we have five axioms:

Quoting:

1. A line can be drawn from a point to any other point.

2. A finite line can be extended indefinitely.

3. A circle can be drawn, given a center and a radius.

4. All right angles are ninety degrees.

5. If a line intersects two other lines such that the sum of the interior angles on one side of the intersecting line is less than the sum of two right angles, then the lines meet on that side and not on the other side. (also known as the Parallel Postulate).

In fact, the last axiom is equivalent to say the sum of the interior angles of any triangle is π (180 degrees).

Hundred of mathematicians claimed that the fifth axiom is not independent and can be deduced from the first four axioms.

But all failed to prove that claim.

Those attempts were the base stone to create new geometries such as the ( elliptic geometry), (hyperbolic geometry),etc, and hundreds of Non- Euclidean geometries where the sum of the angles of the triangle in some of such geometries is greater than π and in other geometries, it is less than π.

Nowadays facts and observations about the surrounding universe proved that our universe is Non-Euclidean one. So trying to prove axioms may lead to new creative ideas that change the whole axiomatic system into a new more efficient model. Why not?

Best regards

Perhaps the best way to think of a set of axioms is that it becomes a definition. Thus, the "axioms of a group" define the meaning of "group," etc. The important questions are: "Which (of equivalent axiom sets) is the most convenient to work with (e.g., ease of verification for significant examples)?" and "Which concepts are most useful and provide the most insight?" Think of giving a definition of "rate".

Imagine a situation that you wish to build a house. Unless you lay solid foundations, whatever you create will fall apart. In mathematics, or better say, in any mathematical theory, the role of foundations is played by system of axioms. By its very nature, mathematics is a deductive science, that is, whatever you develop or construct should follow from preceding developments. You cannot go backward all the time, otherwise you will not develop anything. You need to stop at some moment and accept that certain claims as true statements, and then move forward based on your axioms. Of course, if you change sets of axioms, you usually get different theories.

The story of axiomatization is not new, in fact it is very old and reaches back to Euclid and Aristotle. In Euclid's Elements, the second most popular book in the history of mankind (the first is the Bible) axioms are called postulates (which means 'statements assumed to be true'). To go into further details, I advise to read any textbook on logic and/or foundations of mathematics.

Dear Colleagues,

Thanks so much for your answers.

I believe that some axioms can be proved. For example, the Kolmogorov axioms which have led to development of the modern Probability Theory are consistent with the notion of relative frequency. They can be proved easily, but there are other axioms that can't be proved, although they are obviously true. Why?! If they are really obvious, why can't they be easily proved!

Thanks

@Richard Epenoy: ... an axiom is, by definition, [is] taken to be true and is used as a starting point for further reasoning.

In other words, to build a theory in mathematics, it is helpful to start with a minimal set of obvious assertions that are considered as self-evidentally true and not requiring proof. The guiding rule in an axiomatic approach is that each structure introduced in the subsequent theory must satisfy the axioms for the theory.

For followers of this thread, perhaps the OED definition of an axiom will be of interest:

axiom /"aksI@m/

· n.

1 an accepted statement or proposition regarded as being self-evidently true.

2 chiefly Mathematics a statement or proposition on which an abstractly defined structure is based.

– DERIVATIVES axiomatic adj. axiomatically adv.

– ORIGIN C15: from Fr. axiome or L. axioma, from Gk axiZma, from axios ‘worthy’.

Axioms are the starting point for developing a theory. Can we have any theory which starts from nothing? From our intuition and experience we assume something true without proof and then try to develop the subject around this/these assumption(s). I think the aim of true scientific research is to reduce the set of axioms.

Existing structures of mathematics are built on what are called axioms, statements that are intuitively true. Axioms, definitions, theorems, proofs and corollaries are among the main fabrics and sequence of a mathematical structure. It is therefore not by choice we include axioms in building structures of mathematics but by necessity, necessity of truth.

Pl. follow the link http://www.friesian.com/space.htm

An axiom is true because it is self evident, it does not require a proof. What requires a proof is the subsequent statements we make based on axioms. The axioms of integers do not require proofs as they are trivially fundamental or self evident in their validity, and number theory as a big structure of mathematics, any theorem that is proposed or claimed to be valid requires proof.

Set of axioms as building blocks of a mathematical structure in which an absent or removal of an axiom will create void and inconsistencies in the structure. Consider the axioms of integers and that lead to the development of number theory or real analysis for that matter. Which axiom(s) can be removed so that we have the same structure we use?

In fact axioms are like the foundations of a high rise building. You can see the foundations but the standing structure. Structures of mathematics are high rises of truth in which the foundations are the invisible axioms.

What I understand is: Mathematics is the tree which for the first time rose on the terra of the pure philosophy, ie logic. Logic is the mental representation of the laws of the association of events became experienced by the cause-effect relationships when the human species developed this ability. But the deeper knowledge of logic is the representation of these laws of the interaction of events with measurements, that is, the invention of numbers. Numerical theory and its sets are therefore the basis of all representations that represent measurements, and so the axioms of number theory and their sets seem to are a reflection of an unaltered world of mind. Therefore, every good, perfect definition, must be based on the theory of numbers and their sets by means of imaging, as is a differential equation in the final analysis, and not in sensorical or instinctive means. For example, this progression of numerical definitions extended the clasical Geometry, because the numbers, eg the four numbers of every points of the four-dimensional space-time, revealed to us the Generalization of Euclidean Geometry, ie the existence of the Multitude, as it is the curved space-time.

Indeed, the axioms are basic statements that we have agreed to take for granted - simple as that. One can also refer to them as a convenience. In our daily work as mathematticians, we do not really reflect upon them - they are just there for us as building blocks, just like our offices are built by raw material. So let's then agree that the axoims are the raw materials in our work. :-)