What is the difference between THEOREM, AXIOM and HYPOTHESIS?

Question

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Corrected:
What is the difference among THEOREM, AXIOM and HYPOTHESIS?

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Answer 1

All right. Take the Riemann Hypothesis, a famous unsolved problem in mathematics. It's conjectured that every non-trivial zero of the Zeta function lies on the complex line of the form 1/2+ix, and by computer, it's shown to be true for the first billion such zeros. But since there is no PROOF of this conjecture, it remains an hypothesis. If the day should come that it's ever proven, then it would become a theorem, and one of the most celebrated and important theorems of mathematics. A theorem is a statement that has been proven to be true, such as the Pythagorean Theorem, which states that the sum of the squares of the sides of any right triangle equals the square of the hypotenuse. It's a theorem because it has already been proven, and proven in literally hundreds of different ways. It's a "mathematical fact" now.

Now, but in ORDER to prove the Pythagorean Hypothesis (let's say that it has not yet been proven, so it's not yet a theorem), we have to know what's meant by a "right triangle". Well, it's easy to say, "You know, a triangle with a 90 degree angle in it". But not so fast! Even if a triangle has got a 90 degree angle in it, it is not NECESSARILY true that the sum of the squares of the sides add up to the square of the hypotenuse! How can this be, you might ask? If the right triangle was drawn on the surface of this spherical planet, so that it's kind of warped, the Pythagorean Hypothesis wouldn't be true. The missing piece of the puzzle is the concept of the Euclidian plane, or a flat plane, and order to define such a flat plane, an assumption has to be made, which is that we can have parallel lines which never meet. Or, more precisely, given any line, and any point not on it, there exists one and only one line through it that never meets the first. This may seem obvious to you, but it's not provable in of itself, and so it must be PRESUMED to be true, so we can get on with the business of doing Euclidian geometry, and prove that the Pythagorean Hypothesis is true, whereby it then becomes the Pythagorean Theorem. An axiom, then, is a statement that we PRESUME to be true. All fields of mathematics begins with such axioms, it hasn't been shown possible to even develop any branch of mathematics without at least some foundational definitions and axioms.

In science, however, the terms "theorem, axiom, hypothesis" takes on slightly different meanings. While hypothesis remains to mean "an undemonstrated conjecture", "theorem" is replaced by "theory", and "axiom" is replaced by "law". Moreover, a "theory" is just a body of hypotheses, principles, evidence, explanations, conclusions, on a given subject, such as "Theory of Plate Tectonics".

Answer 2

Hypothesis Theorem

Answer 3

Axioms are statements that are understood to be true and which a whole system may be founded on but they are so basic that there is no formal proof for them within the system. An example would be that in Euclidean geometry, parallel lines never cross. Any definition of 'parallel' in a 2D system would seem to already require an understanding that the lines never cross. A theorem is a logical sequence of ideas which comes to a conclusion (ie a proof) which is then frequently used without deriving it again eg Pythagoras Theorem. A lemma is a theorem which is used to prove another theorem(s). Euclid's lemma is often used in number theory stuff with primes. For advanced math, a piece of work will list the required lemmas because the reader might not already be familiar with them.

Answer 4

A hypothesis is a new concept.
An axiom has been an accepted hypothesis for a long time.
They cannot be proved.
A theorem can be proved often to be true or false.

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Answer 5

http://en.wikipedia.org/wiki/theorem =proved to be a general true
(based on axioms)
http://en.wikipedia.org/wiki/axiom =received as true
(cannot be proved but can be stated based on our experience, similar to postulates)
http://en.wikipedia.org/wiki/hypothesis =tested for truth
(if a hypothesis can be proved to be true based on the axioms that have been accepted as able to describe a reality OR previously proved theorems, then this hypothesis turns into a theorem describing a phenomenon in this reality. The consequences of a general theorem are called corollaries).

Answer 6

Those are three words, right? So, wouldn't you like to know what differences there are among them? 'Between' is for two, right?