Mathematics on the other hand deals in absolute truths. It doesn't matter whether you do your calculations on Earth, in outer space, under the sea or in a theoretical virtual environment on your computer. As long as you perform the operations correctly, the result will always be the same. This is where the difference between scientific theories and mathematical theorems arises. Theories are rigorous and generally accepted attempts at explaining natural phenomena; theorems are descriptions of mathematical properties, which can be proved to be logically true.
A mathematical theorem essentially has three parts. The first is the statement that the theorem asserts. For example, Pythagoras' Theorem tells us that the sum of the squares of the legs of a right-angle triangle is equal to the square of the hypotenuse. That is its statement, because it asserts this property to be true, for all such triangles (in an inner product space, where the idea of angle actually makes sense). The second part of the theorem is the proof. A theorem is useless, unless it can be shown to be correct; the proof uses known theorems and logical connections to show that the theorem logically follows from what is known. For example, if we know that the law of cosines is true, we can use this knowledge to infer Pythagoras' Theorem from it. Furthermore, we can establish the law of cosines, simply by using the definition of the cosine (we call this sort of proof one that arises from first principles). The third part of a proof is its scope. This is actually wholly encompassed in the statement, however it is an extremely important part. The scope of a proof is the collection of all the prerequisite conditions that make a theorem true. If we take Pythagoras' Theorem as an example, what it asserts is not true for arbitrary triangles, so its scope is limited to a very specific family of triangles; those triangles which have two perpendicular sides.
To make the above more obvious, we look at the following example of a Theorem:
Euclid's Theorem: There are infinitely many prime numbers.
Proof: Assume, for contradiction, that there are finitely many prime numbers, 2, 3, ..., p, where p is the largest prime. Then let S = (2)(3)...(p) + 1. Clearly, S is not divisible by any of the prime numbers up to p. Then, either S is prime, or there is some prime factor of S that is larger than p. Both of these cases contradict the assumption that p is the largest prime and therefore the assumption that there are finitely many primes.
Q.E.D.
In this theorem, the text in green is the statement of the theorem and the text in yellow is its proof. The scope of the theorem are the prime numbers. The theorem does not tell us much about any other number sets (except indirectly). The letters Q.E.D. at the end, stand for "Quod Erat Demonstrandum", which is Latin for "thus it is shown". They are often used to mark the end of a proof.
Euclid's Theorem seems like a simple statement, that might seem intuitively true and therefore not need any proof. This way of thinking gives rise to errors. Many things that seem intuitively true, can be shown to actually be false. A mathematician requires proof, before accepting a new idea; and equally, if the proof is correct, the idea is never rejected. Euclid's Theorem has been true for millennia and was never successfully contested, because its proof is logically valid and will continue being so. Newton's Theory of Gravitation (as discussed before) has been shown to be false (although a very good approximation) and is therefore now not used as a high-level model of Gravitation.
I would like to end this post with a few more terms that someone may encounter in connection with theorems and their proofs:
Proposition: A small, stand-alone proven statement, usually not as important or interesting as a theorem. Think of it as a baby theorem.
Lemma: A proven statement which is generally proved on its own and then used to prove a theorem. Think of it as a part of a theorem which deserves its own subsection but is not actually big enough to be a theorem in itself. (Note that this is not always the case; some Lemmas give information on very fundamental concepts and are much more interesting and important than most theorems.)
Corollary: A secondary proven statement that follows directly from a theorem. Usually this is a smaller result, which requires less work to obtain than the original theorem.
Conjecture: An unproven statement. This is where mathematicians allow some intuition; conjectures are usually open problems waiting for someone to prove or disprove them. They are not accepted as true. Fermat's Last Theorem was actually a conjecture until the mid-90's, when it was proved, even though it was called a theorem.
Contradiction: An impossibility, as it were. This is actually a proof technique, where we assume the opposite of what we want to prove. We then use the assumption and logic to arrive at an impossibility. This shows that the assumption must be wrong, which in turn means that what we want to prove (the opposite of the assumption) is true.
Counterexample: A situation where a statement is shown to be false. This is sometimes used with contradiction proofs, or to disprove proposed theorems. A counterexample to a theorem must conform to all its requirements, but fail to conform to its result. If there were a right-angle triangle for which Pythagoras' Theorem did not hold, that would be a counterexample which would disprove the theorem. Of course, no such triangle can be constructed in Euclidean space.
Induction: A proof technique that relies on abstraction. A statement is shown to be true for some base case and then a proof is given which shows that if the statement is true for simpler or smaller cases, then it is also true for larger or more complex ones.
