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I have doubts about the construction of mathematical elements. There are proofs, that are proven using other theorems (corollaries) and/or axioms or definitions, such as Fermat's Last Theorem, the "weak" Goldbach theorem and thousands more.

I recently read proof of the "weak" Goldbach theorem, and it really is amazing the amount of resources, theorems, definitions and general aspects of mathematics that were used to prove that. My question is, why can not this proof become a law?. In fact, I've never heard the word "law" in mathematics, except for definitions like "closure law" or similar. I assume that the axioms act as "laws", or something like that.

Someone once told me that:

mathematics is a house of cards: if the bases are broken, the whole structure will fall.

I can not understand how such a large and complex structure could be compared with that example, but I assume that just because the proofs are not absolute laws.

In conclusion, my specific questions are:

  • What is the difference between laws and theorems?
  • can ever become law a theorem?

Sorry about my ignorance, and thanks!

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  • $\begingroup$ A "law" may be either an axiom or a theorem. It's not a terribly specific term. Sometimes things that start out being considered theorems are later taken as axioms because it turns out to be easier or "nicer" to build up that part of mathematics that way. But mathematicians generally like simple axioms. Those who sort of like complicated ones engage in an activity called "reverse mathematics", which is another story. $\endgroup$ – dfeuer Dec 14 '13 at 0:54
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    $\begingroup$ There's no answer to your two questions because "law" is not a term that mathematicians use generally, so there's not going to be any generally accepted definition for what a "law" is. What exactly do you mean by "law"? $\endgroup$ – Jack M Dec 14 '13 at 1:08
  • $\begingroup$ Something well established and tested things that can not be changed. $\endgroup$ – Tomi Dec 14 '13 at 1:27
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    $\begingroup$ @TomiSebastiánJuárez :mathematicians rarely use the word "law" at all. $\endgroup$ – Stefan Smith Dec 14 '13 at 1:33
  • $\begingroup$ @TomiSebastiánJuárez Well then, theorems are laws, I should think. They've been established to be true by a mathematical proof. Why do you suspect otherwise? $\endgroup$ – Jack M Dec 14 '13 at 1:36
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Mathematical theorems are all built on previous established theorems and axioms and derived using strict logical deductive way. So all theorems depending on previous fake ones, though the proof is absolutely right, will fail.

A well known example is the Euclid's fifth axiom in geometry, which states that two parallel lines don't intersect. Many of theorems about planar geometry are based on this. For example, there's only one line perpendicular to the given line passing through a given point out of the line. The sum of interior angle of a triangle always adds up to $\pi$ and so on. Later when people found that the axiom is not always true, the derived theorem fails. On a sphere, there will be no parallel lines. Given a point (say pole) out of some line of latitude, there are infinite lines of longtitude lines perpendicular to that line. And the sum of interior angle of a triangle is greater than $\pi$. However, the fortunate thing is, we can construct different maths based on different systems of axioms only if the system is self-consistent.

The law in mathematics is different from that in physics because there's no unified nature telling us the truth. Albert Einstein stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." The law has different observations in different systems of axioms. Usually, the law in mathematics are some facts about method of computation, some equations related different concepts and quantities. For example, associative law, commutative law and distributed law refers to method of computation; parallelogram law connects diagonal with edges for parallelogram. And there's one law which is kind of particular. That is law of large numbers, which we have experiments to support that.

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Mathematics is structured as follows: at the basis you have axioms. those are statements that you assume to be true. An example of a set of axioms is ZFC, upon which most (all) of modern mathematics is based upon. It contains statements which are somewhat "obvious" and really basic, such as "If two sets contain exactly the same elements, then they are equal".

Then there are definitions. Those group together objects with similar properties giving them a common name. An example of that is the definition of what is a group, or the definition of what is a function.

Finally, there are the propositions. You often see them labeled as "lemmas" or "theorems", depending on their importance (usually "lemmas" are auxiliary propositions used to prove a deeper "theorem"). Those are statements that are a consequence of the axioms. They are proven using only things you already know to be true (such as the axioms or previously proven propositions). An example of proposition is Brower's fixed point theorem, which states that a continuous function from the closed unit ball in $\mathbb{R}^2$ to itself must have a fixed point.

You were asking about laws, but you didn't specify what you mean by a "law". I think that you could classify as such the union of axioms and theorems, i.e. all the things you know for sure to be true. Then the difference between laws and theorems is that a theorem is a law, but a law is not necessarily a theorem.

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  • $\begingroup$ Somebody downvoted my answer. I'd like to know why. $\endgroup$ – Daniel Robert-Nicoud Dec 14 '13 at 16:11
  • $\begingroup$ Agree. So did on mime and even this not bad question. Somebody may play his privilege. $\endgroup$ – Shuchang Dec 15 '13 at 0:47

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