What does strength refer to in mathematics? My professors are always saying, "This theorem is strong" or "There is a way to make a much stronger version of this result" or things like that. In my mind, a strong theorem is able to tell you a lot of important information about something, but this does not seem to be what they mean. What is strength? Is it a formal idea?
 A: Suppose you have a theorem that says "If $X$, then $Y$."  There are two ways to strengthen such a theorem:


*

*Assume less.  If you can reduce the number of hypotheses, but still prove the same conclusion, then you have proved a more "powerful" result (in the sense that it applies in more situations).

*Prove more.  If you can keep the same hypotheses, but add more information to the conclusion, then you have also produced a more "powerful" result.


Here is an easy example from Geometry.

Let $ABCD$ be a (non-square) rectangle.  Then the internal angle bisectors of the vertices intersect at four points $WXYZ$, which are the vertices of a rectangle.

(You need the condition that $ABCD$ is not a square because if it is a square then all four angle bisectors coincide at a single point.)
Here are a few ways to strengthen the theorem:


*

*The hypothesis "$ABCD$ is a (non-square) rectangle" can be relaxed to the more general "$ABCD$ is a (non-rhombic) parallelogram".  The conclusion that $WXYZ$ is a rectangle still holds.

*Alternatively, you can keep the original hypothesis that $ABCD$ is a (non-square) rectangle, and strengthen to the conclusion to say that $WXYZ$ is not just a rectangle, but a square.

*Having done that, you can then strengthen the conclusion of the theorem even more, by noting that the diagonal of square $WXYZ$ is equal in length to the difference of the lengths of the sides of $ABCD$.

*Once you know that, you can now strengthen the theorem even more by (finally) removing the hypothesis that $ABCD$ is non-square, and including the  case in which the four angle bisectors coincide at a single point as forming a "degenerate" square with a diagonal of length zero.

A: In logic, saying that $\text{statement }x\text{ is stronger than statement }y$, is equivalent to saying that:
$$[\text{statement }x\text{ implies statement }y]\text{ but [statement }y\text{ does not imply statement }x]$$

For example, suppose we conjecture the following statements on a set $S$:


*

*Conjecture $x$: all the elements in $S$ are divisible by $4$

*Conjecture $y$: all the elements in $S$ are divisible by $2$


It is obvious that conjecture $x$ is stronger than conjecture $y$.
A: There are actually theorems in probability that use "strong" and "weak" in their names, for example: 


*

*The Law of Large Numbers. The Strong Law of Large Numbers, actually, implies the Weak Law of Large Numbers.

*The Central Limit Theorem (CLT). I can't find it online at the moment, but one version (the "strong" version) requires that the random variables are iid and have finite mean and variance. The "weak" version requires that the MGFs of the iid random variables exist, but this assumption can be relaxed to give the "strong" CLT.


Quid's comment on this post uses "stronger" like my example 1. From my comment above: "Usually, when I hear that Theorem B is "stronger than Theorem A" in mathematics, I think of that Theorem B uses less stringent assumptions than Theorem A for a similar result," my example 2 is of this type. 
A: In mathematics I would interpret "stronger" as "more general". In fact, being a result very specific reduces its "power" since it cannot be applied outsite exactly that case. For example, if I show that two continuos functions summed give another continuos function is "weaker" than showing that the sum of any two continuos functions is still a continuos function. This because I can apply the first result only to the two functions I considered, while the second one can be applied to any generic couple of continuos functions (even if the are not actually specified). All this is also the reason of the attention in not losing of generality when you try to prove some result.
A: Terry Tao (Ask yourself dumb questions – and answer them!):

For instance, given a standard lemma in a subject, you can ask what happens if you delete a hypothesis, or attempt to strengthen the conclusion

To strengthen a conclusion is to say more.
We could have some lemma (or theorem) that says $p \to q$. To attempt to strengthen the conclusion is to see if we can say more than just $q$ so we would try to see if we could say $p \to q_1$ where $q_1$ is some proposition s.t. $q_1 \to q$.
For instance $x=1$ is stronger than $x=0$ or $x=1$. The former implies the latter.
So if we have some assumption that implies the conclusion '$x=0$ or $x=1$', oh let's say, '$x^2 = x$', we would try to see if we could try to strengthen the conclusion to '$x=1$'. We cannot because it is possible that '$x \ne 1$' while '$x^2 = x$' (namely when '$x=0$').
Let's try using a different assumption:
It is true that '$x+1=2$' implies '$x=0$ or $x=1$'. Here, we can strengthen the conclusion to $x=1$.
A: Claims are said to be stronger or weaker, depending on the amount of information you can imply from the claim. For example, if $x$ is a positive solution of the equation $x^2 = 2$, then the claim $x > 0$ is weaker than $x > 1$ (the second is more precise).
Sometimes we try to make these implications strongest possible. For example, $1$ is the strongest integer lower bound on such an $x$.
