I'm looking for a formal explanation that won't involve calculus (if possible) that would explain why for every group of numbers X (such that all numbers in X are integers), Max(X) = Min(-X) (where -X denotes changing all values in group X to - the values).
Any explanation would be greatly appreciated.
What you are implicitly asking about is the property of the Least Upper Bound and the Greatest Lower Bound, and the relationship between the two. Though I won't bother you with the details in this answer, you should read up on them if you are interested.
Now to the formal answer of your question, as requested:
Proposition: For any $X \subseteq \mathbb{Z}$, $\max{(X)} = \min{(-X)}$, where $-X = \{x \in X \mid x \cdot (-1) \}$
Proof:
Take $x \in X$ s.t. $x = \max(X)$. Now take $-x \in -X$. Suppose $-x$ is not the minimal element of $-X$. Then there exists $-y \in Y$ s.t. $-x > -y$ and $-y = \min(-X)$. Now consider the elements of $X$ that correspond to $-y$ and $-x$: because we have $-y \in -X$, there must exist a $(-y \cdot (-1)) \in X$. Similar for $-x$, as we know.
But if $-x > -y$, then $x < y$. But we know $x$ is the largest element of $X$, and $y>x$ implies that $y = \max(X)$, which is a contradiction. So our supposition that $-x$ is not the minimal element of $-X$ must be false. So by contradiction, $-x = \min(-X)$.