I am just beginning to study proofs. What is the difference between modus ponens and implies/if-then when relating to proofs?
I figure that we are using modus ponens to advance the proof. If we were using $P \implies Q$, couldn't I start with a false P in order to falsely "prove" Q? In the limited number of proofs I have seen it seems like we just use $P \implies Q$. I am thinking it is either shorthand for modus ponens if we assume what we start out with is true?
How should I understand $\implies$ in proofs?
Modus ponens, specifically is an inference rule. An inference rule is says that a conclusion is true if its premises are true. In other words, the inference rules of a system tell you what manipulations you are allowed to perform that don't introduce erroneous conclusions.
Here's modus ponens symbolically.
$$ \frac{A \to B \;\; \text{and} \;\; A}{B} $$
$A \to B$ is a premise. $A$ is a premise as well. $\text{and}$ exists outside the formal language and just tells us that we have both premises. $B$ is the conclusion. The line also exists outside the formal language and tells us how the premises and the conclusion are related.
Modus ponens allows you to take a conditional that you already know and its premise and infer its conclusion. This rule is fundamental to how conditionals are used.
Implies, which I've written $\to$, is how you represent the meaning of if A, then B. It is equivalent to the following:
$$ A \to B \;\; \text{is equivalent to} \;\; (\lnot A) \lor B $$
Another way of looking at it is that $A \to B$ is false if and only if $A$ is true and $B$ is false.