Why do we write proofs "forward?"

One main problem with writing an argument backwards, especially for a student beginning to learn about proofs, is that it would be much more difficult to keep track of what is an assumption and what is a goal. In a proof that $A\implies B$, we should never along the way assume that $B$ is true, otherwise we are being circular; but if the statement of $B$ is written down on your paper already, you might get confused and think you'd already demonstrated it to be true. I'm not saying this will always happen, just that it is a greater risk.

While it's true that "thinking backwards" can sometimes be a useful strategy for attacking a problem, and explaining your strategy to the reader can be a good addition to a formal proof, it is not a substitute; one should always be able to explain the argument starting from your given information and axioms, and proceeding to the desired statement completely "forwards". It is essential to get sufficient practice with phrasing your argument this way.

Because a lot of logical implications are one way, writing things backwards can be confusing. We work backwards to know where we're going, but we write forwards to make sure everything actually works.

However, it is not always the case that proofs proceed from assumptions to goals. Here are two typical exceptions to the rule of start at the beginning and end at the end:

Theorem: XXX

proof. First, we observe that to prove XXX, it suffices to prove YYY, and proving YYY is equivalent to proving ZZZ....

or

Theorem: XXX

First, we have the following lemma:

Lemma YYY

With the lemma, we can prove the theorem as follows....

Proof of lemma. (proof goes here)

In both cases, the first step in the proof is showing we can move our goal to something simpler.

However, there are a few caveats to this style of proof. First, because lots of logical implications go only one way, you need to make sure that you are writing down things which imply your conclusion and NOT just things that follow from your conclusion. Second, because you are not proceeding in a simple order from things you know to things you don't, it is much easier to make mistakes with circular reasoning.

Third, and perhaps most important, while working backwards can make things easier for discovering a proof, it is difficult to read a long proof that is written entirely backwards. The decision to put part of the end at the beginning (or in general, to do anything out of the standard forwards order) must only be done when it improves clarity of exposition. The main reason it might improve clarity is because you have to spend a significant amount of time working towards something that seems off topic, unmotivated, or intermediate. Putting the end of the proof first in these cases means that the reader knows what they are working towards and why they are working towards it.

Please note that putting the end of a proof at the beginning and then jumping to the beginning is very different from doing the proof backwards. Until you appreciate the difference, and until you are sure that you have a very good reason for doing so and have seen enough examples to know how to do so clearly, this is not a proof-writing technique that I would recommend. Yes, if done right, it makes things clearer. However, if done wrong, it either makes things more complicated or introduces logical errors.