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I am aware that this might turn into a discussion, but I have a feeling this might have an answer (maybe something historical?) instead. I'm hoping that those with speculations keep it in the comments.

I have started to work on formal proof writing this quarter, and I discovered that the key to getting to some of them is to think of the problem "backwards." But, alas, when I wrote my proof starting with this, my professor said I shouldn't do it. But why not? It gives the reader a sense of what motivated this type of proof and allows for more understanding, doesn't it?

Mods: Feel free to close, if this turns out to be too much of a discussion. I will be in chat for those willing to discuss this.

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"my professor said I shouldn't do it." - you should maybe have asked him/her for his/her reason, no? –  J. M. Dec 31 '11 at 18:07
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When giving directions, do you start with the destination and work backwards? –  Santiago Canez Dec 31 '11 at 18:08
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I think the short answer is that a good way to think about a proof, or to arrive at a proof, is not necessarily a good way to write a proof. –  Cam McLeman Dec 31 '11 at 18:10
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I can think of one example where the stubbornness of some mathematicians to write the proof "forward" has done more to confuse students than to help them: $\epsilon-\delta$ proofs. –  Raskolnikov Dec 31 '11 at 23:33
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The most important thing to keep in mind when writing a proof is correctness and readability. I mean, the two most important things to keep in mind are correctness, readability, and clarity. I mean, the three most important things to keep in mind are correctness, readability, clarity and ... (with apologies to Monty Python). –  Arturo Magidin Dec 31 '11 at 23:36

6 Answers 6

up vote 42 down vote accepted

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.

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I agree, and let me emphasize that beginners in particular (e.g. undergraduates in their first course that involves writing proofs) are the people for whom the warning "Don't write proofs backwards!" is most relevant. Experienced mathematicians writing for other experienced mathematicians might be able to get away with more flexibility (provided what they write is correct, well written, and well organized). –  idmercer Dec 31 '11 at 21:50
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Great answer. Even after writing proofs for many years, I'll still think that I've gotten something completely nailed down but it is a mess on the page. When I go to compose it forward I'll realize I have a problem. The act of putting something that flowed more easily into a rigorous forward direction proof is an essential skill to have in my mind (even if professionally you don't always make that your final draft). –  Matt Jan 1 '12 at 0:16

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.

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In my opinion, when you write a proof you aren't trying to give the reader a sense of what motivated the proof, or allow for more understanding. The goal of a proof lies in showing that either something follows purely by logic in some theory, or convincing your audience that something follows purely by logic in some theory (and hopefully one of your audience could really show it, or get a machine to provide information to show it, if such a person cared to do so, had the time, and the resources to do so). Or at the very least, without doing this, you aren't providing a proof, you're doing something else... and this can't get said of other things. So, in some sense the theorems proved become a necessary part of the theory, or if they don't, you have to work by other logical rules and/or principles than the theory takes for granted. Theorems aren't pretty paintings (motivation) or proverb writers (understanding). They come as more comparable to tools, and you need to know that a tool will actually work or you've wasted resources, and proofs (at least hopefully) do this.

Also, I think it relevant to mention that you don't seem to have a definition of a proof. The only place I know of a precise definition of what a proof consists of, comes from formal logic. In formal logic, it comes as perspicuous that you can't have backwards proofs in general, because all proofs consist of a sequence of some sort. Though in mathematical discourse, as many other have excellently pointed out here, you don't have proofs appearing exactly in the right sequence were they to get written in some formal logical system, it doesn't seem much of a problem to rewrite the basic information in such a sequence.

So, if you write backwards proofs in your notes, I suggest you then immediately turn things around and re-sequence the proof in the proper order. I would believe that you would learn more by seeing the information both ways. In other words with respect to your question one might respond as follows:

"Why not write them forwards AND backwards, instead of just writing them one way?"

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I agree with Cam McLeman. May be your professor's concern is this: When you explain a proof backwards, the proof won't look cool any more :P. The thing is some proofs can be dumbfounding when you go A => B => C => D without giving a hint of what transpired in your mind. Apparently Gauss did that a lot and thats one of the reasons he is the prince of Mathematics and not say, Euler(my late professor's words). But if you are a good guy(a guy who wants people to understand the proof and not just appreciate your genius), you can prove this way sometimes(and this is what I assume you mean by thinking backwards).

D <=> C <=> B <=> A

No professor should be able to complain about this. This is a logically airtight proof worthy of publication. In cases where the equivalence does not hold, at least you will get some conditional equivalence to generalise the proof. But what I advise is, you may think in this way, explain to people in this way, but when you present it do the same ol A => B => C => D.

Like Zev said the reader should not lose track of your goal while still knowing exactly what is already established. May be now the B => C is a difficult-to-see proposition, but tomorrow it may be an 'Earth is round' proposition.

So in conclusion if you want to be the Duke or Knight of Logic or Mathematics, provide as little information as possible in your proof.

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If you want to provided as little information as possible in your "proof", just write: "Proof: Obvious". I've seen this before, but in no way would I acclaim such an author! –  Doug Spoonwood Jan 1 '12 at 12:05
    
Oops looks like I hurt a lot of people with that line about Gauss. This is a quote from Gauss's wiki page: Gauss usually declined to present the intuition behind his often very elegant proofs—he preferred them to appear "out of thin air" and erased all traces of how he discovered them. End of Quote I assure you I did not edit this article of the wiki. Also apologies for the casual attitude I might have taken in writing the earlier response. I did not mean to say you should reveal 'as little information as possible' in the proof. What I meant was something similar to the tone of above quote. –  ada Jan 2 '12 at 0:05

Sometimes, the intuitive reasoning where you think of the problem "backwards" is well captured in a proof by contradiction.

Suppose you're trying to prove that $A \Rightarrow Z$. Thinking backwards, you figured out that $Y \Rightarrow Z$, so it's enough to prove $Y$. You've also figured out that $X \Rightarrow Y$, so actually proving $X$ would be good enough, and so on.

Now, you can write the proof as follows: Assume that $Z$ is false. This implies $Y$ is false, hence $X$ is false too. Continuing in this fashion, you conclude that hypothesis $A$ must be false, which is a contradiction.

No professor should object to that (unless he/she is an incorrigible positivist).

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The other answers miss something important.

If I want to prove that $A$ implies $B$, I can argue as follows.

Suppose $A$.


Then $X$. Thus $Y$. Therefore $Z$. Thus $B$.

Therefore, $A$ implies $B$.

However, I can also argue "backwards", by proving the contrapositive and arguing forwards.

Suppose $\neg B$.


Then $\neg Z$. Thus $\neg Y$. Therefore $\neg X$. So $\neg A$.

Therefore, $\neg B$ implies $\neg A$, or in other words, $A$ implies $B$.

In fact, I can mix forwards and backwards approaches. The trick is to structure the argument by contradiction.

Suppose $A$ and $\neg B$.


Since $A$, thus $X$, therefore $Y$. Since $\neg B$, then $\neg Z$, therefore $\neg Y$. Contradiction.

Therefore, $A$ implies $B$.

In summary, you don't ever need to argue backwards; at least, not in classical logic.

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