Motivated Proofs

This post focuses on how we can, and why we should, take the time to craft motivated proofs, i.e. proofs which do not contain any puzzling steps.

Imagine that you’re a mathematician and you have just proven a new theorem. You’re excited to show your proof to colleagues around the world, not just to get credit, but to share your knowledge and contribute to the advancement of your discipline. However, to help other mathematicians get the most out of your work, you need to do more than “just” write a correct proof. After all, even if other mathematicians can check that your proof is correct, they may still find parts of it baffling! So, how can you make sure your proof doesn’t bamboozle your colleagues? To answer this, let’s first consider two different ways they might get confused.

The first way your colleagues may get stuck is in figuring out the purpose of a given step in your proof. For example, suppose you introduce a clever construction to derive your final result. Another mathematician may read it and say “Ah, I see that this works. But how does it work?” A second source of confusion can arise if your colleague tries, but fails, to figure out what led you to introduce a given step. For example, suppose you are able to choose a particular value of a variable that makes your proof work out perfectly. A reader may get to that step and wonder “What led you to choose that value?” , wanting a more compelling answer than, for example, “I just guessed!” Let’s call proofs that cause such bafflement unmotivated. Your goal, then, is to write a motivated proof, one which helps other mathematicians to grasp the purpose of each of its steps, as well as what led you to select them.1

But, short of writing down in prose what each step does and what prompted you to introduce it (something that is frowned upon in mathematics papers!) how can you ensure you write a motivated proof? First, we should acknowledge that it is almost impossible to write a proof that will be motivated for everyone. Your colleagues will have all kinds of different mathematical backgrounds, so while one may have no trouble figuring out what each step does and what led you to select it, another may find it much harder to do so. This doesn’t mean that trying to write a motivated proof is hopeless, though. Rather, it just means that you should focus on making your proof motivated for a particular audience, who share a common mathematical “know-how.” So, the first step in writing a motivated proof is to identify your audience!

How, exactly, does identifying your audience help you to write a motivated proof? One big way is by guiding your decisions about how and when to present information to them. As an obvious example, perhaps members of your intended audience are unaware of a particular piece of information that they need to figure out how you came to introduce a particular step or the purpose that it serves. If you don’t include that piece of information in your proof, then it likely won’t be motivated for your intended audience.

As a concrete example of this, we can consider a standard proof of the Cauchy Schwarz Inequality, which works by applying another inequality where it is as sharp as possible.2 Although the proof doesn’t mention this, applying inequalities where they are sharp is generally a good idea, because doing so allows you to squeeze the most information out of them. However, many students aren’t aware of this and so have difficulty grasping how the proof writer came up with the idea of applying the inequality in that manner. Consequently, that proof is not motivated for typical students!

However, there are also more subtle ways in which decisions about the presentation of information can make your proof more or less motivated.3 For example, if certain information isn’t directly relevant for a particular step, it’s generally a good idea to “hide” it.4 That’s because hiding information removes potential red herrings that can confuse a reader over the purpose of a step or how the writer came up with it. At the same time, hiding information highlights what is not hidden, which should be more important. This strategy is very helpful, especially when the audience has limited experience with the tools and concepts used in the proof. Such audience members will generally have difficulty identifying what is or is not relevant on their own.

One effective technique for hiding information is to choose good notation and, more specifically, to make use of appropriate abbreviations. As a very simple example, if you’re working with an algebraic expression like x^3+14 x^2+51 x+54, you can hide information by introducing the abbreviation f(x) = x^3+14 x^2+51 x+54 and referring to f(x) instead of x^3+14 x^2+51 x+54. Another effective strategy which applies to proofs made up of multiple, intertwined arguments is to separate the arguments by breaking out lemmas.5 This hides information by stopping details that are important to the proof of one lemma from spilling out into another, where they are not needed.

However, while methods for writing motivated proofs are simple in theory, they can be difficult and time consuming to apply in practice. The history of mathematics documents this clearly, as some early proofs were quite confusing, and took considerable time to be transformed into more polished, motivated versions.6 There are at least two reasons for this. First, it is sometimes difficult to put yourself in the position of your audience members, approaching your proof for the first time, especially if you have spent considerable time and energy working on it! But, if you can’t do this, it will be extremely difficult to anticipate, and rectify, potentially confusing components of your work.

Second, if you want to write a motivated proof, you’ll need a deep understanding of your argument and all of its components. Such insight does not come easily! For example, you may not be able to put your finger on what (other than a lucky guess), led you to take a particular, important step in the proof. However, you’ll need to find a reason why taking that step was a good idea and then figure out an effective way to communicate this to the reader if you want to write a motivated proof! Moreover, what at first appears to be important, or even essential, to a proof may turn out to be less central once you’ve obtained a deeper understanding of it. This can cause further complications!

At this point, you might be thinking “Maybe it’s not that important for me to write a motivated proof after all!” However, despite the difficulties involved, taking the time to do so is worth the effort, both for others and for yourself! As we’ve already seen, motivated proofs benefit those who read them, by preventing confusion and promoting understanding. But, writing a motivated proof can also deepen your own mathematical knowledge. Such a deeper understanding will put you, as well as your readers, in a good position to reuse the ideas in your proof in new exciting ways!

Notes

[1] George Pólya and Sanders MacLane both consider the concept of a motivated proof in their respective articles “With, or without, motivation” (jstor link here) and “A logical analysis of mathematical structure” (jstor link here).

[2] For a more detailed discussion of that proof of the Cauchy Schwarz inequality, see my paper “Motivated proofs,” a draft of which is available on request.

[3] Jeremy Avigad and I have discussed some of these issues in our work on Dirichlet’s theorem for primes in an arithmetic progression, open access versions of which are available here and here.

[4] For more on “information hiding” in the form of modularity, see Jeremy Avigad’s paper, “Modularity in Mathematics,” a draft of which can be found on his website.

[5] If we do break out lemmas, we then have to consider when and how to present them to the reader. These issues relate to the notion of “hierarchical organization,” which Wilfried Sieg discusses in his paper “Searching for proofs (and uncovering capacities of the mathematical mind,” a copy of which is available on his website.

[6] I examine some case studies in my dissertation, available here.