Some suggestions on how to present a big result to the community

Basil Rathbone is not a theorist, but one of the best—in my opinion—actors to ever portray Sherlock Holmes. I love his “version” of Holmes—just perfect in my opinion.

Today I plan on talking about how to claim a major result. How to write it up, and how to release it to the world.

Holmes will play a role in this discussion, of course. Besides being a wonderful character, he is a great problem solver. In the movies Holmes often plays his violin while thinking, much to the annoyance of Dr. John Watson. The good doctor, as he is often called, hates the “noise” of the playing. Sometimes it is at these moments when the good doctor makes a passing remark that unlocks the problem. Perhaps we should all play the violin or something like it when thinking. Or perhaps the message is working alone is hard—we all need our Watson.

My Suggestions

You have just proved the greatest result of all times. Okay a good result. Here are a few suggestions that you may wish to consider in thinking about your result, in writing it up, and in announcing it to the world. They are just my suggestions—feel free to disagree or to add you own ideas to my list.

${\bullet }$ Be Humble: The Fundamental Theorem Of Algebra states that every non-trivial polynomial over the complex numbers has at least one root. The famous Carl Gauss’s thesis was one of the first “almost correct” proofs of this theorem. It is noteworthy that even the great Gauss entitled his thesis:

A New Proof of the Theorem that Every Integral Rational Algebraic Function of One Variable can be Decomposed into Real Factors of the First or Second Degree.

Note the word “new” in his title. He later did give correct proofs; today there are perhaps hundreds of papers published, each with a slightly different proof. Some use algebra, some use analysis, some topology, some complex function theory, and on and on.

${\bullet }$ Be Humble: Okay it’s a repeat, but it is a key point.

If a problem is a well-known open problem it is likely that some new ideas are needed to solve it. You may think you have them. That is fine. Perhaps you do. But you may wish to step back and ask: what is the trick that I have seen that eluded everyone else? If your answer is: there is no new idea; it just all works, then you should be extremely skeptical.

I once heard a story that I cannot confirm, but it is so cool that I have to repeat it. Today we have just about every type of liquid sold to us in plastic containers—yes I know we should recycle, but that is another issue. There was a time when milk was sold in plastic containers, like today, but soda was still sold in glass bottles. My understanding is that a clever engineer at a large company that made plastic had the “brilliant” insight: why not sell soda in plastic containers? It would be cheaper—plus it would make money for his company, thus he would get a raise.

The engineer told his co-workers his brilliant idea. He could not understand why no one had done it before. They told him: soda in plastic was impossible. No explanation, just that it was impossible.

He went home and decided he would prove them wrong. So he took a plastic milk container, removed all the milk, cleaned it out, and poured in his favorite soda. Then he placed it in his refrigerator and went to bed.

In the morning he went over to his refrigerator and opened the door. He was shocked. They were right. There was his soda container, only it had increased in size many fold. Clearly, the gas pressure in the soda had made the plastic container expand to fill all the voids. It was like some sci-fi movie—the container had forced its way into every spare inch of space in the refrigerator.

I can only imagine the mess it made getting the container out. Yuck. The problem of putting soda into plastic containers was not trivial—there was a reason that the “obvious” idea did not work.

The story has a great ending, however. Unlike his co-workers this engineer started to work seriously on the problem. He asked “why did the plastic container fail?” Eventually he was able to create a new type of plastic that had the right properties so soda would not destroy the container.

${\bullet }$ Be Recursive: One way is to divide the great paper up into pieces. Each piece should stand on its own, and together they make the whole proof. But each piece by itself is easier to explain, easier to write up, and easier to check. This method of dividing the “secret” will also damp down any popular press issues.

In Sherlock Holmes and the Secret Weapon, a 1943 film staring Rathbone, this very trick is used to prevent the theft of a secret bombsight. A bombsight is a device used, by a crew member, to decide when to release bombs so they will hit a target. This is a real issue: the plane is moving rapidly, there usually is wind, the bomb falls a great distance before it hits the ground, and so on.

Holmes saves Dr. Franz Tobel and gets him, with his invention, to England. There Tobel divides the invention into four parts: each part does not reveal the secret of how his new bombsight works.

Of course Professor Moriarty gets involved. There are codes to be broken, escapes to be made since Holmes is captured by his arch rival, and finally a trapdoor that takes the professor by surprise. One of the best of the Holmes movies made during the war.

${\bullet }$ Be Partial: Okay you can prove a huge result. Perhaps a good idea might be to check that your methods at least extend our current knowledge. For example, if you prove that P${\neq}$NP, perhaps you could first write a paper that proves that SAT cannot be done in linear time. This is open. It would be a great result, and would show off your new methods. But it would be more credible a claim, and would still be a major result. It would likely cause less distractions from the press—that can wait for your full paper.

This happened to some extent in the recent resolution to the Poincaré Conjecture. Grigori Perelman proved much more than “just” the Poincaré, but held out the details on all his results. Experts quickly realized that he may have proved more, but even just proving the conjecture was an immense achievement.

${\bullet }$ Be Computational: I have proved theorems about ${n}$-dimensional space, even though my geometric intuition is close to zero. Once I read a claimed proof of an open problem about ${n}$-dimension space. Unfortunately, some key lemma failed even for the case when ${n=1}$: the simple line.

The suggestion is to try examples of your results. Not all proofs can be checked in this manner, but many can. Many of the great mathematicians have been tremendous calculators. They tried cases, they computed examples, they made tables of values. Some of these calculations were used to discover patterns, but they can also be used to see if your ideas make sense.

One of the neat examples of this is the story of how Major MacMahon discovered the rough growth of the partition function. Recall ${p(n)}$ is the number of ways of writing ${n}$ as a sum, if we do not count order. Thus ${p(4)}$ is ${5}$:

$\displaystyle \begin{array}{rcl} 4 &=& 1 + 1 + 1 + 1 \\ 4 &=& 1 + 1 + 2 \\ 4 &=& 1 + 3 \\ 4 &=& 2 + 2 \\ 4 &=& 4. \end{array}$

Apparently he kept lists of the number of partitions of numbers and eventually noticed that the number of decimal digits in the numbers formed a parabola. This suggested that

$\displaystyle p(n) \approx e^{c \sqrt n}.$

He was right: although the correct approximate formula is a bit more complicated.

$\displaystyle \frac{\exp \left(\pi \sqrt{2n/3} \right)}{4n\sqrt 3}.$

${\bullet }$ Be Clear: Writing mathematics is not easy. But you must do a reasonable job if you hope to be able to see if it works yourself. And of course also if you do this it will help your readers.

Gary Miller, the “Miller” in the Miller-Rabin primality test, has told me his theory of writing mathematics. He says it is definitions, definitions, and definitions—just like location, location, location in real estate. What he means is that get the definitions right. They should be clear and crisp. The more precise they are, the better the chances that your proof can be understood. The other advantage of this insight is even if one of your lemmas has a bug, you might be able to get help fixing it. If the statement of what you tried to prove is clear, then this is possible. If you definitions are murky or even worse not stated at all, there is no hope.

${\bullet }$ Be Uneven: What I mean is a more concrete notion of how to be clear in writing down your ideas. A common issue with many wrong proofs is that they spend too much precious time working on the easy part. This is what I mean by being “uneven.” Spend little time on the standard facts, and spend lots of time on the new ideas, on the parts of the proof that are tricky. On the places where you do something new.

I have seen many papers that fell apart spend huge space and time working out details to either things that are known, or on things that follow routinely from known theorems. It is a shame the writers do this—sometimes they spend huge amounts of energy on detailed LaTeX tables or figures of facts that I would not contest. Then when the big step occurs there is little or no detail.

${\bullet }$ Be Google-smart: Know the literature. If you work on any problem you should try to find out as much as you can on what is known. Almost no major work is done in a vacuum. Use the search engines, talk to experts, send friends and colleagues email, and try to see what is known about your problem.

I must admit that this is harder and harder to do these days. There is so much work going on across the world that certainly you may miss something. But you can find out a huge amount of information from the web and related sources.

When I was a graduate student things were really quite different. There was a time when I flipped through every paper in all of Computer Science. Every one. I sat in the CMU library for months and just looked at every bound journal volume that had anything to do with computing. I will not say I read every paper, but I did scan them all. I loved having the time to do this. Also I loved that I could do it. I suspect today this is hopeless, even trying to do it for a subfield is hard. But the more you know the better your chances will be to succeed.

${\bullet }$ Be ${\dots}$: Add your own suggestion here.

Open Problems

Good luck and I hope you can indeed prove some new wonderful theorems. I hope these suggestions have helped. What do you think? Also check out Lance Fortnow’s view on the same subject.

1. September 12, 2010 2:06 pm

Actually very interesting even if you do not contemplate writing earth-shattering papers. Particularly the part about ‘being computational’: I think many mathematics students feel very condescent towards those who toil with innumerable particular cases (engineers…).
It is good to remember that many results in what is thought as ‘pure’ mathematics have been prepared by working through a lot of examples.

September 12, 2010 2:49 pm

Shouldn’t the thing that you call “recursive” be called “modular”?

I think the thing that keeps people from being humble is their huge fear that someone else will claim the proof first. What is needed is some kind of time stamp, but without publicity.

September 12, 2010 4:53 pm

BrazilNut,

Like modular better. Should have thought of it.

• September 22, 2010 10:03 am

Timestamp without publicity: http://arxiv.org/
I understand this is what many good mathematicians and scientists use for this purpose. Much of what’s on there is probably obscure enough to the general public that a posting will not attract too much unwanted attention; but should still be enough to establish priority within the educated community.

• September 22, 2010 3:59 pm

If you really want your arxiv.org timestamp (priority claim) not to attract unwanted publicity, you could just copy Sir Isaac Newton’s strategy – stake claims using coded messages that only become intelligible after the full disclosure of your peer-reviewed paper.

3. September 12, 2010 3:42 pm

Dick, the ” bottle story” that you told is well-known to engineers, and a Google Books search for the phrase “our little machine groaned and grunted like it never had before” will lead you to the words of the inventor himself: Nathaniel Wyeth … the engineer member of the Wyeth family … whose story is highly entertaining.

As a separate remark, rather than holding up Gauss as an exemplar of “be mathematically humble”, it might be even better to hold Gauss up as an exemplar of “be mathematically clear”.

Here Google Books again is the student’s friend, in providing the full text of Gauss’ Disquisitiones generales circa superficies curvas and its English translation General investigations of curved surfaces of 1827 and 1825.

Gauss’ writing has for 185 years served as a model of clarity in mathematical reasoning, exposition, and typography … in particular these particular articles served (in part) as models for Don Knuth’s design of TeX … and as much as another single mathematician, it is fair to say that Gauss invented our modern norms.

But as for humility … well … Gauss’ main geometric result was called by him the Theorema Egregium (“Wonderful Theorem”) … and recognizing the value of this work, Gauss took the trouble to publish the theorem in three Latin editions, two French editions, and two German editions. The point being, Gauss’ Theorema Egregium *was* wonderful … both as a result and as a model of clarity in reasoning and presentation. Thus it would have been inappropriate not to publicize it, and the editors of multiple distinguished mathematical journals sought Gauss’ permission to do so.

As for suggestions … hmmmm … well … IMHO nowadays it is well for mathematicians to be familiar with the engineering literature and vice versa. This is because both mathematicians and engineers are struggling to encompass an ever-increasing torrent of what Dick’s weblog calls “proof technologies” and engineers call simply “technologies” … both communities are embracing steadily higher levels of abstraction to do so … and that’s why it’s shaping up to be a terrific 21st century for both disciplines! 🙂

4. September 12, 2010 4:12 pm

“Being Computational”: It may even be worth including one or two examples in the actual paper. I read somewhere that Littlewood(?) did this in some paper. People came up to him and congratulated him on his interesting new result. But the result in question wasn’t actually his, someone else had published it earlier but without the examples. Apparently just adding examples made it so much more readable that people failed to notice this…

5. September 12, 2010 6:30 pm

I like the article. However I fear some researchers are a little afraid to share partial results of new methods (like sharing the algebrazation methods that led to IP=PSPACE before proving IP=PSPACE).

6. September 12, 2010 8:06 pm

Excellent suggestions and good helps to all un-successful men who try to finish a big job!
But now I have a little suggestions to successful men(with power) such as you:

1, Sure, successful men are very smart and very able, most of un-successful men are not, but some(maybe a little) of un-successful men also are ! they only lack the opportunity and luck.

2,Every year, many people claim they finish a big job, of course most of them are terribly wrong, but most probabily, a little part of them are right! If all successful men(with power) treat them the same:wrong, then no opportunity for the little part.

3, Be Humble is a very good and logical suggestion. But for a man, he make his best effort to work for a long time,then his job is always rejected, if he still always Be Humble ,he would be crush down.

September 12, 2010 10:15 pm

I have another suggestion: start with a first section (after the introduction) that proves manageable result. Some special case or weaker result, if necessary, but something simple enough that it can be taught to students in a 1-hour lecture.

Only experts will read your full proof of your strong theorem, but many will read the part with the easy proof and present it in class.

September 12, 2010 10:35 pm

This post reminds me of a recent paper by Francisco Santos disproving the Hirsch conjecture. http://arxiv.org/abs/1006.2814

I had no familiarity with the area, but I still found the paper a smooth and interesting read–to me, it’s a great example of mathematical writing. One can see how many of the principles outlined in this post apply here: it’s fully modular; it’s computational (there’s both a “human” proof and an explicit computational construction); it’s clear; and it’s humble (he emphasizes all the open problems that are still left).

September 13, 2010 12:37 am

John Nash (A Beautiful Mind) proved quite a few big results. He seemed to be in close contact with other people involved in the problems he worked on, and was constantly informing them of his progress, leaving manuscripts around the Princeton math department for people to look at, etc. So when he made actual announcements, people were ready for them and believed him.

Me, if I solve P vs NP, I’m gonna announce it anonymously here on this blog, so I avoid embarrassment when the mistake is found 5 minutes later. 😉

September 13, 2010 1:45 am

Love this post; very good suggestions. I will disagree only with your non-mathematical second sentence: the perfect Sherlock Holmes was undoubtedly Vasily Livanov… even British fans watching with subtitles agree. 🙂

September 13, 2010 9:11 am

I agree with ano :))

12. September 13, 2010 9:36 am

Be focused: Don’t ramble on about things that are not related to what you are trying to prove, even if you think they are really neat or they’re part of the path you followed to get there (I have a huge problem with this 🙂

Be calm: It takes a long time to get the results right, and to get them into a format that will be believed and accepted. The inspiration may come quickly, but it’s only a fraction of the work (I always get too excited, too quickly, but I’m learning, slowly 🙂

Be patient: If it’s a big result, it will take some time to get accepted. That’s a good thing, but it can be frustrating if you’re hoping the world will give you instantaneous recognition. It won’t, and you’ll have to find other things to do in the meantime (or better refine you work, so it is easier to understand).

September 15, 2010 10:52 pm

Along the same lines: cut down on the jejune, smug and self-congratulatory purple prose in the introduction (“Ever since the dawn of time, mankind has sought [to solve NP-complete problems efficiently]”). The introduction should be devoted to showing that you know the state of the art and to convince readers that you’re onto something rather than already thinking about posterity.

Also, if you’ve really proved a big breakthrough, chances are there are some revolutionary, unexpected results in your paper. Still, when describing your approach in the introduction, try as much as possible to avoid making it sound like magic. If you’ve seen the AMS T-shirt where the blackboard proof reads “and then something magical happens”, you know what I mean.

September 14, 2010 6:29 pm

I agree Basil Rathbone was a brilliant Sherlock Holmes, but I have to call out Jeremy Brett for playing the role closer to the scripts and also gently but assuredly updating the portrayal as the times required/accepted.

Nice post too 🙂

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