## What if math dies?

Over the years I’ve heard a lot about the apparent complete uselessness and inapplicability of modern mathematics, about how I should always look for applications since without them all I am doing is a pointless intellectual pursuit, blah, blah, blah. I had strangers on the plane telling me this (without prompting), first dates (never to become second dates) wondering if “*any formulas changed over the last 100 years, and if not what’s the point*“, relatives asking me if I ever “*invented a new theorem*“, etc.

For whatever reason, everyone always has an opinion about math. Having never been accused of excessive politeness I would always abruptly change the subject or punt by saying that the point is “*money in my Wells Fargo account*“. I don’t even have a Wells Fargo account (and wouldn’t want one), but what’s a small lie when you are telling a big lie, right?

Eventually, you do develop a thicker skin, I suppose. You learn to excuse your friends as well meaning but uneducated, journalists as maliciously ignorant, and strangers as bitter over some old math learning experience (which they also feel obliged to inform you about). However, you do expect some understanding and respect from fellow academics. “*Never compare fields*” Gian-Carlo Rota teaches, and it’s a good advice you expect sensible people to adhere. Which brings me to this:

#### The worst idea I’ve heard in a while

In a recent interview with Glenn Loury, a controversial UPenn law professor Amy Wax proposed to reduce current mathematics graduate programs to one tenth or one fifteenth of their current size (start at 54.30, see also * partial transcript*). Now, I get it. He is a proud member of the “intellectual dark web“, while she apparently hates liberal education establishment and wants to rant about it. And for some reason math got lumped into this discussion. To be precise, Loury provoked Wax without offering his views, but she was happy to opine in response. I will not quote the discussion in full, but the following single sentence is revealing and worth addressing:

If we got rid of ninety percent of the math Ph.D. programs, would we really be worse off in any material respect? I think that’s a serious question.

She followed this up with “*I am not advocating of getting rid of a hundred percent of them*.” Uhm, thanks, I guess…

#### The inanity of it all

One is tempted to close ranks and ridicule this by appealing to authority or common sense. In fact, just about everyone — from Hilbert to Gowers — commented on the importance of mathematics both as an intellectual endeavor and the source of applications. In the US, we have about 1500-2000 new math Ph.D.’s every year, and according to the AMS survey, nearly all of them find jobs within a year (over 50% in academia, some in the industry, some abroad).

In fact, our math Ph.D. programs are the envy of the world. For example, of the top 20 schools worldwide between 12 and 15 are occupied by leading US programs depending on the ranking (see e.g. here or there for recent examples, or more elsewhere). Think about it: math requires no capital investment or infrastructure at all, so with the advent of personal computing, internet and the arXiv, there are little or no entry barriers to the field. Any university in the world can compete with the US schools, yet we are still on the top of the rankings. It is bewildering then, why would you even want to kill these super successful Ph.D. programs?

More infrastructurally, if there are drastic cuts to the Ph.D. programs in the US, who would be the people that can be hired to teach mathematics by the thousands of colleges whose students want to be math majors? The number of the US math majors is already over 40,000 a year and keep growing at over 5% a year driven in part by the higher salary offerings and lifetime income (over that of other majors). Don’t you think that the existing healthy supply and demand in the market for college math educators already determined the number of math Ph.D.’s we need to produce?

Well, apparently Wax doesn’t need convincing in the importance of math. “*I am the last person to denigrate pure mathematics. It is a glory of mankind*…” She just doesn’t want people doing new research. Or something. As in “enough already.” Think about it and transfer this thought to other areas. Say — *no new music is necessary* — Bach and Drake said it all. Or — *no new art is necessary — *Monet and Warhol were so prolific, museums don’t really have space for new works. Right…

#### Economics matters

Let’s ask a different question: why would you want to close Ph.D. programs when they **actually make money**? Take UCLA. We are a service department, which makes a lot of money from teaching all kinds of undergraduate math courses + research grants both federal, state and industrial. Annually, we graduate over 600 students with different types of math/stat majors, which constitutes about 1.6% of national output, the most of all universities.

Let’s say our budget is $25 mil (I don’t recall the figures), all paid for. That would be out of UCLA budget of $7.5 billion of which less than 7% are state contributions. Now compare these with football stadiums costs which are heavily subsidized and run into hundreds of millions of dollars. If you had to cut the budget, is math where you start?

#### Can’t we just ignore these people?

Well, yes we can. I am super happy to dismiss hurried paid-by-the-word know-nothing journalists or some anonymous YouTube comments. But Amy Wax is neither. She is smart and very accomplished: summa cum laude from Yale, M.D. cum laude from Harvard Medical School, J.D. from Columbia Law School where she was an editor of *Columbia Law Review*, argued 15 cases in the US Supreme Court, is a named professor at UPenn Law School, has dozens of published research papers in welfare, labor and family law and economics. Yep.

One can then argue — she knows a lot of other stuff, but *nothing* about math. She is clearly controversial, and others don’t say anything of that nature, so who cares. That sounds right, but so what? Being known as controversial is like license to tell “the truth”… er… what they really think. Which can include silly things based on no research into our word. This means there are numerous other people who probably also think that way but are wise enough or polite enough not to say it. We need to fight this perception!

And yes, sometimes these people get into positions of power and decide to implement the changes. Two cases are worth mentioning: the **University of Rochester** failed attempt to close its math Ph.D. program, and the **Brown University fiasco**. The latter is well explained in the “*Mathematical Apocrypha Redux*” (see the relevant section here) by the inimitable Steven Krantz. Rating-wise, this was a disaster for Brown — just read the Krantz’s description.

The Rochester story is rather well documented and is a good case of study for those feeling too comfortable. Start with this *Notices* article, proceed to *NY Times*, then to protest description, and this followup in the *Notices* again. Good news, right? Well, I know for a fact that other administrators are also making occasional (largely unsuccessful) moves to do this, but I can’t name them, I am afraid.

#### Predictable apocalypse

Let’s take Amy Wax’s proposal seriously, and play out **what would happen if 90-93% of US graduate programs in mathematics are closed on January 1, 2020**. By law. Say, the US Congress votes to deny all federal funds to universities if they maintain a math Ph.D. program, except for the top 15 out of about 180 graduate programs according to *US News*. Let’s ignore the legal issues this poses. Just note that there are various recent and older precedents of federal government interfering with state and private schools (sometimes for a good cause).

Let’s just try to quickly game out what would happen. As with any post-apocalyptic fiction, I will not provide any proofs or reasoning. But it’s all “reality based”, as two such events did happened to mathematicians in the last century, one of them deeply affecting me: the *German “academic reforms”* in late 1930s (see e.g. here or there), and the *Russian exodus* in early 1990s (see e.g. here or there, or there). Another personally familiar story is an implosion of mathematics at Bell Labs in late 1990s. Although notable, it’s on a much smaller scale and to my knowledge has not been written about (see the discussion here, part 6).

**First**, there will be huge exodus of distinguished mathematics faculty from school outside of the 15 schools. These include members of the National Academy of Sciences, numerous ICM speakers, other award winners, etc. Some will move overseas (Canada, Europe, Japan, China, etc.), some will retire, some leave academia. Some will simply stop doing research given the lack of mathematical activity at the department and no reward for doing research.

**Second**, outside of top 15, graduate programs in other subjects notice falling applications resulting in their sliding in world ranking. These include other physical sciences, economics and computer science. Then biological and social sciences start suffering. These programs start having their own exodus to top 15 school and abroad.

**Third**, given the sliding of graduate programs across the board, the undergraduate education goes into decline across the country. Top US high school students start applying to school abroad. Many eventually choose to stay in these countries who welcome their stem excellence.

**Fourth**, the hitech, fintech and other science heavy industries move abroad closer to educated employees. United States loses its labor market dominance and starts bleeding jobs across all industries. The stocks and housing market dip down.

**Fifth**, under strong public pressure the apocalyptic law is repealed and all 180 Ph.D. programs are reinstated with both state and federal financial support. To everyone’s surprise, nobody is moving back. Turns out, destroying is much faster and easier than rebuilding, as both Germany and Russia discovered back in the 20th century. From that point on, January 1, 2020 became known as the day the math died.

#### Final message:

Dear Amy Wax and Glenn Loury! Please admit that you are wrong. Or at least plead ignorance and ask for forgiveness. I don’t know if you will ever see this post or have any interest in debating the proposition I quoted, but I am happy to do this with you. Any time, any place, any style. Because the future of academia is important to all of us.

## The status quo of math publishing

We all like the **status quo**. It’s one of my favorite statuses… The status quo is usually excellent or at least good enough. It’s just so tempting to do nothing at all that we tend to just keep it. For years and years which turn into decades. Until finally the time has come to debate it…

Some say the **status quo on math publishing** is unsustainable. That the publishers are much too greedy, that we do all the work and pay twice, that we should boycott the most outrageous of these publishers, that the University of California, German, Hungary, Norway and Swedish library systems recent decisions are a watershed moment calling for action, etc. My own institution (UCLA) is actually the leader in the movement. While I totally agree with the sentiment, I mostly disagree with the boycott(s) as currently practiced and other proposed measures. It comes from a position of weakness and requires major changes to the status quo.

Having been thinking about this all for awhile, I am now very optimistic. In fact, there is a way we can use our natural position of strength to achieve all the goals we want while keeping the status quo. It may seem hard to believe, but with a few simple measures we can get there in a span of a few years. This post is a long explanation of how and why we do this.

#### What IS the current status quo?

In mathematics, it’s pretty simple. We, the mathematicians, do most of the work: produce a decent looking .pdf file, perform a peer review on a largely volunteer basis (some editors do get paid occasionally), disseminate the results as best as we can, and lobby our libraries to buy the journal subscriptions. The journals collect the copyright forms, make minor edits to the paper to conform to their favorite style, print papers on paper, mail them to the libraries, post the .pdf files on the internet accessible via library website, and charge libraries outrageous fees for these services. They also have armies of managers, lawyers, shareholders, etc. to protect the status quo.

Is it all *good* or *bad*? It’s mostly good, really. We want all these basic services, just disagree on the price. There is an old Russian Jewish proverb, that *if a problem can be solved with money — it’s not a real problem but a business expense* (here is a modern version). So we should deal with predatory pricing as a business issue and not get emotional by boycotting selective journals or publishers. We can argue for price decreases completely rationally, by showing that their product lost 90%, but not all its value, and that it’s in our common interest to devalue it, but not kill it.

#### Why keep the status quo?

This is easy. We as a community tend to like our journals more than we hate them. They compete for our papers. We compete with each other to get published in best places. This means we as a community know which journals are good, better or best in every area, or in the whole field of mathematics. This means that each journal has composed the best editorial board it could. It would be a waste to let this naturally formed structures go.

Now, in the past I strongly criticized top journals, the whole publishing industry, made fun of it, and more recently presented an ethical code of conduct for all journals. Yet it’s clear that the cost of complete destruction of existing journal nomenclature is too high to pay and thus unlikely to happen.

#### Why changing the status quo is impractical?

Consider the alternatives. Yes, the editorial board resignations do happen, most recently in the *Journal of Algebraic Combinatorics* (JACO) which resigned in mass to form a journal named *Algebraic Combinatorics *(ALCO)*. * But despite laudations, the original journal exists and doing fine or at least ok. To my dismay and mild disbelief, the new Editorial Board of JACO has some well-known and wildly respected people. Arguably, this is not the outcome the resigners aimed for (for the record, I published twice in JACO and recently had a paper accepted by ALCO).

Now, at first, starting new journals may seems like a great idea. Unfortunately, by the conservative nature of academia they always struggle to get off the ground. Some survive, such as *EJC* or *EJP*, have been pioneers in the area, but others are not doing so well. The fine print is also an issue — the much hyped *Pi* and S*igma* charge $1000 per article for “processing”, whatever that entails. Terry Tao wrote that these journals suggest “alternatives to the *status quo”. *Maybe. But how exactly is that an improvement? (Again, for the record, I published in both *EJC*, *EJP*, and recently in *Sigma*. No, I didn’t pay, but let me stay on point here — that story can wait for another time.)

Other alternatives are even less effective. Boycotting selective publishers gives a free reign to others to charge a lot, at the time when *we need a systemic change*. I believe that it gives all but the worst publishers the cover they need to survive, while the worst already have enough power to survive and remain in the lead. There is a long argument here I am trying to avoid. Having had it with Mark Wilson, I know it would overwhelm this post. Let me not rebut it thoroughly point-by-point, but present my own vision.

#### What can we do?

**Boycott them all!** I mean all non-free journals, at all times, at all cost. By that I don’t mean everyone should avoid submission, refereeing, being on the editorial board. Not at all, rather opposite. Please do NOT boycott anyone specifically, proceed with your work, keep the status quo.

What I mean is this. **Boycott all non-free journals as a consumer!** Do NOT download papers from journal websites. I will give detailed suggestions below, after I explained my rationale. In short, every time you download a paper from the journal website it gives publishers leverage to claim they are indispensable, and gives libraries the fear of faculty revolt if they unsubscribe. They (both the publishers and the libraries) have no idea how little we need the paid journal websites.

#### Detailed advice on how to boycott all math journal publishers

Follow the following simple rules. On your side as an author, make **every(!) **paper you ever wrote freely accessible. Not just the latest – all of them! Put them on the * arXiv*, viXra, your own website, or anywhere you like as long as the search engines can find them. If you don’t know how, ask for help. If you can read this WP blog post, you can also post your papers on some WP site. If you are afraid of the copyright, snap out of it! I do this routinely, of course. Many greats have also done this for all their papers, e.g. Noga Alon and Richard Stanley. Famously, all papers by Paul Erdős are online. So my message for all of you reading this: if you don’t have all your papers free online, go ahead,

**just post them all!**Yes, that means

**right now!**Stop reading and come back when you are done.

Now, for reading papers the rules are more complicated. Every time you need to download an article, don’t go to *MathSciNet*. Instead, google it first. *Google Scholar* usually gives you multiple options on the download location. Choose the one in the * arXiv* or author’s website. Done.

If you fail, but feel the paper could be available from some nefarious copyright violating websites, consider using Yandex, DuckDuckGo, or other search engines which are less concerned about the copyright.

Now, suppose the only location is the journal website. Often, this happens when the paper is old or old-ish, i.e. outside the 4 year sliding window for *Elsevier. *As far as I am concerned, this part of the publisher is “free” since anyone in the world can download it without charge. Make sure you download the paper without informing your campus library. This is easy off campus — use any browser without remote access (VPN). On campus, use a browser masking your ip address, i.e. the *Opera*.

Now, suppose nothing works. Say, the paper is recent but inaccessible for free. Then email to the authors and request the file of paper. Shame them into putting the paper online while you are at it. Forward them this blog post, perhaps.

Suppose now the paper is inaccessible for free, but the authors are non-responsive and unlikely to ever make the paper available. Well, ok — download it from the journal website then via your library. But then be a mensch. Post the paper online. Yes, in violation of copyright. Yes, other people already do it. Yes, everyone is downloading them and would be grateful. No, they won’t fight us all.

Finally, suppose you create a course website. Make sure all or at least most of your links are to free version of the articles. Download them all and repost them on your course website so the students can bypass the library redirect. Every bit helps.

#### Why would this work? I. Shaming is powerful.

Well, in mathematics shaming is widespread and actually works except in some extreme cases. It’s routine, in fact, to shame authors for not filling gaps in their proofs, for not acknowledging priority, or for not retracting incorrect papers (when the authors refuse to do it, the journals can also be shamed). Sometimes the shaming doesn’t work. Here is my own example of shaming fail (rather extreme, unfortunately), turned shaming success on pages of this blog.

More broadly, public shaming is one of the key instruments in the 21st century. *Mathbabe* (who is writing a book about shaming) notably shamed Mochizuki for not traveling around to defend his papers. Harron famously shamed white cis men for working in academia. Again, maybe not in all cases, but in general public shaming works rather well, and there is a lot of shaming happening everywhere.

So think about it — what if we can shame ** every** working mathematician into posting

**their papers online? We can then convince libraries that we don’t**

*all**need*to renew all our math journal subscriptions since we can function perfectly well without them. Now, we would still want the journal to function, but are prepared to spend maybe 10-15% of the prices that

*Springer*and

*Elsevier*currently charge. Just don’t renew the contract otherwise. Use the savings to hire more postdocs, new faculty, give students more scholarships to travel to conferences, make new Summer research opportunities, etc.

#### Why would this work? II. Personal perspective.

About a year ago I bought a new laptop and decided to follow some of the rules above as an experiment. The results were surprisingly good. I had to download some old non-free papers from publisher sites maybe about 4-5 times a month. I went to the library about once every couple of months. For new papers, I emailed the authors maybe the total of about once every three months, getting the paper every time. I feel I could have emailed more often, asking for old papers as well.

Only occasionally (maybe once a month) I had to resort to overseas paper depositaries, all out of laziness — it’s faster than walking to the library. In summary — it’s already easy to be a research mathematician without paying for journals. In the future, it will get even easier.

#### Why would this work? III. Librarian perspective.

Imagine you are a head librarian responsible for journal contracts and purchasing. You have access to the download data and you realize that many math journals continue to be useful and even popular. The publishers bring you a similar or possibly more inflated date showing their products in best light. Right now you have no evidence the journals are largely useless are worried about backslash which would happen if you accidentally cut down on popular journals. So you renew just about everything that your library has always been subscribing and skip on subscribing to new journals unless you get special requests for the faculty that you should.

Now imagine that in 2-3 years your data suggests rapidly decreasing popularity of the journals. You make a projection that the downloads will decrease by a factor of 10 within a few more years. That frees you from worrying about cancelling subscriptions and gives you strong leverage in negotiating. Ironically that also helps you keeps the status quo — the publishers slash their price but you can keep most of the subscriptions.

#### Why would this work? IV. Historical perspective.

The history is full of hard fought battles which were made obsolete by cultural and technological changes. The examples include the “war of the currents“, the “war” of three competing NYC subway systems, same with multiple US railroads, the “long-distance price war“, the “browser war” and the “search engine war“. They were all very different and resolved in many different ways, but have two things in common — they were ruthless at the time, and nobody cares anymore. Even the airlines keep slashing prices, making services indistinguishably awful to the point of becoming near-utilities like electric and gas companies.

The same will happen to the journal publishing empires. In fact, the necessary technology has been available for awhile — it’s the culture that needs to change. Eventually all existing print journals will become glorified versions of arXiv overlay publications with substantially scaled down stuff and technical production. Not by choice, of course — there is just no money in it. Just like the airline travel — service will get worse, but much cheaper.

The publishers will continue to send print copies of journals to a few dozen libraries worldwide which will be immediately put into off-campus underground bunker-like storages as an anti-apocalyptic measure, and since the reader’s demand will be close to nonexistent. They will remain profitable by cutting cost everywhere since apparently this is all we really care about.

The publishers already know that they are doomed, they just want to prolong the agony and extract as much rent as they can before turning into public utilities. This is why the *Elsevier* refuses to budge with the UC and other systems. They realize that publicly slashing prices for one customer today will lead to an avalanche of similar demands tomorrow, so they would rather forgo a few customers than start a revolution which would decimate their journal value in 5 years (duration of the *Elsevier *contract).

None of this is new, of course. Odlyzko described it all back in 1997, in a remarkably prescient yet depressing article. Unfortunately, we have been moving in the wrong direction. Gowers is right that publishers cannot be shamed, but his efforts to shame people into boycotting *Elsevier* may be misplaced as it continues going strong. The shaming did lead to the continuing conversation and the above mentioned four year sliding window which is the key to my proposal.

#### What’s happening now? Why is Elsevier not budging?

As everyone who ever asked for a discount knows, you should do this privately, not publicly. Very quietly slashing the prices by a factor of 2, then trying to play the same trick again in 5 years would have been smarter and satisfied everybody. To further help *Elsevier *hide the losses from shareholders and general public, the library could have used some bureaucratic gimmicks like paying the same for many journals but getting new books for free or something like that. This would further confuse everybody except professional negotiators on behalf of other library systems, thus still helping to push the prices down.

But the UC system *wanted* to lead a revolution with their public demands, so here we are, breaking the status quo for no real reason. There are no winners here. Even my aunt Bella from Odessa who used to take me regularly to Privoz Market to watch her bargain, could have told you that’s exactly what’s going to happen…

Again, the result is bad for everybody — the *Elsevier* would have been happier to get some money — less than the usual amount, but better than nothing given the trivial marginal costs. At the same time, we at UCLA still need the occasional journal access while in the difficult transition period.

#### AMS, please step up!

There is one more bad actor in the whole publishing drama whose role needs to change. I am speaking about the *AMS*, which is essentially a giant publishing house with an army of volunteers and a side business of organizing professional meetings. Let’s looks at the numbers, the 2016 annual report (for some reason the last one available). On p.12 we read: of the $31.8 mil operating revenue dues make up about 8%, meetings 4%, while publishing a whopping 68%. No wonder the AMS is not pushing for changes in current journal pay structure — they are conflicted to the point of being complicit in preserving existing prices.

But let’s dig a little deeper. On p.16 we see that the journals are fantastically profitable! They raise $5.2 mil with $1.5 mil in operating expenses, a 247% profit margin. With margins like that who wants to rock the boat? Compare this with next item — books. The *AMS* made $4.1 mil while spent $3.6 mil. That’s a healthy 14% profit margin. Nice, but nothing to write home about. By its nature, the book market is highly competitive as libraries and individuals have option to buy them or not on a per title basis. Thus, the competition.

If you think the *AMS* prices are lower than of other publishers, that’s probably right. This very dated page by Kirby is helpful. For example, in 1996, the *PTRF* (Springer) charged $2100, the *Advances* (Academic Press, now Elsevier) $1326, the *Annals *(Princeton Univ. Press) $200, while *JAMS *only $174. Still…

What should be done? Ideally, the *AMS* should sell its journal business to some university press and invest long-term the sale profits. That would free it to pursue the widely popular efforts towards free publishing. In reality that’s unlikely to happen, so perhaps some sort of “Chinese wall” separating journal publishing and the *AMS* political activities. This “wall” might already exist, I wouldn’t know. I am open to suggestions. Either way, I think the *AMS* members should brace themselves for the future where the *AMS* has a little less money. But since the *MathSciNet *alone brings 1/3 of the revenue, and other successful products like *MathJobs* are also money makers, I think the *AMS* will be fine.

I do have one pet peeve. The *MathSciNet,* which is a good product otherwise, should have a “web search” button next to the “article” button. The latter automatically takes you to the journal website, while the former would search the article on *Google Scholar* (or *Microsoft Academic*, I suppose, let the people choose a default). This would help people circumvent the publishers by cutting down on clicks.

#### What gives?

I have always been a non-believer in boycotts of specific publishers, and I feel the history proved me more right than wrong. People tend to avoid boycotts when they have significant cost, and without the overwhelming participation boycotts simply don’t work. Asking people not to submit or referee for the leading journals in their fields is like asking to voluntarily pay higher taxes. Some do this, of course, but most don’t, even those who generally agree with higher taxes as a good public policy.

In fact, I always thought we need some kind of one-line bill by the US Congress requiring all research made at every publicly funded university being available for free online. In my conspiratorial imagination, the *AMS* being a large publisher refused to bring this up in its lobbying efforts, thus nothing ever happened. While I still think this bill is a good idea, I no longer think it’s a necessary step.

Now I am finally optimistic that the boycott I am proposing is going to succeed. The (nearly) free publishing is coming! Please spread the word, everybody!

UPDATE (March 19, 2019): Mark Wilson has a blog post commenting and clarifying ALCO vs. JACO situation.

## What we’ve got here is failure to communicate

Here is a lengthy and somewhat detached followup discussion on the very unfortunate Hill’s affair, which is much commented by Tim Gowers, Terry Tao and many others (see e.g. links and comments on their blog posts). While many seem to be universally distraught by the story and there are some clear disagreements on ** what** happened, there are even deeper disagreements on

**happened. The latter question is the subject of this blog post.**

*what should have***Note:** Below we discuss both the *ethical* and *moral* aspects of the issue. Be patient before commenting your disagreements until you finish the reading — there is a ** lengthy disclaimer** at the end.

#### Review process:

- When the paper is submitted there is a very important email acknowledging
. Large publishers have systems send such emails automatically. Until this email is received, the paper is not considered submitted. For example, it is not unethical for the author to get tired of waiting to hear from the journal and submit elsewhere instead. If the journal later comes back and says “sorry for the wait, here are the reports”, the author should just inform the journal that the paper is under consideration elsewhere and should be considered withdrawn (this happens sometimes).*receipt of the submission* - Similarly, there is a very important email acknowledging
. Until this point the editors ethically can do as they please, even reject the paper with multiple positive reports. Morality of the latter is in the eye of the beholder (cf. here), but there are absolutely no ethical issues here unless the editor violated the rules set up by the journal. In principle, editors can and do make decisions based on informal discussions with others, this is totally fine.*acceptance of the submission* - If a journal withdraws acceptance
*after*the formal acceptance email is sent, this is potentially a serious violation of ethical standards.**Major exception:**this is not unethical if the journal follows a certain procedural steps (see the section below). This should not be done except for some extreme circumstances, such as last minute discovery of a counterexample to the main result which the author refuses to recognize and thus voluntarily withdraw the paper. It is not immoral since until the actual publication no actual harm is done to the author. - The next key event is
of the article, whether online of in print, usually/often coupled with the transfer of copyright. If the journal officially “**publication***withdraws acceptance*” after the paper is published without deleting the paper, this is not immoral, but depends on the procedural steps as in the previous item. - If a journal
the paper*deletes**after*the publication, online or otherwise, this is a gross violation of both moral and ethical standards. The journals which do that should be ostracized regardless their reasoning for this act.**Major exception:**the journal has*legal reasoning,*e.g. the author violated copyright laws by lifting from another published article as in the Dănuț Marcu case (see below).

#### Withdrawal process:

- As we mentioned earlier, the withdrawal of accepted or published article should be extremely rare, only in extreme circumstances such as a major math error for a not-yet-published article or a gross ethical violation by the author or by the handling editor of a published article.
- For a published article with a major math error or which was later discovered to be known, the journal should not withdraw the article but instead work with the author to publish an
or an*erratum*Here an erratum can be either fixing/modifying the results, or a complete withdrawal of the main claim. An example of the latter is an erratum by Daniel Biss. Note that the journal can in principle publish a note authored by someone else (e.g. this note by Mnёv in the case of Biss), but this should be treated as a separate article and not a substitute for an erratum by the author. A good example of acknowledgement of priority is this one by Lagarias and Moews.*acknowledgement of priority.* - To withdraw the disputed article the journal’s editorial board should either follow the procedure set up by the publisher or set up a procedure for an ad hoc committee which would look into the paper and the submission circumstances. Again, if the paper is already published, only non-math issues such as ethical violations by the author, referee(s) and/or handling editor can be taken into consideration.
- Typically, a decision to form an ad hoc committee or call for a full editorial vote should me made by the editor in chief, at the request of (usually at least two) members of the editorial board. It is totally fine to have a vote by the whole editorial board, even immediately after the issue was raised, but the threshold for successful withdrawal motion should be set by the publisher or agreed by the editorial board
*before*the particular issue arises. Otherwise, the decision needs to be made by consensus with both the handling editor and the editor in chief abstaining from the committee discussion and the vote. - Examples of the various ways the journals act on withdrawing/retracting published papers can be found in the case of notorious plagiarist Dănuț Marcu. For example,
*Geometria Dedicata*decided not to remove Marcu’s paper but simply issued a statement, which I personally find insufficient as it is not a retraction in any formal sense. Alternatively,*SUBBI*‘s apology is very radical yet the reasoning is completely unexplained. Finally, Soifer’s statement on behalf of*Geombinatorics*is very thorough, well narrated and quite decisive, but suffers from authoritarian decision making. - In summary, if the process is set up in advance and is carefully followed, the withdrawal/retraction of accepted or published papers can be both appropriate and even desirable. But when the process is not followed, such action can be considered unethical and should be avoided whenever possible.

#### Author’s rights and obligations:

- The author can withdraw the paper at any moment until publication. It is also author’s right not to agree to any discussion or rejoinder. The journal, of course, is under no obligation to ask the author’s permission to publish a refutation of the article.
- If the acceptance is issued, the author has every right not go along with the proposed quiet withdrawal of the article. In this case the author might want to consider complaining to the editor in chief or the publisher making the case that the editors are acting inappropriately.
- Until acceptance is issued, the author should not publicly disclose the journal where the paper is submitted, since doing so constitutes a (very minor) moral violation. Many would disagree on this point, so let me elaborate. Informing the public of the journal submission is tempting people in who are competition or who have a negative opinion of the paper to interfere with the peer review process. While virtually all people virtually all the time will act honorably and not contact the journal, such temptation is undesirable and easily avoidable.
- As soon as the acceptance or publication is issued, the author should make this public immediately, by the similar reasoning of avoiding temptation by the third parties (of different kind).

#### Third party outreach:

- If the paper is accepted but not yet published, reaching out to the editor in chief by a third party requesting to publish a rebuttal of some kind is totally fine. Asking to withdraw the paper for mathematical reasons is also fine, but should provide a clear formal math reasoning as in “Lemma 3 is false because…” The editor then has a choice but not an obligation to trigger the withdrawal process.
- Asking to withdraw the not-yet-published paper without providing math reasoning, but saying something like “this author is a crank” or “publishing this paper will do bad for your reputation” is akin to bullying and thus a minor ethical violation. The reason it’s minor is because it is journal’s obligations to ignore such emails. Journal acting on such an email with rumors or unverified facts is an ethical violation on its own.
- If a third party learns about a publicly available paper which may or may not be an accepted submission with which they disagree for math or other reason, it it ethical to contact the author directly. In fact, in case of math issues this is highly desirable.
- If a third party learns about a paper submission to a journal without being contacted to review it, and the paper is not yet accepted, then contacting the journal is a strong ethical violation. Typically, the journal where the paper is submitted it not known to public, so the third party is acting on the information it should not have. Every such email can be considered as an act of bullying no matter the content.
- In an unlikely case everything is as above but the journal’s name where the paper is submitted is publicly available, the third party
*can*contact the journal. Whether this is ethical or not depends on the wording of the email. I can imagine some plausible circumstances when e.g. the third party knows that the author is Dănuț Marcu mentioned earlier. In these rare cases the third party should make every effort to CC the email to everyone even remotely involved, such as all authors of the paper, the publisher, the editor in chief, and perhaps all members of the editorial board. If the third party feels constrained by the necessity of this broad outreach then the case is not egregious enough, and such email is still bullying and thus unethical. - Once the paper is published anyone can contact the journal for any reason since there is little can be done by the journal beyond what’s described above. For example, on two different occasions I wrote to journals pointing out that their recently published results are not new and asking them to inform the authors while keeping my anonymity. Both editors said they would. One of the journals later published an acknowledgement of retribution. The other did not.

#### Editor’s rights and obligations:

- Editors have every right to encourage submissions of papers to the journal, and in fact it’s part of their job. It is absolutely ethical to encourage submissions from colleagues, close relatives, political friends, etc. The publisher should set up a clear and unobtrusive
*conflict of interest*directive, so if the editor is too close to the author or the subject he or she should transfer the paper to the editor in chief who will chose a different handling editor. - The journal should have a clear scope worked out by the publisher in cooperation with the editorial board. If the paper is outside of the scope it should be rejected regardless of its mathematical merit. When I was an editor of
*Discrete Mathematics*, I would reject some “proofs” of the Goldbach conjecture and similar results based on that reasoning. If the paper prompts the journal to re-evaluate its scope, it’s fine, but the discussion should involve the whole editorial board and irrespective of the paper in question. Presumably, some editors would not want to continue being on the board if the journal starts changing direction. - If the accepted but not yet published paper seems to fall outside of the journal’s scope, other editors can request the editor in chief to initiate the withdrawal process discussed above. The wording of request is crucial here – if the issue is neither the the scope nor the major math errors, but rather the weakness of results, then this is inappropriate.
- If the issue is the possibly unethical behavior of the handling editor, then the withdrawal may or may not be appropriate depending on the behavior, I suppose. But if the author was acting ethically and the unethical behavior is solely by the handling editor, I say proceed to publish the paper and then issue a formal retraction while keeping the paper published, of course.

#### Complaining to universities:

- While perfectly ethical, contacting the university administration to initiate a formal investigation of a faculty member is an extremely serious step which should be avoided if at all possible. Except for the egregious cases of verifiable formal violations of the university code of conduct (such as academic dishonesty), this action in itself is akin to bullying and thus immoral.
- The code of conduct is usually available on the university website – the complainer would do well to consult it before filing a complaint. In particular, the complaint would typically be addressed to the university senate committee on faculty affairs, the office of academic integrity and/or dean of the faculty. Whether the university president is in math or even the same area is completely irrelevant as the president plays no role in the working of the committee. In fact, when this is the case, the president is likely to recuse herself or himself from any part of the investigation and severe any contacts with the complainer to avoid appearance of impropriety.
- When a formal complaint is received, the university is usually compelled to initiate an investigation and set up an ad hoc subcommittee of the faculty senate which thoroughly examines the issue. Faculty’s tenure and life being is on the line. They can be asked to retain legal representation and can be prohibited from discussing the matters of the case with outsiders without university lawyers and/or PR people signing on every communication. Once the investigation is complete the findings are kept private except for administrative decisions such as firing, suspension, etc. In summary, if the author seeks information rather than punishment, this is counterproductive.

#### Complaining to institutions:

- I don’t know what to make of the alleged NSF request, which could be ethical and appropriate, or even common. Then so would be complaining to the NSF on a publicly available research product supported by the agency. The issue is the opposite to that of the journals — the NSF is a part of the the Federal Government and is thus subject to a large number of regulations and code of conduct rules. These can explain its request. We in mathematics are rather fortunate that our theorems tend to lack any political implications in the real world. But perhaps researchers in Political Science and Sociology have different experiences with granting agencies, I wouldn’t know.
- Contacting the AMS can in fact be rather useful, even though it currently has no way to conduct an appropriate investigation. Put bluntly, all parties in the conflict can simply ignore AMS’s request for documents. But maybe this should change in the future. I am not a member of the AMS so have no standing in telling it what to do, but I do have some thoughts on the subject. I will try to write them up at some point.

#### Public discourse:

- Many commenters on the case opined that while deleting a published paper is bad (I am paraphrasing), but the paper is also bad for whatever reason (politics, lack of strong math, editor’s behavior, being out of scope, etc.) This is very unfortunate. Let me explain.
- Of course, discussing math in the paper is perfectly ethical: academics can discuss any paper they like, this can be considered as part of the job. Same with discussing the scope of the paper and the verifiable journal and other party actions.
- Publicly discussing personalities and motivation of the editors publishing or non-publishing, third parties contacting editors in chief, etc. is arguably unethical and can be perceived as borderline bullying. It is also of questionable morality as no complete set of facts are known.
- So while making a judgement on the journal conduct next to the judgement on the math in the paper is ethical, it seems somewhat immoral to me. When you write “yes, the journals’ actions are disturbing, but the math in the paper is poor” we all understand that while formally these are two separate discussions, the negative judgement in the second part can provide an excuse for misbehavior in the first part. So here is my new rule:
**If you would not be discussing the math in the paper without the pretext of its submission history, you should not be discussing it at all.**

#### In summary:

I argue that for all issues related to submissions, withdrawal, etc. there is a well understood ethical code of conduct. Decisions on who behaved unethically hinge on formal details of each case. Until these formalities are clarified, making judgements is both premature and unhelpful.

Part of the problem is the lack of clarity about procedural rules by the journals, as discussed above. While large institutions such as major universities and long established journal publishers do have such rules set up, most journals tend not to disclose them, unfortunately. Even worse, many new, independent and/or electronic journals have no such rules at all. In such environment we are reduced to saying that this is all a failure to communicate.

#### Lengthy disclaimer:

- I have no special knowledge of what
*actually happened*to Hill’s submission. I outlined what I think*should have happened*in different scenarios if all participants actedand**morally**(there are no legal issues here that I am aware of). I am not trying to blame anyone and in fact, it is possible that none of these theoretical scenarios are applicable. Yet I do think such a general discussion is useful as it distills the arguments.**ethically** - I have not read Hill’s paper as I think its content is irrelevant to the discussion and since I am deeply uninterested in the subject. I am, however, interested in mathematical publishing and all academia related matters.
- What’s ethical and what’s moral are not exactly the same. As far as this post is concerned, ethical issues cover all math research/university/academic related stuff. Moral issues are more personal and community related, thus less universal perhaps. In other words, I am presenting my own POV everywhere here.
- To give specific examples of the difference, if you stole your officemate’s lunch you acted immorally. If you submitted your paper to two journals simultaneously you acted unethically. And if you published a paper based on your officemate’s ideas she told you in secret, you acted both immorally and unethically. Note that in the last example I am making a moral judgement since I equate this with stealing, while others might think it’s just unethical but morally ok.
- There is very little black & white about immoral/unethical acts, and one always needs to assign a relative measure of the perceived violation. This is similar to criminal acts, which can be a misdemeanor, a gross misdemeanor, a felony, etc.

## ICM Paper

Well, I finally finished my **ICM paper**. It’s only 30 pp, but it took many sleepless nights to write and maybe about 10 years to understand what exactly do I want to say. The published version will be a bit shorter – I had to cut section 4 to satisfy their page limitations.

Basically, I give a survey of various recent and not-so-recent results in *Enumerative Combinatorics* around three major questions:

**(1)** What is a formula?

**(2)** What is a good bijection?

**(3)** What is a combinatorial interpretation?

Not that I answer these questions, but rather explain how one *could answer* them from computational complexity point of view. I tried to cover as much ground as I could without overwhelming the reader. Clearly, I had to make a lot of choices, and a great deal of beautiful mathematics had to be omitted, sometimes in favor of the Computational Combinatorics approach. Also, much of the survey surely reflects my own POV on the subject. I sincerely apologize to everyone I slighted and who disagrees with my opinion! Hope you still enjoy the reading.

Let me mention that I will wait for a bit before posting the paper on the arXiv. I very much welcome all comments and suggestions! Post them here or email privately.

P.S. In thinking of how approach this paper, I read a large number of papers in previous ICM proceedings, e.g. papers by Noga Alon, Mireille Bousquet-Mélou, Paul Erdős, Philippe Flajolet, Marc Noy, János Pach, Richard Stanley, Benny Sudakov, and many others. They are all terrific and worth reading even if just to see how the field has been changing over the years. I also greatly benefited from a short introductory article by Doron Zeilberger, which I strongly recommend.

## How to write math papers clearly

Writing a mathematical paper is both an act of recording mathematical content and a means of communication of one’s work. In contrast with other types of writing, the style of math papers is incredibly rigid and resistant to even modest innovation. As a result, both goals suffer, sometimes immeasurably. The * clarity* suffers the most, which affects everyone in the field.

Over the years, I have been giving advice to my students and postdocs on how to write clearly. I collected them all in ** these notes.** Please consider reading them and passing them to your students and colleagues.

Below I include one subsection dealing with different reference styles and what each version really means. This is somewhat subjective, of course. Enjoy!

****

**4.2. How to cite a single paper.** The citation rules are almost as complicated as Chinese honorifics, with an added disadvantage of never being discussed anywhere. Below we go through the (incomplete) list of possible ways in the decreasing level of citation importance and/or proof reliability.

(1) “*Roth proved Murakami’s conjecture in* [Roth].” Clear.

(2) “*Roth proved Murakami’s conjecture *[Roth].” Roth proved the conjecture, possibly in a different paper, but this is likely a definitive version of the proof.

(3) “*Roth proved Murakami’s conjecture, see* [Roth].” Roth proved the conjecture, but [Roth] can be anything from the original paper to the followup, to some kind of survey Roth wrote. Very occasionally you have “*see* [Melville]”, but that usually means that Roth’s proof is unpublished or otherwise unavailable (say, it was given at a lecture, and Roth can’t be bothered to write it up), and Melville was the first to publish Roth’s proof, possibly without permission, but with attribution and perhaps filling some minor gaps.

(4) “*Roth proved Murakami’s conjecture* [Roth], *see also* [Woolf].” Apparently Woolf also made an important contribution, perhaps extending it to greater generality, or fixing some major gaps or errors in [Roth].

(5) “*Roth proved Murakami’s conjecture in* [Roth] (*see also* [Woolf]).” Looks like [Woolf] has a complete proof of Roth, possibly fixing some minor errors in [Roth].

(6) “*Roth proved Murakami’s conjecture* (*see* [Woolf]).” Here [Woolf] is a definitive version of the proof, e.g. the standard monograph on the subject.

(7) “*Roth proved Murakami’s conjecture, see e.g. * [Faulkner, Fitzgerald, Frost].” The result is important enough to be cited and its validity confirmed in several books/surveys. If there ever was a controversy whether Roth’s argument is an actual proof, it was resolved in Roth’s favor. Still, the original proof may have been too long, incomplete or simply presented in an old fashioned way, or published in an inaccessible conference proceedings, so here are sources with a better or more recent exposition. Or, more likely, the author was too lazy to look for the right reference, so overcompensated with three random textbooks on the subject.

(8) “*Roth proved Murakami’s conjecture* (*see e.g.* [Faulkner, Fitzgerald, Frost]).” The result is probably classical or at least very well known. Here are books/surveys which all probably have statements and/or proofs. Neither the author nor the reader will ever bother to check.

(9) “*Roth proved Murakami’s conjecture.*^{7} Footnote 7: *See* [Mailer].” Most likely, the author never actually read [Mailer], nor has access to that paper. Or, perhaps, [Mailer] states that Roth proved the conjecture, but includes neither a proof nor a reference. The author cannot

verify the claim independently and is visibly annoyed by the ambiguity, but felt obliged to credit Roth for the benefit of the reader, or to avoid the wrath of Roth.

(10) “*Roth proved Murakami’s conjecture.*^{7} Footnote 7: *Love letter from H. Fielding to J. Austen, dated December 16, 1975.*” This means that the letter likely exists and contains the whole proof or at least an outline of the proof. The author may or may not have seen it. Googling will probably either turn up the letter or a public discussion about what’s in it, and why it is not available.

(11) “*Roth proved Murakami’s conjecture.*^{7} Footnote 7: *Personal communication.*” This means Roth has sent the author an email (or said over beer), claiming to have a proof. Or perhaps Roth’s student accidentally mentioned this while answering a question after the talk. The proof

may or may not be correct and the paper may or may not be forthcoming.

(12) “*Roth claims to have proved Murakami’s conjecture in* [Roth].” Paper [Roth] has a well known gap which was never fixed even though Roth insists on it to be fixable; the author would rather avoid going on record about this, but anything is possible after some wine at a banquet. Another possibility is that [Roth] is completely erroneous as explained elsewhere, but Roth’s

work is too famous not to be mentioned; in that case there is often a followup sentence clarifying the matter, sometimes in parentheses as in “(*see, however,* [Atwood])”. Or, perhaps, [Roth] is a 3 page note published in *Doklady Acad. Sci. USSR* back in the 1970s, containing a very brief outline of the proof, and despite considerable effort nobody has yet to give a complete proof of its Lemma 2; there wouldn’t be any followup to this sentence then, but the author would be happy to clarify things by email.

UPDATE 1. (Nov 1, 2017): There is now a video of the MSRI talk I gave based on the article.

UPDATE 2. (Mar 13, 2018): The paper was published in the *Journal of Humanistic Mathematics*. Apparently it’s now number 5 on “Most Popular Papers” list. Number 1 is “My Sets and Sexuality”, of course.

## Fibonacci times Euler

Recall the *Fibonacci numbers* given by 1,1,2,3,5,8,13,21… There is no need to define them. You all know. Now take the *Euler numbers (OEIS)* 1,1,1,2,5,16,61,272… This is the number of alternating permutations in with the exponential generating function . Both sequences are incredibly famous. Less known are connection between them.

(1) Define the *Fibonacci polytope* to be a convex hull of 0/1 points in with no two 1 in a row. Then has vertices and vol This is a nice exercise.

(2) (by just a little). For example, . This follows from the fact that

and , where is the g*olden ratio*. Thus, the product . Since and , the inequality is easy to see, but still a bit surprising that the numbers are so close.

Together with Greta Panova and Alejandro Morales we wrote a little note “Why is π < 2φ?” which gives a combinatorial proof of (2) via a direct surjection. Thus we obtain an indirect proof of the inequality in the title. The note is not a research article; rather, it is aimed at a general audience of college students. We will not be posting it on the arXiv, so I figure this blog is a good place to advertise it.

The note also explains that the inequality (2) also follows from Sidorenko’s theorem on complementary posets. Let me briefly mention a connection between (1) and (2) which is not mentioned in the note. I will assume you just spent 5 min and read the note at this point. Following Stanley, the volume of is equal to the volume of the *chain polytope* (=*stable set polytope*), see Two Poset Polytopes. But the latter is exactly the polytope that Bollobás, Brightwell and Sidorenko used in their proof of the upper bound via polar duality.

## The power of negative thinking, part I. Pattern avoidance

In my latest paper with Scott Garrabrant we disprove the *Noonan-Zeilberger Conjecture*. Let me informally explain what we did and why people should try to disprove conjectures more often. This post is the first in a series. Part II will appear shortly.

#### What did we do?

Let *F ⊂ S _{k}* be a finite set of permutations and let

*C*(

_{n}*F*) denote the number of permutations

*σ ∈ S*avoiding the set of patterns

_{n}*F*. The

*Noonan-Zeilbeger conjecture*(1996), states that the sequence {

*C*(

_{n}*F*)} is always

*P-recursive*. We disprove this conjecture. Roughly, we show that every Turing machine T

*can be simulated by a set of patterns*

*F,*so that the number

*a*of paths of length n accepted by by T is equal to

_{n }*C*(

_{n}*F*) mod 2. I am oversimplifying things quite a bit, but that’s the gist.

What is left is to show how to construct a machine T such that {*a _{n}*} is not equal (mod 2) to

**any**P-recursive sequence. We have done this in our previous paper, where give a negative answer to a question by Kontsevich. There, we constructed a set of 19 generators of

*GL(4,Z)*, such that the probability of return sequence is not P-recursive.

When all things are put together, we obtain a set *F* of about 30,000 permutations in *S _{80}* for which {

*C*(

_{n}*F*)} is non-P-recursive. Yes, the construction is huge, but so what? What’s a few thousand permutations between friends? In fact, perhaps a single pattern (1324) is already non-P-recursive. Let me explain the reasoning behind what we did and why our result is much stronger than it might seem.

#### Why we did what we did

First, a very brief history of the NZ-conjecture (see Kirtaev’s book for a comprehensive history of the subject and vast references). Traditionally, pattern avoidance dealt with exact and asymptotic counting of pattern avoiding permutations for small sets of patterns. The subject was initiated by MacMahon (1915) and Knuth (1968) who showed that we get Catalan numbers for patterns of length 3. The resulting combinatorics is often so beautiful or at least plentiful, it’s hard to imagine how can it not be, thus the NZ-conjecture. It was clearly very strong, but resisted all challenges until now. Wilf reports that Richard Stanley disbelieved it (Richard confirmed this to me recently as well), but hundreds of papers seemed to confirm its validity in numerous special cases.

Curiously, the case of the (1324) pattern proved difficult early on. It remains unresolved whether *C _{n}*(1324) is P-recursive or not. This pattern broke Doron Zeilberger’s belief in the conjecture, and he proclaimed that it’s probably non-P-recursive and thus NZ-conjecture is probably false. When I visited Doron last September he told me he no longer has strong belief in either direction and encouraged me to work on the problem. I took a train back to Manhattan looking over New Jersey’s famously scenic Amtrack route. Somewhere near Pulaski Skyway I called Scott to drop everything, that we should start working on this problem.

You see, when it comes to pattern avoidance, things move from best to good to bad to awful. When they are bad, they are so bad, it can be really hard to prove that they are bad. But why bother – we can try to figure out something awful. The set of patterns that we constructed in our paper is so awful, that proving it is awful ain’t so bad.

#### Why is our result much stronger than it seems?

That’s because the proof extends to other results. Essentially, we are saying that everything bad you can do with Turing machines, you can do with pattern avoidance (mod 2). For example, why is (1324) so hard to analyze? That’s because it’s even hard to compute both theoretically and experimentally – the existing algorithms are recursive and exponential in *n*. Until our work, the existing hope for disproving the NZ-conjecture hinged on finding an appropriately bad set of patterns such that computing {*C _{n}*(

*F*)} is easy. Something like this sequence which has a nice recurrence, but is provably non-P-recursive. Maybe. But in our paper, we can do worse, a lot worse…

We can make a finite set of patterns *F,* such that computing {*C _{n}*(

*F*) mod 2} is “provably” non-polynomial (Th 1.4). Well, we use quotes because of the complexity theory assumptions we must have. The conclusion is much stronger than non-P-recursiveness, since every P-recursive sequence has a trivial polynomial in

*n*algorithm computing it. But wait, it gets worse!

We prove that for two sets of patterns *F* and *G*, the problem* “C _{n}*(

*F*) =

*C*(

_{n}*G*) mod 2 for all n” is undecidable (Th 1.3). This is already a disaster, which takes time to sink in. But then it gets even worse! Take a look at our Corollary 8.1. It says that there are two sets of patterns

*F*and

*G*, such that you can never prove nor disprove that

*C*(

_{n}*F*) =

*C*(

_{n}*G*) mod 2. Now that’s what I call truly awful.

#### What gives?

Well, the original intuition behind the NZ-conjecture was clearly wrong. Many nice examples is not a good enough evidence. But the conjecture was so plausible! Where did the intuition fail? Well, I went to re-read Polya’s classic “*Mathematics and Plausible Reasoning*“, and it all seemed reasonable. That is both Polya’s arguments and the NZ-conjecture (if you don’t feel like reading the whole book, at least read Barry Mazur’s interesting and short followup).

Now think about Polya’s arguments from the point of view of complexity and computability theory. Again, it sounds very “plausible” that large enough sets of patterns behave badly. Why wouldn’t they? Well, it’s complicated. Consider this example. If someone asks you if every 3-connected planar cubic graph has a Hamiltonian cycle, this sounds plausible (this is Tait’s conjecture). All small examples confirm this. Planar cubic graphs do have very special structure. But if you think about the fact that even for planar graphs, Hamiltonicity is NP-complete, it doesn’t sound plausible anymore. The fact that Tutte found a counterexample is no longer surprising. In fact, the decision problem was recently proved to be NP-complete in this case. But then again, if you require 4-connectivity, then *every* planar graph has a Hamiltonian cycle. Confused enough?

Back to the patterns. Same story here. When you look at many small cases, everything is P-recursive (or yet to be determined). But compare this with Jacob Fox’s theorem that for a random single pattern π, the sequence {*C _{n}*(π)} grows much faster than originally expected (cf. Arratia’s Conjecture). This suggests that small examples are not representative of complexity of the problem. Time to think about disproving ALL conjectures based on that evidence.

If there is a moral in this story, it’s that what’s “plausible” is really hard to judge. The more you know, the better you get. Pay attention to small crumbs of evidence. And think negative!

#### What’s wrong with being negative?

Well, conjectures tend to be optimistic – they are wishful thinking by definition. Who would want to conjecture that for some large enough *a,b,c* and *n,* there exist a solution of *a*^{n} + *b*^{n} = *c*^{n}? However, being so positive has a drawback – sometimes you get things badly wrong. In fact, even polynomial Diophantine equations can be as complicated as one wishes. Unfortunately, there is a strong bias in Mathematics against counterexamples. For example, only two of the Clay Millennium Problems automatically pay $1 million for a counterexample. That’s a pity. I understand why they do this, just disagree with the reasoning. If anything, we should encourage thinking in the direction where there is not enough research, not in the direction where people are already super motivated to resolve the problem.

In general, it is always a good idea to keep an open mind. Forget all this “power of positive thinking“, it’s not for math. If you think a conjecture might be false, ignore everybody and just go for disproof. Even if it’s one of those famous unsolved conjectures in mathematics. If you don’t end up disproving the conjecture, you might have a bit of trouble publishing computational evidence. There are some journals who do that, but not that many. Hopefully, this will change soon…

#### Happy ending

When we were working on our paper, I wrote to Doron Zeilberger if he ever offered a reward for the NZ-conjecture, and for the disproof or proof only? He replied with an unusual award, for the proof and disproof in equal measure. When we finished the paper I emailed to Doron. And he paid. Nice… 🙂