How to describe relatives with up-down notation

It is very clear that a lot of children (and even adults) struggle with the nomenclature for relatives, and understanding where relatives fit onto the family tree. What do I call my cousin’s daughter? How am I related to my father’s cousin? Which of my relatives are a “second cousin once removed”, yes there are two! A solution to these problems is to view your relatives as located at the vertices of a binary tree, and use the mathematical fact that there is a unique path from any one vertex on the tree to another. Then you just need to describe how many steps from you up to a common ancestor, and then down to your relative. For example, with this naming scheme your cousin is called 2-up-2-down because you must go up two steps (parent, then grandparent) to a common ancestor, and two steps down (to aunt or uncle), and then to their child (your first cousin).

Here is a small dictionary: a sibling is 1-up-1-down, a parent is 1-up, a son/daughter is 1-down, a grandson/daughter is 2-down and a grandmother/father is 2-up. A second cousin is 3-up-3-down and a third cousin is 4-up-4-down.

If you go down more than up, you add “once removed” or “twice removed” or “thrice removed” etc. Hence 2-up-3-down is a first cousin once removed, as you are one generation lower than your first cousin 2-up-2-down. Similarly, 3-up-4-down is a second cousin once removed. Makes sense, right? But 3-up-2-down is sadly, and confusingly, also called first cousin once removed. Yes, two different relatives have the same name, see Table of Consanguinity (source Wikipedia). Thus specifying 2-up-3-down or 3-up-2-down avoids any ambiguity: having the same name for different relatives, who are 5 = 2+3 steps away from you, is bound to confuse. Once you feel comfortable with up-down notation, you will realise that 1-up-1-down is your 0th cousin. In general, N-up-N-down is your (N-1)th cousin.

Genealogists know relationships intuitively, and tend not to describe them via the position on a binary tree. They take a lot of words to distinguish between 3-up-2-down and 2-up-3-down. I prefer to use this compact unambiguous up-down notation to describe complicated relationships. I am not suggesting to call your sibling 1-up-1-down or your parent 1-up, but relations beyond cousins would be fair game, especially if you want to be unambigous and understandable. Up-down notation is also gender-neutral. In English the gender neutral word sibling (sister/brother) is well-known; however, nibling (niece/nephew) and pibling (aunt/uncle) and other variants are less well-known.

In summary, convential language is better than up-down notation if the relations are simple. If they are more complicated, then up-down notation clearly wins. How often have you hear people talk of their great-great-great-great grandfather, only to ask yourself: “How many greats was that?”. However, 6-up makes it clear. I get annoyed when hearing sentences like: “the number of atoms in the universe is 100-million-trillion-trillion-trillion-trillion-trillion-trillion”. It is more helpful to have good notation and say “1 followed by 80 zeroes” or 1080 if the audience can remember scientific notation.

If you want to specify the gender of a relative, as many genealogist do, then words like “aunt” are shorter than my female 2-up-1-down, and “nephew” is shorter than my male 1-up-2-down. A good compomise is to use a mixture of English names and up-down notation: choosing the one that is brief, unambiguous and easy to understand. The Table of Consanguinity will help to learn the up-down system. Enoy!

Why so few open access journals?

Mathematicians are good at seeing how to solve problems, but how good are they at implementing the solutions?

In 2012 Tim Gowers wrote a blog that went viral. He described how mathematics was being negatively impacted by Elsevier, and it did not need to be. Mathematicians can set up high quality journals which are free to the user, or very low cost. Your university library has likely cut the number of journals it offers because it can not afford the astronomical prices of journals offered by large publishers such as Elsevier/Taylor-Francis.

One of the oldest electronic journals, founded in 1994 by Herbert S. Wilf and Neil Calkin, is The Electronic Journal of Combinatorics. Since then there have been others including:

  • Enumerative Combinatorics and Applications,
  • Algebraic Combinatorics,
  • Advances in Combinatorics,
  • Discrete Analysis,
  • Ars Inveniendi Analytica.

I am surprised that the list of Open Access, arXiv overlay journals is so small and that commercial publishers abound. There are many commercial journals (as opposed to society journals) with very low academic standards and fewer with high academic standards which are very expensive and lock up mathematical research via expensive paywalls and, at times, lengthy copyright agreements.

If you do not know about the many benefits, and few challenges, associated with Open Access publishing, then you may be interested in Open Access Directory.

For an informative (if lengthy) historical account of background to this area, see the letter from Donald E. Knuth to fellow members of the Journal of Algorithms editorial board outlining the problem, describing the open-access solution.

A list of journals whose editorial boards have resigned can be found here.

So my questions are: Why are there so few Open Access journals? and Why do ruthless commercial publishers like Elsevier/Springer have such a strong hold on academic publishing (in mathematics).

Of course academics trade in reputation/status, and if commercial journals are perceived to have higher status than society journals (because they cost much more, for example), then there will be some academics who will wish to submit to, referee for, and act as editors for, commercial journals even if it is to the long-term detriment of mathematics. In a separate blog I will consider best practice academic publishing: how are errors/updates handled? when non-trivial computer programs form part of a proof is computer code stored? is mathematical research immediately available? Can mathematics flourish with 5, 10, 20 year copyright agreements? etc

How should we handle errors?

It is no surprise that there are many more mathematical papers published now than previously. Data from the International Association of Scientific, Technical and Medical Publishers (STM) from 1968-2018 can be seen here. They show a threefold increase from 1980 to 2012 and a five-sixfold increase from 1975 to 2018. It is reasonable to assume that the number of mathematical errors have increased proportionally, or maybe at a greater rate as the number of publications per mathematician grows.

Some mistakes are worse than others. The simplest mistake is a typo, and this is can commonly be detected and corrected, but occasionally the typo occurs in a complex mathematical expression and correction is not straightforward. Sometimes authors give incomplete arguments and while the reader may believe the claimed result, the proof might be have a gap. Of course there are examples also of results/statements that are just plain wrong. The last two examples raise questions of what should be done for the paper concerned and for the papers that rely on erroneous or incomplete arguments.

Over the last century many databases of groups have been published (p-groups, simple groups, primitive groups, groups of ``small’’ order). Many had errors of omission, of duplication, or incorrect entries commonly due to typos or transcription errors. For example, the number of groups of order 1024 was believed to be 49,487,365,422 for 20 years. However, recently David Burrell rechecked the original computer calculations and found that 1,867 groups had been omitted from the correct total of 49,487,367,289. Moreover, software can have bugs, known to experts but not fixed due to lack of resources. For example a bug in GAP, involving isomorphism of groups of order 512, has not yet been fixed since April 2020. Volunteers are often not rewarded/regognized for their service.

Proofs that depend on long and complicated arguments, or on complicated computer calculations, are naturally more prone to error. Worse, such results are less likely to be rechecked by others, so errors can go undetected for a long time as illustrated above. I do not believe that computer-free proofs are less or more error-prone than computer-assisted proofs. A simple computer check can be more convincing than a very long and complicated hand calculation. Complexity is the problem. But there is another problem: many mathematical publications do not have accompanying computer code for a referee to check. Of course this is not needed for simple computer checks, but for complicated checks it certainly is, and mathematicians have not collectively insisted that a complete proof (including supporting calculations) should be published.

Mathematics builds on itself it a way that is unique in the sciences, and authors commonly trust the correctness of the published literature, so it is conceivable that if mistakes proliferate that in the future Mathematics could face a crisis of reliability. A replication crisis has been observed in the Medical Sciences, Psychology and other fields since roughly 2010. These fields have taken a number of steps to address this problem, see here. The arbitrary choice of p < 0.05 for significance, and the attendant problem of p-hacking, is less of a problem for pure mathematics. However, other biases such as lack of availability of qualified referees, pressure to publish, confirmation bias, etc can play a role. Should mathematics place more emphasis on reproving central results whose veracity has been checked only by one or two people ever? A rare example of this is the “The Classification of the Finite Simple Groups” by Gorenstein, Lyons, Solomon. At the time of writing 8 of the 12 proposed volumes have been written. It is unclear whether the surviving authors (Lyons and Solomon) will complete the proof or whether another proof strategy will prevail. It would be a rude shock if a new finite simple group was discovered (but nobody suspect this).

Mathematics has undergone crises in the past, for example the foundational problems in set theory and problems of rigor in algebraic geometry. It is possible that future mathematics may inadvertently build large bodies of knowledge based on some erroneous results.

There are many issues: who rechecks results? should MathSciNet be updated to include errors? what if canonical names for groups in databases change because of errors? what about publications that depend on flawed results?

Published papers can in principle be updated by using Crossmark. This says: Research doesn’t stand still: even after publication, articles can be updated with supplementary data or corrections. It’s important to know if the content being cited has been updated, corrected, or retracted - and that’s the assurance that publishers can offer readers by using Crossmark. It’s a standardized button, consistent across platforms, revealing the status of an item of content, and can display any additional metadata the member chooses. Crucially, the Crossmark button can also be embedded in PDFs, which means that members have a way of alerting readers to changes months or even years after it’s been downloaded.

Journal of Conjectures and Proof Strategies

Academics want publications. I will talk about academic mathematicians because I know them best. Lots of publications improve your chances as an applicant for a job, or for a promotion. Junior mathematicians are advised to research safe topics: perhaps were established methods can be applied to a new problem that has a good chance of success. If the problem has applications, or is interesting, then the fact that the methods are established may not be an impediment to publication. Many established mathematicians follow this lower risk approach.

Should Mathematics encourage more risk-taking? If so, there had better be publications as a reward. It is not uncommon for a mathematical paper to include a conjecture or question, after a series of established theorems/propositions/lemmas. We tend to be cautious about stating a conjecture: if it can be solved quickly with little effort, we will feel embarrassed. A mathematician may well have many reasons for believing a conjecture and maybe has an outline of a proof but some part can not be completed. The evidence for a conjecture and strategies for its proof are commonly not published, so different researchers who try to solve the conjecture may well repeat the same work. If Mathematics had respected avenues to publish what could be regarded as incomplete work, then the field may advance more quickly. There are caveats:

  1. the work should be on a conjecture of interest to a broad community of mathematicians;
  2. the proposed methods/ideas should make notable progress, ideally offering some hope of success;
  3. the research should be written in a general manner so that other researchers can build on the results to either prove the conjecture or significantly close the gap;
  4. the paper may include computational evidence that the conjecture is true (ideally with well written supporting computer code), or the paper may reduce the problem to a different problem requiring specific expertise.

Is MathOverflow a suitable avenue for such risk-taking? Yes and no. A posting on MathOverflow is not a refereed journal article, and may not lead to one. A mathematician may posts 70% of a solution on MathOverflow (I know that is hard to quantify), and someone may contribute the remaining 30% in a single author publication generously acknowledging your post. You can doubtless think of examples where a joint author paper would be more appropriate. In the competitive world of academia there will be winners and losers and gracious and ungracious authors. Having a published partial result is less risky, and more attractive (especially to junior mathematicians). A 70% proof of the P = NP conjecture may well be easy to get published, but for less important (but still significant) problems I think you could find it hard to get your work published by a respectable journal. I wonder whether fictional journals such as “Journal of Conjectures and Proof Strategies” could help mathematics advance more rapidly, by reducing the amount of replication, and by building on the shoulders of other researchers?