# Oπnions: Can a horse have an Erdős number?

Lucy Rycroft-Smith reflects on the use of this well-established measurement

Are you a mathematician? Care to quantify that?

For decades, mathematicians have been having fun by measuring the collaborative distance between themselves and the most prolific mathematician (in terms of number of individual papers published) to date, Paul Erdős (although he may only be number two; this is disputed).

Your Erdős number is calculated by the shortest collaborative distance between you and Erdős, where 0 is ‘you are Erdős’ and 4 is ‘you published a paper with someone who published a paper with someone who published a paper with someone who published a paper with Erdős.’ He was a favourite amongst people who tell colourful stories about mathematicians and people who write mathematical quotes on coffee mugs alike, widely quoted as saying: “A mathematician is a device for turning coffee into theorems.” Erdős died in 1996, meaning the number of people with an Erdős number of 0 or 1 is now finitely bounded. (Unless you believe in ghostwriting.)

The criteria used to define ‘paper’ and ‘published with’ vary, as you can imagine, but the Erdős Number Project at Oakland University in Michigan, USA, suggests:

Our criterion for inclusion […] is some research collaboration between them resulting in a published work. Any number of additional coauthors is permitted. Not normally included are joint editorships, introductions to books written by others, technical reports, problem sessions, problems posed or solved in problem sections of journals, seminars, very elementary textbooks, books on history, memorial or other tributes, biography, translations, bibliographies, or popular works.

Erdős wrote or co-wrote 1475 papers by this definition. How many people do you think there are with
Erdős number 1? What about 2? See if you can make a reasonable conjecture before I tell you.

There are approximately 511 people with Erdős number 1, and around 11,002 people with exactly Erdős number 2. Were you close? Now, what do you think happens to the distribution of these values as the Erdős numbers increase? See if you can predict the results for Erdős numbers 3 and 4. What do you predict to be the mean and median Erdős number for published mathematicians? What about non-mathematicians (however you define them) —w

What sort of numbers would you expect from them?

A plot of the ‘Erdős function’ using approximate (and restricted) data from the Mathematical Review database in the US. Was it the sort of distribution you anticipated? The mean Erdős number from this data comes in at 4.65 and the median Erdős number is 5.

The Erdős number project at Oakland University has some intriguing things to say, including the revelation that at least one non-mathematician (a medical doctor) has an Erdős number of at most 9 —and that someone once auctioned off an Erdős number of 5 on eBay. There is even a suggestion that there may be a horse with an Erdős number of 3… I went in search of that horse, of course.

Recent work by animal psychologists showing that not only primates but even birds have surprising intellectual capacities, opens the question as to whether non-humans can boast an Erdős number. The answer is indeed yes; the story is however curious.
Jerry Grossman at Oakland University contributed an article to a Bridge magazine, jointly with Smarty, his wife’s horse. As Grossman (Grossman, Jerrold Wayne) has an Erdős number of 2, the horse achieved an Erdős number of 3.’ — The Extended Erdős Number Project

So: if we relax the criteria enough to include almost any publication, then yes, at least one and probably more horses have an Erdős number (please write in if you know of more—the Extended Equine Erdős Number Project sounds like a brilliant plan). This would also mean that the numbers quoted previously would be significantly higher—but only, presumably, above the Erdős number of 1, because Erdős himself didn’t publish much in ‘lowbrow’ places. Does the measure become meaningless if we relax the constraints to this degree? We can turn to several parallels for ideas.

The Bacon project computes an actor’s ‘distance’ from Kevin Bacon in the much the same way, using IMDB. This means that it uses data from a single database of credits on films and TV shows—an imperfect but fairly defined line (does it include cameos? singers? writers and directors?).

Looking further afield, we come to the general principle of ‘six degrees of separation’, originally coined by Frigyes Karinthy in 1929—the idea that in any kind of highly connected network of human relationships or endeavours, the ‘average’ shortest path is six. Watts and Strogatz showed that in ‘small-world networks’, the average path length between two nodes is equal to $\ln N/\hspace{-2pt}\ln K$ where $N$ is the total number of nodes, and $K$ is the number of acquaintances per node.

For example, in 2011 the average distance between people (by number of connections) was 4.74 on Facebook, and 4.67 on Twitter. Of course, here we have pretty discrete rules about connection: the idea of ‘friending’ or ‘following’ on social media is slightly less nebulous than ‘knowing’ in real life. Should we use some kind of index to tell people how connected they are to an arbitrary person of value on social media? It’s likely this would have very little meaning; those sorts of connections, unlike publication credits, are easily obtained and cheaply won.

So it seems that by most reasonably strict definitions, any mathematician worth their salt would expect an Erdős number of around 4 or 5, right?

(I’ve been working up to something. Did you notice?)

Source: 4th Estate

I can remember exactly when I first heard of Erdős. It’s February 2000. I’m 16 years old and for my birthday, someone gets me a book with a weird title: The Man Who Loved Only Numbers. I’m not particularly into men and feel pretty much the same way about numbers (perhaps I’m pi-curious), so I put it into a pile and forget about it for a while. Then, one evening, casting around for new reading material, I pick up the purple and turquoise paperback out of idle curiosity.

I don’t sleep that night. I devour the book like it’s chocolate. I’m thrilled by the portrait of the highly eccentric, highly prolific mathematician who just arrives on each international doorstep with a suitcase and a catalogue of problems to solve, sure of a welcome and a cabal of willing minds. I’m utterly intrigued by the religious overtones of the book; to Erdős, mathematics is inseparable from a kind of fanaticism that inextricably connects him to a higher being who he suggests has written a book of all the perfect proofs in the world (called, somewhat underwhelmingly, The Book). His life’s mission: to discover as many of them as possible. His motto: “my mind is open”.

When he said someone had ‘died’, Erdős meant that the person had stopped doing mathematics. When he said someone had ‘left’, the person had died.

A teenage me, proficient but not amazing at mathematics, dreams of an Erdős number to call my own and all that it might bring: validation, acceptance into a community, that magic label “mathematician”. I’m tired of people not taking me seriously, of calling me a “silly girl”. I have no authority in my little world, and the tales of Erdős and the reverential welcome he gets wherever he goes seem far, far out of reach.

It’s October 2017. Winter is setting in. I’m cold and grumpy. I stumble into work and fire up my computer, opening my email inbox.

Then I sit up straight in my chair and punch the air, thinking ‘what a time to be alive’. This is the email I received on that fateful day:

I have just received my copy of Mathematics Today and see that you have very kindly added my name to your brilliant article. You now have the privilege of having an Erdős number 3, so shout about that! My PhD supervisor published with Erdős when the latter visited Reading many years ago. I have a couple of publications with my supervisor, so I’m a 2!

Let me be clear—this was not a ‘brilliant’ mathematics research paper—it was a very ordinary review article from a conference, where I happened to have credited a kind colleague who read it through and suggested amendments with an entirely earned co-authorship. Nowhere in my wildest dreams would I have imagined that this would be the result. Thrilled is something of an understatement. Somehow, the time and space between me and this ‘real’ mathematician, respected and revered, had been shrunk. I was breathing the same air as Erdős. (Please don’t write in about that, either.)

But of course, after the euphoria had died down, I saw the real truth: while I might feel happy about this numerical value placed on my metaphorical value as a mathematician, it’s in many ways meaningless. The stuff that suggests a person has contributed to the field can be as intangible as a patient explanation to a child or a quick sketch of a unifying structure in the air with a finger. While formal publication is important, it’s not all or even most of what mathematicians do. The focus on getting this type of validation is pretty telling of a group of people who feel insecure, and I’d put actors and mathematicians right at the top of that list, in my humble experience.

What about the veritable army of maths teachers out there painstakingly explaining quadratic relationships for the fiftieth time to the same class? What about the pubfuls of people playing with puzzles and games and origami every month? What about the women and LGBT people and BAME people and disabled people and countless others who don’t get to study to the highest level they like, don’t get picked for the jobs they’re qualified for, don’t get to publish their ideas where they deserve?

This isn’t just an ‘only measuring in one dimension’ problem—it’s a problem of thinking that the product, not the process, is the thing. Translated into education contexts, it’s easy to see why we have generations of children thinking not getting a perfect score on a mathematics exam means they are not a (‘natural’) mathematician.

Put simply: you are not a mathematician because of an Erdős number. Sometimes, you are a mathematician in spite of it.

Do we need a better measure? Nope. We need to stop measuring.

Erdős is widely quoted, and often misattributed—see, for example, the first page of this article, where I suggested he said “A mathematician is a device for turning coffee into theorems”.

Another famously misattributed quote (this time to Einstein) would seem to apply here:

Not everything that can be counted counts, and not everything that counts can be counted. — William Bruce Cameron

Lucy Rycroft-Smith is a writer, researcher, speaker and designer in mathematics education. She is studying for her PhD at the University of Cambridge, considering the idea of knowledge brokering—how, what, when, where and why we communicate research to mathematics teachers. Her other research interests include gender, sexualities and identity, feminism, decolonisation, curriculum design, board gaming, teaching discrete mathematics, representations of mathematics, maths and the arts, and professional development. She is the author of The Equal Classroom: Life-Changing Thinking About Gender (2019) and co-editor of Flip the System UK: A Teachers’ Manifesto (2017).

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