As a response to comments from muriel who sent me a link to the Human Behavior and Evolution Society website, I republish my old post with my own modest contribution to evolutionary psychology. A fresher version of the same text can be found in my book.

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to begin right off with something very strong,

in order to attract the reader’s attention at once,

so he wrote that a cat had got on a tram-car,

and then went back to the episode with the severed head.

Michael Bulgakov, The Master and Margarita

This is my first posting [rememeber, it is written on 15 August 2006], and I want to use it to illustrate my approach to the study of mathematical practice.

I start with some observations and conjectures which may appear to be bizarre and out of tune from the usual discourse on mathematics. However, I tested some of them in a warm-up talk that I have given at a forum discussion “Where do mathematicians come from?”, part of a very peculiar conference, that of the World Federation of National Mathematics Competitions (WFNCM). It was held a couple of weeks ago in Cambridge, England. On my way from Manchester to Cambridge, four hours by train, I had seen three people solving Sudoku. In one case, a lady of middle age shared a table with me and I had a chance to watch, in all detail and with a growing fascination, how she was solving an elementary level Sudoku puzzle. Her actions followed a certain rhythm: first she inspected the puzzle row by row and column by column until she located a critical cell (the one which value had been already uniquely determined by the already known values in other cells) , then, with obvious agitation, checked that this was indeed the case, happily wrote the digit in, smiled betraying a childish satisfaction, relaxed for a few seconds, and, after a short pause, started the search again.

Next day, in my talk at the conference, I pointed out that, from a mathematical point of view, solving a (elementary level) Sudoku puzzle is nothing more than solving a triangle system of Boolean equations by back substitution, something very similar to what we do after a Gauss-Jordan elimination in a system of simultaneous linear equations. But has anyone ever seen people on a train solving systems of linear equations from a newspaper? Why is Sudoku popular, and systems of linear equations are not? (Actually, I was slightly wrong: at the time of my talk, I was unaware of Kakuro which combines linear and Boolean equations. But one still has to see whether Kakuro beats Sudoku in popularity.)

Still, why is Sudoku popular? I believe the answer is in a rhythm of repeated cycles of operations each of which engages our brains just up to a right and most pleasurable level of intensity. As a student, I experienced a soothing, relaxing effect of carrying out a recursive algorithm, like long division, or Euclid’s algorithm or a diagonalisation of a lambda-matrix. Later, in my research work, I felt a similar emotional impact of inductive arguments in finite group theory: you start with a minimal counterexample to the theorem, and then simplify it step by step, like removing layers from an onion, until you pin-point the core contradiction and destroy the counterexample. My teacher, Victor Danilovich Mazurov, expressed the principle of a “minimal counterexample” using a line from a Russian fairy tale:

Тут старший брат спрятался за среднего, средний за младшего, а младший на колени упал, ручки поднял и пощады просит.

(The biggest brother hid behind the back of the younger one, the younger one hid behind the youngest one, and the youngest brother fell on his knees, raised his hands and pleads for mercy.)

I would not write this posting now if the audience of my talk at the WFNMC conference did not immediately agree with, and approve of, my comparison of execution of certain types of recursive algorithms with the bubble wrap popping. I should perhaps explain that the audience included some of the best experts on mathematical education in the world, especially on advanced and non-standard aspects of mathematics teaching. They definitely knew everything about “recreational mathematics”, puzzles, brainteasers and conundrums of every possible kind. Their support allows me to be quite confident in my comparison of Sudoku with the bubble wrap popping. In any case, the lady on the train was doing her Sudoku in an immediately recognisable bubble wrap popping rhythm.

So, with the authority of the conference on my side, I dare to formulate my thesis:

We will not understand the psychological and neurophysiological roots of an important aspect of mathematical practice until we figure out why is bubble wrap popping so addictive and pleasant activity. Why does it comfort and help to relax? Why is it soothing?

Actually, some years ago I have formulated a rather embarrassing conjecture that the attraction to bubble wrap popping is genetically determined. Buble wrap triggerss in humans archaic instincts linked to an ape-like behavior: grooming (and even more importantly, mutual grooming) and destruction of lice. In apes and monkeys, mutual grooming is an important part of social bonding, which explains its soothing, comforting, relaxing effect.

In my search on the web for a confirmation of my conjecture I have not managed to get further than websites devoted to virtual bubble wrap popping, like this one. It is very difficult to google for anything containing the words “bubble wrap”, since almost everything sold on the Internet is mailed in a bubble wrap packaging, and, as the result, Google produces 9,090,000 hits for “bubble wrap”. I offered the problem to my colleague Gregory Cherlin, who was more internet savvy and carried out a successful search – see his notes on The Bubble Wrap Gene. I take the liberty to reproduce them here:

“Obsessively popping bubble wrap is apparently an increasingly common activity. Sasha Borovik has suggested this is related to normal grooming behavior in primates.

We give some references for others interested in this question.

- Description of the gene HOXB8, NIH
- Controls normal grooming behavior. Disruption in mice leads to obsessive grooming behavior.
- “
Summary:This gene belongs to the homeobox family of genes. The homeobox genes encode a highly conserved family of transcription factors that play an important role in morphogenesis in all multicellular organisms. Mammals possess four similar homeobox gene clusters, HOXA, HOXB, HOXC and HOXD, which are located on different chromosomes and consist of 9 to 11 genes arranged in tandem. This gene is one of several homeobox HOXB genes located in a cluster on chromosome 17. Hoxb8 knockout mice exhibit an excessive pathologic grooming behavior,leading to hair removal and self-inflicted wounds at overgroomed sites. This behavior is similar to the behavior of humans suffering from the obsessive-compulsive spectrum disorder (OCD) trichotillomania.”- Trichotillomania Learning Center
- Discussion of grooming-related disorders in humans
- Houston Zoo Conservation Program
- Discussion of bubble-wrap popping in primates (p. 8)
- Virtual bubble wrap

Meanwhile, my own search for buble wrap popping on Google Scholar led me to the book

*Teens Together Grief Support Group Curriculum. Adolescence Edition; Grades 7-12* by Linda Lehmann-Norquist, Shane R Jimerson and Ann Gaasch, Psychology Press (UK) 2001, 165 pp. ISBN 1583913025 .

I have not seen the whole book, but, apparently, page 57 contains sufficiently revealing words:

… Bubble Wrap: Give the teens a square of bubble wrap to pop for one of their breaks. They really get into the sound and action of popping the bubbles. …

As I suspected, the soothing and comforting effect of bubble wrap popping is indeed well-known to practicing psychotherapists.

Yes, it appears that HOXB8 is indeed the Bubble Wrap Gene and is responsible for Sudoku being attractive to humans. I would rather hear more on that from geneticists and neurophysilogists.

At last I am in position to formulate the moral of this fable. I believe that a real understanding of one of the key issues of mathematical practice (and especially of mathematics teaching):

- why are some objects, concepts and processes of mathematics are more intuitive, “natural”, or just more convenient and acceptable than others?

cannot be achieved without taking a hard and close look at the very deep and sometimes archaic levels of human mind and human neural system. Indeed, Stanislas Dehaene said in his book The Number Sense that

We have to do mathematics using the brain which evolved 30 000 years ago for survival in the African savanna.

If particular, should we be surprised if it would be confirmed indeed that the most comfotable pace of execution of a recursive algorithm is set by a gene responsible for grooming behavior?

I very much value a “global” outlook at mathematical practice (best represented by David Corfield’s Philosophy of Real Mathematics), but I personally prefer to concentrate on the “microscopic” level of study.

“For immediate popping. At your own risk, insofar as local statutes allow.”

Personally, I would add to your list of pleasurable activities calculations of cohomology groups of sheaves (writing the long exact sequence of cohomology coming from a ses of sheaves), as well as computing residues. But I wouldn’t call these relaxing; rather, I used to find them to some extent exciting. (In our early university years me and my colleagues thought of the residue theorem as a synonym of intellectual orgasm, and I think that line of thought can be important; I think the pleasure we gain from certain mathematical activities definitely borders the sensual. ) It seems all of these activities involve an element of chase, but to a large extent you know the outcome – somewhat of a cat-and-mouse game. Probably that’s why with time you get less excitied with these.

But I’ve been asking myself – aren’t all these examples simply instances of conditional reflexes? Sequences effort-reward. Feeling good because you “score”. Feeling good because you’ve developed a skill, a tactical ability.

By:

Another Peteron June 1, 2008at 6:52 pm

Was my previous comment insulting to someone or was it eaten by the spam filter?

Dr.Borovick, please feel free to erase this comment if the previous one just got into the filter.

By:

Another Peteron June 1, 2008at 7:22 pm

Another Peter:

Non-pathological (like in various versions of neurosis) condtional reflexes are formed only for actions which are natural in execution, even if serve different purposes. Look at circus animals: seals juggle balls, while hares beat in drums, but have you ever seen a hare balancing a ball on its nose? Animals who can walk (even a short distance) on hind legs and climb trees (humans, apes, bears) can be trained to ride a bycicle.

I am in complete agreement with you, long division is a conditional reflex. But what are its underlying neurophisiological mechanisms?

By:

Alexandre Borovikon June 2, 2008at 5:36 am

Another Peter — My apologies: for some reason your comments get stuck in the spam filter. Since they got finally through, al yur further comments (which are most appreciated) should get published without delay.

By:

Alexandre Borovikon June 2, 2008at 2:04 pm

Dr. Borovik, I admit, as a current undergraduate, I feel don’t have much to add to this conversation (or blog in general, though I greatly enjoy reading it). However, I couldn’t help but laugh when you made your systems of equations vs. Sudoku comment, so I think you’ll find this slightly amusing.

More than a year ago, I was in a linear algebra class. As per usual, we started out with systems of linear equations and Gaussian elimination. When I showed my friend (non-mathematics, but science major) what we were doing, she wanted to try one out for size. Long story short, she (almost) forcibly made me give her these equations daily, so she could do them at night before she went to bed! Knowing this, I sneakily gave her one that couldn’t be solved. It suffices to say that the look on her face the next day was perfect.

I realize she’s probably the exception to the rule. Personally, I find canceling like terms and simple algebra to be very gratifying (especially some of the tiny tricks with factorials). I’m not really sure why either. I also enjoy bubble wrap (the virtual one was very unsatisfying though…maybe if the differences between the two were studied something interesting might pop out?). Interesting questions.

By:

Billyon June 2, 2008at 8:53 pm

[…] For some reason, I find it fun to observe this sensitive dependence using an ordinary calculator. Try calculating something like the golden mean , and hit it with over and over and watch the parade of integer parts go by (a long succession of 1’s until the precision of the arithmetic finally breaks down and the behavior looks random, chaotic). For me this activity is about as enjoyable as popping bubble wrap. […]

By:

Continued fraction for e « Todd and Vishal’s blogon August 4, 2008at 1:16 am

[…] at thinking, in my experience, is similar to the pleasures which people gain by doing crosswords or doing Sudoku puzzles; it may also be a form of mathmind. References: Mark Evan Bonds [2006]: Music as Thought: […]

By:

What is music for? at Vukutuon August 28, 2010at 2:49 pm