Tuesday, 21 November 2017

On 73

73 is a permutable (or anagrammatic) prime, a prime that can have its digits' rearranged (in base 10) in any permutation and still be a prime number. 73 is also an emirp, a number whose reverse, 37, is also prime — a property also evident in terms of the respective ordinal positions of 73 and its emirp partner, 73 being the 21st prime [1] while 37 is the 12th prime.  73 is a Sophie Germain prime and is palindromic in binary 1001001 (interestingly, all Fermat primes and Mersenne primes are subsets of the binary palindromic primes).  73 is also an octal palindrome 111 and the only octal prime repunit.  73 is, moreover, and for some of the reasons given here, Sheldon Cooper's favourite integer in The Big Bang Theory — as was first referenced in the show’s 73rd episode.  (Jim Parsons, incidentally, the actor who plays Sheldon Cooper, was born in 1973.)

However, after listening to Charles Yang, associate professor in the University of Pennsylvania Department of Linguistics, in this episode of the superb Tell Me Something I Don't Know podcast, hosted by Stephen Dubner of Freakonomics fame, it is another little number-73-centred-nugget-of-information that has prompted this post.  Namely, that once an English speaking child has learned to count to 73, they realise they can keep going, and thus ‘understand’ the concept of infinity (NB the 1st two comments at the end of this blog).

Having delved into Yang's fascinating research, let me try and elaborate:  If there is a linguistic rule, a generalisation in other words that can be applied to a set of N words, but within this set of N words there is a subset of words, e [2], that do not follow this rule and that must therefore be memorised, then Yang (2005, cf. 2015) has shown that:

\[e < {\theta _N}\;where\;{\theta _N}: = \frac{N}{{\ln \left( N \right)}}\]

This model, 'dubbed' the Tolerance Principle by Yang (2005, cf. 2015 pp16-17), asserts that for the generalisation to be ‘productive’, i.e. for the rule that generates words to tolerate the exceptions and thus be applied instead of the pure memorisation of all N, e must not exceed / ln(N).  It delineates, in other words, a threshold value for e (as a function of N) at which a word-forming rule ceases to be a productive mechanism for the learner to apply, and thus provides a ‘sufficiency measure’ for such linguistic generalisation.

By way of illustration, Yang (2015, pp948-949) gave this example of regular and irregular verbs: ‘If a typical English speaker knows e = 120 irregular verbs, the productivity of the -ed rule is guaranteed [my emphasis] only if there are many more regular verbs. Specifically, there must be N verbs, including both regulars and irregulars, such that θN = / ln(N) ≥ 120.  The minimum value of N is 800.  In other words, if there are at least 680 regular verbs in English, the -ed rule can tolerate 120 irregular verbs’.

In his Tell Me Something I Don’t Know appearance, Yang described applying this Tolerance Principle to how we as children learn to count.  We know how many of our counting words are exceptions to the rule, those numbers that is that ‘bear no relation to their quantity, [that are] are completely arbitrary’, those numbers to put it another way that ‘won’t tell you anything about how numbers are forming' (Yang, in Berger, 2017) — in the case of children counting in English, there are 17 of these exceptions, e = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 20, 30, 50} [3].

Once we have memorised, ‘by brute force,’ the number words one to ten, we start to observe and learn rules for continuing.  For example, we start to see that fourteen is formed from ‘four and ten’, sixteen from ‘six and ten’, twenty-four from ‘two tens and four’, 'twenty-six from 'two tens and six', etc.  But there remain of course those exceptions to the rule — why is eleven not one-teen, or twelve not two-teen, etc. —  where ‘memorisation [is] the sole educational mechanism’ (Yang, in Berger 2016).  To this end, using the Tolerance Principle, we can find the least amount of words that we need to learn ‘to overcome the [17] exceptions we have to memorise’.

The smallest value of N such that θN = 17 is 73:

\[\begin{array}{l}\begin{align}N &= 73\;\\\because17 &= e < {\theta _N} = \frac{{73}}{{\ln \left( {73} \right)}}\\\;where\;{\rm{ }}\frac{{73}}{{\ln \left( {73} \right)}} &= 17.014...\end{align}\end{array}\]

Or in other words, once a child has learned to count to 73, they have learned the rules of the game sufficiently to overcome the cognitive need for memorisation [4].  Once children have passed this 'tipping point', when they come to a number that they are unaware has a word outside of the system they have subconsciously generalised, a thousand for example, they make up a word that fits their generalised system.  Thus, once an English speaking child can count to 73, they can count forever.

Notes & (Select) Links:

[1]  Incidentally, also the product of multiplying 7 and 3.

[2]  Not to be confused with Euler's number, e.

[3]  Yang noted that in Chinese, e = 11, meaning that a Chinese speaking child need only count up to 42 before they are able to continue and thus appreciate the concept of infinity, some one whole year before English speaking children.  

[4]  Welsh may be one of the languages with the smallest threshold at θ= 36.  The numbers 1-10 in Welsh are the only 'exceptions': Un (one), Dau (two), Tri (three), Pedwar (four), Pump (five), Chwech (six), Saith (seven), Wyth (eight), Naw (nine), and Deg (ten).  All numbers beyond this are generalised from them, for example eleven is un deg un (one ten one), twelve is un deg dau (one ten two), seventy three is saith deg tri (seven ten three), etc.


  1. Thanks to @JKBye who quite rightly suggested through his tweet (https://twitter.com/JKBye/status/936663473374642176) that 'implicit understanding of generativity for number words is neither necessary nor sufficient for an explicit formalization of infinity.' Of course, a child’s understanding of ∞ is not a function of the generativity for number words in their language. I think the 'significance' of 73 is more about it removing the need for ‘brute force’ memorisation as children learn to count.

    @JKBye also goes on to generously give a point of comparison, namely that 'it wasn't until Chomsky in the '50s that the infinite generativity of language was properly formalized (see his famous debate w/ Skinner), and yet we frequently generate novel linguistic utterances implicitly.' To find out more about what is known as the 'debate that changed psychology,' go to these links: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2223153/ https://youtu.be/FlyU_M20hMk

  2. You may also be interested in this 2017 paper by Siegler and Lortie-Forgues (http://www.psy.cmu.edu/~siegler/2017-SieglerLortieForgues-HardLessons.pdf), heard through @dylanwilliam who tweeted 'It is sometimes claimed that Chinese and Korean students excel in math because their number and fraction naming systems are simpler. It doesn't seem to help Welsh students, who have identical systems...'