Abstract Confusions

Complexity is not a cause of confusion. It is a result of it.

On Counting, Countability, Uncountability and Confusions thereof

The very basics of math skills are not taught by any one, it is inculcated from birth. In my childhood days, I remember how I got surprised by my grand mother’s mathematical (rather counting) skills. She was not taught in any school, she has to manage the household stuff, few farm works. She could never read a word, but still she could precisely count the number of coconuts, add subtract multiply or tally the rupee notes and settle the account. Counting is something that comes naturally to one. In fact I read some where, counting is not exclusive to humans, birds like crows can count till five. Another news item claims chimpanzees can even count better than humans.

Asian Advantage in Counting

Another article even attributes geographical / language factors for mastering the mathematical ability of remembering number. If you haven’t read it, here is the summary: the article advocates the certain languages (Asian in this case) have inbuilt advantage in manipulating numbers in mind. Because, the words for the numbers are smaller and easy to store, retrieve, manipulate.

Chinese number words are remarkably brief. Most of them can be uttered in less than one-quarter of a second (for instance, 4 is ‘si’ and 7 ‘qi’) Their English equivalents—”four,” “seven”—are longer: pronouncing them takes about one-third of a second. The memory gap between English and Chinese apparently is entirely due to this difference in length. In languages as diverse as Welsh, Arabic, Chinese, English and Hebrew, there is a reproducible correlation between the time required to pronounce numbers in a given language and the memory span of its speakers. In this domain, the prize for efficacy goes to the Cantonese dialect of Chinese, whose brevity grants residents of Hong Kong a rocketing memory span of about 10 digits.

It could not help me but to compare this with the counting in my mother tongue Tamil.  In English, we have to count fourteen, sixteen, seventeen, eighteen and nineteen, so one would think that we would also say one-teen, two-teen, and three-teen. Not the case. It is little bit in a different form: eleven, twelve, thirteen, and fifteen. Compare that to Tamil counting, 11 is pathinoonnu (பதினொன்று or pathu+one; ten+one), 18 is pathinnettu (பதின்னெட்டு or pathu+ettu; ten+eight).

As a result the article concludes Asian children have advantage of storing more numbers in their mind and manipulate them faster.

On that being said, let me point you that, though counting is one of basic skills, easily acquired, gifted, it is the one which is widely misunderstood too. The notion of counting is different from highly debated.

Counting and Ordering

Intuitively when you count something, you will be labeling them with numbers, a one-to-one mapping between the object being counted and the natural numbers. When my grandmother counted coconuts she would have used her fingers to keep track and folded one finger after another, a one-to-one mapping.

Ordering is a different concept.  Some how, even small children knew two candies is better than 1 candy. They can order 2 candy and then 1 candy. The ordering is known as “Natural Ordering”. You know which number is bigger to another number fairly to some extend.

Is there a connection between ordering and counting? There seems to be deeper connection then what we could imagine.

Countability

Simply put, countability is the ability to count. Not your ability, but the ability of  object being counted. Consider this, in young days children often challenge others to count the sand particles in handful of sand or fistful of hair in head or even the number of stars in a quite bright night sky. Most the smaller kids would blink and could not reply. But as you grow older, you would would reply with some huge number and counter-challenge to disprove it by counting. The point to take is, these objects are countable, just not easily. Is a litre of milk countable? It is measurable, but again measurability  in mathematics is another field worth separate discussion.

Fine, let us get more abstract, are the natural numbers countable? Assume a bag that contains 10 balls each numbered with one number between 1 to 10 unique. You would pick all of them, not in any particular order, but after picking 10 of them, you could confirm that you have picked all of the balls from bag without seeing into the bag to see whether any thing more is left. How? You would just need to count the balls as you pick, mind you, count without any order. So, are they countable? the fact that you are able to count them one, two, three using your fingers prove they are countable. Are the whole of integers countable? Yes, the same argument finger in the hand applies, only but you have remember the sign for negative numbers. Are the rational numbers (fractions) countable? This is where it gets tricky. Yes, they too. You can always think there could be innumerable rational numbers between 0 and 1, but still the entire set of rational numbers are countable.

The famous mathematician, Cantor, has given the following brilliant demonstration. You have to understand that by counting you mean to can assign an order to objects and make sure nothing gets lost from being numbered, or rather from counted. If by some how you can introduce such an order in the rational numbers (not just lesser then, greater to), and make sure no other number is missed between any given two, the countability of rational numbers can be established.

Ordering of Rational Numbers - Countable Rational Numbers

Ordering of Rational Numbers - Countable Rational Numbers

Cantor used the above pairing method. Start with writing with number 1/1, 1/2, 1/3 ... in the first column and 2/1, 2/2, 2/3 ... and so on. Now in the above zig-zag pairing, you will be able to comb all the number (just like counting bag of 10 balls using fingers). Such a zig-zag pairing will not leave any number from being missed out. Though this is not the natural ordering, but this ordering will make sure rational numbers are countable.

Uncountability

Do un-countable numbers exist? They do. Consider the easiest and familiar-est of all: “real numbers”.  Consider the below sequence of number (refer below, for better, imagine S1 as big number with all digits separated by comma). S2, S3 and others numbers which are written in some order. The idea is so that, what ever order the numbers are ordered, there will be a number which can be shown not be part of such an arrangement. All it needs to show is, such number arrangement will always leave some numbers uncounted for, and hence uncountable.

s1 = (0, 0, 0, 0, 0, 0, 0, …)
s2 = (1, 1, 1, 1, 1, 1, 1, …)
s3 = (0, 1, 0, 1, 0, 1, 0, …)
s4 = (1, 0, 1, 0, 1, 0, 1, …)
s5 = (1, 1, 0, 1, 0, 1, 1, …)
s6 = (0, 0, 1, 1, 0, 1, 1, …)
s7 = (1, 0, 0, 0, 1, 0, 0, …)
s0 = (1011101, …)

Now, S_0' is constructed by taking the diagonal numbers, i.e., 1st digit from S_1 , 2nd digit from S_2 , 3rd digit from S_3 and so on. Call this as S_0' . Find the complement (replace all 0s with 1 and 1s with 0) and call it as S_0 . Now check whether such a number S_0 is present in our collection. It would have not. Since our construction is such it would have taken care, (remember that we took the diagonal number and reversed it, so if such a number had existed in the collection, we would have picked the diagonal digit and reversed it).

If you are not sure, add this number to the collection, call it S_8 , and start over again. If you are not able to get the idea, take a paper and pencil, imagine you know only four numbers, and do this exercise.

This method is known as Cantor’s Diagonal argument and was first demonstrated by Georg Cantor (refer here for his interesting rather troublesome life). Diagonal argument is such a profound method it paved in way to much more brilliant ideas and proofs later. Russell’s paradox which shock the base of set theory is based on diagonal argument. Later Alan Turing’s answer to the Entscheidungsproblem and Gödel’s incompleteness theorems which spelled the end for David Hilbert’s grand formalization of mathematics, were also developed based on diagonal argument demonstration.

All of the above methods would require detailed explanation and a separate detailed post. Hope you understood it is easy to just keep counting. ;)

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2 responses to “On Counting, Countability, Uncountability and Confusions thereof

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