The Magic of Numbers – Serial 2

We need to learn a language before we can count

When you were a child, you were taught to associate words and the sounds from your voice box with images. Below our parents showed us what a cat is by simply saying and pointing “That is a cat”. When you were a baby, you cannot write or read the Latin alphabet or number symbols yet. So you unconsciously associated the sound “cat” with the image below:

Language is about associating spoken words with the images we see in our environment. The associations are stored in the memory. You as a baby, being imitative and want to talk to your mother, you might babble unintelligible sounds at first but as you grow you can pronounce words more correctly.

Then you can better imitate her mother speaking. You as a baby knew by trial and error the only way to get your mother to understand you is to imitate what she is saying. Thus before you can learn numbers and counting, you have to learn your mother tongue first.

Millions of years ago, humans learnt that they can associate things with sounds. So they created an arbitrary (unique) sound for everything they see. Also the sound “cat” can be substituted with the sound “neko” without affecting the association.

Consonants are the distinct sounds we can make with our voice box while vowels are sounds that can connect consonants to form a word. Being conscious and creative beings, we created languages from those sounds.

Body language in the form of facial expression and other body movements was also a form of communication that enhances how the sounds are expressed. Those first group of humans (friends) agreed on a common system of communication using voices (words created from combining vowels and consonants) and body language.

Children rely on body language of any person around them to guess the meaning of a sound of what another person is saying. To use language, we have to agree on the meanings of the words. We are already agreeing to the meaning the moment we ask what a word means or look up the meaning in the dictionary. This behavior is very similar to imitating your mother speaking when you were a baby. Body language helps greatly in imitating.

Writing and reading was also not popular in the past 500 years and more. So written symbols did not make sense to those that cannot read and write. Writing started when we can make abrasions on the sand, soil or use paint. Eventually paper was invented, so we used ink. Then computers came and we can use the keyboard. Reading cannot happen without writing. The purpose of writing is that somebody can read it or as a reminder to yourself. The words you write must be spelt correctly.

As a survival instinct, we were first able to make sense of size by comparing surrounding objects with the size of our body, as discussed extensively in the previous article. So the first humans realized that they can make sense of size by counting with their fingers.

Why we need to learn counting and numbers?

We won’t be able to use the clock, measurement systems, and money. We also won’t be able to compare objects in situations where comparative vocabulary is imprecise. Numbers are an aspect of group consciousness due to systems of agreement.

Those systems have numbers and counting as a foundation, so if we don’t have the idea of numbers and counting, they won’t exist.

By having a properly defined sequence of numerals, just as we have defined the sequence of the Latin Alphabet by singing it, we were able to have the idea to count. So numbers are just like temporary names we assign to things based on our ability to imagine things big and small.

Learning how to sing the Latin Alphabet, we were taught to how to organize words in alphabetical order. As you know, the sequence of the Latin Alphabet in your computer keyboard is muddled up, so that is not the defined sequence.

The beginning of the sequence is defined as small and as we continue the sequence, it get bigger in our imagination. The number sequence is defined below.

We were taught or conditioned to think: Small must be closer to the beginning of the sequence and Big must come after it. This is the foundation for star rating systems for like and dislike we frequently encounter in online stores.

We also know that our house is big and our planet Earth is very big (compared to the size of our body). Without an organized sequence of numbers, we won’t be able to answer the question: How big?

So numbers help in comparing.

Associating numerals with fingers open based on a sequence.

Your mum teaches you numerals by making sounds as your mum opens her fingers with based on a sequence. You as a baby, associated the number of fingers open with the sounds, just like how you associated the image above with the sound “cat”.

The sounds your mother is saying as she opens her fingers are called numerals. So the sounds (numerals) below are arbitrary associations based on the number of fingers open.

You as a baby, realized you have a set of fingers in your hand. You start with a closed fist and imitate your mother opening her fingers. You tried to imitate the sequence of the fingers opening.

Our hands have the thumb, index finger, middle finger, ring finger and little finger. You as a baby progressively learnt the sounds “one” to “ten” based on the fingers sequentially opening:

Opening your fingers has to follow a proper sequence as to properly associate the numeral. Associations will be confusing if we are mixed up with the sequence: “Start: all fingers on right hand closed. Open thumb. Open Ring finger and so on”.

It is difficult for the average person to first open the thumb and then open the ring finger, so the sequence of opening fingers in the table above is the easiest. It is easier to open the the index finger and then open the middle finger.

Numerals are vocabulary associated with the number of finger open. The word “numeral“ and “number” are sometimes used interchangeably but for this article we will reserve the word “numeral” for the spoken or written word such as “eight” in English or “huit” in French. The word “number” will only be used for the written symbols such as “8”. Today we use the same symbols regardless of the language.

For semantic clarity, numerals are more associated with spoken and written language while numbers are more associated with symbols in this article. Numerals were first learnt as part of the vocabulary for kindergarten and then the symbols called numbers.

It is safe to conclude that numerals were conceived before numbers because babies first imitated the fingers opening before they even heard of the number symbols.

Associating numbers with numerals

Your mother enrolled you to a kindergarten. The teacher taught the child how to write and read aloud the Latin alphabet and some written symbols associated with number of fingers open based on the sounds her mother taught her.

You were taught that the written symbols are called “numbers”. You as a little child realized it was very convenient to write those symbols rather than writing the whole word. In the table below, you were also taught that writing sticks or tally marks can represent fingers open.

Associating symbols with numerals is also arbitrary. The symbol “5” can be associated with the numeral “eight” but everyone has to agree on a single association so that everybody can easily understand each other.

Learning How to Count

Now that you have become acquainted with numerals in spoken and written language and symbols called numbers in writing, the teacher taught them how to count. To count, the child’s association with numerals and numbers should be solid. So ultimately, counting is based on our ten fingers.

You as a child was taught that numbers follow a defined rhythm and sequence. Without that rhythm and order, you cannot count properly. It was easy to memorize the Latin alphabet by making a child hear a song about the Latin alphabet and making you recite the alphabet by singing it. You imitated how the teacher recites the rhythm of counting.

The proper sequence of numerals is “one, two, three, four, five”. Not “two, four, one, five, three”. The proper sequence is important for assigning numerals to objects. The correct sequence of numerals is helpful because it can help us differentiate with what we have counted and what we have not counted in images or objects we see. Without a proper sequence of numerals or numbers, it won’t be clear if something is increasing or decreasing.

The symbols called numbers should also follow a proper sequence based on the numerals associated with those symbols. The proper sequence is “1, 2, 3, 4, 5”. Not “2, 4, 1, 5, 3”.

This proper sequence too is arbitrary, but everyone has to agree on a common sequence to be able to understand each other. Counting is useless without a common sequence and the proper association of numerals and numbers.

The teacher teaches you to point (focus on) to objects as she counts. You subconsciously use your ability to recognize and identify patterns (described in the introduction). This ability is possible when the five senses and memory work together. Then the teacher gave the image below to exercise your counting as a class.

The class was taught to speak a numeral as they point an object with their fingers. The objects to pointed on or focused at must look similar. We can only focus one object at a time due to linear and sequential thinking. Recognizing objects is essential to counting.

They were also taught that nothingness or emptiness is associated and indicated with the numeral “zero” or the symbol “0”. The teacher teaches them, “Since you see no dogs in the blackboard, you can say ‘No dogs’ or ‘zero dogs’ “.

The teacher asked “How many cats are in the picture below?”. Then the children started counting in the sequence their teacher taught them. This is how they count:

We can denote focusing on objects with the green circle. They start by focusing on the grey cat, they say “one”.

They focus on the orange tabby cat, they say “two”.

They focus on the orange striped cat, they say “three” and so on.

They use the next numeral in the sequence as they point the next similar object. The pointing stops when there are no more to point to. They counted five cats. As they speak the numerals they visualize their fingers opening. They are told not to point or focus on the object they have already counted. The position of the objects in their eyes helps in this.

The child was also taught to use ordinal numerals to help in descriptions and to better differentiate similar objects. Example: “What is the color of the third cat?”, the teacher asks the class. The class replies “orange and white”. Ordinal numerals are used repeated tasks as you can see in the infographic below.

Imagining size

Our ability to imagine sizes is essential to counting. As explained above, as we continue counting, we visualized things getting bigger. We will use flowers as an example to demonstrate how we imagine size. Flowers vary in sizes, so we will use one kind of flower. Below we will imagine smaller things fitting into bigger things.

We can represent a flower by using an abstract geometric idea called a circle, because this can simplify things our mind. In the previous essay, we also represented trees as circles too.

We can imagine big and small circles in our mind. We imagine a big circle, so to make sense of the big circle we count how many small circles that can fit in to the big circles.

How do we imagine size in graphic form? We recalled the size of a small circle (green):

Then we visualized another circle in our mind bigger than the previous circle (blue):

Our minds now imagines both circles. We assign 1 to the green circle. The numbers that follow after 1 (2,3,4,5,6…) in the defined sequence we learnt in kindergarten are defined to be bigger than 1.

Since we have already assigned 1 to the smaller (green) circle, we cannot assign 1 to the bigger (blue) circle. If we do this, it is not logical based on the defined number sequence.

Instead we can assign an number bigger than 1, because we see in our eyes that it is looks like two small circles combined into a big circle. Let’s assign 4, since four circles seen to be able to fit it at first sight.

Thus we have selected the small circle (green) as our reference point. Four (4) small circles can fit in. Imagining things fitting into a container (water bucket) is also one of the ideas behind measurement.

Counting beyond ten (10)

The child was taught that they can count beyond ten (10) by closing their fingers after they have recited the numerals one to ten, and then opening them again but using different numerals in the proper sequence to indicate they have already used the numerals one to ten:

So the numeral “thirty-seven” means that we have closed our fingers three (3) times and we have opened seven fingers again. We repeat the sequence of numerals each time we close our fingers, but we add a marker (thirty) to the left of the numeral to indicate the number of fingers closed.

Higher order numerals also exist in vocabularies in various languages. In English, they are hundred, thousand and million. It will be defined based on the lower order numerals we have revised so far.

Hundreds means you have closed your fingers ten times already during counting. While you were reciting the numerals one (1) to hundred (100), a friend helped you by also counting with his fingers how many times you have closed all your fingers, in case your counting goes wrong. it will be illustrated in the infographic on the next page.

When Mr. Hundred has counted that your fingers have closed a hundred (100) times already, it means thousands (1000). Mr. Hundred counts very similarly to how you have counted from 1-100.

We realized we can continue counting forever because we can create numerals that have higher order numerals and lower order numerals combined like: “five hundred and thirty-seven”, “three thousand, two hundred and forty-two”, “seven hundred and sixty-seven” and so on.

This is how we count beyond thousand: “forty-thousand, two hundred and forty-six”, “ five hundred fifty five thousand, four hundred and seventy six”. Eventually we reach “one thousand thousand, or million. Million is defined as one thousand thousand.

Eventually we can count beyond million: “four million, six hundred thousand and one hundred” and “three hundred million”, this eventually led to the idea of infinity and the continuum of numbers.

But numerals are created only within language via sounds and associations with imagery and written symbols called numbers, so they are only objects of thought or ideas and do not exist in nature.

The only useful thing about numbers is that they help compare or differentiate and be more precise, after all to make sense of our surroundings, we compare surrounding objects with the size of our body, or compare new things from what we remember.

The next serial will deal with how we write larger numbers or symbols because large numerals such as those above are cumbersome to write. Even representing numbers in stick form is difficult to write, such as those below. A combination of numbers is easier to write using the positional notation or place value.

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