The Magic of Numbers – Serial 1

This serial discusses that comparing is based on what we first observe with our five senses and setting reference points from our memory. To appreciate and remind ourselves of the usefulness of counting and numbers, we have to imagine life without measurement systems and mathematics, which the scope of this serial.

Imagine life without measuring instruments and mathematics

The purpose of our five senses is to observe what is happening in our surroundings and we also have the ability to remember what we have observed, so without remembering our observations, we cannot have something to compare with.

So our ability to remember is crucial for making sense of what is happening around us. Nothing of what we observe makes sense without comparing. The tricky part is that comparing sometimes happens subconsciously that we don’t notice we are comparing.

Example: We don’t need to think about comparing the exact facial features of 2 people, we just know by simply looking or remembering someone you have met before and comparing the facial features of the new person with someone you have remembered. More examples of subconscious comparing in real life will be given below.

Remembering things is a key part of learning. So the first things you have learned  as a baby is important for making sense of newer things to learn. So we cannot learn without points of reference or point of comparison, the first point of reference is the size of our body. We usually recall the points of reference from our memory.

In this article we will take a look at what life is like without measuring instruments, number/counting systems and mathematics below. In fact we sometimes, in everyday life, without noticing, forget to use them, so we sometimes rely solely on our five senses.

But we rely on the clock every day as our measuring instrument that we cannot imagine living without it, because the sun is not always visible indoors and at night. This will be explained in a later serial.

The racial references or town/city references in this article were used just to make explanations clearer.

Examples of how we compare with our five senses

How we perceive size

First of all, babies knew how big they were by looking (using the sense of sight they are born with) at the surroundings and comparing the size of their body with some objects (e.g their toys) in the surroundings. When we were born we did not know how to use yet the number system, mathematics and measuring instruments.

When we look at the soccer field we immediately noticed with our sense of sight that it is big. Why do we call it big? We unconsciously compared, via instinct to the size of our body. We recalled our size from our memory because it is the first thing we felt and saw. Thus it is our very first reference point.

Let’s assume we were born with a blank memory. So by our feelings we committed our size to our memory. Without this feeling of size, the objects around us won’t make sense. With our memory blank at first, the first size we knew was the size of our body.

If you say in everyday language “the soccer field is big” or “the basketball is small” you will be understood since we compare the size of things unconsciously to our body.

This means we don’t need to say: “the soccer field is big compared to my body” or “the basketball is small compared to our body”.

Distance of objects

How do we know if a place is far? In the playground, a child (Finnish girl) saw two trees, one palm tree in front of her and another mango tree behind the palm tree. Unconsciously comparing how far the two trees are from the position of her body, she says that the palm tree is nearer and and mango tree is further.

Also because we always move, we know we can move away from the palm tree. Thus she concludes unconsciously that comparing how far her body from the palm tree is not reliable. Knowing plants don’t move, she can treat the palm tree as a reference point. She now says in her mind, with respect to the position of the palm tree, the mango tree is near.

But she also notices other trees, apart from those two trees. The child also notices there is an orange tree just on the left of the mango tree, separated by a wide walkway. The child recalls the positions of the palm tree, mango tree and orange tree. At first she uses how far apart the palm tree and the mango tree. Judging from that, she notices that the orange tree is further with respect to the distance between the palm tree and mango tree. The diagram below illustrates the positions of the objects.

Now when she asks her father how if she can go to her friend’s house to play. She just introduced herself to another playmate and has had a good time with that playmate. The father replies, “It is far from our house, so you cannot go”. The girl having just gained the ability to compare distances based on her memory of the positions of the trees, the word “far” now makes sense to her, so she does not feel so good about her father’s reply. She also does not feel good because of her dad’s body language.

The girl has not even heard of the meter rule or other devices that measure distance and has not even heard of the word “precision” or even practiced mathematics, but the positions of the trees in the playground has helped her made sense of “near” and “far”. She has not even heard of numbers and counting systems yet, as tools for precision.

She only had her sense of sight and her ability to compare based on her memory. Even the size of her body has helped in making sense of distance and the meanings of the word “near” and “far”. She had already made sense of her size based on the surrounding objects: the play ground and trees are bigger than her. She says unconsciously in her mind: “For something to be far, the place has to be really big as well, much bigger than me (my body)”.

Our sense of sight has the components of field of vision and visual acuity (how clear objects are from afar). We know when an object is far if we see familiar objects (like trees on the road) more clearly than other objects, like a road sign. We know a road sign is far when the letters don’t read clearly. How do we know it is far? We compare the positions of the road signs, the trees before it and the position of our vehicle with our sense of sight.

Getting pleasure from our senses and how it alerts any danger

When we get pleasure from our five senses we tend to want some more of it, that we start to like it. We have have fond memories of something delicious we have tasted, or the fragrant smell of the a flower. As we all know, our five senses helps us detect something dangerous too. Our senses can also detect the intensity of the stimulus: is the smell of the flower too strong, or is the music too loud?

But we won’t know something is too loud if we don’t have the memory of something sounding softer or if we don’t have the survival instinct of avoiding music/sounds that damages our eardrums. We tend to move away from sharp objects the instant we see them. Below is an example of we react to stimuli based on our memory or survival instincts.

Even animals tend to compare the size of the things they see with the size of their body. Mountain lions, also called cougars, tend to run away if their eyes see a creature that is bigger than them. Their survival instincts tell them that anything bigger than their body is probably more dangerous than them.

Comparing the size of objects in our surroundings with the size of our body is a survival mechanism too because dangerous objects, such as moving boulders can be dangerous.

Hot and Cold (Temperature)

How do we know if something is too hot or too cold? Our sense of touch simply feels it because temperature is vital for the regulation of biological process, so temperature detection vital for the survival of the animal body. So homeostasis of temperature cannot happen without the temperature nerve endings everywhere in our body. Those sensors are part of the sense of touch.

Irritation to the hottest and coldest places are a survival instinct. We shiver if we feel too cold or sweat if we feel the searing heat.

But we won’t know the difference in temperature of two hot rooms if we don’t have a first reference point. Since we were born, the first temperature we have ever felt is room temperature, which depends on the local climate. To make sense of the temperature of the sauna or park on sunshine, without using a thermometer for precision, we simply feel from our sense of touch that it is hotter than the room. The homeostasis of the body makes us aware of the temperature. Then we store how we have experienced the outside heat in our memory.

Below is an example of a scenario of a person that has forgotten to use a thermometer and has to rely on his memory.

One day Average Joe was browsing a travel guide, Joe becomes curious of a colder place, having experienced the searing heat of the sun every day. Joe was stimulated to take a break from the searing heat when he found a country on the guide that has a cooler climate. The guide must have reminded him of the temperature of air conditioned places. The guide has convinced him to take a break in a cooler climate. He then marks on a calendar the best time to travel.

With respect to temperature, Joe has two points of reference that are vital to his travel decision: He experiences every day the searing heat of the sun and goes to bed every day with the air-conditioner on, making the room cool. We can see here that three things were compared.

Due to the linear and sequential thinking, two objects were compared at a time. First the room temperature and the park temperature at sunshine, second the room temperature and the air conditioned bedroom. Again while deciding why to travel, he did not use any number systems or mathematics because the guide simply did not state the temperatures in Celsius. Instead the travel guide evoked his experiences in temperature from his memory.

With respect to his sense of sight, he wanted to see different places but wanted to see one with a cooler climate. The picture of the place in the travel guide was convincing.

Fragrant flowers in a garden

If we have smelled only one type of flower (e.g, roses) in our lives we tend to treat it as the general smell of a flower, as if one kind of flower exists. To avoid this problem we must smell different kinds of flowers (e.g, lilies, hibiscus, jasmine and bougainvillea). Every flower has a distinct smell the nose can detect. Not only do we differentiate flowers with their smells but with what they look like (sense of sight).

Again, even when differentiating between flowers we need a reference point, usually it is the first flowers we have ever smelled as children. When a child sees a new flower she have never seen at all, her eyes and nose can detect instantly the similarities and differences between the first flower (jasmine) she saw and the new flower (red rose). The child may even say it is the best flower she has ever smelled, based on the experiences of the flowers she saw and smelled.

Here is how our nose can differentiate between the intensity of a smell without number systems or measuring devices: A child goes to a room, she sees many roses, she smells something good. The smell of the rose and the intensity of the smell is recorded to her memory as an arbitrary reference point. Then she goes to another room of the same size and finds less roses and she also noticed that the roses smells weaker there. The child concludes that the smell of a flower will be stronger if there are more flowers.

Public establishments and tools

How satisfied we are with public establishments or tools, is also based on our ability to compare. This is why we have to try things we have never felt before. This is why advertisements keep convincing us to try something new (tools). Software companies even let you try their software for free but with restrictions. Without trying something, we won’t know which is better. This is why it is best not to judge a book by its cover.

Schools and operating systems

Any student who has stayed in a private school all his study life may not know what the best school is because he is reluctant to change school. A student who has been in a different school can tell which school is better.

Through our memory we can recall how many times did we change private schools during our elementary or high school life. Only through experiencing different schools, we can compare which school is the best. If we stay only in 1 school throughout, we won’t have anything to compare with.

This is the same with a computer operating system: If a user used Windows all his adolescent life, he won’t have an idea if other operating systems (Mac OS X and Linux) are better, unless he tries one.

Comparative Vocabulary and the Origins of Measurement

We use these every day comparative vocabulary when we don’t need precision or need to think quickly or don’t carry any measuring instruments. Here are some of the comparative vocabulary we use every day:

Same, similar, equal to, different, more than and less than, quite.

We will visualize some of these terms using the diagrams below. The problem with comparative terms is that they are purely arbitrary. To demonstrate this arbitrary nature, we will use playgrounds with trees as a setting.

One playground is in Szczecin (Stettin in German) and the other is in Reykjavik. The Polish boy in Stettin does not know what is happening in Reykjavik and Helsinki, so the Polish boy cannot share the points of reference he chose with the girl in Reykjavik.

The playground maps below have the same map scale. In the two cities, the trees are spaced out differently in all cites as we can see in the diagram.

There are three perspectives here: You, the reader, the Polish boy and the Icelandic girl. Each have their unique points of reference.

From what the Icelandic girl in Reykjavik can see (perspective):

From what she can see (perspective):

– The spacing between the apple tree and the oak tree is less than the cherry tree and oak tree.
– The cherry tree is the same size as the apple tree.
– The oak tree is bigger than my body.

To make sense of the word “far” or “near” next time, she chooses an arbitrary reference point. She chooses the spacing between the apple tree and the oak tree. Making sense of distance is already explained with the example of the Finnish girl above.

From what the Polish boy in Stettin can see (perspective):

From what he can see (perspective):

– The spacing between the mahogany tree and the pine tree is more than the maple tree and pine tree.
– The size of the maple tree is more than the pine tree.
– All of the trees are bigger than my body.

To make sense of the word “far” or “near” next time, he chooses an arbitrary reference point. He chooses the spacing between the pine tree and the maple tree. Making sense of distance is already explained with the example of the Finnish girl above.

From both of their perspectives, we can see that measuring distances (spacings) is comparing in itself. So the word “measure” is based on “compare”.

The reader is not in the playground but has two maps that have the same scale. Only you, the reader of this article can see, based on the maps, that the reference point of distance the children chose are different, if the reference points of the Polish boy and Finnish girl are compared side by side.

The children have the degree of freedom of choosing a reference point. Due to this degree of freedom, the reference points are purely arbitrary.

You, the reader wants to make sense of size and knows that how close the children are to the trees can affect their perception of space. You want to be more precise and exact, so you choose from these arbitrary points of references:

– The spacing between the cherry tree and the apple tree
– The spacing between the apple tree and the oak tree
– The spacing between the oak tree and the cherry tree

– The spacing between the mahogany tree and the pine tree
– The spacing between the maple tree and the pine tree
– The spacing between the mahogany tree and the maple tree

When you compare the position of the all trees in the two city maps by simply looking at the maps, you notice with your eye that the trees are spaced out differently. To know if they are all truly different, you take one of the arbitrary reference points.

Not having any ruler available at the moment and forgetting the number systems, you start comparing the distances above by taking a small piece of paper (sticky note) and takes note of spacing between the cherry tree and the apple tree by marking on the paper two points with a pencil. The sticky note made it easy for you to compare all the spacings in the maps. This will be illustrated below.

You mark with a pencil the spacing between the apple tree and the cherry tree in the maps: Using the sticky note, you compare the spacing between the mahogany tree and the pine tree with the marked spacing:

You conclude: “The spacing between the mahogany tree and the pine tree is more than the spacing between the apple tree and the cherry tree.” This is how you use comparative vocabulary if you don’t have measuring instruments with you.

Using the sticky note, you compare the spacing between the apple tree and the oak tree with the marked spacing:

You conclude: “The spacing between the apple tree and the oak tree is less than the spacing between the apple tree and the cherry tree. So the spacings are different.”


Thus it is impossible to be precise and exact without choosing an arbitrary reference point. So measurement is also impossible without an arbitrary reference point. Our eyesight detects distance and it can let us choose any arbitrary reference points.

So precision is just a mental artifact based on choosing a reference point. In real life many different reference points are available to choose from and just one reference point will make sense of distance or size:

As in the example above, just one spacing used as a reference point (the spacing between the apple tree and the cherry tree) is enough to make sense of the other five spacings.

When we need precision and comparative vocabulary does not work anymore for a certain situation, i.e engineering situations, we developed counting/number systems.

It is difficult to make sense of precision without counting/number systems. Those systems eventually became the foundation of mathematics. Number systems will be discussed in the next series of articles. Measurement systems will be unreliable without a reliable number system.

Numbers and measurement systems are intertwined. If we have not conceived counting and numbers from our consciousness (ideas), we would not have developed measurement.

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