Math and Our Brain

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How did human brains become capable of discovering math ideas?

Human brains have been developed by evolution to make sense out of patterns.

Patterns of Magnitude & Shape

Two patterns which were recognized early were magnitude (as size) and shape.

The sense of magnitude or size is very essential for all living beings to survive in the world. The idea of size comes in various forms. Estimating the breadth of a stream to be jumped over, the size of an opponent or herd of animals, the distance &amp; speed for catching a prey are all instances of such survival skills.

The sense of shape helps us to differentiate between friend &amp; enemy, edible &amp; poisonous and to maintain our sense of direction while travelling.

All living beings have acquired basics of these skills through the process of evolution. If a species had not acquired this skill they would have died out.

The level of development of these skills depends on the environment in which the being lives. For instance, the sense of smell in animals is far superior to that in humans.

The sense of magnitude developed at a faster rate &amp; to a sophisticated level in humans due to superior social, economic and cultural practices developed by them. A developed human society cultivates, hoards, trades &amp; fights with other cultures. Hence it develops the necessary math to plan battles &amp; seasons for cultivation and document trade transactions. It also helped them to dig canals, measure produce and build sophisticated structures.

Our brains are &ldquo;meaning-making&rdquo; machines. We do not easily accept that any pattern could have occurred by chance or randomly. Hence, we look for causes. We try to connect the cause and the effect by some logic. This is the beginning of intellectual development.

The human brain developed the intellectual ability to identify, extend and reproduce patterns that they saw in the environment around them. Patterns help the brain to categorise sensory data and organise it for understanding. Through this, the brain developed a unique ability to understand abstract ideas. Understanding patterns also helps in predicting and avoiding dangers. Hence pattern recognition is vital for life.

Measurable &amp; Countable Magnitudes

We sense magnitude in 2 different ways - as a measurable quantity or a countable quantity. These can also be understood as answers to “how much?” and “how many?”.

Human history tells us that sensing magnitude in terms of size, weight, length &amp; height has been with us from our pre-human stages. These can be called &ldquo;measurable quantities&rdquo;. Volume or weight of a bag of rice, area of a farm, distance to the next village, the time remaining for sunset are examples of measurable quantities. In all living beings &ldquo;measurable size&rdquo; is the more easily understood aspect of magnitude. This is apparent in the many words in daily language which deal with size like big, small, large, tall, short, bulky etc.

The idea of measurable quantities gradually developed into ideas like volume, length, area and weight. We could say that though a lake was longer than another, its area was smaller. Or that though one tree was taller than another, it was thinner. Most of these estimates were either subjective or based on measures related to individuals. A bag of rice that a child could lift was lighter than one which it could not lift.

But there is yet another means by which magnitude is measured – as a countable quantity.

The world around us is full of objects which are distinct. The idea of a set emerges from our experience of the world around us. Even when some cows are standing together, we see each cow as a separate distinguishable object. This distinctness is not observable in case of clouds & flowing water.

Humans in fact give different names to sets of different objects. We have a "herd" of cattle, a "flock" of birds and a "pride" of lions.

Humans also saw that sets had independent properties. Some sets were more in 'countable quantity" than other sets. There were various kinds of sets having the same quantity. For example all animals had four legs & two eyes.

All sets had a property which was eventually called cardinality or numerosity.

And the idea of a number emerges from the cardinality of a set. The idea of a number is the numerosity or cardinality of a set.

Numerosity of small sets can be “perceived” and such numbers are called Perceptual numbers.

Numerosity helps us to sense that we have more fingers than eyes. But numerosity or &ldquo;countable quantity&rdquo; developed much later than the sense of size. Hence a sense of number (as in seeing that we have five fingers) is a recent development and may be not more than 15,000 years. Hence the idea of number is more difficult for children to grasp than the ideas of size.

Language also adds to this problem. The number words in languages dealing with numerosity are far less than the number of words dealing with measurable quantities. Some of these phrases and words are "how many", "many" & "few". Even numerosity & cardinality are words which are not used that much in real life.

Research has proved that a rudimentary number sense or “how many” develops in children as early as six months. Children of age three & four have the ability to perceive quantities up to five, perceptually, without counting! Hence numbers one to five are also called Perceptual numbers.

We interact with our environment, more in terms of "how much" than "how many". Possibly the need for a judgment of "how many" did not arise until our possessions increased. But in the last century, with the development of trade and economy, the need for understanding "how many" has increased very rapidly. So we are still struggling to shift from perceptual numbers to understanding very large numbers. May be that is one reason, math is a difficult subject.

Words used to describe &ldquo;measurable quantities&rdquo; are much older. When the concept of &ldquo;countable quantities&rdquo; developed, many of the same words were used to denote both measurable and countable quantities and the exact meaning had to be derived from the context. For example the use of &ldquo;more&rdquo; can mean both measurable (serve me more rice) as well as countable (give me more fruits).

It takes some experience &amp; maturity for a child to realise that a basket of few coconuts is &ldquo;less&rdquo; than a handful of nuts in terms of numerosity! The coconuts may be &ldquo;more&rdquo; in terms of weight but they are &ldquo;less&rdquo; in terms of number!

The idea of countable quantities is a very powerful aspect of intellectual development. It is this which developed into number sense, counting, numbers, operations and the entire discipline of math!

Though numbers were developed later, they, through the math of measurements &amp; measuring units, gave exact meaning to measurable quantities! The idea of more and less was sharpened by using numbers to specify and compare them. A bag of rice which was a measurable quantity was converted into a countable quantity like 5.2 kgs! A bag weighing 5 kgs was understood to be less (lighter) than a bag weighing 10 kgs.

Parallelly identification by shapes, led to the study of their properties and relations. This developed into geometry.

In summary, the evolutionary skill of sensing magnitude as size or shape developed in humans into a sense of numbers, measurement and shapes and developed into the discipline of mathematics.

Human Brains & Arithmetic

Evolution of language abilities also seems to have primed the human brain for arithmetic! There seems to be consensus among scientists, however, that only humans can mentally represent numbers precisely and with symbols, and that we need some kind of education to do so.

Many higher math skills, including arithmetic, depend on the use of language—a symbols-based system—where quantity-based judgments are pre-verbal. Indeed, arithmetic is difficult to do if one does not have the language for it. This could also explain the fact that in many cases dyslexia and dyscalculia occur together.

Number Sense in Animals

Number Sense is the ability of the human brain to develop a sophisticated sense of numerosity using numbers. We will see this in detail in another chapter.

Recently many experiments conducted with animals have proved that they also have a rudimentary sense of numbers and some of them even seem capable of simple arithmetic operations.

These experiments prove that number sense is an evolutionary gift to many living beings. It is not a special gift to humans. Of course, humans have built an entire discipline of math on this gift of number sense.

Mathematicians opine that Number Sense is the most important skill for acquiring a mastery over mathematical concepts.

Math, Language & the Brain

We should also see here a vital difference between mathematics and language. Language changes from place to place and can be learnt only by listening to others. But ideas in math develop from our observation of our body and the environment. Mathematics developed by different civilisations tend to be similar, even though there have been no interactions between them. It is an internally developed discipline helped by logical thinking. It is a product of our human evolutionary development.

In that sense, math should be easier to learn than languages!

A study published in the journal Current Biology (February 2022) is revealing more about the way the brain processes math. It processes math & language in different parts of the brain. While our brains process ordinary language in the left hemisphere, math triggers neurons in both hemispheres.

The neuroscientists say “We found that different neurons fired during additions than during subtractions.” Another confirmed: “Even when we replaced the mathematical symbols with words, the effect remained the same. For example, when subjects were asked to calculate ‘5 and 3’, their addition neurons sprang back into action; whereas for ‘7 less 4,’ their subtraction neurons did.”

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