Math and Our Brain

How did human brains become capable of discovering math ideas?

Brains are Meaning-Making Machines

Human brains have been developed by evolution to make sense out of patterns. Pattern recognition is very necessary for survival of living beings on earth.

The human brain developed the 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.

This is the beginning of intellectual development. Our brains are "meaning-making" 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.

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 & speed for catching a prey are all instances of such survival skills.

The sense of shape helps us to differentiate between friend & enemy, edible & poisonous and to maintain our sense of position & 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.

Historian George Ghevarghese Joseph says "No society, however small or remote, has ever lacked the basic curiosity and “number sense” that is part of the global mathematical experience"

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

Measurable & 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 & height has been with us from our pre-human stages. These can be called "measurable quantities". 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 "measurable size" is the more easily understood aspect of magnitude. This is apparent in the variety of 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.

The Idea of a Set of Discrete Objects

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 from one another. 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 and we also see the cows as a group. This distinctness is not observable in case of clouds & flowing water.

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

Set, Numerosity, Cardinality & Number

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.

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

Recent research seems to even suggest that there is a "numerosity module" in our brain which may also have the rudiments of a "number line" with magnitudes increasing to the right.

Numerosity developed at a slower Rate

But throughout recorded history, numerosity or "countable quantity" developed at a much slower rate in humans than the sense of size.

This is possibly because the sense of "size" was more critical to survival than the sense of "numerosity". And because, to a certain extent, the sense of "size" includes in it the sense of "numerosity".

We can make out the difference between two herds of cattle, just by the sense of size rather than by counting.

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.

Possibly only in the last 15,000 years, after humans developed agriculture, settled down and acquired possessions, that the sense of numerosity started developing.

Hence the criticality of the sense of number is a recent development. Hence the idea of number is more difficult for children to grasp than the ideas of size.

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

Language Difficulties with Numerosity

Language also adds to this difficulty. Since the sense of number was not so critical, there was no need to develop a sophisticated vocabulary related to numerosity. Hence the number words in languages dealing with numerosity are far less than the number of words dealing with measurable quantities.

Words used to describe "measurable quantities" are much older. When the concept of "countable quantities" 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 "more" can mean both measurable (serve me more rice) as well as countable (give me more fruits).

Some of these phrases and words are "how many", "many" & "few". Numerosity & cardinality are words which have come into use only in the last few decades.

Criticality of Numerosity in the Technological Age

But in the last century, with the development of technology, trade and economy, the need for understanding "how many" has increased very rapidly. We have been forced to rapidly to shift from the stage of perceptual numbers to understanding very large numbers.

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

Through the math of measurements & measuring units, numbers gave exact meaning even 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 kg! A bag weighing 5 kg 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 and developed into geometry.

The math curriculum in our schools became complex in a short period of time, before humans have got used to such ideas.

May be that is one reason, math has become a difficult subject.

In short, 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 have the ability to 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.

The Ishango Bone

The Ishango Bone is a bone with notches, about 10cm long, accepted to be around 20,000 years old, which seems to indicate that it was used as a "counting tool." It was discovered in the Democratic Republic of Congo.

Many theories have been proposed about its relation to counting and math. One of these proposes that the notches reveal an understanding of prime numbers. The notches themselves are grouped leading to a speculation about system of representing numerals.

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, deeply rooted in our biological ancestry, to many living beings. It is not a special gift to humans based on our ability to use language

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.”

Math and Language

Initially scholars generally assumed that only societies with writing systems could truly practice math. But with increasing anthropological studies they are discovering societies which have discovered math where no "writing" is involved.

One such example are the people of Vanuatu in the South Pacific Ocean who seem to have used “sand drawing” for thousands of years, as a communication and storytelling tool. Traditionally, it consists of a person drawing a continuous, closed line with their finger in beaten earth, sand beaches or ashes. These sand drawings seem to be related to "graph theory."

In the last 100 years, researchers have discovered mathematical applications in several traditional cultural practices.

Unreasonable Effectiveness of Math

Scientist Eugene Wigner talked, in a famous article, about the unreasonable effectiveness of math in understanding nature. We can try to understand this effectiveness from an Advaitic (Non Dualistic philosophy) point of view.

Our brains have a mathematical structure built into them. Possibly the same mathematical structure exists in the Universe also. May be the structure of the brain has been derived from the structure of the universe.

This could explain why our brain is able to understand the Universe through the medium of math!

The famous Advaitic phrase says Tatvamasi, Thou Art That. Your individual soul is a part of the Universal Brahman.

We can interpret it through science as follows - the math in your brain is the same as the math underpinning the Universe!