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Systems of Abstraction and Conceptualization

Introduction

It is widely asserted that people tend to know more than they can explain because words are symbolic of our inner experiences. How so? Well, let's begin by asking a few philosophical questions.
What is color? What does "tastes good” mean? Is it possible that 10 newborn babies alienated from each other and from all human contact will end up each with concepts of time and counting (or number)? In his book, "The Secret Life of the Mind", Mariano Sigman, a neuroscientist stated: "from the day we are born, we are already able to form abstract, sophisticated representations. Babies have notions of mathematics and social reasoning" (2017, p6). The concept of mathematics is part of our innate intuitions and thus one can say that we were born predestined to create The Mathematics Department. In fact, without an innate mathematical system, no living creature can survive. For instance, the hypothalamus, an organ in the brainstem that regulates the functions of many endocrine organs has a temperature set-point of 37 degrees Celsius from day one (Shier, Butler, & Lewis, 2019). This means that since the day you were born, there is a thermostat in your head that keeps track of your body temperature and works to keep it at about 37 degrees Celsius. Before mathematics was made an artifact of human culture, the hypothalamus knows what 37 degrees is, what "less than" 37 degrees is, and what "greater than" 37 degrees is.
If your body temperature is not maintained at about 37 degrees Celsius, your organs will begin to shut down and die eventually. So, when your body temperature falls below 37 degrees Celsius, the hypothalamus causes the release of certain chemicals to make your blood capillaries dilate (vasodilation) and thus increase the body's metabolism to generate enough heat to push the body temperature back up to about 37 degrees Celsius. This is a negative feedback mechanism that ensures our survival. The opposite (vasoconstriction) kicks in when body temperature sores over 37 degrees. This same organ has osmoreceptors that keep track of the volume of water in the cells of our body. To achieve homeostasis, cells of the body need the right level of hydration or osmotic pressure. When a person is dehydrated, the hypothalamus stimulates the posterior pituitary gland to release a chemical called Antidiuretic Hormone which causes the body to retain water. Followed by secretion of Aldosterone for sodium retention by the adrenal gland. This induces the feeling of thirst.
The entire autonomous nervous system, a system that innervates organs whose muscles cannot be voluntarily or consciously controlled (such as muscles of the heart and digestive systems), when studied, revealed precise mathematical patterns (or algebra).

Number systems everywhere

Number systems are ubiquitous. Every culture and civilization have words for numbers. From the Munduruku hunter-gatherers in the Brazilian Amazon to the nomadic Fulani in Africa, quantitative data is an essential commodity for survival. Challenging questions like "Do we have enough water? Are these grains enough to feed the family? Do we have enough material to build a raft?" buffeted our ancestors. The effort to solve these challenging questions pushed the invention of an abstract tool for taking stock. Counting thus became indispensable.
It is alleged that the formulation of numbers by the indigenes of Papua New Guinea was triggered many centuries ago by elaborate customs of gift exchange (Bellos, 2010).
The prevalence of numbers indicates the human sensibilities and natural notion of amounts yet the intellectual leap from the concrete idea of say "five" kids to the invention of a symbol for the abstract idea of "five", might have taken a bit of time as well as some good measure of the development of the intellect to come about. Humans used numbers for many centuries before the thought of studying the numbers themselves in a systematic way. Rules governing when to use numbers, however, were greatly influenced by culture. The Munduruku for instance forbade the counting of humans. They can take stock of things but not humans. You cannot ask a Munduruku "how many kids do you have"? Nor "how many people live in your town"?
It is interesting to note that the Israelites in biblical times had similar laws that forbade census. According to 2nd Samuel Chapter 24, King David counted the army of Israel and was punished for the deed.

Artifacts of human culture

Ancient civilizations reached a level that provided leisure time to ponder the nature of things. Some began to speculate about the nature and properties of numbers. The positive integer, 1,2,3,4,5,6... also called the natural numbers is man's first mathematical creation. The influential German mathematician and philosopher, Leopold Kronecker (1823-1891) asserted that "The natural numbers are the work of God. All others are the work of mankind." (Abbot, 2010). This curiosity about numbers developed into a sort of number mysticism or numerology, and even today numbers such as 3, 7, 11, and 13 are considered omens of good and bad luck.
The first scientific approach to the study of integers, that is, the origin of the Theory of Numbers, is generally attributed to the Greeks. Around 600 BC, Pythagoras became entranced by the numerical patterns he found in nature. It led him into believing that the secrets of the universe could be understood only through mathematics. He later became the founder and charismatic leader of a mystical sect called the Pythagorean Brotherhood, devoted to philosophical and mathematical contemplation. (Apostol, 1976). He tutored his disciples to revere mathematics. In addition, the disciples had to obey strict bizarre rules such as never passing an ass lying in the street. I can imagine a disciple hurriedly walking home just to use the bathroom and suddenly finding an ass laying at the intersection. This is a real test of faith. Lampoons of such nature were rife in the Athenian comic theater during the beginning of the Christian era.
Pythagoras and his disciples made rather thorough studies of the integers. They were the first to classify integers as:
Even numbers: 2,4,6,8,10,...
Odd numbers: 1,3,5,7,9,11,...
Prime number: 2,3,5,7,13,17,...(a number greater than 1 whose only divisors are 1 and the number itself)
Composite numbers: 4,6,8,9,10,...(numbers that are not prime except 1 which is neither prime nor composite).
The square of a number, n, was algebraically expressed by Pythagoras as the sum of the first n odd numbers. This means that, for instance, the square of 3 is the sum of the first 3 odd numbers (1+3+5); the square of 5 is the sum of the first 5 odd numbers (1 + 3 + 5 + 7 + 9), and so on.
The Pythagoreans also showed keen interest in the study of right triangles with integer sides. Such triangles are now called Pythagorean triangles. The corresponding triple of numbers representing the length of the sides is called a Pythagorean triple. Pythagoras stipulated the following geometric theory: for any right triangle, the square of the length of the hypotenuse is the sum of the square of the lengths of the two legs.
A Pythagorean doctrine that stated that "all numbers could be expressed as a ratio of integers" caused the unfortunate demise of Hippasus.
Hippasus, an early disciple of Pythagoras, (in his musings, I think) realized that the length of the hypotenuse of a right triangle with legs 1 unit each cannot be expressed as a ratio of two integers considering all known properties of numbers. By applying the properties of integers, Hippasus reached a logical impasse showing that the square root of 2 cannot be expressed as a ratio of two integers. The Brotherhood, instead of realizing the need for a larger number system, declared Hippasus heretic and was drowned at sea.
Today we know that Hippasus discovered the existence of irrational numbers.