Intro to Classical Mathematics
Classical mathematics is the science which deals with the logic of quantity, shape, and arrangement in space. The word originally comes from the Greek words for “study”, “knowledge”, and “learning,” capturing the true heart of the subject. Mathematicians working in the field seek and use complex patterns in order to formulate new conjectures. They also help resolve the falsity or truth of conjectures by using mathematical proof.
Mathematics is in everything we do and all around us. It is the building block for almost everything in our everyday lives, such as mobile devices, art, architecture (ancient and modern), engineering, money, and even sports. When mathematical structures are representative models of real-world phenomena, mathematical reasoning can provide predictions or insight about nature. But different kinds of math lead to different insights, as not all math is classical.
Intuitionism and Constructivism vs. Classical Mathematics
Classical mathematics in particular is the kind that most of us are familiar with. It encompasses mainstream math like algebra, geometry, and logic that tend to make a lot of sense for those who explore the field. But there are a couple of other types of math, namely intuitionist and constructive mathematics.
Constructivism concerns itself with finding or constructing mathematical objects to prove that they exist. Intuitionism, which is constructivist in nature, constitutes the philosophy that math is not an actual fundamental part of objective reality. Rather, it is merely a construct of human minds and mental activity. Thus, they are both at odds with classical mathematics which sees math as real and follows different principles for proving existence of objects.
For example, it is valid in classical mathematics to prove the existence of an object by first assuming it does not exist and then deriving a contradiction of that assumption. Constructivism does not allow this derivation, and instead requires proving the actual existence itself. The primary basis of classical mathematics is classical logic and the Zermelo-Fraenkel (ZFC) set theory. These building blocks lead to the most widespread forms of math we know today.
The Basics of Classical Mathematics
The ZFC set theory is an axiomatic system meant to formulate a set theory without any paradoxes. All mathematical systems are valuable combinations of sets of theorems and axioms that could be logically and scientifically deduced from the axioms. Inquiries into the philosophical and logical basis of mathematics often reduce to questions about whether the axioms in a specific system can ensure its consistency and completeness.
Some of the systems that we get from this basis are geometry, algebra, number theory, and mathematical analysis. These all deal with space, structure, quantity, and change, respectively. The general idea in each area of math is to search for patterns that are useful in solving problems.
Before the complex axioms and methods of modern day, mathematics started with basic counting, measurement, calculation, and study of shapes and objects. Even ancient tribes needed a bit of mathematics to calculate the accurate position of the sun as well as the physics of hunting.
Uses Through Time
Since the advent of recorded history, mathematical discoveries have been at the heart of every civilized society, and in great use in even some of the most primitive cultures. Today, we all apply it in virtually all daily activities. For instance, a painter needs to use ratios in the mixing of paints and recording time, among other activities. From simple to complicated mathematics involving algebra and arithmetic to financial calculations such as taxation, time, and the value of trade assets, math surrounds us.
We know that the Babylonians and the ancient Egyptians possessed impressive skills in applying mathematics. In fact, the pyramids may have been much more than just the tombs of the first kings who reigned. Some even speculate that the pyramids were the first computers. The dimensions and alignments of the pyramids are said to have been used in the tabulation of complex mathematics in the same way we make use of calculators and log tables in the modern world. Similar ideas exist about pyramids of other cultures in Mesoamerica in which the pyramids played important roles in astronomy.
It is worth mentioning that the needs of mathematics arose based mainly on the wants of society. This is why the more complicated a society, the more complicated its mathematical needs and abilities. Read on for more details of the evolution of classical mathematics over time.
History of Classical Mathematics
Classical Mathematics in Pre-History
How did mathematics come to be? Initially, where did it get started? The origin of mathematics reveals it as a continuously growing set of expressions relating to subject matter. It is an abstraction that many animals (including and especially us) share with quantities and numbers. What does that mean? Well, the realization that a certain amount of things like four trees and four bananas are the same in quantity.
Quantity and its recurrence is always considered the first abstraction. You don’t necessarily have to see that five objects minus four objects leaves you with one. But subtraction, multiplication, and division naturally came to be. A further step from the early concept is the ability to perceive amounts that are not physical: for instance, time.
Factually, mathematics came into existence before the written scripts and communication. You will find the primary ways of counting, including tallies and tied knots. This evolved into full number systems and mathematical methods.
Several early civilizations including those of India, China, Egypt, Mesopotamia, and Central America contributed to math as we know it today. The first people to develop a basic counting system were the Sumerians. They also developed arithmetic, which covers basic operations, fractions, multiplication, and square roots.
The Sumerians’ popular system advanced through the notable Akkadian Empire around 300 B.C to the Babylonians. In Mesoamerica six hundred years later, the Mayans, who also possessed strong astronomy skills, developed their elaborate calendar systems. The concept of zero was also developed during this time.
A lot of early texts mention Pythagorean triples and hence, by inference, it seems the Pythagorean Theorem is the most widespread and ancient mathematical development after basic geometry and arithmetic.
Elementary arithmetic (such as addition, multiplication, subtraction, and division) first appeared in the archaeological records of Babylonians. In addition, the Babylonians possessed a valuable place-value system as well as the use of the popular sexagesimal numeral system. This system is still used today to measure angles and time.
Mathematicians started to work with geometry as civilizations developed; geometry computes volumes and areas in order to make precise angular measurements and has several practical applications. Note that geometry is highly important in everything from fashion and interior design to home construction.
Euclid introduced the famous axiomatic method around 300 BC, which is still used in math today, consisting of definitions, theorems, axioms, and proofs. His math textbook Elements is commonly considered the most influential and successful textbook of all time. Archimedes of Syracuse (c. 287–212 BC) is considered the greatest mathematician of antiquity.
Islamic Golden Age
During the Golden Age of Islam, in particular during the ninth and tenth centuries, mathematics witnessed several important innovations that built on Greek mathematics. Also, the most valuable achievement of Muslim mathematicians was the development of algebra.
Some other key achievements during the Islamic period are improvements in spherical trigonometry as well as the inclusion of the decimal point to the Arabic numeral system. It is worth mentioning that many notable and influential mathematicians from this golden period were Persian, like Omar Khayyam and Al-Khwarismi.
Fibonacci was among the first European mathematicians. He was very famous for his notable theories on algebra, arithmetic, and geometry. Also, the Renaissance brought important advances that included logarithms, decimal fractions, and projective geometry. Moreover, mathematicians considerably expanded upon number theory and ushered in a modern age of classical mathematics with theories such as analytic geometry, probability, and calculus at its forefront.
In the 17th century, the development of calculus by Leibniz and Newton revolutionized mathematics. The most influential mathematician of the 18th century was Leonhard Euler, and he contributed numerous discoveries and theorems.
Carl Friedrich Gauss was perhaps the famous mathematician of the nineteenth century. He made numerous important contributions to several fields like analysis, algebra, differential geometry, number theory, matrix theory, and statistics.
Fields of Classical Mathematics
We can broadly group elementary mathematics into the following fields or branches.
Also known as number theory, this is the most basic and oldest form of mathematics. It studies integers and their functions, including topics such as prime numbers, rational numbers, algebraic integers, and equations using such numbers. Thus, arithmetic mainly concerns the subtraction, addition, multiplication as well as division of real numbers that are not negative.
Professional mathematicians at times use the term (higher) arithmetic when they refer to more advanced or complex results relating to number theory; however, be careful not to confuse this with basic or elementary arithmetic.
It’s a type of arithmetic where unknown quantities are used along with numbers. Note that these unknown quantities are often represented by a variety of letters of the English alphabet like X, Y, Z, A, and B etc. or at times symbols.
Keep in mind that the use of letters is important as it helps us in generalizing the rules and formulas that you write while also helping you determine the unknown missing values present in algebraic equations and expressions.
Together with analysis, geometry, topology, number theory and combinatorics, algebra is one of the principle branches of pure mathematics.
As we mentioned, math helps us navigate a complex world of numerous movable objects. This is exactly where geometry comes into the picture, as it deals chiefly with the size, shape, and positions of figures and space. This can include the properties and measurements of real lines, points, angles, solids, and surfaces. In this sense it is very useful, hence its appearance in many early civilizations that needed to deal with volume, area, and length.
However, it can also branch into abstract calculations of object in other dimensions with impossible properties. When looking at real-world objects, it often serves as one of the most practical branches of mathematics. It helps us create a myriad of structures and devices that form the heart of modern society.
Calculus was independently developed by Gottfried Leibniz and Isaac Newton in the seventeenth century. It deals primarily with the calculation or measurement of instantaneous rates of change (also called differential calculus) as well as the summation of various infinite small factors in order to determine some whole (popularly called integral calculus). As you can imagine, it has proven an important scientific tool in a variety of disciplines.
A course in calculus often serves as a gateway to more complicated and advanced courses in mathematics that focus on the in-depth study of limits and functions. This is widely known as mathematical analysis. This is why calculus has widespread and diverse applications in science, engineering and economics and can solve numerous problems for which algebra is not sufficient on its own.
Trigonometry involves the measurements and properties of triangles as well as the relationships between their angles and sides. It is similar to geometry, but with the focus on angles that gives us many important functions. Sine, cosine, tangent, secant, and related terms come from this area of mathematics.
Although the historical origins of algebra, arithmetic, and geometry often hide themselves in the dark fog of ancient history, one thing is for certain: trigonometry originated with the 2nd century astronomer called Hipparchus of Nicaea. The word trigonometry is derived from 2 Greek terms trignon (which means a triangle) and metron (which means a measure).
The areas listed above all concern space, structure, quantity, and change and form part of pure mathematics. On the other side of the coin is applied mathematics. It uses the same principles and classical mathematical methods, but applies them to other fields. Engineering, science, business, and other industries all put math to use to solve problems and improve results.
Statistics is an example of the application of classical mathematics. It uses probability, decision theory, and algebra to predict and optimize outcomes in the real world. Similarly, actuarial science uses statistics to assess risk in various economic settings. Many branches of science also heavily depend on applied mathematics to perform calculations and create models.
Classical Mathematics and the Search for Truth
Right from the humble start of tied knots, mathematics has continually advanced into science with its immense impact on various fields of study. As far as we currently know, mathematics affirms every truth. It also takes knowing mathematics to appreciate the beauty of nature. Even areas that we consider vastly different from it, such as art and music, immerse themselves in the same principles of patterns, frequency, time, and geometry. The same goes for art, where proportions are critical in creating both beauty and abstraction.
Thus, math itself is an art containing an inherent beauty that teaches creativity, logic, and problem-solving in life. This makes the study of classical mathematics critical in all parts of every society. It does not only develop more scientists and engineers, but also equips learners with creativity to apply in day to day life.