What is mathematics? There are many definitions, and they’re all correct in their own way, so there isn’t a definite, generally accepted one. However, math, short for mathematics, can be explained as a desire to find and use patterns in order to determine if something is true or false via a mathematical proof. The proof isn’t just a bunch of sheets of paper with numbers and shapes on it, determined to bore you to sleep. It has a real use, and when proven right, can help us determine how objects, shapes, and motions can be understood and even predicted in nature.
Origin of the word
Etymology or the origin of the word gives us a guide on what later become mathematics we know today. It’s no surprise that the word came from Ancient Greece, knowing how many genius mathematicians came from there. Let’s just mention Pythagoras, Archimedes, Euclid, and many others you should remember from elementary and high school. The word comes from “mathema” which means “what one gets to know” or “that which is learned”. But the full word we know and use today actually came from “mathematikos” which means “related to learning” and we have an adjective today – mathematical. In Ancient Greece, even though teachers were called “mathematikoi”, it had a meaning of just that in a school, rather than mathematicians we call them today.
Mathematical terms, notations, and language
Even though we use many terms we consider normal today, they weren’t actually invented or defined until the 16th century. We’re sure it was harder to prove something in the past because they had to use words to explain phenomena and changes. The mathematician people consider brought the biggest contribution was Leonhard Euler, who invented and put in use terms we use today, along with Gottfried Wilhelm von Leibniz. We can thank Leibniz for terms like analysis, abscissa, variable, parameter, coordinate, function, etc.
Thanks to the new language, the studies have developed faster and with bigger success. That’s because with common words we usually have a physical object, while with mathematical terms the object is usually abstract, and doesn’t have a physical representation. Also, one term can describe multiple ideas or operations. Most of those that we all learned in elementary school class weren’t new to us, and we’ve used those words before. However, when applied here, they have a specific mathematical meaning, more than in everyday speech. Just remember words like or, only, if, iff (if and only if), etcetera. This precision of words is called rigor among mathematicians.
Additionally, this allowed for definite proof, because everybody had to follow rigor when publishing a theorem. This is to avoid intuitions and mistakes and only stick with proven terms and definitions. One more term that you probably heard but maybe don’t remember the meaning – axioms. Those are used in mathematics very often, and axioms are self-evident truths, something we accept as it is and because it is true. They require no explanation, which brought a debate and disagreement, but they’re essential for the study even today.
Fields of Mathematics
The field is incredibly vast and can be divided and subdivided into many studies. However, we’ve accepted four subdivisions, and those are a study of quantity, space, structure, and change. In other words, we call them arithmetic, geometry, algebra and analysis studies. Those can be further divided to help connect it with other fields of studies – mathematical logic, foundations, and applied mathematics.
Foundations of mathematics
To clarify the foundations, we developed mathematical logic and set theory. The goal for both was to apply formal logic to other areas, and while the foundations of mathematics are still not completely agreed upon, it’s still an ongoing topic since the 1930s. Mathematical logic deals with setting up a framework, and studying what the implications and consequences of that framework are. Today we have an additional science with the development of computers – theoretical computer science.
Mathematics subdivisions or pure mathematics
We’re finally at a point that should be very familiar to you – you’ve been studying it since you started going to school, and even in the kindergarten in the simplest forms.
Quantity or study of Arithmetics
This is probably the first you encountered when you started school, and it deals with numbers. First, it started with natural, whole numbers like 1, 2, 3, etc, then it progressed to integers like -1, -2, etc. As you progressed through school, you learned about rational numbers or fractions like 3/4, 1.44, etc. Finally, you were ready to accept the concept of infinity, real numbers like -e, roots, number pi, and complex numbers like i.
Structure or study of Algebra
Mathematical objects, usually sets of functions or numbers have consequences manifested with internal structures. Those further can have properties that can be studied, and combinatorics, or combining numbers inside a bracket in a certain order should spark your memory. Additionally, we have a number theory which leads us to shapes you had to draw in your elementary school notebook with numbers on your rectangular coordinate system. Some are more complicated and are studied later on in life – like a group theory (think Rubik’s cube and the variety of colors), and even graph theory and order theory, which you should perfect at college.
Space or study of Geometry
This might be the favorite of people who like to draw, and it combines space and numbers. You had to learn a ton of formulas for rectangles and squares that you should be able to recite right now. Trigonometry branched out from geometry, of course, and it deals with sides and angles of a triangle, as well as trigonometric functions you were required to sear into your brain. If you decide to study mathematics further, you’d learn more about differential geometry and topology (just think of a ring like a donut), and even fractal geometry – those cool snowflake shapes you have on your desktop.
Change of study of Calculus
This is one of the most important tools we have at our disposal today. The ability to track, describe and understand change is incredibly powerful and gives us a possibility to predict a lot of things. The simplest things are functions that we studied in high school, and they’re a central concept of this field of mathematics. We can also study real and complex numbers, as well as spaces of functions, which lead to quantum mechanics, something you hear often about.
This is simply a way to describe mathematical methods used in other branches of science like engineering, industry, and even business, but that’s no surprise – many scientific fields are deeply connected and even depend on each other.
Whether you’re a fan of mathematics or you just went through school only learning the basics, it’s irrefutable how much impact this study has in the real world. It allowed for many problems to be solved and defined, and the simplest of problems like counting apples or measuring the field behind your house now has a way to be written, proven and defined. Without a doubt, we can say the world around us is better because of mathematics, and many other areas were improved year by year.