Some mathematicians offer the explanation that mathematics is a science because it uses the scientific method. They try with renewed attempts, incorporating what they know, to invent new ways to uncover the secrets of nature. This may be true enough on the surface, but it is not what actually goes on.
Mathematics is considered a science because it is a branch of knowledge. When did this knowledge begin? When humans learned how to count. Even that was a struggle. There is a difference between a cardinal and an ordinal number. You may want five pairs of socks to take with you when you travel. It doesn’t matter which ones you will wear first. But when you take them out of the drawer you usually count 1, 2, 3, 4, 5. Although counting 3, 5, 2, 1, 4 would be just as good in order to know you have five pairs of socks in your luggage.
The Greeks invented geometry. We can do marvelous things with geometry. We can sight with our thumb up along the length of our arm and know the height of a pine tree that we know is three hundred yards distant. We can calculate how much tiling will cover the area of a floor in a room by measuring the length of its sides. The Greeks studied conic curves out of curiosity. But we find uses for this once abstract knowledge as time goes on. An ellipse tells us the location of a planet in orbit which has the sun at one of its two focal points. Some people need this precise knowledge. Other people just think of the sun as being at the center of a circle. Parabolas and hyperbolas are conics that have their uses too.
Arab traders got us to thinking about algebra. Since the traveling merchants have to intensely calculate their what ifs before a journey would begin they needed algebraic equations in order to solve for their unknown quantities. They developed rules for solving equations such as: like terms combine with like terms, and what you do to one side of an equation you do to the other side. Perhaps they were a little too motivated by the life struggle back in those days. But these are the rules we still remember in mathematics.
Mathematics is considered to be a science because it is an orderly extension of a branch of knowledge. However if you are actually a mathematician creating new theory you also function as an artist. An artist is more contemporaneous. It takes wile and craft and cunning to use your personality to invent something that will enter the branch of knowledge. It is not merely advancing by placing one foot in front of the other. It is not like as an engineer would do who would publish a paper which has 3 % new ideas in it. It is not even like a physicist who finds everything jhe (he or she) looks for. Physicists use the tools available, and those tools are often previous mathematics. No. When a mathematician creates it is something entirely new. So mathematics is an art as well as a science. The trouble is that sometimes you have to actually be a mathematician to appreciate that it is an art. Beauty is in the eye of the beholder.
To illustrate this point, how many people are even aware that something more than arithmetic, algebra, and geometry exist?
For over one thousand years algebra lay dormant. People could only solve up to the quadratic equation (second degree). Then around the time of the Renaissance the third and fourth degree equations were cracked. A lot of vigor went into cracking the fifth degree. But in the early 1800s Niels Henrik Abel and Evariste Galois proved that one cannot solve the general fifth degree algebraic equation merely by extending by radicals. Since then a whole abstract branch of knowledge, still known as algebra, has sprung up. Who in the general public even has heard of groups, rings, fields, algebras, ideals, ideles, adeles? Who has the faintest idea what class field theory would be? If no one knows this branch of knowledge, can it still be called a science? It is something artistic, though. If you can understand it it is pretty to look at.
Once we knew that pi and e and some radicals were irrational numbers, the game was afoot to invent a new class of numbers known as the real numbers. So the branch of algebra was scientifically extended into the realm of analysis. But problems arose which drew out the artistic nature of mathematicians. Georg Cantor proved that the quantity of real numbers couldn’t even be counted there were so many of them. Who would have thought of that? What does this mean to the general public? If there are so many more real numbers than integers or rationals (fractions), why do we know the names of so few irrationals? But these concepts really matter to the small subclass of the world who are mathematicians. Many of the sciences rely on mathematics as if it is a solid foundation. So mathematicians must be sure that their theories are logically consistent down to the last iota.
Can you imagine the struggle between mathematicians and the general public? In the 1800s the church thought that it controlled infinity. They hated Cantor for his new invention of infinity and persecuted him.
So mathematical knowledge is a science because it is an orderly field of knowledge. And a working mathematician must be an artist in order to create. And also in the past there has been prejudice against mathematicians. It’s only natural. People tend to want what they can use now. Few people are really mature enough to coexist with types they don’t understand.