Labels of an Abstract

 When I was young, I knew mathematics as a sequence of unending numbers. Addition, Subtraction, Multiplication and Division are just ways to play with those numbers. Math is only about numbers and counting.  I even didn’t find shapes as part of Math. Shapes and lines are what I call Art before. Since I noticed the patterns of the numbers, I used to think that math was easy and bearable. It just has to live with its own rules. Today, I see mathematics as a complex system of interpreting dimensions, structures, and problems in infinite ways.  
     My interpretation of mathematics widened up a significant magnitude after I had read the book “What is Math Really?” by Ruben Hersh. He seems to define mathematics in numerous ways according to the Mathematicians of before. As I have observed, the book highlights the Philosophy of Mathematics rather than pure Mathematics. I have highlighted a lot of concepts I became interested in, some of them I had already forgotten. The proceedings are what I have learned in the book.
     Mathematics, according to the dictionary long ago, is the science of numbers and figure. It was as simple as my understanding when I was young. Now, Mathematics is defined as a science that involves ‘the groups, rings, and fields of abstract algebra, the convergence structures of point-set topology, the random variables and martingales of probability and mathematical statistics, and much much more’(Hersh ,1997).
     Mathematics is divided as applied and pure. “Pure math stresses proof above result while Applied math stresses results above proof. Applied mathematics is still inseparable to pure mathematics. Technology and science advances are dependent on applied math, which is then dependent on pure math.
     Now, the question is “What is the nature of these mathematical objects?”. The book somehow strongly states that “Math is an abstract”. This means that mathematical objects can neither exist mentally nor physically and it can’t exist apart from them. According to Hersh, it is like music. It is not a biological or physical object yet it exists in these manifestations. And it made sense as a mental entity.  
     However, Platonism regards mathematical objects as definite existing objects independent of us. And since man is empirical, mathematics is only a discovery, not an invention. It is quite logical since the galaxies and all matter exist in a equilibrium of math equations. “ ‘Why and how is mathematics possible?’ Asked Kant. The question was countered, ‘Why should your question have an answer? Mathematics is possible. Its an old saying. What is happening can happen’.” It is like energy, it can neither be created nor destroyed. It is already there.
     However, formalism arguments are set against Platonism. Formalists regarded mathematics as have been created by people. It is further supported by proving that math isn’t discovered indeed. They showed that there is nothing to discover because it hasn’t been created yet (Hersh, 1997).
     Philosophy is compared to mathematics. I might say that Mathematics is completely different or separate from Philosophy. Mathematics, along with Science and Language, is a fixed science-one problem has a single stated answer. Philosophy is not. It generates answers continuously non-stop, and every idea is logical. These are two completely different ways or approaches to define the world. According to Hersh, Philosophy can be part of mathematics. It will turn into a branch of mathematics. It is the Philosophy of mathematics. Philosophy again is a subject of certainty and uncertainty. It is never direct, precise, or straight. Philosophers, I think, may have incorporated mathematics to handle philosophy a tiny bit bearable and precise. Thus, the concepts and mathematical logic we know of today may be primarily due to the mathematical problem solving of philosophical ideas before.  
     The foundation of Philosophy of mathematics was by Pythagoras. He worshiped number as it is religion. It became that their lives was based on numbers. In Plato’s time, geometry where not physical entities yet they are transcendental existing as the human mind perceives. He also believes that ‘Mathematics is central in Philosophy’ (Hersh, 1997) not the other. Another, Neoplatonists believe in the number 7.  According to Cusa, in her Theology of Infinity, God is the absolute maximum or the infinite being and denies that the universe is so too. There are a lot more. Then the Philosophy of mathematics became mainstream.
      Since we had incorporated Philosophy to mathematics, why do we do mathematics? We haven’t been wondering why we use it. Formalism is empirical. It is limited and cannot encompass other ways of thinking. It is just it. We sometimes never knew why we do things. A formalist description of mathematics has a distant resemblance to our actual knowledge of mathematics (Hersh,1997). Let us go back to the definition of mathematics. I have forgotten the word “Proof”. Proof convinces and explains. In research, convincing is primary. In high-school and undergraduate class, explaining is primary (Hersh,1997). Proof is what makes mathematics possible. It is precise, thus does need any logical proof to descend an answer. In words, proof is math and math is proof (Hersh,1997).
      Now, there is also the concept of infinity. Infinity is all or countless possibilities. N is infinite. One divided by zero is infinite. And there are many ways to prove it. Mathematics is proof and abstract. The idea runs down to our culture. Philosophy has great influences to the foundation till the formation of mathematics we understand today.
     In my terms, mathematics has an amazing history. How it came to be, why we use it and how we understand it is giant leap for the advances of mankind. I still find the whole definition complex. That must be how we should understand math since it has a whole lot of history with philosophy.