Philosophies of Mathematics
Written on Sep 14, 2015, when I was taking Math 402 Non-Euclidean Geometry Class.
Mathematics is an important part in people's life. Ancient people studied mathematics to try to understand the world. Throughout history, many thinkers gave opinions on the roles and natures of mathematics, and its systems and theories are developed.
In my opinion, mathematics, this abstract subject's origin is inseparable from another critical subject, philosophy. Both of them are ways people tried to understand the existence in the world.
Throughout the history, there are four main philosophies of mathematics developed: platonism, formalism, logicism and intuitionism. To proceed to further discussion, we can have a general idea of what they are first.
Platonism: It is a kind of "realism", which holds that mathematical entities are abstract and independent from human mind. It exists, it eternal, and waits for human to "discover". In platonism, Göbel's theories play important roles. He states that there is a strong parallelism between plausible theories of mathematical objects and concepts, and plausible theories of physical objects and properties. In my understanding, this is saying that there exist parallel universes, where in one universe physical objects exist and in another universe mathematical object exist. However, in terms of the relationship between themselves and their concepts/ properties, they share a lot of things alike in the relationship.
Formalism: David Hilbert played an important role in developing formalism. The main idea here is that the mathematical results and theories that we study only hold in a certain "game", where a certain set of axioms are assumed to be true. That is to say, these theories are "deducibly" true when a certain axioms hold true. Therefore, formalists view higher mathematics as a study of a "formal axiomatic system". Formalists believe that it is also important to study other systems where other axioms holds.
Logicism: Logicism denies the independent existence of mathematics, stating that math is deductible to logic, and therefore, is just part of logic. Logicism is saying that the nature of mathematics is logic, and theorems are logical deduction. Gottlob Frege, the founder of logicism, developed "Basic Law V", which was found inconsistent by Russell. Recent logicism has also been weakened and lacks purely logical supports.
Intuitionism: It states that mathematics is mental construction. Human have a natural intuition of natural numbers and mathematical theorems are constructed in our mind based on the intuition. Also, since we are all finite being, one can only construct a finite amount of mathematics.
Out of these four, my personal preference are platonism and formalism.
I do believe in mathematical realism. Since I was a kid, I found it intriguing that a lot of shapes and curves in this world, like flowers, leaf, table, and so on, can all be represented by some mathematical functions, if they are of perfect forms; And for everything in this world that we live in, it seems that there are some rules that govern how things work. For example, when something falls from the sky, we can calculate its estimated velocity by using the formula of gravity and taking the friction of the air into account. All of these make me believe that mathematics is a "perfect" form of the reality, and that it has the rules that explain the world.
But this world is not perfect, as there is no perfect straight lines or perfect circles. Therefore, platonism makes sense to me that mathematics is a parallel universe of our world, where things exist in a perfect mathematical form. That universe, in my mind, could help explain our world.
However, I also like the opinions in formalism. If I think that there is different universe exists, where mathematics is a reality, why wouldn't I accept that mathematics is not a single universe, but different universes where different axioms hold and help explain how things work in different situations? This sounds very appealing and interesting to me, and I would like to accept the possibility of that to happen. Also, take an example of Euclidean geometry and non-Euclidean geometry, when non-Euclidean geometry is not discovered yet, people only study Euclidean geometry, which is a system that contains a certain axioms, and a lot of theorems for this system are found and stated as "truth". But when later people realized that there are also non-Euclidean geometry, where one of the axioms needs to be changed, some of the old theorems that hold in Euclidean geometry are no longer true in this new system. Therefore, I do believe that certain mathematical results and theorems only hold in a certain axiomatic systems where there are underlying rules that make those results and theorems deductible. But when you change a situation, the results and theorems may not be true any more. And we cannot prove that there does not exist other universes.
In terms of the prevalent viewpoint taken in the teaching of mathematics, I think it is platonism. We are viewing mathematics as existing theorems and rules for us to discover, and we are often times trying to use it to explain this physical world, rather than thinking it as something that we make up by ourselves in our mind. We are also not exploring other "abstract" universes/games that we cannot observe in real life. Therefore, I think platonism is what the prevalent viewpoint is nowadays.
However, I think it is very important to know those other philosophies of mathematics as well, because to some extent, each one of them has their flow of logic. It is important to get to know them, to think about them critically by oneself and take one's own stand. Even if one cannot take a stand on which one to support, it is also important to understand that what we learn is not completely objective, but based on this type of thoughts, and so we are "designed" to be taught in a certain way about mathematics.
