In last TOK class, we discussed if math was discovered or invented. Personally, I think math is rather invented than it is discovered. Math so far is magnificent method to explain and describe nature and explain things happening. However it is not that we discover what is already out there, math is completely artificial thing, which started from human’s basic ability to think. Math is not there to discover, but our universe is here, all humans are trying is to explain these beautifully by math. As brandon mentioned in the class, math is like a language, it is used to describe our universe. With math, we formed a deeper and sophisticated understanding of nature and we were able to apply it into different areas.
In this sense, math is an abstract concept that does not really exist, however it is there to visualize concepts that are too difficult for our brains to understand. As Mrs.Connah mentioned in class, math is all about its axioms. It is like a tool to define conclusions according to different inputs. Conclusions may vary with different axioms however the axiom has to be definite. In math, it is often logical arguments that a statement is proved/disproved by another logic. However, to prove that a statement is true, it needs other ‘proved’ logics to explain. So it all comes down to these laws and theorem, which are regarded as the absolute truth. If we go even further down, what is left? What makes these laws and theorem the absolute truth? It is the definition- Axioms.
If we are to argue different statements without any definition in the background, it will be pointless, as if they are communicating with different languages while they don’t understand them. This is the need of axiom in math. There has to be premises which CANT be refuted. For example (as Euclidean Geometry defines) : points are simplest form of geometry which don’t have volume,area, width, if these are gathered its called line, it has length but no width, when its gathered it is called side (which has area) and so on. These are the definitions, and these can’t be refuted. Many people confuse this concept that math is very similar to language. For example, If you are talking Euclidean Geometry, it is pretty stupid to say right angle is 90 degrees, because it is defined as it is. This is same thing as saying apple does not indicate fruit that is red. Just like how you can refute an argument that says apple is blue, however you can’t try to disprove that apple is apple (what it indicates) , you can disprove arguments that are made with axioms but you can’t disprove axioms itself. So basically you are pretty stupid to say 1+1 in decimal number does not equal to 2. That is just saying happy should not indicate when people are feeling pleasure or contentment!! Yeah. What a dumb. Though, 1+1 can vary with different axioms depending on which math you are talking.
Imagine these axioms are not absolute, then you would be saying 1+1 = 4 and no one can disprove it.. Math is originally designed to bring up a solid conclusion to one input, answers according to certain input should never be varying with different people. (and thats why we fail at exams..) Book such as Euclidean Geometry through its long long pages define all the axioms that are needed, and there they are, you can’t stand up and say no to that. As I mentioned above, math is an abstract concept, yes they are non-existing but they are to help us with visualize things.
Can you imagine an object that does not have a width, no volume, no length yet existing? If you can, you are already very familiar with mathematical thinking, logically, it does not make sense at all but it exists in math. They call it “point” , if you try to imagine it with a little dot that you make with your pen on the paper, it is not the case because that thing you created still have volume and width and legnth and all that. Even if you want to disagree, it is just true that we are now too familiar with mathematical logics. (not necessarily that we are good at it). because you can’t ever find such thing as line and point that maths say and yet we can still imagine them. This is basically what maths do for us, visualize things that we can’t really imagine. Personally, I think this is the reason why math is extremely difficult for when we were younger, because these things they are saying is absolutely non-sensual things that we could never understand. (although some might argue math is still hard…:))
To sum up, math is defined language just like how we normally speak English. These are non-existing concepts that help us to visualize things that we can’t really imagine. To do this, different axioms that can’t be refuted has to be defined, otherwise it will be disastrous. In this sense, math is ‘invented’ rather than ‘discovered’.