Mathematics: Is it discovered or created? (1 Viewer)

[seanieg89]

I have my own views on the matter which would have come up elsewhere on bos before, but I would be interested in hearing the thoughts of the mixture of (mostly) HS students, HS teachers, and undergraduates who frequent this site (without any bias from reading what I have to say first).

Some of the big questions which every view of the philosophy of mathematics should have at least a partial answer to, are:

1. What IS mathematics?

2. Would the theorems of mathematics "exist" if there were no humans to scribble things in formal language on paper?

3. Would mathematics be different in a hypothetical universe whose laws of physics differ from our own?

4. Can a choice of an axiomatic system to build a branch of mathematics be any more "correct" than another?

5. Are there any deeper truths about the world in which we live that the proof of a theorem reveals?


Feel free to share your thoughts or pose more questions along these lines, I find them very interesting to discuss!

 

[hawkrider]

I don't know how to answer 2,3,4,5 so I'll do 1:

Mathematics is the abstract study of topics such as quantity (numbers), structure, space and, change.

P.S. Mathematics is awesome!

EDIT: You're studying pure mathematics at uni? Wow, that shows how much you like the field of study.

 

[iJimmy]

http://www.youtube.com/watch?v=IJx6W6vAAc0

watch the series

 

[asianese]

1. Humans realised that they needed some form of system of counting that would satisfy the needs of trade, bartering, architecture and economy. This gave birth to arithmetic as we now know it. Humans, being the smart critters they are, began to formulate and play a little game they now call mathematics. Mathematics is a game where humans set some rules and go play by them. They share strategies with each other, sharing techniques and ideas. They attempt to play the game as well as they can. But it is not a game where there are winners or losers. It is for their own enjoyment. Early on, humans realised that in fact their 'arithmetic' was done better and more efficiently if they used strategies from the game that was just 'for fun'. Nowadays people still love playing the game just for the sake of it. Others like to utilise the game for their own purposes. They might want to find at what rate the force of gravity will accelerate a mass, or model how fast the $AUD will fall in the next quarter. Or maybe they are in pursuit of 'perfect' physical objects - what makes a sphere a sphere, and what makes a triangle a triangle? Turns out that our game is very telling of these things.

2. It's merely a game that wouldn't have been invented or played. So it would exist, but no one would no about it.

3. No, they wouldn't. The game could be separate from the physical world.

4. 'Correctness' is subjective. Within the game, there are those who feel that their strategy is superior. They can get a real kick out of looking at their game plans and progression because its 'beautiful' and 'correct'. Others may not understand the rules that well and fudge their way through it. They can still feel as though they are playing the game, but they aren't doing it by the rules ultimately.

5. No doubt our game can be applied to the physical world...sometimes the game predicts something should happen and it does in fact occur in real life. But is this a coincidence? (That is more a question of how the 'laws' of physics 'govern' the physical world - or does the physical world just have a mind of its own and do anything it likes?)



I know I haven't expressed my opinions well - breaking bad is in the background as I scribble this up on my phone

 

[omgiloverice]

Much philosophy anyways I'll give it a go.

1. What IS mathematics?

Mathematics is the direct application of human logic (whether it be deductive or inductive) towards the abstract study of quality (numbers), structure (combinations and permutations), space (geometry) and change (calculus).

And by either creating rules or tools (aka imaginary numbers, argand diagram, cartesian co-ordinates) we seek to find new or interesting properties, that work within the established rules.

Or we can create conjectures then attempt to prove them. (such as: Fermat's Last Theorem)

2. Would the theorems of mathematics "exist" if there were no humans to scribble things in formal language on paper?

This is similar to whether or not numbers exist. According to Wikipedia 'a theorem is a statement that has been proven on the basis of previously established statements'. Therefore relying on this, it is safe to assume that the existence of a theorem depends on whether or not a statement exists. Since statements must be written down in order to exist. Therefore the answer to the question is a no.

Another way of thinking about it, is that the notations we use are simply tools that allow us to have an arbitrary understanding or quality or change or whatever field you are working in.


3. Would mathematics be different in a hypothetical universe whose laws of physics differ from our own?

Difficult to answer, you'll have to be specific in which laws are different. However my intuition says no, but then again our intuition might be fixed to a certain algorithm of a simplistic type of logic. If that is the case then how can we prove that our logic, are absolutely correct? and if they are. Is it possible to a imagine a universe that operates on an complete different set of logic? for example what if 1+1 does not equal 2. Of course I am now just making ridiculous statements, but it is interesting (but probably useless) thing to think about.

4. Can a choice of an axiomatic system to build a branch of mathematics be any more "correct" than another?

Hmm obviously I am not well versed on mathematical logic, but to my understanding an axiom as a statement so obvious that everybody can agree on it. (for example if x = y, and y = z, this implies that x = z).

Another issue that is preventing me from answering this, is what level of 'branch of mathematics' are you talking about. For example the topic of trigonometry exists within the topic of space, which exists in the topic of pure maths, which exists in maths. o.o (according to wikipedia)


5. Are there any deeper truths about the world in which we live that the proof of a theorem reveals?

You need to clarify on the meaning of 'deeper truths', before I can answer that.