Imagination in Mathematics

[God]

I need to come up with ways in which imagination is in Mathematics.

 

I think that axioms were created by people by using their imagination. However, i do not know how to explain how i know imagination is used. Anyone have any ideas to help me explain it? 

 

Axiom: Self-evident truth that requires no proof.

 

For example one of Euclid's Axioms: Things which are equal to the same thing are also equal to one another.

 

Thanks!

[Thrashmaster]

One of the points of ToK is trying to think for yourself... but I like it to much to not give at least one point.

 

Have you ever done a proof or a hard math problem? How do you know where to direct your work? Can you see where it is going before it is there and only your hand has to catch up? That's the way I always think of intuition in math, but I think it can also apply to imagination.

 

You could also say something about imaginary numbers... or at the very least imagining them on the Argand plane, since that is harder to conceptualize than what the normal coordinate plane means.

[Sofia.]

I'd say any new application in mathematics requires imagination - at some point, you have to imagine a new way of calculating something, because the old way doesn't work. Especially if you haven't studied much for a test, then you need to be imaginative to pass  

Like Thrashmaster said, imaginary numbers could be interesting to discuss, or negative numbers, non-Euclidian geometry etc. 

[Vioh]

I need to come up with ways in which imagination is in Mathematics.

I think that axioms were created by people by using their imagination. However, i do not know how to explain how i know imagination is used. Anyone have any ideas to help me explain it? 

Axiom: Self-evident truth that requires no proof.

For example one of Euclid's Axioms: Things which are equal to the same thing are also equal to one another.

 

Personally, I think axiomatization requires very little imagination, because axioms often rely on our intuitions. This can be seen clearly from Euclid's axioms which even a child can understand, as the axioms are so intuitive, and often belong to everyday phenomena.

 

Having said that, I still believe mathematicians need some sort of imaginations for the process of abstraction. In mathematics, abstraction is very very important as it helps us to translate concepts into actual mathematical objects. Let's take a simple example: the number system!!! We represent numbers using different abstract mathematical symbols (such as binary numbers, decimal numbers, etc). What do you see when you think of the number 2? Well, some people actually see the symbol '2' in their head; while some people see 2 fingers in their head..... But none of these things is the actual number itself. Those things are just different representations of the same concept! So it's possible to argue that abstractness requires imagination.

 

Another interesting example is infinity. In our real world, infinity often doesn't exist, and hence our intuition doesn't allow us to actually comprehend that concept. But we can use our imagination to do that. In other words, we can't see infinity, we can only imagine it!!!

 

As Thrashmaster & Sofia have already mentioned, solving a maths problem also requires a bit of imagination as one problem can have so many solutions. A good mathematics often needs an imaginative brain to come up with a new way to think about a problem, thus will be able to invent a faster algorithm to get the final answer.