Why the name "Applied Mathematics" is a misnomer

A common feature of mathematics across the world is the desire by mathematicians to split the subject into categories. The advantages of this are obvious; if you organise a conference on Algebraic Topology, it is relatively clear what kind of lectures a prospective attendee can expect, for example. But there is one categorisation which I believe needs to stop: the notions of pure and applied mathematics.

This is not to say that maths does not have varying degrees of use in the real world - that is of course true. It's not very hard to argue that partial differential equations have had a bigger impact on physics and engineering than commutative algebra. But I feel that the name "applied maths" is a little misleading. It seems to imply that this "applied maths" is in some way more real than so-called "pure maths" which is ridiculous. Mathematics exists entirely inside its own, objectively perfect; it isn't real at all.

To make this point, I will refer to the marvellous (if arrogantly-subtitled) book "The Road To Reality: A Complete Guide To The Laws Of The Universe" by Sir Roger Penrose. In the first chapter, Penrose discusses the concept of mathematical truth. 

Firstly, we can pretty quickly realise that mathematics does not lie in the real world. Just think of a circle drawn on a piece of paper; you may look at it and think "yeah that's a circle" but in reality it is just a representation of a circle. A circle is the collection of points of on specific distance away from a given centre, which is nigh on impossible to draw in reality. 

Interestingly, however, Penrose also argues that mathematics cannot sit wholly inside the human brain either. This is a strange concept until we think about just how many results can be drawn from simple mathematical objects; take prime numbers for example. The concept of a prime number has been around since the ancient Egyptians, and yet they are still extremely relevant in modern life, through their use in RSA encryption; the system that drives internet security. Although it was humans that decided these numbers were interesting, our ancient ancestors could not possible have understood just how rich and surprising they would prove to be. Thus, mathematics must exist in a third world, which Penrose refers to as the "Platonic Mathematical World". Of course, the physical, mental and mathematical worlds are linked and interact with each other, they should not be mistaken for being the same as one another.

So my argument is essentially that the term "applied mathematics" is used incorrectly to imply that this maths is somehow real and relevant to everyday life. While a lot of mathematics is done with application in mind - and this is the reason why I agree with its usage in universities, for example, as it is a good way of signalling the focus and direction of a research group - I disagree with the ideas that "applied maths" somehow exists and that pure mathematics is only useful for its own sake.

This is just the opinion of a maths undergrad; if anyone disagrees or has any points to add please let me know - I don't hold my views too vehemently and if you make good points I will change my mind!