Q:

“Without the assumption of the existence of uniformities there can be no knowledge.” 

So I am to discuss the question with reference to two areas of knowledge.  I chose to talk about mathematics and the arts (including literature, visual arts, music, etc.).  
I put a lot of thought into this, but am still confused in terms of the underlying juxtaposition of the question and the philosophical difficulty.  My teacher said that this is one of the most difficult philosophical questions he has come across, but I fail to understand or see why this statement is difficult, which makes me think that I dont understand it.  

A:

I believe this is one side of the more general Problem of Universals, to use the medieval designation.  
If you take your "fail to understand" as the stimulus for continuing the thinking through of the issue you have set upon, you are already in the same position as the philosophers to answer this.  Individuality pierces everything one comes across, an oak with resin sticking to its autumnal leaves, a person plunged into contemplation of a difficulty in philosophy, the house at the corner of the block.  Each phenomenon is shaped by being part of a class of things, so that one could speak of other cases of each thing.  No individuality without universality.  The pervasive presence of this difficulty is hiding in each nook and corner, pacing back and forth.  

A:

Long comment
Mathematics is not the only topic. 
You have to consider empirical knowledge and the related Problem of Induction: 

As Hume wrote, induction concerns how things behave when they go "beyond the present testimony of the senses, or the records of our memory".  Hume argues that we tend to believe that things behave in a regular manner, meaning that patterns in the behaviour of objects seem to persist into the future, and throughout the unobserved present.  Hume's argument is that we cannot rationally justify the claim that nature will continue to be uniform [...]

Regarding arts (mainly "visual" ones, but also music) , aesthetic perception is based on forms, and thus regularities.  See e.g. Rudolf Arnheim, Art and Visual Perception: A Psychology of the Creative Eye. 
Regarding mathematics, you can see: John Adam, Mathematics in Nature: Modeling Patterns in the Natural World as well as: Ian Stewart, Nature's Numbers: The Unreal Reality of Mathematics.