I think the way that I've come to think about mathematics is becoming problematic and I'm wondering if I should abandon it. When I study mathematics, I find myself trying to compare the mathematical constructs, operations, entities, and even the basic terminology (which I have come to understand is incredibly elegant, precise, and deliberate) to real world, physical, even visible phenomena. I think under the pretense that the things I do in the mathematical world represent real, fundamental structures in this Universe. For example, the fact that terms can 'cancel' out in an equation has profound implications on the workings of the Universe and should be heeded and studied as such.
In other words, I try to make sense of the things I learn in math classes by finding their analogs in the real word, because I assume they must have at least one. Thinking with this frame of mind has led me to appreciate mathematics in a deeply profound and beautiful way, and it's the mindset that I try to share with other people when explaining why mathematics should be studied and why people describe it as beautiful. When I learn something new in a math class, I try to understand and remember that these are not simply tedious equations and formulas that mean nothing and come from nowhere, but that they have real physical and, mostly, intuitive meaning.
All that being said, I'm taking my first liner algebra course this term, and it's becoming harder to utilize this mentality, not simply because linear algebra deals with such things as infinite dimensionality which we obviously have no intuitive way of grasping or visualizing, but really just because the class seems more about computation and calculation than concept and philosophy.
I worry that my thinking has led me astray, primarily because it becomes hard to focus on just doing sheer, brute force calculation without wondering and worrying about what these constructs really mean. This leads me to fall behind in lecture, take hours longer than is probably necessary on the homework, and add to an overall level of frustration that has been building for some time now because of it, which only clouds my understanding even more.
My question is really more of a plea for advice. Should I abandon my way of thinking about mathematics as though it will become increasingly unhelpful in future courses and topics, or is linear algebra truly more about numerical gymnastics than tangible interpretation? Should I focus, currently, on simply learning the algorithms for computation now assuming that the philosophical groundwork will be exposed later on, after which the conceptual work that I'm looking for will yield itself?
I'd really appreciate responses from the people that frequent this site. I've been nothing but overwhelmed at the level of quality, thought, and sincerity in the answers I've read here and throughout the conversations I've eavesdropped so far.
Also, please direct me to similar questions if you know of any, and help me with the tagging of this question, as it is the first one I've ever asked on this site.
Best Answer
No, don't abandon your love of analogies and your search for connections to the "real world". But a caveat: be guided by it, not shackled to it.
A few more remarks.
(1) Linear algebra can be presented sevaral different ways: computationally, conceptually, geometrically, physically, etc. It sounds like you've encountered a mismatch between your course and your personal learning style. Here's a post asking for textbook recommendations: Text recommendation for introduction to linear algebra. (Many textbooks are available free on-line, so browse and sample. This link can get your started.)
(2) Linear algebra is especially rich with connections to the real world: it has a strong geometrical content, many applications to fields like physics, economics, and statistics, and a beautiful conceptual structure. (A book like Hoffman and Kunze emphasizes this conceptual structure; Strang's book highlights the real-world applications, but takes a more computational approach.)
(3) But linear algebra also illustrates the danger of clinging too tightly to our everyday intuition. You mention infinite dimensionality. This is a good example. First, unless you come from another universe, you can't truly visualize anything except three dimensions. Linear algebra got its start (in the nineteenth century) when people realized that the mathematics worked just as well with any value of $n$, even though the visual interpretation demands $n\leq 3$.
That takes us as far as the theory of finite-dimensional vector spaces. (I hope you have seen the abstract definition of a vector space by this point in your course.) In the twentieth century, mathematicians realized that many, but not all results about finite-dimensional vector spaces don't actually depend on the assumption that the space has a basis of $n$ vectors. They recognized that matrices are just a way of representing linear transformations, and they immediately noticed that linear transformations appear all over the place, not just in a finite-dimensional context.
In short, they applied analogy: finite-dimensional results suggest more general results, but you can't blindly generalize.
When I learn a new result in linear algebra (or topology, or many other fields), I try to find an example of it that I can visualize. Sometimes this doesn't work. Some results don't lend themselves to visualization.
So my advice is: (a) accept your own personal learning style; (b) spend some effort trying to accomodate it (ask questions!); (c) when you run into a mismatch, don't bang your head against the wall.
Finally, a bit of somewhat discouraging (and apocryphal) advice from von Neumann, one of the most brilliant mathematicians of the 20th century: "Young man, in mathematics you don't understand things, you just get used to them."