Thinking of mathematics in terms of analogs

Question

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.

Answer 1

The tension between following abstract rules as against intuition has been present in mathematics for centuries if not much longer. From the time of Newton and Leibniz onwards mathematics became more algebraic due to the calculus. For example, the eighteenth century mathematician Lagrange played a critical role in moving away away from diagrams towards equations.

The Mécanique analytique summarised all the work done in the field of mechanics since the time of Newton and is notable for its use of the theory of differential equations. With this work Lagrange transformed mechanics into a branch of mathematical analysis. He wrote in the Preface:-

"One will not find figures in this work. The methods that I expound require neither constructions, nor geometrical or mechanical arguments, but only algebraic operations, subject to a regular and uniform course."

(source MacTutor history of mathematics)

But other eminent mathematicians were concerned about losing meaning. For example the nineteenth century Augustus de Morgan wrote

The moving power of mathematical invention is not reasoning but imagination.

One of my favourite mathematical quotes is due to him. He compares solving a mathematical problem to solving a jigsaw puzzle of a picture of a map. Those mathematicians who follow rules to transform from one line to the next are like people who do a jigsaw upside down, caring only for the forms of the pieces without regard to the big picture underneath.

a person who puts one of these together by the backs of the pieces, and therefore is guided only by their forms, and not by their meanings, may be compared to one who makes the transformations of algebra by the defined laws of operations only: while one who looks at the fronts, and converts his general knowledge of the countries painted on them into one of a more particular kind by help of the forms of the pieces, more resembles the investigator and the mathematician.

(see, e.g., Mathematics in Victorian Britain on google books)

I'm not a gifted mathematician, unlike some on this site, so you may take my personal experience with a pinch of salt. Nevertheless I've got a deal of sympathy with your experience. I've often gotten frustrated and bogged down when I couldn't understand the big picture or get a feel for what was going on. For example, some years ago I struggled when I studied a waves and diffusion course presented in a very abstract fashion, and only "got it" when I bought a Dover book on PDEs for Scientists and Engineers. If that rings a bell, then I'd encourage you to seek out books, websites, tools and other resources to relate the abstract to the practical.

The second thing for me is that the more I study science and mathematics, the more I find that they are like a language. I once studied cell biology in great depth, which is a hellish in terms of memorizing facts, and intuition seemed nowhere to be found. It was only once I had memorized everything for the exam that things seemed to slot into place and I could see the patterns. Mathematics, although less about learning endless reams of facts, is, I find, a little like that. You have do the exercises, practise the equations, learn the vocabulary of the subject area you are studying and then, only once you know the vocabulary, are you in a position to form the intuition to connect things together. So you need to have some patience and faith and work through the grind of the calculations in the knowledge that once you've mastered the techniques you will be in a position to have an intuitive understanding. Sometimes you need the vocabulary to see the poetry.

My final thought is that there are great people in all walks of life who think abstractly, and great people who think intuitively. The important thing is to know who you are, know your own strengths and weaknesses, find out what you like and what you don't -- and where you need to do the latter adapt rather than abandon your approach.

Answer 2

You've written this post as a leading question, you'll get much too bloomy answers. My advice is to search for the motivations behind the introduction of this and that concept - knowing that these exist will let you concentrate on the plug and chuck you need to get your answers.

I want to add that not all mathematical object are physical things - e.g. the mathematics of games or economy isn't concerned with "things in the world" but with concepts which are made up themselves. Unless you want to argue that the rules of chess and compound interests are something in the world. My point is that made up rules are hardly ever arbitrary, otherwise people wouldn't be interested in publications on these subjects. In any case, of all subjects, I'd say linear algebra is the one where the visualization is actually easier - it deals a lot with geometry and transformation.

Answer 3

Linear algebra is by no means about computations over concepts. There's actually a rather precise dichotomy that approximates that between computations and concepts in linear algebra, namely that between matrix algebra and the theory of abstract linear transformations. It's from the latter perspective linear algebra most naturally displays its ability to express intuitive geometric concepts (simple examples include rotations and dilations of space or lines, planes, and hyperplanes in higher-dimensional space.) The decomposition theorems like the Jordan normal form and singular value decomposition also have fundamentally conceptual content, but normally aren't reached until the end of a first course in the simplest cases, and a second course for the most interesting ones.

More generally, I don't think you should try to restrain yourself from thinking about interpretations of mathematical objects you meet, especially if you're interested in becoming a mathematician. I think that by and large (pure) mathematicians are very much interested in concepts, and in many cases find computations boring-you're likely to find less, rather than more, focus on computation as you continue on in math, although this will depend on which courses you take. That said, it's almost as dangerous to have too little computation as too much, so when you ask whether you should consider learning the algorithms in your courses and trusting that the conceptual content behind them will become the later, the answer is probably "yes, for now." It's certainly fine to ask people about what the things you're working on mean even now, but if you get excessively caught up in such questions you might find yourself later on understanding linear algebra intuitively but kicking yourself for not learning how to row reduce and compute eigenspaces fluently when you had the chance.

Answer 4

Linear algebra (and also functional analysis to some extend) are fields where it's still possible to have geometric interpretations. Linear transformations (in particular, matrices) can represent reflections, rotations and scalings which transform vectors.

You lose some exact graphical interpretation when you move from 2- and 3-dimensional vector spaces to 4- or 5-dimensional, or even infinite-dimensional spaces, but having an idea what would happen in 2d or 3d has always helped me. For instance, I like to view the graph of a linear transformation from $V$ to $W$ as a line - the x-axis the vector space $V$, the y-axis is the vector space $W$. I'm aware that in most cases, $V$ and $W$ are not one-dimensional, but are possibly infinite-dimensional, but visualizing a projection is better than no visualization at all.

Answer 5

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."

Answer 6

Get Friedberg, Insel and Spence's "Linear Algebra" 4th edition, sit down, and do every non-computational problem in the first three chapters. It's my favorite math book. It introduces linear algebra via an axiomatic approach that you'll probably see often as you go on in math.

It's rigorous, it provides intuition and a meaningful framework to linear algebra, and best of all, it tells you what matrices really are (hint: check out Theorem 2.21, and look at the diagram on pg. 105). It also has some extremely useful procedures for computational applications.