Are all Math proofs just the application of Definition?

Question

I'm going to try and keep this concise. I am a second year undergraduate studying Mathematics & CS, although I am taking some third and fourth year math courses (Abstract algebra & advanced number theory). I used to really enjoy mathematical proofs but I am starting to feel as though they have patterns and are repetitive. It seems that all proofs, at undergraduate level, involve taking a few definitions from the hypothesis, and manipulating them and applying a few logical steps to reach the conclusion. I can't count how many times I read a question, wrote down the hypothesis and the definitions associated with it, and instantly figured out how to reach the conclusion. At this point, only a few proof questions interest me since they involve some novel ideas but the rest seem so repetitive.

So my question is, will this ever change? When is the point when we move from proving everything from the basic definitions onto more advanced and novel proofs?

Note: This question doesn't apply to non proof problems. I am only talking about proof questions.

Answer 1

It absolutely changes. Most fourth-year courses encounter something more complex by the end of the class - it may be that you're just early in the year. Bear in mind that these courses are intended to teach, not just show, which means they have to drag their heels a little and make sure that everyone's got the definitions down before they start making things complicated.

Graduate-level courses, on the other hand, are almost exclusively considerably more involved proofs. It is possible to take these as an undergraduate, so you may want to look into it; in many universities, you will need the instructor's permission, but most instructors are happy to allow undergrads.

Answer 2

1) It absolutely will change. Undergrad math is really not representative at all of what mathematicians do, and you shouldn't get discouraged by it. Math should not be boring and uncreative and pedantic. It sounds like you're not being challenged; take more advanced classes, or start taking seminars and colloquia instead of standard lectures. There's a lot of nontrivial math on the Internet: this board, the arxiv, public lecture notes, etc. For example, Hatcher's Algebraic Topology is available online for free from the author.

2) Although it isn't a very useful one, the most accurate answer for when that point comes is that it varies. I would say that by the second year of grad school--- that is, the point where you've taken the generic required classes and probably finished qualifying exams--- the work should be more interesting than just straightforward symbol manipulation or deduction from hypotheses.

That is, the cutoff point above is classes aimed toward second-year grad students. You don't have to wait until you're actually a second-year grad student to take them.

3) To explain where the lack of novelty is coming from, I think there are several related factors in play.

• There's a difference between doing math and using math. A lot of the undergrad curriculum is designed to give toolboxes to people in other fields, and the proofs offered are desultory and uninspired. (It's a common exercise in calculus classes, for example, to ask students to verify the mean-value theorem by hand for various functions. Why bother?) I think it's a bit like physics: Undergrad Newtonian physics is about using rules like F = ma, conservation of momentum, etc. to solve contrived problems; modern physics is about figuring out what the rules actually are to produce the physics that we've observed. At the undergrad level, you have the definitions given to you. Beyond that, you have to figure out the right definitions that give interesting math.
• There's a much shallower tree of prerequisites for undergrad math, which means that the proofs involved largely follow the definitions, just because there's not much else to work with. If you want to prove something in, say, an introductory real-analysis course, then there aren't many tools available beyond the definitions in the text and some introductory point-set topology. Artin's Algebra has a chapter on Galois theory but only handles the finite, characteristic $0$ case, which means that a lot of the more interesting field theory involved is shoved under the rug.
• Undergrad math is designed for people who are still learning how to be mathematicians (as opposed to the math itself): common proof techniques (e.g., induction), how to construct a proper proof, the difference between rigor and pedantry, etc.
• Undergrad math is old and well-trodden. There are ideas in more advanced math like the Whitney trick that seem incredible at first but become ordinary after using them again and again. The corresponding ideas in less advanced math have already been seen and used again and again, and their novelty has worn off by now.
• Undergrad math focuses on more familiar objects. A lot of the complexity of topology, for example, comes from looking at more pathological spaces. In undergrad math, you really only have vector spaces and some basic manifolds to deal with.
• There actually are some clever proofs at the undergrad level. Check out Proofs from THE BOOK, for example.

Answer 3

Surely, Math wouldn't be nearly as interesting and challenging to the millions of folks who do it if all it took was applying definitions.

It depends on the particular topic how quickly you'll escape these "massage the definition" questions, but definitely by the end of your undergraduate career you should see far fewer of these questions.

That is not to say these kinds of questions are unimportant. They are included in most textbooks to ensure you truly grasp the concepts being defined, which can get quite difficult as things become more abstract and complex.