A few mathematical study skills... Reading Theorems
by Ashley Reiter Ahlin
In almost any advanced math text, theorems, their proofs, and motivation
for them make up a significant portion of the text. The question then
arises, how does one read and understand a theorem properly? What is
important to know and remember about a theorem?
A few questions to consider are:
We might ask more questions about the proof of theorem:
- What kind of theorem is this? Some possibilities are:
- A classification of some type of object (e.g., the classification
of finitely generated abelian groups)
- An equivalence of definitions (e.g., a subgroup is normal if,
equivalently, it is the kernel of a group homomorphism, or its left and
right cosets coincide)
- An implication between definitions (e.g., any PID is a UFD)
- A proof of when a technique is justified (e.g., the Euclidean
algorithm may be used when we are in a Euclidean domain)
- Can you think of others?
- What's the content of this theorem? I.e., are there some cases in
which it is trivial, or in which we've already proven it?
- Why are each of the hypotheses needed? Can you find a counterexample
to the theorem in the absence of each of the hypotheses? Are any of the
hypotheses unnecessary? Is there a simpler proof if we add extra hypotheses?
- How does this theorem relate to other theorems? Does it strengthen a
theorem we've already proven? Is it an important step in the proof of
some other theorem? Is it surprising?
- What's the motivation for this theorem? What question does it answer?
Note that, in some ways, the easiest way to read a proof is to check that
each step follows from the previous ones. This is a bit like following a
game of chess by checking to see that each move was legal, or like
running a spell-checker on an essay. It's important, and necessary,
but it's not really the point. It's tempting to read only in this step
by step manner, and never put together what actually happened. The
problem with this is that you are unlikely to remember anything about how
to prove the theorem, if you've only read in this manner. Once you're
read a theorem and its proof, you can go back and ask some questions to
help synthesize your understanding. For example:
- Can you write a brief outline (maybe 1/10 as long as the theorem)
giving the logic of the argument -- proof by contradiction, induction on n,
etc.? (This is KEY.)
- What mathematical raw materials are used in the proof? (Do we need a
lemma? Do we need a new definition? A powerful theorem? and do you
recall how to prove it? Is the full generality of that theorem needed, or
just a weak version?)
- What does the proof tell you about _why_ the theorem holds?
- Where is each of the hypotheses used in the proof?
- Can you think of other questions to ask yourself?
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