[July 18, 2007] Editor's note, I found this in my files one day. I must have written this back around Sophomore year of college. I fixed it up and put it online for your enjoyment:


Guide to Being a Student of Mathematics

or "How I fooled my professors into thinking that I'm a being of reasonable intelligence"

by Siwei Zhu
200...4?


I Introduction

 The world of Mathematic is full of the perils of ignorance and stupidity. If one does not tread carefully, one could make a mistake and be forever thought of as a person of moderate or below-average intelligence (POMOBAI for short), truly a fate worse than death. Because I am a kind, compassionate, altruistic, and generally excellent man, I have prepared this wonderful article to serve as a shining beacon of guidance to help poor, confused, ignorant, and just plain stupid students (hint: that's you) of today drink from the fountain of knowledge. In here you shall learn all the necessary skills needed to survive an upper-level or graduate mathematics class.



II Lecture

 1. Staying awake

To begin with, you ought to have a healthy sleeping schedule. Ask yourself, is it truly necessary for the enjoyment of your life to stay up until 5AM playing DDR, Guitar Hero, or whatever that weird DJing game is called? If you inevitably find yourself nodding off during class, however, (and this is especially likely for afternoon classes) you should immediately stop listening to whatever your professor is saying about Russell Crowe, and start doing something productive in your notebook. For example, you may make a sketch of your hand (which is the most difficult thing to draw), or if you may compose a "Guide to Being a Student of Mathematics". The point is to keep your pen moving. Your professor will think that you are diligently taking notes (and give you brownie points for that), and you should occasionally look up at the board to encourage this misconception.

If all else fails, you may, as a last option, consume methamphetamines. If anyone asks, you are following the footsteps of the great Paul Erdos.

2. What to do if you are called upon to answer a question

You should avoid being called upon like the plague, for such an encounter has the potential to expose to your professor the fact that there's nothing between your ears but cotton candy. As a preventative measure, you should give off an aura of attentiveness. From time to time, ask a trivial question to show that you are following along with the proof. For example, "is that an i or a j?" "Are you saying this is true for ALL quasi-linear nondeterministic systems?" And at the end of a proof or lemma, "is the converse necessarily true?" If you do this right, your professor will say "that is an excellent question" and give you brownie points. If you run out of questions, make random comments: "I think that won't be easy to prove," "that's really not obvious," "this statement has far-reaching consequences"; the trick is not to make a comment that actually has originality or insight, but to say something fairly obvious before your classmates get to it. However, you should not say something TOO obvious, or your professor will deduce that you are a dumbass: "I don't understand why all Normal spaces are immediately Hausdorff, I mean, isn't it possible that..." would give the game away. It is best to make general statements that could apply in any context and which makes no specific allusions to the current discussion.

Now, should this elaborate defense fail, and the dreaded moment comes upon you when the professor actually calls on you to answer a question, you should defer. Don't refer to specific entities: "I think Frank Han would like to answer this for me" is almost surely doomed to fail. "I don't think it's fair for me to give away the answer all the time, professor. Why not give the first-years a chance to think about it?" works better.

If this, too, somehow fails, make a statement which is true (this part is important!), and which is remotely related to the current discussion. Then hope that the professor, seeing that you are on the right track, will say something about how that's not really what he's looking for, and move onto another victim. For example, "can you tell us why in every finite partially ordered set the minimum number of chains covering the set is equal to the maximum size of an antichain?" "Well you see, we all know that by Zorn's Lemma, if a nonempty partially ordered set has the property that every chain has an upper bound, then it has a maximal element." -notice that, towards answering the question, this reply is less helpful than a fat kid on a ultimate Frisbee team with asthma and amputated legs and also a fear of Frisbees, but it is tactfully on the subject of chains, and more importantly, it is a statement which the professor cannot refute unless he rejects the Axiom of Choice (which he should, were he sensible). Now, if this trick works, the professor will look a little baffled, and say "well, ok... but that's not really what I'm looking for... anyone else?" At the very least, he will give you hints and try to set you on the right path. However, you will know that this trick failed miserably if you hear something like "What the hell does that have to do with anything??? We're talking about partially ordered sets here, not Frisbee, you ninny!" in which case you should just shut up and bask in your stupidity while your classmates snicker at you.



III Homework

 1. Writing proofs

This also applies to the papers that you will later write as an established, upstanding mathematician.

Approach proofs and public speaking with abundant conceit. Remember that the main ingredient in a good proof (and indeed, the secret to being a good mathematician) is not clarity or brevity or even insight, but an unjustified sense of superiority caused by an overestimate of one's abilities and intelligence. Right away you should attempt to establish an atmosphere of arrogance and erudition to intimidate the readers. Gloss over difficult points to show that you can effortlessly see the reasoning behind it, thereby demonstrating your genius, and say "clearly and obviously" when you do, but elaborate on fairly simple parts, and impress upon the reader the weight of their stupidity, which is so onerous that they would be lost on such elementary arguments were it not for your charitable kindness. In particular, frequently use the phrases "in particular," "we see that," "it can be shown that," and ESPECIALLY "hence". I cannot stress how important it is to use "hence"; no proof sounds proper without this word. As a mathematician, you should also practice saying it in real life: "I stubbed my toe during the night, hence I have a limp."

2. Managing the problems

Sometimes you will come upon a problem which is really very easy and simple, but since you have mathematician's block, you're unable to come up with a satisfactory explanation immediately. The correct way to handle this is to dismiss it as "trivial". Since it is so simple, your professor won't doubt your ability to solve it-if you wanted to. Instead, he will applaud your independence and sympathize with your wanting to skip the boring little exercises and skip ahead to the interesting, challenging stuff.

Example: "Show that there exist certain decision problems for which certificates can be verified in polynomial-time, but which elude solution by a general polynomial-time algorithm." Answer: "trivial."

3. Tricks of the trade

Other times you will come upon hopeless difficult problems, and be stuck on them for hours without any visible lead. How do you overcome this hurdle? You may begin by trying the two oldest tricks that they teach you -- induction and the pigeonhole's principle.

Example: "Prove that the integral of the curvature of a knotted curve is always at least 4pi." "We perform induction on the length of the curve. The case n=1 is trivial..."

Should these prove insufficient, there are a few other things you can do.

i. You can claim the result, and use a totally unrelated theorem as justification. For example, "we know that by the Jordon Closed-Curve Theorem, G has a 1-factor." It is usually good to use an important theorem to add credibility to your cause. Amongst the most highly recommended are Fundamental Theorem of Algebra/Caculus/Local Theory of Curves/Arithmetics/Field Theory, Biquadratic Reciprocity, Navier-Stokes, Poincare's Theorem...

ii. You can construct a statement which has absolutely zero mathematical validity, and bury it in the middle of a long list of homethecies, moebius inversions, lorenz transformations, taylor expansions, changes of variables, complete coverings, liftings, projections, embeddings, commutative diagrams, energy arguments, fourier transforms, modus ponenses, syllogisms, generating functions, and just general verbosity. Hopefully, by the time your professor or TA arrives at the point of singularity (where mathematical reasoning doesn't hold and all hell breaks loose-but carefully constructed so as to direct the "proof" towards the desired outcome), he would have been so confused and tired that all he really wants to do is finish it up and goto bed.

iii. You can claim the result and just not write any justifications, clean and simple.



IV Conclusion

 You now have the necessary tools to proper study mathematics. I leave you with some general principles by which to conduct your lives.




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