travel: my advice to the young.

lately i found myself giving advice to younger grad students about graduate school. sometimes they ask, other times it arises in conversation, and often it is about the summertime.

in any case, i say much of the same thing:

1. make sure to take a full month off from math, or any serious work whatsoever, especially if you'll be teaching next fall for the first time. it is important to be well-rested, if only for prepare for when we cannot be .. say, during the fall and winter terms.
   
   
2. if you're planning to take a qualifying exam at start of fall, unless you really don't know the subject, don't spend the entire summer studying for it. you'll get the jitters and start second-guessing habits which would have served you perfectly well if you had simply trusted them.
   
   
3. travel. attend a conference or a summer school.
   

i'll explain further the third suggestion, below. it's full of obvious observations, but i just felt like writing them all down.

besides, it's my blog. (:

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years ago the advisor told me (and the rest of an audience of 50+ people) that young people should attend conferences [1] which is good advice.

i think it a good thing when students first observe current mathematical research in an unfiltered way, if only to see what impressions they form.

correct or appropriate or otherwise, these impressions are honest and are worth something, much like the fleeting fancies and imagination of childhood.

as every adult knows, childhood is a naive time and a foolish time, but it is not without value. it is the most honest time in our lives and where our potential, though not yet realised, is greatest. when we are children, we are at our most possible; by not having the burdens of experience, we can do anything. [2]

but despite such latency, such potential, as children we make silly choices. even if we can choose whom we want to be, we may not choose well, and this is where the analogy breaks. a new graduate student is not a child, and has the capacity to make the most of his/her nascence.

a graduate career is predictable in its steps.

graduate students will choose advisors, and over a span of years they will learn more than the methodology of mathematical research. they will adopt from their advisors viewpoints and philosophies about mathematics and academia in general, even mannerisms and rivalries.

but a graduate student may never develop a healthy skepticism in the sociology of mathematics. (s)he may never learn to question what (s)he has been taught holistically by an advisor and a department; the default is accepting that reality is what you've experienced, and that your experience is true and special.

mathematics may lie in the realm of platonic Forms and embrace canonical choices, but the sociology of the world is not canonical. it does not fit so conveniently into a single worldview, even if that is the worldview of an assorted collection of mathematicians in a single locale.

this is why it is useful to travel. what graduate students learn from their department may be gospel and serve them well for a lifetime of inquiry, but it does no harm to permit a reasonable skepticism.

questions do not erode good foundations; they only show us which parts are worn and rotten, so that we may build better foundations ..

.. and even if the foundation breaks, then how well-built could it have possibly been?

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writing all of this, i didn't say precisely how traveling and 'worldliness' are related. in all honesty, i can't say that i'm fully sure of the connection.

one outcome of travel, especially attending conferences, is that we can quickly encounter ideas and techniques that we have never seen before. this is an invaluable resource, but as for what ..

we will be confused by something .. not that it would be a new feeling for any graduate student, of course, but the source of the confusion arises as something novel from the daily grind. so travel can be a source for novelty and further possibility.

human minds dislike confusion, because their thoughts will run less smoothly in spite of it. an academic mind will seek to understand and resolve the confusion, because of firm convictions in the rationality and consistency of human thought.

when we seek to resolve confusion, we then begin to ask questions out of individual curiosity. so in travelling, we learn to ask our own questions.

it is important to ask questions. some people, despite attending school and college, never learn this.

many graduate students solve research problems given to them by their advisors, and that is fine. graduate school doesn't last forever, however, and students will become independent researchers.

there will come a time when our inquiry is our own responsibility, and it will matter whether we can ask good questions: questions that are interesting and nontrivial, yet questions that we have some chance of answering.

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having made a short story long, let me be brief:

graduate school is not just a road to a ph.d. thesis; it is also becoming an independent researcher. an advisor is invaluable in helping you along the road of research, but you will not always have an advisor.

so learn to be independent: form your own ideas and ask your own meaningful questions. traveling is one means of developing such skills out of circumstance, with the convenient bonus of novelty and possibility.

[re-reads]

wow. i can go on and on, can't i?
at this rate, i'm probably become an ideologue.

[1] then again, he said this before i became his student and before i started graduate school, and i never figured out exactly how old or young is "young." but i agree with the idea.

[2] well, not exactly: there are always constraints upon us, but if you never learn there are constraints, then are they really there? i'll leave that question to the philosophers.