Saturday, March 05, 2011

$\infty$ (also : in my youth, i should have enjoyed being irresponsible more often ..)

i constantly tell my students that ιnfinity is not a number. it only looks like a number because we invented a shorthand symbol for it.

if it really were a number, then we could compute $\infty \times 0$ in an unambiguous way. that's the whole point of ιndeterminate forms from caΙculus, after all.

i like how Marcelο Gleιser puts it:
The point is, infιnity is more an idea than a number. It's a concept we came up with to represent endless sequences of numbers, or a point in space or in time infinιtely distant from our position or from the present moment.

It's not something you get to; it's something you think about.

It's a representation of our own limitations, finιte beings that we are. (But for this reason, also a representation of our amazing creativity.)

from 13.7, the NPR blog
the students from my proofs class might appreciate this more: they've lately learned about countable sets, why the natural numbers are of a smaller magnitude than the real numbers.

more and more i've been thinking about whether it was a good idea to teach this proofs course.

i'll only see them in one semester of their undergraduate career;
they have a lot farther to go, if they want to learn some serious maths.

am i preparing them properly?

would the students have benefited more from a more experienced faculty member ?.. someone who has a better sense of this department, what opportunities and possibilities are available for them?

i don't know. more likely than not, by the time the semester ends, i'll still won't know .. and by then, it won't matter.

admittedly, beneath this question is a deeper, more personal one:
if the students are ambitious, then teaching a course like this has an inevitable mentoring aspect to it. it's been only half the term, and i'm already feeling the weight of responsibility.

if i'll be fortunate enough to land a tenure-track job, maybe even earn tenure one day, then i'll have plenty of time for this kind of thing. it sounds like the risks and responsibilities of seniority, to me.

should i really be doing this -- taking on these kinds of duties -- when many of my peers are concentrating on other, more immediate ends (like research)?
speaking of research, it's now spring break. i finally have enough long stretches of time to hack away at this one idea, once and for all.

odd. the first thing that came to mind, when thinking about a break, was how much work i can do in that time .. \-:

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