First Thoughts

It is impossible to begin. Every sentence must contradict itself.

Every thought that rushes up is beaten back down immediately by others, equally vaporous. I need to think clearly. When I learned I had failed quantum mechanics, and will not graduate, I wrote to the professor the most honest statement I was capable of producing.

Professor C-,
[The registrar] informed me I haven't done enough work to pass your class.  I'm
sure this is true.  The work I turned in was:
Midterm (I got about 25%)
HW 1
HW 2
Final (I answered six questions before time ran out).

I've attempted HW 3 a few times, but I still don't understand it.  I've
tried reading the lecture notes and other homework assignments but have
not made any significant headway.  I admit I did not learn more than a
fraction of the material for the course, and that this has become a
pattern for me over the last two years.  The total work I turned in for
professor M-'s course was similar, except I was unable to do any of
the problems on his final.

I don't understand the fundamentals from ph125a and ph125b, or peripheral
knowledge like ph106ab or ACM95.  Ph125c is mostly incomprehensible to me.
I should have attended the lectures earlier this year, studied
independently while I wasn't enrolled, and entered third term prepared.  I
don't have a good explanation for why I did not.  The only poor
explanation I have is that at the time, I mistakenly (perhaps
delusionally) believed I could catch up in a week or two of concerted
effort.  On other occasions I had the feeling, on opening the text to an
advanced chapter, that the magnitude of elementary material I would have
to study before I could make any progress on the assigned work was so
great that the effort was futile, and so I set it aside and did something
else.

Ph125c is my last requirement for graduation.  I no longer have
aspirations towards a career in physics, but I do realize that any career
path will become nearly unnavigable if I never finish my degree.

I'm trying to think of a reason why, if you were to allow me to make up
the missing work over the summer, I would actually get the work done.  It
would involve radically different behavior patterns than any I've
exhibited in the past for longer than a couple of weeks.

I believe I would get the work done, but I have always, in the past,
believed I was going to get work done, and I cannot explain why I believe
this.  I've learned that drawing up schedules and making commitments for
myself provides me with no psychological incentive if I do not believe
they are fundamentally important.  Only a complete shift in my mentality
could suffice to change my habits.

I'm unable to come up with a reason why this summer should be different
than last that is convincing to me.  The closest I can come is that this
summer I am more jaded and less inquisitive, and therefore less likely to
abandon my work in favor of other diversions.

If you think it could be productive, I'd like to meet with you in person
to discuss Ph125c and whether a viable option remains to finish the
course.

I apologize for the long and personal nature of this email, but I thank
you for your time and attention.

Mark

There was the hope that, if he knew failing the class was going to hold me here another year, he might decide 30% or so was sufficient, after all. But I didn't actually believe this email would accomplish anything. Nothing material. I felt a need to simply state the case. To acknowledge a professor as a human, and to acknowledge my own academic struggles as those of a human.
I have not collaborated on homework more than a handful of times over the course of my entire Caltech career. I have hardly ever sought help from TA's or professors. I never understood learning as an interpersonal process. I only understood academics as something I must do.

Am I, now, by stating it so plainly and so publicly, crying a pathetic mewl in hopes of sympathy? Am I "coping"? I hope not. I hope I am beginning.

I need a radical and deep alteration of the fundamental patterns of my cognition. In these first moments it is impossible to speculate. When I think, my only thoughts are how pointless all endeavors are. They only end in failure. We're six billion pinwheels flapping in the breeze. We dream of giant wind turbines, but we generate no power. Only a little noise.

Of course I was angry - the was enough work turned in it wouldn't have been immoral to pass me, by any stretch. But this anger was impossible to sustain under the waves of self-loathing, washing up regularly in a rising tide, each crest bowling me over with the force of its impact.

I watch them walking, driving, talking, yelling. What is their point? What do they think they can ever accomplish with such a trivial thing as existence? I cannot understand how this enduring pattern began. How I got to where I am now, tottering on the edge of the nest, realizing I have wings but no feathers.

I don't know in what direction even to wander. My belief, my hypothesis, is that I am at war with my ego. In moments of quiet solitude, when I am not thinking about myself, I am the happiest person I know (also, at that moment, the only person, but that is another thing). My mind can fly off in incommunicable flights of joy. I can sit by myself, laughing uncontrollably at the idea of a joke I do not know how to articulate, nor imagine, nor describe. It's simply a shape and a rhythm bouncing back and forth between my neurons. It emerges unbidden and enraptures me. Sometimes.

But I kill it. An innate, insane irrationality consumes me as soon as I think about the future, and my place in it. I'm important. Therefore, I must do important things. I can't even tell you right now what one important thing is. This will pass, in a few days. But will there be any lasting impression? Will I cease defeating myself with my life of fits and starts?

I am even watching myself typing, laughing at the predictability of my petty emotional response. I am mocking myself for being so crestfallen by something so trivial. Pitying myself for being a simple cognitive automaton, no different from any of millions of others of the same mold.

And I wonder why I recognize intrinsic meaning in such an image. Why should a sentient being be downtrodden simply because it is insignificant? Aren't the glories of life made more vibrant when reflected off the shining friendly faces of others?

The Harmonic Series

I've been daydreaming about how I would teach an introduction to calculus in just two hours. It's hard because when I start thinking about calculus, which I generally take for granted, I begin to see holes and gaps in my understanding. (What precisely is an "infinitesimal" differential element supposed to mean, anyway?)

Just now, instead of another hole, I found this nice smooth pebble, my first for this blog. It's a simple proof that the harmonic series diverges. The harmonic series is this one:




The sum "goes to infinity", meaning that if you take any number at all, after enough terms the harmonic series grows bigger than it. The proof proceeds via reductio ad absurdum.

Assume the limit is some number S. Then break the sum into two smaller sums, one of the odd terms and one of the even terms, like so:










All the terms from the original sum are there in one of those two smaller guys. Also, s₁,the sum of odd terms, must be greater because each individual element is greater (i.e. 1> 1/2, 1/3 > 1/4, etc.)
So algebraically:





Now for the trick. Just multiply s₂ by 2.







If s₂ is half of S, then s₁ must be the other half.




Wait! s₁ is supposed to be bigger, but now they're equal to the same thing. To check, plug (8) and (9) into (6), and you're left with:




Which is a contradiction. Therefore, the hypothesis is wrong, and the sum is not a finite number. It is positive, since each of its terms is, and so the sum simply gets bigger and bigger without bound.

After a quick googling, it turns out my proof is number 8 on a long list (continued here) of proofs, but that no one found it until 1979!