## Tuesday, 9 July 2013

### Today I Learned: It's All Goodstein

I came back from China last Thursday and I've found it surprisingly difficult to restart this blog.

Maybe it's the thought that now that I've finished my semester with my Accelerated Maths tutorial class, I'll no longer have a real human audience for this writing? Or maybe it's just sloth. =P

So, I was away in China for a little over two weeks. It's summer there and suffocatingly hot and humid.

The first week or so I spent with my grandfathers: they live in the same apartment building, one floor away from each other. It was good to see that my (paternal) gramps was much healthier looking than the last time that I went back, although his attention span has grown somewhat evanescent. He would sit in his wicker chair, nude from the waist up, brows poised delicately on the fulcrum of furrowedom and unfurrowedness. The clacking of the rotating mechanical fan and his rhythmic laboured breathing lending his countenance an ethereal air. Except when we spoke, I was never sure if he was with me, or dreaming.

And the second week I spent in Shanghai. Yea, Shanghai's pretty chill, just not weatherwise. But let's start with the actual journey to China.

My flight was delayed by half an hour, so I sat in frustration (*) reading Harer and Penner's Combinatorics of Train Tracks. Nearing the original (undelayed) boarding time, I hear some commotion behind me and turn around to have a look: a Chinese girl had approached the metallic security door that you go through to get onto the plan. She was feeling...no...she was...groping every crevice of the door, until a few of the elderly aunties came up to her and told her that boarding had not yet commenced. She seemed content with that answer, strolled back to the waiting area, before swivelling about and running back to the gate, flailing at imaginary buttons that might magically open this door.

The buttons, they just weren't there.

And soon, the Chinese aunties came hobbling towards her once more, and so we saw the repetition of a few more cycles of this explanation/flailing-loop. Throughout this process, the airport staff are struggling to communicate with her, but to no avail. In between their attempts to talk to her, she makes a phone call to her dad (I found this out later), frantically crying out in fumbling, tumbling Chinese: "I'll listen to you", "The steel door is closed. It's closed" and "I want to go home; I want to go home; I JUST WANT TO GO HOME".

I just want to go home.

At this point the staff are asking for someone to help them translate, and I happily acquiesce, and we independently arrive at the (obvious) conclusion that she is probably a higher-functioning Asperger's syndrome-sufferer (**). Soon, security is called, and as they gently drag her from the security gate, I call out to her and begin explaining to her that I understand her desire to be on the plane since it's nearing 11. At this, she seemed relieved. I then ask if she received the firm instructions that she must be on the plane by 11 from a family member like her mother or her aunt (I didn't know that it was her father at this point) in the hopes that I could convince her to call them again and that I might then explain to them the actual situation (my guess is that she probably miscommunicated to the plane situation, and that perhaps they could give her new instructions). Maybe she was troubled at my line of questioning, maybe I just reminded her of her family and homesickness, but whatever the case she ran off.

Immediately, the security guards asked me to help talk to her, so I race around the crowd, get to her and by the time that I reach her she is in a state of hysteria. She is surrounded by a flock of chanting aunties, and has begun dry-sobbing. I offer her my hands (***), she takes them, and I begin anew my attempt to get her to call her father (by this time I've been informed by someone that it was her father).

She does (****) calm down.

But then I'm interrupted by a forceful middle-aged uncle who tells her that she must calm down and a middle-aged lady with an infant in her arm saying "look, my baby isn't afraid, so you shouldn't be scared either". And....and...I back away, entrusting them with the situation. And returning to my seat.

I..just walked away. =/

And then security took her away.

I mean, fair enough - security was probably always going to take her away; even if I'd managed to get her to contact her father, explain the situation, and have him relay to her new instructions regarding plane boarding, how might the airline staff trust her to behave sufficiently "normally" on the long flight back to China? A part of me keeps trying to find ways out - could her father have assigned me as a temporary "guardian" on this flight? Indeed, completely ridiculous "solutions" like these.

And after boarding, the empty seat next to mine on a mostly filled out plane did not help relieve me of these haunting thoughts: I kept wondering if this was her seat, the seat of that poor, crying, sobbing girl flailing herself against the hard, cold, metallic barrier: wanting to go home, but hindered by her own frets and incomprehension.

I guess, as a Christian, there's a parable here - there is, for me, a reminder that often we Christians, Jews and Muslims (worshippers of the same God) struggle and flail and cry and shout to "go home", but are stumbled by our own incomprehension. We do not see that Heaven/Paradise is to come to know, love and even befriend the God of the Jews, the Father of Abraham/Ibrahim, the Creator (*****) of the universe. And instead, miss the point.

So...erm...in remembrance of my inability to communicate with this girl...let's...let's try to communicate beyond the limitation of language. Let's try to do today's maths without words:

$$1=2^0.$$
1. $$\leadsto 3^0-1=0.$$
$$\Rightarrow G(1)=1.$$

$$2=2^1.$$
1. $$\leadsto 3^1-1=2=2\cdot 3^0.$$
2. $$\leadsto 2\cdot 4^0-1=1=4^0.$$
3. $$\leadsto 5^0-1=0.$$
$$\Rightarrow G(2)=3.$$

$$3=2^1+2^0.$$
1. $$\leadsto 3^1+2^0-1=3^1.$$
2. $$\leadsto 4^1-1=3=3\cdot 4^0.$$
3. $$\leadsto 3\cdot 5^0-1=2=2\cdot 5^0.$$
4. $$\leadsto 2\cdot 6^0-1=1=6^0.$$
5. $$\leadsto 7^0-1=0.$$
$$\Rightarrow G(3)=5.$$

$$\Rightarrow G(4)=\ldots 7?$$

$$4=2^2.$$
1. $$\leadsto 3^3-1=28=2\cdot 3^2+2\cdot 3^1+2\cdot 3^0.$$
2. $$\leadsto 2\cdot 4^2+2\cdot 4^1+2\cdot 4^0-1=41=2\cdot 4^2+2\cdot 4^1+4^0.$$
3. $$\leadsto 2\cdot 5^2+2\cdot 5^1+5^0-1=60=2\cdot 5^2+2\cdot 5^1.$$
4. $$\leadsto 2\cdot 6^2+2\cdot 6^1-1=83=2\cdot 6^2+6^1+5\cdot 6^0.$$
5. $$\leadsto 2\cdot 7^2+7^1+5\cdot 7^0-1=109=2\cdot 7^2+7^1+4\cdot 7^0.$$
6. $$\leadsto 2\cdot 8^2+8^1+4\cdot 8^0-1=139=2\cdot 8^2+8^1+3\cdot 8^0.$$
$$\ldots$$
1. $$\leadsto 384^2+20\cdot 384-1=155136=384^2+19\cdot 384+383.$$
$$\ldots$$
$$\ldots$$
$$\vdots$$
$$\ldots$$
1. $$\leadsto 0.$$
$$\Rightarrow n=G(4)=3\cdot 2^{77}(2^{3\cdot 2^{77}}-1)+44>10^{100000}.$$

!!!

$$G(6)=???\Rightarrow$$ $5. =) Note that this is all made possible thanks to Goodstein's theorem, as in: it's not completely obvious that $$G(n)$$ is finite for every positive integer $$n$$. And I'm not the one paying out$5 for computing $$G(6)$$, it's Ronald Graham.

*: My frustration did not stem from the delay of my flight, it was more because I was stuck on one of the steps in one of their proofs.
**: May be highly non-pc, but I will call it suffering, and I won't bother justifying myself in this footnote. =/
***: Let's not romanticise this in the most literal interpretation of "romanticise", she was no Rita Leeds.
****: At least...she does in my memories - which I don't completely trust, although I've written most of this down in a diary, so it should be fairly accurate.
*****: Whatever creation means.