Author’s note: This article is unstructured, so handle with care as things won’t transition or segue as nicely as they could. I didn’t bother very hard to change that, it’d have been more work than it’s worth to corral everything into greater coherence. Regrettably, I made the decision to not pack a journal this trip, so I found myself forced to painstakingly ration my phone’s battery. It was just as glaringly obvious then as it is now why the written record—through pen and paper—sits nigh uncontested in the pantheon of man’s greatest inventions. All that said, everything below is inspired by my raw recollection of various trains of thought over that weekend; unsurprisingly, not every train safely made its way to a destination in the weeks that have since passed. Either way, let’s begin!
Last month, I went on a backpacking trip in Zion National Park with some friends. Over three nights, we hiked Angel’s Landing and the Narrows top-down, spending a night camped along the river bank before resuming our extraterrestrial trek early morning the next day in drysuits. Despite the long drive across states (11 hours one-way in this case) and the foregone tennis playing, I value the disconnection from the portal to the internet contained within my phone. What’s more, I get to hang out with friends who I adore very much, and almost as importantly, get structured time to think. And when I say think, I mean really think: to think the sort of think that can only be thunk when you’ve nothing else to distract you.
On the second night, in keeping with tradition, I woke up for my nightly pee-and-stroll. What I clearly remember is having first looked up at the stars in the 5am dark, and reflecting on whether or not it was the first time I’d ever seen three stars vertically stacked atop each other. Calvin and Hobbes best describes my feelings then.
Payments in AR
As I paced about, I thought about a couple things according to an agenda of sorts. The first thing on said agenda was AR glasses, “arGlasses”, but from a techno-optimistic lens. For whatever reason, payments specifically came to mind.
I don’t quite know why I’m into payments. I’m certainly impressed (and rather terrified) by the durable rent-extracting duopoly created by Visa and Mastercard, plus it’s been beaten into my head that “payments is hard”. Mingled in there too might be some long-lasting feeling of rejection: I’d listened to Patrick Collison at MIT, thought “wow, what an amazing story!” only to later be rejected by Stripe for a SWE internship my undergraduate junior year (whose Dutch competitor I’m now coincidentally an investor in).
Even just as an acquisition play, I imagine the question of how to seamlessly pay in an AR world is worth thinking about. I came up with more questions and room for exploration than conclusions however. I figured that QR codes (or even barcodes) could be used to trigger payment flows along with other conditions, e.g. gestures or voice commands–if beacons are used, perhaps the point-of-sale system too could act as a trigger. As far as authentication and fraud goes, retinal scanning seems plausible, or perhaps the flow (which includes authentication) would proceed more seamlessly on a companion device with existing implementations. Whatever the details of the flow could be rather straightforward, but maybe there’s opportunity in the latter part of the user journey? Here my instinct is that various deals, discounts, bundling and such can be surfaced more compellingly in AR so that it converts and upsells far more efficiently than a webpage on a phone or laptop screen—particularly for in-store purchases. Another fundamental, perhaps obvious, idea I toyed with is that arGlasses could give us a lot of room to leave considerable parts of nature and our physical environment untouched, and thus becomes interesting from an ecological preservation standpoint.
Norse poetry
The above ticked off only the first part of a three-part thinking agenda. Next for the night was poetry. That I like words is an understatement, I love words. A couple things had been on my mind that made me go “aha, a potential poem!” before the trip. I’d just finished a Netflix animated series called “Twilight of the Gods” inspired by Norse Mythology and there’s a line in the series that led me to the following verse in the Havamal (a collection of Old Norse poems attributed to the Norse God Odin), it goes:
Know that I hung on a windy tree
Nine long nights,
Wounded with a spear, dedicated to Odin,
Myself to myself,
On that tree of which no man knows from where its roots run.
There’s something about “myself to myself”, the implied offering of a god to himself, that so held my attention—it’s circular and feels paradoxical … it’s irresistibly intriguing! Like many others, I see the parallels here between Odin and Jesus’ crucifixion in the Christian tradition. Either way, the show had me thinking of Valhalla and this idea that you fight in this world for the honor of fighting again (and again and again and again…) in the next. When I combined the aforementioned with my love for competition and its tumor-like presence in my mind, the draft of a poem materialized in my mind that night.
The Ocean’s surface is a Valhalla of sorts,
For what dies on its surface
Sinks to its depths,
Where it bloats and rises back up again,
As the ocean offers itself to itself
That’s where I ended with that. I didn’t like the imagery and thought it too direct, even if that lent itself quite nicely to the circularity of Valhalla and self-offering. As such, my mind moved to thinking about the Earth and competition. Once again, this is a draft, so a lot of the lines could use some work.
Conjured from the Earth
All things do according to their nature,
Vying in competition,
To the Earth, winners, losers are all the same,
Upon their death, welcome and blessed with a monument.
Except for the cowards, they are rejected and cast away,
Nothing grows where they fell,
For how could the soulless give rise to another soul?
To not partake in nature, in competition,
Is to either be a coward or wise,
But the wise know not they are, so then if you are sure
Verily you are a coward,
And so shunned will you be.
Welcome by the Earth is they who try, win or lose.
It sucks, I’ll be the first to admit it; although I’m sure there are some interesting enough things that I’ll revisit or steal for another poem or short story down the line. But as it stood, I gave up thinking about it in that night’s pacing. The stars were very beautiful to look at and I could spare no more time in my head.
The mind’s eye
Or at least that was my genuine intention, but alas, I was a prisoner to my own agenda. I briefly considered two more things as a final hurrah: (1) a mental reminder to later think deeply on formal verification again, that is, in the context of AI systems and their outputs, and (2) Eueler’s polyhedra formula which I tried very hard to reason through and recall that night. I wrote down with the 5% charge left in my phone:
“The relation between a polyhedron, its number of faces, its edges and its points. Tried recalling it. I think for some n where n is the number of points, we can write a formula for the number of faces and number of edges.”
Sitting in front of a computer with Safari open, I’m reminded that the formula is written \(F + V = E + 2\) Where $F$ is the number of faces, $V$ the number of vertices, and $E$ the number of edges. Remarkable.
What are the motivations to the above? For (1), I’m simply interested in formally verifying AI systems and their outputs in a scalable manner. With regards to (2), it’s two-reasoned, for one, invariants, particularly topological invariants, are incredibly cool, and secondly, I’ve been trying very hard to practice visualizing and rotating shapes in my mind. I suck at it. I don’t think in pictures or videos like I know plenty folks do. I think almost strictly in words and concepts, and that happens in blank nothing. I’ve found however, that besides being very tired, I can sort of see things in my mind when I focus on doing so, especially when I close my eyes and gesticulate wildly about with my hands. I’ve talked to a couple folks about this at various times (including on this trip) and it’s always fascinating what some people are capable of seeing in their “mind’s eye”. A common thing I ask whenever this discussion comes up is to imagine an apple and the “resolution” of it. I’ve gotten answers that have ranged from a blurry 280p (a la old school YouTube) to 4k (I have to choke down my jealousy and quickly wipe away my tears). I then ask them to imagine 10 apples, what do they see? A million apples? Here I’m often told they either see a grid of apples or apples scattered all over the floor as far as they can see. An infinite number of apples? Their response doesn’t change from the million case. How about me? Well in all cases, I don’t really “see” an apple, I think of an apple, much like I don’t “see” the number 3 in my mind, nor do I see the letter “A” either. They simply just are, concepts in my mind that are entirely useless unless further prompting breathes into them some life. If I’m now told to imagine 10 apples, it’s not 10 apples that occupy my mind, but rather the instruction that the concept of 10 should now be connected to the concept of an apple. Once again, both are meaningless and don’t take any mental space until they’re made to do something interesting, e.g. “a boy is juggling 10 apples”. My mind then is a collection of many such lazily initialized instructions linking concepts. As further evidence of this, I’ll also ask folks to imagine a dog, hold it in their mind, and describe it to me with as many details as possible. Most folks I’ve asked can convincingly describe the dog as if it sits before them! they’ll tell me “yes it has a collar”, “no, it’s not leashed”, etc. I, on the other hand, if asked the same, would make up descriptions on-the-fly in order to appease you given that I can’t describe what was never there to begin with. This visually impaired style of thinking might explain a little why I’m so embarrassingly terrible at navigating without visual aids; which I might mention are really only useful to me if I can also physically manipulate and rotate them to fit any desired orientation.
Phew, that was all in one night. I’ll turn to the final thoughts on that trip. This was daytime thinking, replete with many “Gedankenexperiments” (a term for conceptual thought experiments popularized by Albert Einstein) that happened the next two days while wading through the Zion narrows. I’d be remiss if I didn’t mention that there was also some rather pointless thinking, for example, trying to recall what Napoleon and Alexander the Great’s horses were named. (Answer: Marengo and Bucephalus, respectively. I remembered Alexander’s but not Napoleon’s.) No matter, I bothered my friends with it all, pointful and pointless.
Physics and such
I thought about the velocity profile of water, namely whether one could write an equation relating the height of a point in water as measured from its surface with the velocity of the water at that point (and since velocity is a vector, for any direction). I began thinking about this on first thinking about whether a homogenous (without pre-existing cracks or fractures) rock fully submerged in a stream with unidirectional constant flow could ever become anything other than one smaller rock, that is to say, could this rock become multiple rocks solely through erosion. On this question, I concluded no, as the constant flow of water on a fully submerged rock would cause mechanical weathering that could only break the rock’s surface into smaller fragments, affecting its shape and size, but never (assuming unidirectional flow) multiply the number of rocks as we visibly determine it. For there to be more than one rock, there must be a persistent and dramatic difference in the rate of weathering between some collection of points on the rock. (Under normal circumstances, a rock can become another rock due to biological activity, pre-existing fractures or cracks, chemical weathering, temperature, tumbling of the rock and collisions with other rocks, etc.) Given this belief, I then considered whether a rock underwater is, at all the points buffeted by water, possibly undergoing different rates of weathering due to differences in the speed of incoming water. Going off the previous thinking, I concluded no, that the rate of water is at all its points the same within some negligible margin. Here I was picturing the water like a bulk mass, much like a car: consider a car traveling down a road at constant 10 mph relative to some stationary observer. It should be the case that every point along its non-rotating surface also moves at 10 mph or extremely close to it. I had to rethink all this however when I observed a leaf traveling on the surface of the narrows and later still a leaf traveling beneath its surface. I attempted to be empirical here and test in the same location with roughly similar leaves their speed upon and below the water’s surface. I was not able to discover anything conclusive here but it was too late, doubt was sown. When I later had access to a charger and the internet, I found that this is well-studied. The flow velocity does indeed vary with depth and is properly called the velocity profile. There’s a lot of mathematics about it that I’ve been meaning to read but haven’t yet found the time to. I’ll need to revisit the rock multiplication question when I have.
I thought too about indistinguishable particles. I’ve heard before that all electrons and protons are identical and indistinguishable from the next. Taken directly from my notes (once again, when I was able to charge and turn on my phone at dinner the next day).
“I posed the question of if given two “identical” atoms, can you determine which is which. Now, for identical objects like, 3 and 3 which both denote the number 3, or that number arrived by 1 + 1 + 1 or some myriad of other ways, I can identify either three by saying there’s a “left 3” and a “right 3”. That’s to say I’m distinguishing particles by some characteristic as to their placement in space, something extrinsic to these particles. (Which for certain particles might be impossible to measure precisely, quantum mechanics being to blame and all that). I can, however, pose the question more strongly so as to avoid such cheating, namely, given two identical particles, insofar identical as far as you can tell, can you arrange some procedure or experiment such that, when being them one at a time, may identify which is which?”
My reasoning about why this must be true was mostly because I imagined the consequences of its negation (cheating, yes). I know that particles being identical and interchangeable is of great consequence in statistical mechanics and anything else whose proof lay upon probabilistic foundations. Assuming that statistical mechanics and its results are mostly true given all that we’ve achieved with it, I concluded so too must be this assertion (nested implications). My friend insisted however that there should be some experiment that can allow one to discern between particles even in the strong-form question. Here I attempted unconvincingly to make another argument. Namely, to conduct an experiment or process to distinguish two particles, one must be able to “mark” either particle and have that mark persisted over time. This would imply that such particles possess memory and through it could theoretically behave differently to another particle in similar circumstances. The rest of the argument would be explaining why this implication is false. Given that I’m neither a physicist nor am I caught up with the latest literature on the matter, I couldn’t finish this line of thought. On these questions too I haven’t had the time to do much research, but of the cursory research (by which I mean reading the Wikipedia page on “Indistinguishable Particles”) I have done, I’m pleased to see that some of my intuition is in line with what we currently know. Once again, there’s some math here that I’d like to understand.
A not so well written conclusion
So ends the big things I recall thinking about on that trip. I quite frankly wish I’d thought some more. Or been more careful with my phone’s battery, or better yet, have brought a journal and a pen so I could jot things down. Next time.