Notes

In English

A boring, colloquial way.

"Almost all the eggs are gone!" (as half a dozen remain)

Not-Finite-ness

A slightly-less-boring mathy way.

"Almost all prime numbers are odd!"

There are infinitely many primes. There is exactly one even prime number (2). Infinity minus one is... infinity.

"Almost all natural numbers are larger than one-hundred thousand quadrillion quadrillion vigintillion! (10^83)"

There are infinitely many natural numbers. There are infinitely many natural numbers larger than one-hundred thousand quadrillion quadrillion vigintillion. No practical difference between 10^83 and 1 (other than that one is an upper bound on the number of atoms in the universe).

You formalize this with sets: if you have some (infinite) set, a subset whose complement has a finite cardinality encompasses almost everything in the set. We call this cofiniteness.

Probability ~One

A way to quantify surety.

"Almost all the people will lose money at the casino!"

I'm not sure what the exact rates are, but I'd bet money on this being true.

"Almost all numbers are composite!"

It is well known that the number of prime numbers below $N$is approximately $N/\ln N$. In the limit as $N$ goes to infinity, the ratio of primes to non-primes numbers goes to approximately zero. So if you choose a random positive integer, I would bet my life savings that it's composite.

"Almost all graphs are asymmetric."

An intuitive explanation: if you take all possible combinations of nodes and vertices and let the number of each tend towards infinity, you should expect chaos to triumph over order. (sorry Ramsey). This does depend on a certain definition of symmetry, however, and a clearer statement would be "almost all graphs have only one automorphism."

The Lebesgue Measure is Not-Zero

A straightforward-yet-strange way for real numbers.

The Lebesgue measure is what you get when you try to generalize length, area, and volume to $n$-dimensions. It forms the basis for our current understanding of integration, and helps us figure out how big stuff is.

Something with zero volume basically doesn't exist anyway.

"Almost all real numbers are irrational!"

Well yes, all the rationals have measure zero. All countable sets have measure zero, and the rationals are countable.

(Measures are nice: they're a neat generalization of physical scales (mass, volume, etc.) to arbitrary mathematical objects. They're particularly useful for dealing with continuous things we love Lebesgue measures)

"Almost all real numbers are noncomputable!"

Well yes? Of course an arbitrary real number can't be computed to arbitrary precision by a finite algorithm? Computable numbers are countable, remember? Nevermind that basically all the numbers we deal with are computable, this is obvious.

(Measure zero stuff basically doesn't exist, even if they're the only things we use on a daily basis)

"Almost all real numbers aren't in the Cantor set!"

Well yes, of course! Even though the Cantor set is uncountably infinite, it still has measure zero! It's a weird pseudo-fractal embedding of the real line that somehow manages to lose everything in translation but still keep all the relevant information.

(Idk, the Cantor set is weird)

It is Contained in a Nonprincipal Ultrafilter

A filter $\mathcal{F}$ on an arbitrary set $I$ is a collection of subsets of $I$ that is closed under set intersections and supersets. (Note that this means that the smallest filter on $I$ is $I$ itself).

An ultrafilter is a filter which, for every $A\subseteq I$, contains either $A$ or its complement. A principal ultrafilter contains a finite set.

A nonprincipal ultrafilter does not.

This turns out to be an incredibly powerful mathematical tool, and can be used to generalize the concept of "almost all" to esoteric mathematical objects that might not have well-defined or intuitive properties.

(One of the coolest uses of nonprincipal ultrafilters is in the construction of the hyperreals, post forthcoming).

Let $\mathcal{U}$ be a nonprincipal ultrafilter over the natural numbers. It obviously contains no finite sets, but we run into a slight issue when we take the set 

$$E=2,4,6,8,\dots$$

 and its complement 

$$O=1,3,5,7,\dots .$$

 By the filter axioms, only one of these can be in $\mathcal{U}$, and one of them has to be in $\mathcal{U}$. And thus, we can safely say:

"Almost all natural numbers are even."

At a rationality workshop I ran an activity called “A Thing.” Not only because I didn’t know what to call it, but because I didn’t know what to expect. In retrospect, I've decided to christen it "Directed Babbling."

It was borne out of a naive hope that if two individuals trusted each other enough, they’d be able to lower their social inhibitions enough to have a conversation with zero brain-to-mouth filter. I thought this would lead to great conversations, and perhaps act as a pseudo-therapeutic tool to resolve disputes, disagreements over emotionally charged topics, and the like. However, it turns out this isn’t necessarily the best use case for a conversation where you simply say the first thing that comes into your head.

As with any writing trying to describe social dynamics, this may be somewhat inscrutable. However, I will try my best to explain exactly what I claim to be a useful conversational tool, for use-cases  from “solving hard technical problems with a partner”, to “diving off the insanity deep end”. 

Background

Alice and Bob are having a conversation. Alice says X, which Bob responds to with *Y, * in the context of the conversation (the previous things that Alice and Bob have said to each other) and the context of the world (Bob’s priors). Typically, Bob’s System 1 formulates Y and Bob’s System 2 “edits” it (for lack of a better term) - in most cases, the final output has more to do with System 1 than System 2. However, most of the time in discussion is spent with these System 2 “add-ons” - formulating ideas into sentences, making sure that the vocabulary is appropriate for the conversation, etc. 

Hypothesis: if you intentionally remove the System 2 filters from the conversation between Alice and Bob, then you get a rapid feedback loop where the System 1 responses are simultaneously much faster and shorter than the original, which lets the conversation have a much higher idea density. 

Setup

We paired participants and asked them to come up with a topic to start their conversation on. Following that, their instructions were to say “the first thing that came into their head” after hearing their partner, and see where this led. After fifteen minutes of this, we checked in and had a discussion on how this went. Repeat about 6 times. 

Observations

Individuals did not report a loss in the ease of communication or a lack of nuance - rather, they reported being much more tired than normal and that their perception of time was quite dilated. Someone likened it to “having a 40 minute conversation but feeling that only five minutes had passed.” Generally, sentiment was extremely positive at this being an alternative method of communication.

A few concrete things that some individuals did:

• Consciously refused to talk in sentences, and only in key words
• Chose a “spiciness” level before hand (think hotseat 1-10)

Other variations included choosing to talk about either technical or emotional topics, focusing on responding to the last thing the person said vs. the first thing that came into their head, etc. 

Use Cases

The most surprising outcome was that this method of conversation seems to be quite useful for technical discussions when both individuals have similar levels of intuition on the subject. It was counterintuitive for me when I tried this: I expected technical conversations to be driven much more by System 2 than System 1, especially when compared to other types of conversations. But when discussing some mathematical proof, it turns out the System 1 responses represent much more the *motivations *for certain logic than the actual logic itself, and this is what allows for the partner to understand better. See here.

As an introspection tool, it also seems quite useful. If both you and your partner are interested or confused by some social phenomenon, lowering your System 2 filters removes a lot of the implicit restrictions we place on our speech with regards to social contexts, and it opens the door for more valuable conversations.

Failure Modes

The obvious failure mode is a conversation in which  Alice and Bob decide on a topic they both feel strongly about, disagree, and then one or both leaves feeling hurt/ having a worse opinion of the other. While you can’t eliminate this risk entirely, some safeguards make it much less likely:

• Setting a “spiciness” level for the conversation before it begins.
  • Mutually arriving at what exactly a level “7” means is probably necessary.
• Only talking about emotionally charged topics with individuals you trust to handle it maturely.
• Having extremely low barriers to exit the conversation. Making it a social norm to get up and leave one of these at any moment is the bare minimum.

A subtler failure mode is a conversation which waffles between topics without any substance being exchanged between the participants. Such as:

Alice: "Do you prefer London or New York?"

Bob: "Purple."

Alice: "Clouds."

Bob: "Steak."

… and so on. I personally find these to be very entertaining, but it is a good idea to set expectations beforehand of exactly how unhinged you would like the conversation to be (some calibration is necessary).

Directed Babbling seems to have much higher idea density and not much information loss compared to typical conversation. I would recommend that you try this with someone sometime, especially if you’re stuck on a technical problem with a partner. If you do end up trying this, please let me know how it went! My sample size currently is quite small, and more data is always great!

Hydrogen is the fuel of the future. It’s the most abundant element in the universe, is incredibly mass-efficient, and can be produced without much fuss. It can be used for both large and small scale energy production (think fusion and fuel cells respectively), and has virtually no emissions, carbon or otherwise. However, there’s just one problem.

It’s absurdly volume-inefficient.

One liter of hydrogen can produce 0.01 MJ of energy at STP (standard temperature and pressure, 273K and 1 atm) compared to 34 MJ/L for gasoline. That’s 0.02% of the energy output for liter. Granted, things improve drastically for liquid hydrogen, where the comparison is 8 MJ/L vs 34, but this requires maintenance of temperatures below −252.8°C, only a few degrees above absolute zero.

Gaseous hydrogen isn’t that easy to store either: it requires containers pressurizable up to 700 bar (700x atmospheric pressure). That’s 10,000 PSI. And even then, it takes 5L of hydrogen to match up to 1L of gasoline. The Department of Energy estimates that to meet most lightweight vehicular driving ranges, between 5-13 kg of hydrogen need to be carried onboard the vehicle. If we do some quick calculations, that means that vehicles need to carry between 85 - 294 L of liquid hydrogen (some multiple of this for gaseous hydrogen) to go anywhere. For all you Americans, this is roughly between 22-77 gallons.

(For reference, 5-13kg of hydrogen converts to 20-70 liters of gasoline, in terms of energy efficiency. I also make no claim as to the specific accuracy of the numbers, these are napkin calculations, and this doesn’t take into account energy efficiencies created by recycling hydrogen or fuel cells being more efficient than combustion engines, etc.)

It would not be feasible to have stereotypical sedans with fifty gallon gas tanks in a world of solely hydrogen fuel. Additionally, the extreme conditions liquid hydrogen must be stored as bottlenecks its production and transportation.

The Department of Energy set standards for hydrogen storage to be met by 2020 in order for hydrogen fuel to become feasible for portable power and light-duty vehicular applications. These were:

• 1.5 kWh/kg (overall system performance)
• 1.0 kWh/L (overall system performance)
• $10/kWh(translatesto$333/kg for stored hydrogen)

So far, none of these goals have been met, and I have doubts about the scalability of present research (although I’m open to criticism on this take). Let’s take a look at what’s being done to fix the issue.

Short-term Solutions

The majority of short-term solutions to hydrogen’s issues as a widespread fuel consist of creating inexpensive, high pressure storage solutions for H2 gas. From the Department of Energy’s website again, this means developing fiber-reinforced composites that can be produced cheaply and withstand 700 bar pressures. As these don’t necessarily address the underlying volume-inefficiency problem, we can move on to the long-term solutions.

Long-term Solutions

Long-term solutions take two forms: higher-density gaseous storage and materials-based hydrogen storage. The former develops vessels that can compress H2 gas more, while the latter seeks to manufacture materials that have better volumetric hydrogen ratios. This will mainly focus on the materials based approach.

The main research avenues for materials-based hydrogen storage are metal hydrides, adsorbents, and chemical hydrogen storage materials. Let’s look at each.

Metal Hydrides

Metal hydrides are compounds in which metal atoms form ligands with hydrogen. The strength and nature of these bonds vary widely with the metal, but they allow compounds to act as hydrogen carriers without getting decomposed themselves, which is useful for materials cycling (and drastically improves efficiency).

The issue is, the sorts of complex hydrides that have the most promising properties cannot be produced at scale cheaply in any amount, let alone the quantities required for hydrogen to become a serious competitor to gasoline. However, if these compounds were able to scale, then I would be very excited about our energy futures.

Adsorbents

Adsorption is the process by which molecules stick to the internal or external surface of something. Attaching hydrogen gas to some compound would allow it to retain its original molecular form and simultaneously compress it (by virtue of gases being quite volume inefficient). The issue with these is that they don’t compress from the original that much, and are expensive to make.

Chemical Hydrogen Storage Materials

These are perhaps the most straightfoward: find materials that you can either hydrolyze or pyrolyze to release hydrogen, and hope that they have less volume than liquid hydrogen and that they’re easier to store. Well, lo and behold, it turns out there’s an entire class of molecules like this: Amine-boranes.

Amine-boranes are essentially ammonia molecules complexed to boron with extra hydrogen thrown in. The simplest amine-borane is ammonia borane, or borazane, with the chemical formula NH3BH3. It hydrolyzes and pyrolyzes well, has a higher molar density of hydrogen than hydrogen itself, and is a stable solid at room temperature. What more could you ask for?

Well, it’s absurdly expensive to synthesize. I suspect the cost can be reduced at scale, but it does not seem feasible at the moment.