isomorphismes

Lucas’ “rational expectations” revolution in macroeconomics has been tied to the ending of stagflation in the world’s largest economy, and to the reintroduction of “psychology” into finance and economics. However, I never felt like the models of “expectation” I’ve seen in economics seem like my own personal experience of living in ignorance. I’d like to share the sketch of an idea that feels more lifelike to me.

First, let me disambiguate: the unfortunate term-overlap with “statistical expectation” (= mean = average = total over count = ∑ᵢᴺ•/N = a map from N dimensions to 1 dimension) indicates nothing psychological whatever. It doesn’t even correspond to “What you should expect”.

If I find out someone is a white non-Hispanic Estadounidense (somehow not getting any hints of which state, which race, which accent, which social class, which career track…so it’s an artificial scenario), I shouldn’t “expect” the family to be worth $630,000. I “expect” (if indeed my expectation is not a distribution but rather just one number) them to be worth $155,000. (scroll down to green)

Nor, if I go to a casino with 99% chance of losing €10,000 and 1% chance of winning €1,000,000 (remember the break-even point is €990,000). “On average” this is a great bet. But that ignores convergence to the average, which would be slow. I’d need to play this game a lot to get the statistics working in my favour, and I mightn’t stay solvent (I’d need to get tens of millions of AUM—with lockdown conditions—to even consider this game). No, the “statistical expectation” refers to a long-run or wide-space convergence number. Not “what’s typical”.

Not only is the statistical expectation quite reductive, it doesn’t resemble what I’ve introspected about uncertainty, information, disinformation, beliefs, and expectations in my life.

A better idea, I think, comes from the definition of Riemann integration over 2+ dimensions. Imagine covering a surface with a coarse mesh. The mesh partitions the surface. A scalar is assigned to each of the interior regions inscribed by the mesh. The mesh is then refined (no lines taken away, only some more added—so some regions get smaller/more precise and no regions get larger/less precise), new scalars are computed with more precise information about the scalar field on the surface.

NB: The usual Expectation operator 𝔼 is little more than an integral over “possibilities” (whatever that means!).

(In the definitions of Riemann integral I’ve seen the mesh is square, but Voronoi pictures look awesomer & more suggestive of topological generality. Plus I’m not going to be talking about infinitary convergence—no one ever becomes fully knowledgeable of everything—so why do I need the convenience of squares?)

I want to make two changes to the Riemannian-integral mesh.

First I’d like to replace the scalars with some more general kind of fibre. Let’s say a bundle of words and associations.

(You can tell a lot about someone’s perspective fro the words they use. I’ll have to link up “Obverse Words”, which has been in my drafts folder for over a year, once I finish it—but you can imagine examples of people using words with opposite connotation to denote the same thing, indicating their attitude toward the thing.)

Second, I’d like to use the topology or covering maps to encode the ignorance somehow. In my example below: at a certain point I knew “Rails goes with Ruby” and “Django goes with Python” and “Git goes with Github” but didn’t really understand the lay of the land. I didn’t know about git’s competitors, that you can host your own github, that Github has competitors, the more complex relationship between ruby and python (it’s not just two disjoint sets), and so on.

When I didn’t know about Economics or Business or Accounting or Finance, I classed them all together. But now they’re so clearly very very different. I don’t even see Historical Economists or Bayesian Econometricians or Instrumental Econometricians or Dynamical Macroeconomists or Monetary Economists or Development Economists as being very alike. (Which must imply that my perspective has narrowed relative to everyone else! Like tattoo artists and yogi masters and poppy farmers must all be quite different to the entire class of Economists—and look even from my words how much coarse generalisation I use to describe the non-econ’s versus refinement among the econ’s.

These meshes can have a negative curvature (with, perhaps a memory) if you like. You know when you think that property actuaries are nothing at all like health actuaries that your frame-of-reference has become very refined among actuary-distinguishment. Which might mean a coarse partitioning of all the other people! Like Bobby Fischer’s use of the term “weakies” for any non-chess player—they must all be the same! Or at least they’re the same to me.)

Besides the natural embedding of negatively-curved judgment grids, here are some more pluses to the “refinement regions” view of ignorance:

1. You could derive a natural “conservation law” using some combination of e.g. ability, difficulty, how good your teachers are, and time input to learning, how many “refinements” you get to make. No one can know everything.
   
   (Yet somehow we all are supposed to function in a global economy together—how do we figure out how to fit ourselves together efficiently?
   
   And what if people use your lack of perspective to suggest you should pay them to teach you something which “evaluates to valuable” from your coarse refinement, but upon closer inspection, doesn’t integrate to valuable?)
2. Maybe this can relate to the story of Tony—how we’re always in a state of ignorance even as we choose what to become less ignorant about. It would be nice to be able to model the fact that one can’t escape one’s biases or context or history.
3. And we could get a fairly nice representation of “incompatible perspectives”. If the topology of your covering maps is “very hard” to match up to mine because you speak dialectics and power structures but I speak equilibria and optima, that sounds like an accurate depiction. Or when you talk to someone who’s just so noobish in something you’re so expert in, it can feel like a very blanket statement over so many refinements that you don’t want to generalise over (and from “looking up to” an expert it can also feel like they “see” much more detail of the interesting landscape.)
4. Ignorance of one’s own ignorance is already baked into the pie! As is the beginner’s luck. If I “integrate over the regions” to get my expected value of a certain coarse region, my uninformed answer may have a lot of correctness to it. At the same time, the topological restrictions mean that my information and my perspective on it aren’t “over there” in some L2-distance sense, rather they’re far away in a more appropriately incompatible-with-others sense.

In conclusion, I’m sure everyone on Earth can agree that this is a Really Nifty and Cool Idea.

I’ll try to give a colourful example using computers and internet stuff since that’s an area I’ve learned a lot more about over the past couple years.

First, what does ignorance sound like?

• (someone who has never seen or interacted with a computer—let’s say from a non-technological society or a non-computery elderly rich person. I’ve never personally seen this)
• “Sure, programming, I know a little about that. A little HMTL, sure!”
• “Well, of course any programming you’re going to be doing, whether it’s for mobile or desktop, is going to use HTML. The question is how.

OK, but I wasn’t that bad. In workplaces I’ve been the person to ask about computers. I even briefly worked in I.T. But the distance from “normal people” (no computer knowledge) to me seems very small now compared to the distance between me and people who really know what’s up.

A few years ago, when I started seriously thinking about trying to make some kind of internet company (sorry, I refuse to use the word “startup” because it’s perverted), I considered myself a “power user” of computers. I used keyboard shortcuts, I downloaded and played with lots of programs, I had taken a C++ course in the 90’s, I knew about C:\progra~1 and how to get to the hidden files in the App packages on a Mac.

My knowledge of internet business was a scatty array of:

• Mark Zuckerberg
• “venture capital”
• programer kid internet millionaires
• Kayak.com — very nice interface!
• perl.
  
  
• mIRC
• TechCrunch
• There seem to be way more programming going on to impress other programmers than to make the stuff I wanted!
• I had used Windows, Mac, and Linux (!! Linux! Dang I must be good)
• I knew that “Java and Javascript are alike the way car and carpet are alike”—but didn’t know a bit of either language.
• I used Alpine to check my gmail. That’s a lot of confusing settings to configure! And plus I’m checking email in text mode, which is not only faster but also way more cooly nerdy sexy screeny.
• Object-Oriented, that’s some kind of important thing. Some languages are Object-Oriented and some aren’t.
• “Python is for science; Ruby is for web”
• sudo apt-get install
  
• I had run at least a few programs from the command line.
• I had done a PHP tutorial at W3CSchools … that counts as “knowing a little PHP”, right?

So I knew I didn’t know everything, but it was very hard to quantify how much I did know, how far I had to go.

A mediocre picture of some things I knew about at various levels. It’s supposed to get across a more refined knowledge of, for example, econometrics, than of programming. Programming is lumped in with Linux and rich programmer kids and “that kind of stuff” (a coarse mesh). But statistical things have a much richer set of vocabulary and, if I could draw the topology better, refined “personal categories” those words belong to.

Which is why it’s easier to “quantify” my lack of knowledge by simply listing words from the neighbourhood of my state of knowledge.

Unfortunately, knowing how long a project should take and its chances of success or potential pitfalls, is crucial to making an organised plan to complete it. “If you have no port of destination, there is no favourable wind”. (Then again, no adverse wind either. But in an entropic environment—with more ways to screw up than to succeed—turning the Rubik’s cube randomly won’t help you at all. Your “ship” might run out of supplies, or the backers murder you, etc.)

Here are some of the words I learned early on (and many more refinements since then):

• Rails
• Django
• IronPython
• Jython
• JSLint
• MVC
• Agile
• STL
• pointers
• data structures
• frameworks
• SDK’s
• Apache
• /etc/.httpd
• Hadoop
• regex
• nginx
• memcached
• JVM
• RVM
• vi, emacs
• sed, awk
• gdb
• screen
• tcl/tk, cocoa, gtk, ncurses
• GPG keys
• ppa’s
• lspci
• decorators
• virtual functions
• ~/.bashrc, ~/.bash_profile, ~/.profile
• echo $SHELL, echo $PATH
• “scripting languages”
• “automagically”
• sprintf
• xargs
• strptime, strftime
• dynamic allocation
• parser, linker, lexer
• /env, /usr, /dev,/sbin
• GRUB, LILO
• virtual consoles
• Xorg
• cron
• ssh, X forwarding
• UDP
• CNAME, A record
• LLVM
• curl.haxx.se
• the difference between jQuery and JSON (they’re not even the same kind of thing, despite the “J” actually referring to Javascript in both cases)
• OAuth2
• XSALT, XPath, XML

This is only—as they say—“the tip of the iceberg”. I didn’t know a ton of server admin stuff. I didn’t understand that libraries and frameworks are super crucial to real-world programming. (Imagine if you “knew English” but had a vocabulary of 1,000 words. Except libraries and frameworks are even better than a large vocabulary because they actually do work for you. You don’t need to “learn all the vocabulary” to use it—just enough words to call the library’s much larger program that, say, writes to the screen, or scrapes from the web, or does machine learning, for you.)

The path should go something like: at first knowing programming languages ⊃ ruby. Then knowing programming languages ⊃ ruby ⊃ rubinius, groovy, JRuby. At some point uncovering topological connections (neighbourhood relationships) to other things (a comparison to node.js; a comparison to perl; a lack of comparability to machine learning; etc.)

I could make some analogies to maths as well. I think there are some identifiable points across some broad range of individuals’ progress in mathematics, such as:

• when you learn about distributions and realise this is so much better than single numbers!
  
  
  
  
• when you learn about Gaussians and see them everywhere
  
• when you learn that Gaussians are not actually everywhere
  
  
  
• in talking about probability and randomness, you get stuck on discussions of “what is true randomness?” “Does randomness come from quantum mechanics?” and such whilst ignorant of stochastic processes and probability distributions in general.
• (not saying the more refined understanding is the better place to be!)
• A brilliant fellow (who now works for Google) was describing his past ignorance to us one time. He remembered the moment he realised “Space could be discrete! Wait, what if spacetime is discrete?!?!?! I am a genius and the first person who has ever thought of this!!!!” Humility often comes with the refinement.
  
• when you start understanding symbols like ∫ , ‖•‖, {x | p} — there might be a point at which chalkboards full of multiple integrals look like the pinnacle of mathematical smartness—
  
• but then, notice how real mathematicians’ chalkboards in their offices never contain a restatement of Physics 103!
  
  
  A parsimonious statement like “a local ring is regular iff its  global dimension is finite” is so, so much higher on the maths ladder than a tortuous sequence of u-substitutions.
• and so on … I’m sure I’ve tipped my hand well enough all over isomorphismes.tumblr.com that those who have a more refined knowledge can place me on the path. (eg it’s clear that I don’t understand sheaves or topoi but I expect they hold some awesome perspectives.) And it’s no judgment because everyone has to go through some “lower” levels to get to “higher” levels. It’s not a race and no one’s born with the infinite knowledge.

I think you’ll agree with me here: the more one learns, the more one finds out how little one knows. One can’t leave one’s context or have knowledge one doesn’t have. And all choices are embedded in this framework.

1. This was the first time I felt I actually did anythingin life. All the writing papers in school or doing problem sets was like performing rites to the rain gods: going through some motions that are supposed to be important and very held up by society, but actually are not. And even similarly with jobs I had had within organisations, entering with a CV (another cultural rite or ritual with perhaps little objective significance), acting within such a large bureaucracy that I was sheltered from the “rubber meeting the road” at the bottom line, once again felt like more of a rain dance than “no tree no shade” where
   • if you don’t insulate your house it gets cold in the winter;
   • if you don’t cover up your bike it rusts in the rain;
   • if you don’t cook then ∄ dinner;
   • if you get tired of packing up your things moving from flat to flat and take a nap, then when you wake up the stuff still hasn’t moved itself.
2. So that changed my perspective very much on “They should have this”. I heard a customer say one time “I’ve been saying they should have this for so long!” I kept my composure outwardly but inside I was venomous. Oh, really? You‘ve been saying that “they” should “have” that? How amazing of you to conceive of the concept that somebody should do something! I’ve got a lot of f***ing news for you: the only reason this exists is because of ME. If I stop rowing this boat at any time it’s going to sink. The rant went on but now “They should have that” has become my little personal keyword for ignorant people who think that the built world comes to them through some exogenous force rather than from people taking action. The positive flip side to hating that kind of behaviour is that I felt less distance between me and anybody who makes anything happen. (Although in reality there’s a lot of distance between someone running a micro cap business and the CEO of a billion dollar chemical company. But at least the distance was no longer magical or incomprehensible.)
3. I wish I had known more about the law (just the different ways to incorporate, basically) and about accounting (tax accounting; a little QuickBooks).
4. Spreadsheets and economic theory were mostly not-useful. They were a little bit useful but not on direct things I had to do. Also I got a lot of stupid ideas from economic theory and things didn’t work out like I thought they would based on what I thought were reasonable assumptions. I’m sure the spreadsheet would be more useful for people who have to communicate more, and accounting more useful in an org where someone has to keep track of more things. No maths to speak of. (For the business, that is. I did some OCW courses in my free time.)
5. It does not take a genius to understand that £200 is better than £100. And on the cost side I’m not sure any formulas can tell you whether the £80 discount product is better for you than the £100 regular product. I think I was distracted by various “higher learning” ideas and it probably would have been worse if I had more degrees.
6. One of my enduring difficulties was how to figure out what to do based on various people who knew more than me, but were giving me conflicting information, told me. There’s no way I could have known or learned (certainly not fast enough) all the stuff I needed to know. So I would just ask other people. But I always got different answers and I felt like this must be a central problem/necessary skill of management in general. You’re never going to know, so how do you combine these differing reports given what you know about the bias and the incentives and the knowledge exposure of the various people who know more than you but mutually disagree.
7. For me, my management style was basically an extension of my personality. Flaws and all. There was no “second layer” of here’s what I’m going to say to get you to do something, versus here is what I mean. It was just: “Look, if you do this, I’m going to be f***ed. So please don’t do that.” No fake getting angry either, I would fume at home and kick stuff when an employee was not around if they did something that screwed me over (i.e., cost me money or time). I did, though, always keep my second layer around customers or journalists or people who basically had no clue what was up.
8. Politicians’ work consists of spending all their time with f***ing idiots (the other side) and having to compromise over to their stupid and detestable views. Or just going over their inane and stupid opinion and listen to their egotistical blather night after night. I had to work with branches of various municipal governments, including putting forward new legislation in some local councils. Some of the people were legit. And watching politics in person was instructive. (Again, “it’s just people”. That’s a meaningless phrase but I’ll go into it some other time.) But there were some atrocious people (both elected officials and bureaucrats) that I couldn’t stand to work with on a daily basis. I would fly across the room and choke someone. Or at least yell. Definitely not respectfully and politely trim the sails and subtly guide them into my way of thinking whilst letting them think they came up with it themselves. So now I disrespect the view that “all politicians are bad” or whatever. No, politicians have to work with intolerable borderline-evil nincompoops = the people who disagree with me.
9. A lot of other small businesspeople helped me a lot. Like they would just do free things for me. Or would always be nice if I called to ask for advice. Or they would tell me personal details about their past or what they really thought about things—as if by putting in all the effort I was doing, I had joined a club where people who live in the land where if the money stops flowing then everything sinks (as opposed to the way salary people think), where everything needs to be perfect because it’s your baby, and all the other associated things—I had gotten onto a same level with people who were much older.
10. Speaking to those other people’s businesses, by the way, I had a lot of fun learning how they did their do, but there’s no freaking way I could have run or even worked for them in their lines of operation. So when various self-styled Entrepreneurs on the Web (probably just trying to get pageviews or “think aloud” through their own stuff) start saying “it’s like this”, and “you have to do that” or “business is this or that” or whatever, I’m like, you’re either fake or I have no clue how you could be so arrogant. Maybe I’m more credulous if it’s Pleased But Not Satisfied, but if it’s some 20- or 30-something trying to say how the world works … yeah…emphatically: shut up. (NB: This doesn’t apply to people who talk about their own line of business. Just those trying to generalise with no right to.)
11. People gave me way too much authority or respect. (Converse of thinking that those who work in minimum wage jobs must be stupid or irresponsible.) Sort of like my first time standing at the front of a class and all of a sudden I (same me as always) am the teacher. Some would ask me for advice or questions on matters I had no clue about. But just because I was labelled a “boss” or “entrepreneur” when in fact I can’t tell them whether their idea is a good one or not. Maybe it was reasonable in that certain things did start to sound really stupid to me and that may have been based on being closer to “reality” if they were floating in their bureaucratic job.
12. I wouldn’t say I was particularly happy. I felt respected, I felt like a fountainhead, and I was busy. But if people asked “Are you happy?” I would honestly say I didn’t consider happiness as an objective. My objectives were to accomplish A, B, C.
13. You don’t merely need to be offering something that other people want. You also need to be able to extract dinero from them. People who view capitalism as the only necessary beneficent social force forget this. If my business generates a larger consumer surplus than yours, nobody cares. Very profitable businesses do not necessarily generate more total surplus than sort of profitable businesses, because the very profitable businesses might just be better at keeping more of the trade surplus for themselves. Then there’s an incentive for someone running the high-consumer-surplus business to get out of it and spend their life doing something that will reward them more for their work. A soup kitchen for the homeless seems like it could be an example. An almost infinite consumer surplus to give food to someone who is starving but they don’t have any money to give you, so it’s a bad business.
14. It was completely not mysterious where wealth comes from. Economists debate about supply side versus demand side and it seemed very ethereal before I did this business. Value is created at every one of my transactions when one of my consumers paid me and experienced the utility bump from doing so. All of the people who supplied me contributed but did not create wealth per se. In this sense the “sure thing” businesses that one would want to lend to / invest in (selling pickaxes to miners) are not the value creating ones because they do not take the frontline of risk of will consumers actually want this. And most of my customers did not have a latent desire for what I was selling, I just thrust it in their face and they said yes/no to the deal.
15. Related to that: the paint people mix together something like 25 kinds of paint. They make my specific colour of paint. This is more valuable to me than the price I pay (that’s my “consumer surplus”). But how do I notate it on my balance sheet? To me it’s worth much more than £100 because it’s exactly what I need to touch up my stuff and I can keep everything perfect (convex returns to approaching perfection—that’s a theme; notice the stores that mark up a lot always have everything else impeccable). But its resale value is nil because who wants my paint colours? Kind of interesting mathematically and maybe that observation is important to understanding where value comes from and how it flows through the economy?

It was a very micro cap business I started with a credit card and £1000 borrowed from a friend. I had just a few recurring expenses and one big initial investment. Ended up with about 8 part-time employees by the peak. A lot of people think I shut it down because of problems with the government. But actually it was because I took an unrelated outside risk—investing time in an Internet startup—which didn’t pay off and took too much time away from my main business, so I had to shut it down.

My three motivations or main reasons I started it were:

• It seemed better than doing an MBA. I thought I would probably hire someone who had actually managed a company rather than someone who blew $150k on some classes about how to manage a company.
• I was and still am into development economics. This is essentially trying to solve the problems of the world’s poor. After realising that there is less need for economic theorists than just for people who Actually Do Stuff (ag scientists, road builders) and for Money, I decided my best way to fight poverty would be to create well-paying jobs—even if it’s just a few.
• The “piano lessons” theory of doing stuff. When I took piano lessons I was taught to practise songs slow at first or slow and isolating just one hand at a time. Then as my skill increased I could increase the speed. More or less the same idea: start in an insulated market, doing a small market cap, copying almost exactly a business I had worked at before, very low costs, very modest goals, and just make sure I could accomplish the small thing before dreaming big.

So I did not make a lot of money doing this business, which by definition makes it not a great business. But I did support myself, I did accomplish my goals (to learn, to not lose money, and to employ at least some people at a notably higher wage than they could make elsewhere), and I did make my community a more vibrant place, during the time I was working at it.

During the period I was doing that I definitely felt “Anybody could do something like this. Not necessarily make a lot of money but add something new to their community. Sure, it’s a lot of butt busting work, but at least I’m giving an effort.” Maybe what I missed in that “lecture” was that most people in fact do not want to work their butts off for not so much money. It’s in fact much more attractive to have a guarantee that no matter how bad things get you will still get this much money traded for also fixed hours. And maybe that is amplified in rationality because mainly the companies offering variable pay are usually some startup that will fail and therefore they offer “sweat equity” i.e. unquantified ownership of a worthless company. Or maybe I was actually just quite lucky to have experience working at a company that I guessed I might be able to replicate without a lot of up-front capital. That was an assumption I was aware was flawing my “lecture” even at the time.

The other reason it’s probably not smart to encourage others to “Just do something” is that planning, thinking things through beforehand, and developing expertise that can shut down competitors must have more value in a more difficult market or a higher market cap.