isomorphismes

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.

• “X does something whilst preserving a certain structure”
• “There exist deformations of Y that preserve certain properties”
• “∃ function ƒ such that P, whilst respecting Q”

This common mathematical turn of phrase sounds vague, even when the speaker has something quite clear in mind.

Smeet Bhatt brought up this unclarity in a recent question on Quora. Following is my answer:

It depends on the category. The idea of isomorphism varies across categories. It’s like if I ask you if two things are “similar” or not.

• “Similar how?” you ask.

Think about a children’s puzzle where they are shown wooden blocks in a variety of shapes & colours. All the blocks that have the same shape are shape-isomorphic. All the blocks of the same colour are colour-isomorphic. All the blocks are wooden so they’re material-isomorphic.

In common mathematical abstractions, you might want to preserve a property like

• still a group
• still a vector space
• still a simplicial complex
• still a plane
• still similar to the original triangle
• still sum to the same constant (symplectic)
• deforming the path without pushing it over a singularity doesn’t change the contour integral

after some transformation φ. It’s the same idea: “The same in what way?”

As John Baez & James Dolan pointed out, when we say two things are “equal”, we usually don’t mean they are literally the same. x=x is the most useless expression in mathematics, whereas more interesting formulæ express an isomorphism:

• “Something is the same about the LHS and RHS”.
• “They are similar in the following sense”.

Just what the something is that’s the same, is the structure to be preserved.

A related idea is that of equivalence-class. If I make an equivalence class of all sets with cardinality 4, I’m talking about “their size is equivalent”.

Of course the set  is quite different to the set  in other respects. Again it’s about “What is the same?” and “What is different?” just like on Sesame Street.

Two further comments: “structure” in mathematics usually refers to a tuple or a category, both of which mean “a space” in the sense that not only is there a set with objects in it, but also the space or tuple or category has mappings relating the things together or conveying information about the things. For example a metric space is a tuple . (And: having a definition of distance implies that you also have a definition of the topology (neighbourhood relationships) and geometry (angular relationships) of the space.)

In the case of a metric space, a structure-preserving map between metric spaces would not make it impossible to speak of distance in the target space. The output should still fulfill the metric-space criteria: distance should still be a meaningful thing to talk about after the mapping is done.

I’ve got a couple drafts in my 1500-long queue of drafts expositing some more on this topic. If I’m not too lazy then at some point in the future I’ll share some drawings of structure-preserving maps (different “samenesses”) such as the ones Daniel McLaury mentioned, also on Quora:

• Category: Structure-preserving maps; Invertible, structure-preserving maps
  
  
• Groups: (group) homomorphism; (group) isomorphism
  
• Rings: (ring) homomorphism; (ring) isomorphism
  
• Vector Spaces: linear transformation, invertible linear transformation
  
• Topological Spaces: continuous map; homeomorphism
  
• Differentiable Manifolds: differentiable map; diffeomorphism
  
• Riemannian Manifolds: conformal map; conformal isometry