Is Mental Math Part of the Current Meta?

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Is Mental Math Part of the Current Meta?

There comes a point in most people's math education where they suddenly discover that, with no notice whatsoever, "good at math" has turned from a question of rapidly and accurately performing arithmetic to navigating increasing layers of abstraction. There's a lot of fun to be had here, both in the form of getting more confidence that the formulas you've memorized are meaningful and in learning where they apply.

In one sense, this is a nice graduation from math-to-impress-teachers to Real Math™, but it can be disconcerting one if you're good at math-to-impress-teachers and not Real Math™. In the other case; if you find out that you hated math because the part you hate is the part calculators are good at, and the part you like is more beautiful, that's a great feeling.

I think there are quite a few people who come to share that great feeling—after all, we’ve seen amazing mathematicians occasionally make elementary errors. For example, here's Taleb saying that Mandelbrot "was unable to add/subtract". Maybe there was a time when this kind of calculating skill was a literal prerequisite for doing anything more sophisticated, but if we have calculators, Excel, Wolfram Alpha, and numpy, maybe it's okay to struggle. A private equity fund manager who takes a little while to calculate a 15% tip shouldn’t be concerning, since the IRR they're calculating for an investment is going to be done in Excel with lots of cross-checks anyway.

In gaming terms, the "current meta"—the general decisions that most reasonably good players make most of the time—suggests that getting good at mental math is a poor use of time

A good model of the don't-bother-with-mental-math argument is that there are two ends of the continuum for calculations you'd want to make: at one end, there are questions that are so unimportant that they don't need an answer. And at the other end, there are questions so important you'd never want to introduce human error into your attempt to answer. What's left is a narrow slice indeed.

But that model only argues that mental math shouldn’t be used to generate final answers. I asked about this on Twitter using the time-honored technique of just asserting an answer so people would be motivated to correct me. And the response revealed a meta category: while mental math is not a useful skill for getting answers, it is a great one for rapidly iterating on questions.

Examples abound:

• In some careers it’s handy, and not just obvious ones like market making or advantage gambling (if you insist on enforcing some arbitrary separation between the two). Negotiation is sometimes a matter of increasing degrees of freedom until you hit on a formula both sides can solve for, but every new negotiating point creates costs and benefits for both sides. Being able to assess the impact of some new contractual terms while still keeping a running tally of how close you are to a satisfactory deal is a valuable tool for keeping a negotiation moving, and time-to-close is often a determining factor in getting a good deal. This is not just because people like to hold to verbal agreements, but because it's a fast way to travel through the space of possible deals to find one that's good for both sides. If you can keep saying "that works, as long as we can also agree that..." without accidentally agreeing to lots of terms that are cumulatively slanted towards the other side, you can come up with a deal outline that's different from where you started but better for both sides.
• It’s a good auditing tool, and speeds up a breadth-first research process. Looking at a growth company is basically like the process of canonizing a saint: you have some basic criteria to check off, but you spend most of your time on assessing the truth value of a handful of extraordinary claims. Finding which companies are miraculously able to avoid commoditization, overshooting growth, technological obsolescence, malinvestment pursued in a misguided attempt to avoid obsolescence, hubris, etc. is partly a matter of identifying which metrics and drivers seem to make the biggest difference. And that process is faster if it's easier to mentally assess lots of claims about what makes a company special and estimate whether or not they make a material difference to the outcome. (For example: over a five-year period, the difference in revenue growth between a company growing total customer count 100% with no change in revenue per customer, and one growth customer count and revenue per customer at 50% apiece is 80% higher revenue for the latter case.) This is also a good tool for sanity-checking numbers—extrapolate consensus growth rates, compare that to the size of the market, and you may find that the thesis is less about a company taking share and more about it creating an entirely new market.
• Mental math is a partial inoculation against scope insensitivity. It sometimes seems like the number system employed by political commentators (and politicians themselves) has a special number called the “illion,” which is a large but non-specific quantity. While it's nice to just practice a zen approach to any outrage whatsoever, it's good to have a backup plan.
• Mental math is a good idle process compared to other things you can do when you're bored. (One systematic approach to mental math was invented by a mathematician to pass time in a concentration camp.)
• Actually practicing mental math is a good way to get intuition about operations research, compilers, and other domains where juggling lots of variables and minimizing the use of memory/storage are important. Spend a little while playing around on Math Trainer and you'll find that there's an evolution from focusing on pure computation to focusing on quickly inventing and executing strategy, and within that there's some quick mental juggling over the tradeoff between calculation speed and extremely short-term memory. (If you're subtracting two six-digit numbers, doing it in two chunks of three digits instead of three chunks of two is faster, but three chunks instead of two will be more accurate.)

On its surface, the argument for mental math is that it speeds up the exploration process in a few domains, but doesn't end up affecting the outcome that much. But there's a Jevons Paradox-adjacent dynamic at work: we all naturally select out of tasks that require things we're bad at and select into the ones we're good at. Steve Yegge pointed out something similar years ago about typing speed: people who can't type fast will prefer to write terse emails, and when a discussion is happening in email form, they're less likely to participate. A slow typer might argue that they spend more of their time thinking than typing, but that's partly because they don't have the opportunity to think out loud by writing, or convene a discussion with friends. Quick mental math means a slower setup time for some kinds of research, and means quicker turnaround between consuming raw information and finding something useful to dig into. And once that's an option, it will tend to push the work you do in a direction that makes more use of mental math skills.

At some point, we may reach a scary-for-white-collar-workers utopia where all the world's useful information is being constantly crawled, analyzed, and recombined by AIs, which will spot interesting patterns with superhuman speed and accuracy. Until then, the low-latency anomaly spotting skill of rapid mental arithmetic will still be a valuable one.

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Elsewhere

Diseconomies of Scale

Twitter is now a wholly-owned subsidiary of X Holdings I, Inc., best known for its founder, Elon Musk. There's already been some excitement—firing senior managers (for cause, in order to avoid paying severance), spot-checking employees' code1, and bringing in outside assistance. And he's already getting results: GM has paused advertising on Twitter, and there are concerns that Tesla's China sales will make it harder for Twitter to address disinformation from China's state-run media.

A point that sometimes comes up about the super rich is that they're not as aggressively outspoken as most of us would be with that level of financial security. If no one can touch you economically, why not run your mouth a little bit? One reason for this is that there's a selection effect: the people who find controversy most appealing don't get as rich; if a ten million dollar net worth is enough to speak your mind regardless of the economic consequences, you probably won't be famous as a billionaire, just mildly infamous as a troll. (And there are a few such people; Tim May had time to write some highly entertaining trollish essays because he made enough money working for Intel to retire at 35.) But the other reason for circumspection among the rich is that their net worth is usually in equity, not cash, so they actually have more to lose than the average person (this is true even in cases where their net worth is future royalties, not stock). Musk has generally been willing to support a free-for-all exchange of views, albeit with some limitations. Twitter as a company owned by a well-funded billionaire has more freedom from moderation pressure; Twitter as a company tied to a highly levered owner of a car company and a spaceship company has a lot less.

The Diff covered what Musk can do with Twitter ($) back in April.

Licensing

Modern SaaS companies are a nice case study of the Nexus of Contracts theory of the corporate form: they're buying lots of tools and infrastructure from other companies, packaging the results, and selling those to end users. When the bundle is internal—Zoom or Slack for communications, for example—sellers of inputs don't have much pricing power. There's a cost to switching to a cheaper provider, but the company can measure it and decide whether or not it's worth it. But when the bundled product is external, a switch can seriously inconvenience users. For example, Adobe licensed some colors from Pantone, but under new license terms it's charging for them and some users are seeing an error message: "This file has Pantone colors that have been removed and replaced with black due to changes in Pantone’s licensing with Adobe." It's convenient to have a trusted brand-name supplier like Pantone, but this reliance leads to opportunism.

Pantone's parent company's parent company, Danaher, was written up in The Diff last year ($). Danaher is very good at acquiring low-growth businesses and then finding ways to squeeze efficiency gains of this one. When you look at their 17% compounded return over the last 20 years, it's partly the result of instances like this.

Inflation

The Economist looks at the countries where central banks raised rates early and aggressively ($, Economist) and finds that they're still facing high inflation—with an average inflation rate of just under 10%, compared to 8.2% in the US. Emerging markets are in a difficult position, where they have to be faster to change policy because they're more vulnerable to shocks, but a world where rising rates are slowing global growth is also a world where more capital is fleeing to the US, depreciating currencies and thus raising prices everywhere else. The story notes a grim possibility: when there's more awareness of inflation, it's stickier, since workers and firms are both making decisions that assume the current level of inflation will persist—and if everyone bakes in a certain level of inflation, that's roughly the natural level we'll hit.

Normalization

Cathay Pacific Airways will start using Russian airspace for some flights again after suspending that practice in February. Countries can lose their pariah status suddenly when there's a change in policy or, more likely, a change in leadership. But they can also lose it by increments as conflicts get less attention and the dollar benefit of ignoring them grows.

Meanwhile, Russia has exited a deal allowing grain exports from Ukraine ($, FT), might make the war more salient again. Russia is the world's biggest wheat exporter, and Ukraine is #5. While the wheat market is big, it's also inelastic—especially in developing countries, high food prices don't lead people to consume less, but do make them more likely to support protests and coups. A globalized economy makes it easy to import and export many things, and political instability is one of them.

Ads: Content and Context

Reddit is building more self-serve tools for small advertisers. In recent years the small company advertising playbook has heavily relied on Meta's properties; with excellent targeting and infinite inventory, they were the best place to build and scale ad campaigns. Weakening user-level signals raises the relative importance of contextual ad signals, and one place Reddit excels is in legible context. Since comments are such a common interaction—the top 20 posts on Reddit's homepage have a minimum of 250 comments apiece, and many have thousands—Reddit has an abundance of textual information for matching ads and audiences. The company was planning an IPO late last year, and has delayed it due to the weak market. When the IPO window opens again, it's entirely possible that they will be able to point to gains in market share as the market has shifted, and will be able to articulate an argument for why those gains are durable in the face of increasingly strict privacy norms.

Diff Jobs

Companies in the Diff network are actively seeking talent! If you're interested in exploring growth opportunities at unique companies, please reach out. Some top current roles:

• A well funded early stage startup founded by two SpaceX engineers is building the software stack for hardware companies. They're seeking a frontend engineer who can build powerful visualization tools for monitoring real-world devices. (Los Angeles)
• A new service that's trolling the dating market with a better product and better monetization is looking for a full-stack founding engineer. (Los Angeles)
• A company that helps emerging tech businesses identify and hire top talent—and that gets paid in equity—is looking for new talent. A great way to build a startup portfolio and a great network without an in at a VC firm. (US, multiple locations)
• A systematic investing firm that offers customized strategies to an institutional and high net worth client base is looking for a senior engineer who can implement and optimize complex strategies across asset classes. (NYC)
• A crypto data company is looking for experienced DevOps people who can build systems to quickly ingest and analyze on-chain data. (US, remote)

If you’re looking for talent, we should talk! Diff Jobs works with companies across fintech, hard tech, consumer software, enterprise software, and other areas—any company where finding unusually effective people is a top priority.