Category Archives: Nonfiction

Recommendations for Improving Teaching in Brazilian Jiu Jitsu

I’ve been doing Brazilian Jiu Jitsu (BJJ) for about a year and a half now, and, like many people do, I’ve fallen in love with the sport/art. It’s elegant, technical, and a great workout. But, I’ve noticed that even though BJJ is very advanced as a grappling art, its teaching traditions and orthodoxies have not advanced beyond Helio’s heyday. As such, I thought I’d write this essay to give an idea of how to advance the art of teaching BJJ, and hopefully give something back to the sport that’s given so much to me.

First, a bit about my credentials. I don’t believe I am qualified to write this article because I’m excellent at BJJ. I’m pretty good for my experience level, but that’s not saying much. I am, however, very qualified to write this as an educator. In my professional life, I work both as a private tutor and private teacher. As a tutor, I prepare adults for some of the most difficult exams that exist (the GMAT, the GRE, and the LSAT), and I have been very successful at it. As a teacher, I teach philosophy classes over the Internet to Chinese students in massive online courses. These courses can contain anywhere from a few hundred to a few thousand students.

The difficult thing about my professional life is that I sell my teaching ability entirely on its effectiveness. I give out no grades, the subject matter I teach isn’t fun, and there’s really no strong sense of community over the Internet. Moreover, as I teach adults, students are paying me themselves. If they aren’t getting value out of what I teach, then there’s no point to them paying me. Then they leave and stop paying me. As such, I’ve had to devote a lot of time, research, and experimentation towards becoming an engaging educator. I constantly have to make sure my students are learning, and I constantly have to make sure that they continue showing up. If I don’t, then it won’t be much time before I have nobody left to teach at all.

This is different from a BJJ class. A BJJ class has other draws to keep people coming besides just learning how to grapple effectively, like exercise, community, competition, and the desire to achieve a higher belt. These are all good things, and I don’t want to eliminate them from a BJJ class. However, it has allowed for antiquated teaching methods to continue. Or, to put it frankly, people often don’t leave BJJ gyms even if the teaching sucks, so the teaching has no reason to get better.

In my home gym, which I think is representative of gyms in general, classes are an hour long and usually follow this structure: 20 minutes of warmup using BJJ-ish moves (like shrimps or forward rolls), 25 minutes of drills based on whatever the coach feels like teaching that day, 15 minutes of free rolling with whomever. This is not an effective way of learning. Frankly, if I didn’t spend my free time looking up stuff on reddit and Youtube, I would not have a solid foundation for my jiu jitsu.

Now, given the state of BJJ’s teaching, and my credentials as a teacher, I’d like to state my recommendations for improving teaching in BJJ.

Creating a structure for learning

1. Explicit hierarchical structure and progress tracking

For any big subject, students need to have a mental model of what the overall field looks like, and how what they’re learning right now fits into the big picture. BJJ is no exception. Students should be made aware from the very beginning the positions of BJJ: full guard, half guard, etc. The general concepts of each position need to be made explicit, like staying on your side in half guard. Then, when specific techniques are taught, each student should be aware of that technique’s place in the BJJ hierarchy.

Here’s a visual example of what I mean.

In classes, this structure should be referred to explicitly and repeatedly. Every student should understand why
they are learning a certain technique, and what its its purpose is in the BJJ hierarchy. To the greatest extent possible, students should never be confused about the options for a position or, even worse, be unfamiliar with a position entirely.

This explicit structure also allows for progress tracking. Ideally, students have a variety of options from every position (like using a kimura from bottom half guard if the underhook is shut down). Being familiar with all the options from a position is a crucial step towards mastering it. Progress tracking is: a motivational tool for students, as they can see what they’ve learned and feel happy about it, an analytical tool for students, as the weaknesses or holes in their game are apparent, and a planning tool for students, as they can see what they should learn next.

2. Goals (short, intermediate, long)

BJJ has a huge problem with attrition. One of the biggest reasons for that is that there’s not a clear sense of structure to the learning. While that’s somewhat addressed in 1, it’s also important to have clear, universal goals set by the instructor. These goals need to be universal in the sense of “here’s what any person should learn in sequence”.

I prefer setting goals by sequences to setting goals by belt level, as awarding belts is a subjective decision by instructors. This is especially problematic when it comes to blue belts, who have an incredibly wide range of skill sets and knowledge bases. It is almost impossible to generalize “what a blue belt should know”, but it’s easy to generalize “what someone who’s done two years of BJJ should know”.

Once again, it’s important that these goals are explicit. They don’t need to be super specific, as in “learn a kimura by class 3”. They can be general, like “learn two sweeps from closed guard, then learn one submission from mount”. But the goals do need to be explicit, and there needs to be doubts or ambiguity about their content or when students should tackle them.

If it’s not feasible for the instructor to state the goals in class, the goals should be written, displayed, and made known among every student. Every student should be able to immediately answer two questions: what they’ve learned, and what they should learn next. This prevents the “White Belt Overwhelm” and the “Blue Belt Blues”.

No student should be made to feel discouraged or ashamed about being slow to progress in sequence. People have different skills and engagement in the sport, and progressing slowly (or even regressing in extreme circumstances) is better than quitting. However, going to class night after night and not knowing what to learn is not a good thing.

Improving the in-class experience

3. Repetition/callback

Along with structure and goals, repetition is a key part of learning. Each lesson should be structured in the beginning as “This is what we are going to learn”. In the middle: “this is what we are learning”. In the end: “this is what we have learned”.

From an instructor’s perspective, the repetition seems unnecessary and boring. But that is because the instructor doesn’t have to get over the shock of unfamiliarity. To a student, BJJ is like a new language, and the movements are like hearing the words of a new language. Even after hearing them, it’s very difficult to say them.

Repetition also helps orient the students as to where they are in the structure. If students understand the movements well enough to tune out the instructor’s repetition, then they are able to think more deeply about the move, and situate it logically.

This repetition includes repeating over a period of weeks or months. Calling back to earlier lessons helps students understand the deeper connections of BJJ (like all the uses of a kimura), and is both satisfying and allows for a deeper understanding of the move learned previously and the move being learned currently.

4. Takeaways from lesson

Even if a lesson is well taught in the moment, it is too easy to forget it as soon as you go home and the distractions of everyday life come roaring back. Takeaways from the lesson give something for people to hold onto once the lesson is over.

If you want to give someone a takeaway to remember, keep it to one sentence. The idea isn’t to give them all the details you gave in the lesson, but instead to give them a trigger for recalling. If you want to give them all the details, provide it in video or document format. Once they remember their trigger, they can use the video or document to remember the details (see 9. Supplementary Material for more information).

5. During lesson, vary levels of guidedness

During a lesson, it’s up to the teacher how best to teach a technique or movement. If it’s something that’s very technical and being taught for the first time, the teacher should be precise and detail exactly how each body part should move. If it’s more of a concept or feeling, the teacher should give goals, and the student should move according to the teacher’s goals or guidelines. If the teacher is testing a reflex, the student should move without any guidelines from the teacher.

Many BJJ teachers have trouble with intermediate levels of guidedness. When they teach technique, they want to dictate exactly how the student should move. When they let the students do it on their own, they give them zero guidance. However, this is a very hard gap for a student to bridge on their own. There’s a big difference between seeing someone do something (or being told to act in a certain way), and doing it yourself, especially against a resisting opponent.

To give a concrete example, when first teaching the kimura from halfguard, the teacher should show the move directly, and give step by step instructions. Then, when students break out to do it on their own, the teacher should repeat the step by step instructions as the students copy the movement. Once the students are comfortable with that, the teacher should start the students in the correct position, then ask them to finish the kimura themselves. Finally, the student should complete the entire kimura sequence by themselves.

The idea here is to reduce the cognitive load of the students. Remembering an entire move sequence and performing it correctly is hard. Transition from being instructed to doing it on their own needs to be gradual. What ends up happening is that students don’t quite do the moves correctly, then proceed to practice them incorrectly. This either results in them not remembering what’s taught, or, even worse, remembering it incorrectly, giving them problems down the road.

6. Give your students a chance to fail in controlled ways

Going along with teacher guidedness is the idea of giving students a chance to fail. This might either sound ridiculous or obvious, depending on your view of teaching, but first I need to clarify what I mean by this. Failure is when you try, and your try obviously doesn’t work.

So, misremembering a move and being corrected is a failure. Compromising your base in guard and getting swept is a failure. Getting caught in an unfamiliar submission is not a failure, because it’s unclear where your error was, or what you should have done to correct it.

As you might imagine, much like guidedness, failure lie on a spectrum from 100% teacher controlled (like misremembering a technique) to 0% teacher controlled (like getting swept during a free roll with another student). The intermediate levels of teacher-controlled failure(like getting swept while doing positional drilling) are likewise lacking in BJJ education.

Failure should be regular in any class. If students do not fail, they will not learn. And so the BJJ class must be a place where failure is welcomed and encouraged. This is difficult for students, because failure is not as fun as succeeding. The instructor must strive to create an atmosphere where students are, if not happy to fail, then not unhappy to fail.

Improvements in Communication

7. Narrative

During class itself, the easiest way to teach is by a narrative structure. Not in the sense of creating a complex story, but in the sense of creating clear reasons why we are performing moves in a certain way. This allows for deeper understanding of the move, and aids with retention.

For example, when teaching the kimura from half guard, the teacher could explain that the person on bottom wants to control the wrist of his opponent. The opponent does not want his wrist controlled. It’s a simple wording change from simply “control the wrist of your opponent”, but it transforms the action from a static movement to a dynamic one, in which wrist control is actively fought for. Furthermore, it leads the student to useful questions: “why should I control the wrist?” “How do I control the wrist?” “How will my opponent prevent me from controlling his wrist?”

These can be answered by the instructor. Of course, the instructor can share these details without the questions being asked, but it’s easier for the student to remember and understand an answer when the student is motivated to know the answer. People like motivations.

8. Compliment liberally, criticize gently

One of the easiest ways to motivate a student, create a welcoming learning environment, and just make people like learning is to compliment them. Compliments are key to a healthy learning environment, but it is amazing how seldom teachers use them.

For example, one of BJJ’s biggest problems is keeping around the bottom 25% of the class in terms of ability. Women who are small, for instance, have a tough time staying motivated in BJJ. Once they begin rolling, they learn that size and strength play unfortunately big parts in rolling, and they rarely can get legitimate taps, even from people less experienced than them. This leads to discouragement, and even a belief that size and strength are the only things that matter in BJJ.

Compliments help keep people around, and working on what the teacher thinks is appropriate. Criticisms are also useful (like for letting people know they’ve failed), but they’re dangerous when used casually. It’s easy to discourage or anger people with criticisms, which leads to unproductive learning environments. Criticisms should be obvious but gentle. So, if someone performs a technique incorrectly, the teacher can say, “Not quite. Try doing this.” It’s obvious that it’s a criticism, and that the student failed, but it’s not framed in a discouraging manner.

After Class

9. Supplementary material and long term memory retention

It’s easy for a teacher to fall into the trap of thinking that, if they point students towards supplementary material, the students will stop taking the teacher seriously, or even stop coming altogether. This is far from the case. The enemies of teaching are forgetfulness and apathy. Students will not quit BJJ (or a BJJ gym) because they learn too much. They will quit because they forget too much, and they don’t want to put in the effort to remember again.

So supplementary material is crucial. There are many videos and breakdowns available online for every position, but most students will not seek these out on their own. They need explicit instruction to seek them out, and they need to be tested on the material. This testing should, of course, be friendly. As before, failure must be a welcome part of the BJJ learning environment.

Supplementary material is perhaps the only way that students can assist their long term memory and their progress tracking. If a move is used regularly, it remains in short term memory, and is easily recalled. But, if a move is not used regularly, then it is unlikely a student will remember it again months later, unless they have been reviewing it through supplementary material.


BJJ is a wonderful sport and a wonderful art. It is one of the few athletic activities in the word that’s universal and timeless. Our ancestors from thousands of years ago would be able to understand an ankle lock or a rear naked choke, and, as can be seen in old carvings and writings, used them themselves.

But, just as modern BJJ has gone far beyond the techniques of even a few decades ago, so too should the teaching of BJJ advance beyond the teaching of a few decades ago. Advancing teaching is not just about fancy tools or even the Internet, but about using proven techniques that incorporate the tools and encompass them in a broader philosophy.

The 9 recommendations outlined above will make any BJJ class more pleasant, more productive, and ultimately allow any BJJ instructor to better pass on the noble art. And, in turn, this will lead to greater retention of students, more rapid advancement of students, and a benefit to the art as a whole.


The flow of money

Warren Buffett is probably one of my favorite people to read about business. He’s not just a great investor, but he writes clearly and accessibly, even when dealing with difficult topics. Normally, he does most of his writing in his annual letter to the shareholders of Berkshire Hathaway, his company. Every CEO writes a letter to the shareholders, but I’m not aware of any that write like Buffett does.

I’ve been influenced a lot over the years by Buffett’s writings, including his discussions of how he chooses companies to invest in, which I’ve discussed previously. There’s a different topic I want to discuss today which Buffett has discussed before. This is the idea of “float”.

Float is an insurance term meaning the amount of money you have that is really other people’s. In insurance, you collect money from your customers as soon as they sign up in the form of premiums. You pay the money out in response to claims. Usually, people sign up for insurance because they expect or fear that they will need it. Or, in other words, that the insurance company will pay them more in claims than the customer has paid in premiums. Otherwise, the customer would just keep the premiums in a bank account, and pay all expenses out of pocket.

If the customer is right, this is bad news for the insurance company. No business can be run by giving more money to customers than they give you. One option for the insurance company to make money, therefore, is just that the customer is wrong. This is your “underwriting profit”. If, on average, the customer pays you, the insurance company, more than they receive in claims, and that extra amount is enough to cover all your costs of doing business, then you’ll have a nice business for yourself. After all, there are few businesses better than dealing directly with money, without the added inconvenience of other stuff getting in the way.

But there’s also another way of making money, even if you pay out more to the customer than you receive. There will always be some amount of time between when the customer first pays you a premium, and you have to pay that first claim. It might be a month, a year, or a decade, but during that time you have all the money they promised you, and they have none of the money you promised them. If you are good at making money from other money, then you can do very well on this money that they lent to you. One of the main sources of the cash that Buffett uses to buy whole companies is from this float, and he relies on the companies he buys to return the money back to him in time to pay out claims.

What’s interesting is that this model doesn’t just apply to insurance companies. It applies everywhere, to anyone who holds money that they eventually have to pay to someone else. For instance, Amazon sells products on their website and then pays the vendor 24 days later, with the money they got from the consumer. Even if a vendor just uses Amazon as a warehouse (“Fulfillment by Amazon”), the vendor has to wait 2 weeks to get their money, along with the fees they pay Amazon to stock their items.

These 2-3 weeks can make a ton of difference, if you know how to use that cash. You can use that to make your biweekly payroll, invest in a short-term investment vehicle to make 1% or 2% on it, or just hold onto it to smooth out any unexpected interruptions in your cash flow. It’s like free, unlimited payday loans, or maybe just your dad always willing to give you some cash when you need to pay rent.

If you’re interested in business yourself, any business where there’s a lag between when you get paid and when you have to pay out is a good one. If you’re writing a book, take pre-orders. If you’re holding an event, take deposits. Try your best to avoid selling tickets at the door whenever you can.

It’s not impossible to make alternative business models work, of course. Every business that accepts credit cards or checks is accepting some delay in payment, and people who sell through Amazon (or worse, supply to Anheuser Busch) accept a lot of delay in payment. You just have to plan for it, and be confident that payment will arrive when you need it to. However, if you need cash in the meanwhile, you’ll have to pay for it, and you’ll have to pay through the nose. Those companies with float, however, will be floating high and sitting pretty.

How to valuate a company

There are few things more interesting to me than how we spend our money. Money is the clearest, easiest way of valuing things. If you spend $10 on a thing, you value it more than the thing you weren’t willing to spend $10 on. There isn’t necessarily a deep meaning to this, because there’s a million different ways in which we spend and use money, or, to use the terms of Wittgenstein, “money games”. However, you can also translate it to time, which is also how we can measure our life. If you spend $10 on something, and you make $10/hour, you value something at 1 hour of your time, or some small fraction of your life.

In general, spending money is not an easy thing to think about logically or philosophically. You end up with strange comparisons, like poor people value a Big Mac at more hours of their life than rich people. Some people find buying a boat and keeping it in their garage worth a year of earnings, while other people aren’t willing to spend that same amount to keep themselves alive. It’s unclear how far we can go, therefore, just talking about how people spend their money.

However, we can certainly compare spending money in one specific circumstance. One of humanity’s favorite pastimes is spending money to make more money. Whether it’s gambling, investing, or trading, it’s easy to make an “apples-to-apples” comparison. Whichever strategy consistently ends up with the most money is the right strategy.

Of course, we still have the entirety of Hume’s problem of causation. We’re asking what strategy causes us to make the most money, and it can be quite tough to see if there’s a necessary connection or not. We can try to solve it in our normal ways, like with Popper’s method, but at the end of the day, it’s a difficult problem to solve.

But, putting that aside for now, I want to talk about one specific way of using money to make more money: by investing in companies. This is how a significant portion of America (and an even more significant portion of America’s money) spends its day. We can argue about how useful it is to have so much of America focused on allocation of capital, to use the economic term, but for now let’s just look at how it’s done.

The most interesting thing about investing in companies is how much smart people disagree about how to do it. Unlike medicine, engineering, or research, investing is a field with many fundamental disagreements about how it’s done. This is because it’s particularly hard to see the relation between cause and effect in investing. It’s possible to make or lose a lot of money accidentally, just by buying a certain stock. In medicine, it’s quite hard to accidentally cure a person, and in engineering, it’s quite hard to accidentally build an airplane, but it is entirely possible in investing for a random guy off the street to achieve the same results as an expert.

I want to examine two strategies for investing in companies. Traditionally, they’d probably be divided into “value” and “growth” strategies, with the idea of buying shares in a company because it’s cheap vs. buying shares in a company because it’s likely to grow. However, I think there’s also a philosophical division between them, which should become obvious once we go through them.

The first strategy, the “value” strategy, is the “intrinsic value” strategy of Warren Buffett, the famous “Oracle of Omaha”. This strategy is based a lot on past trends and examining a company’s financials. Without going too much into detail, Buffett believes that there are three components to determining the value of a company: the business the company is in, the financials of the company, and the management team.

First of all, Buffett tries his best to understand the business. His specialty is insurance companies. When he’s looking to buy an insurance company, he’ll look at the specific metrics for that company that matter for insurance, like premiums and payouts, and compare them to the industry as a whole. He’ll also consider the long term prospects of the entire industry, and see if there are any trends to watch out for. Buffett refuses to invest in industries that he doesn’t understand and can’t perform this sort of analysis on.

Second, Buffett looks at the financials of the company. He cares about how money moves around the company and what it’s used on. Cash, in the long run, is the lifeblood of any company, and a company’s ability to grow and respond to changing market conditions depends on the cash that it has and can use. Also, as an owner of the company, which a shareholder is, cash is what you ultimately want to get out of the company. It’s fun to own a share of Coca Cola, but you also need to pay your rent at the end of the month.

Thirdly, Buffett looks at the management team. He sees how intelligent they are at using cash, how honest they are, and how emotionally stable they are. A good management team should know when and how to use cash, including when to give it to shareholders. They should also always be honest with shareholders, especially when admitting mistakes. Finally, they should be able to resist bad decisions, even if the rest of their industry is doing it.

Philosophically speaking, Buffett’s investing philosophy is somewhat deductive, somewhat inductive, and a bit intuitive. He bases his idea of what financials should be on deductive assumptions, including mathematical calculations and logical assumptions about the way money is used. His knowledge of a business and what a good business should be is inductive, based on his familiarity with certain industries, like the insurance industry. His ideas of management are inductive, based on his experience with bad management, and deductive, based on logical ideas of how cash should be used, but they are also intuitive. They’re intuitive because reading people is always intuitive, in that it’s something almost all humans are born with, and those that aren’t (like autistic people), find it impossible to learn.

It’s also important to note that Buffett strongly believes in a “margin of safety”. Once he evaluates a company based on his three criteria, he comes up with an “intrinsic value” for the company, which is the amount he would be willing to pay for the company. To compensate for the assumptions that go into this intrinsic value, he discounts it, and the amount between what his calculations tell him he should pay for the company and what he will actually pay for the company is the margin of safety.

Now let’s look at our other model. Andreesen Horowitz is perhaps the most famous venture capital firm operating out of Silicon Valley right now. As a venture capital firm, they cannot rely on investing in businesses with a solid track record, or that are currently undervalued. Instead, they have to find businesses with the capability to grow. Their business model requires specifically for them to find businesses with the capability to grow rapidly. To use a baseball analogy, Buffett’s business model requires him to consistently hit singles and doubles. Andreesen Horowitz’s business model requires them to occasionally hit home runs.

A16Z, as they call themselves (see if you can figure out why), have 5 criteria that they look at when trying to choose companies to invest in. They look at leadership, ideas, market potential, monetization, and how the A16Z team can help the companies. Let’s take a look at each.

Leadership they divide into 3 categories: the ability to effectively direct people, charisma, and brilliance. While these are all easy enough to understand, they’re hard to describe directly in the way that Buffett can describe his management qualities directly. You can imagine people disagreeing over who is an effective, charismatic, or brilliant leader, although it’s possible to come up with examples for each.

For ideas and market potential, they just say they want “breakthrough” ideas and big market potential. This is sort of obvious, but what is frustrating is that they hedge this by saying ideas might not seem breakthrough at first, and market potential might not seem big at first, but they should be breakthrough and big eventually. In other words, if a company has already changed the world, A16Z says they’re classified as world-changing, which isn’t exactly a bold prediction.

The ability to make money is not important, according to A16Z, but it is important that the companies are in categories which the A16Z team can help with. Specifically, A16Z only invests in companies which rely on software, which A16Z believes is their core competence. Along with investing in software companies, A16Z helps them practically with expansion issues.

A16Z’s model is almost entirely intuitive. Without specific criteria of how to look at leadership, ideas, or market potential, A16Z relies on whatever seems to them to fit based on their feelings. The only part that’s inductive is how well the companies fit in with the A16Z team’s competencies, which is based on their experience developing previous software companies.

To compare the two models, while neither is scientific, Buffett’s comes much closer. Because Buffett assumes most of his predictions will come true, it’s possible to attack his model if his predictions tend not to come true. A16Z assumes most of their predictions will not come true, and only a few will succeed, which means it’s harder to know when their model no longer works.

Also, because Buffett’s model mostly relies on deductive and inductive thinking, it is much easier to teach it to others, and it can then be judged based on how well it performs in other’s hands (which would take away the possibility that it relies on Buffett himself, rather than the model). A16Z’s model relies on intuition, especially with regards to understanding the management team, which mostly cannot be taught. Therefore, it would be very difficult for A16Z to teach their model to others, and still harder to evaluate it.

This isn’t to say that A16Z’s model is useless, but philosophically speaking, it’s much harder to tell if their model is good. In a practical sense, if you’re interested in using money to make more money, and you’re looking to do so by investing in companies, I’d advise you to rely on Buffett’s model. If you’re skeptical, evaluate it for yourself by seeing how other people have done with it, or by forming hypotheses based on it and attempting to disprove them. It’s much harder to evaluate A16’s model, and therefore much more dangerous to rely on it.

Thinking about big numbers

People are bad at comparing large numbers. Anything past 100 and everyone’s eyes glaze over. One thing that I notice this a lot in is business journalism, ironically enough. We’re told what we should learn from different companies: what to do, what not to do, and how to think. Some business leaders are called visionaries, others are called disasters.

But we’re never given much of a basis of comparison. We get arrows pointing diagonally up to the right, telling us something’s increased over time, or arrows pointing diagonally down to the right, telling us something’s decreased over time. The scale matters, though. Comparing a large company’s victories with a small company’s victories is like talking about a brilliant NBA player versus a good high school basketball player. There’s a comparison to be made, but we have to be aware of where our comparison falls short.

As you know, I care a lot about how to think, and comparisons are part of that. When we report on business, we report on how most of the world spends its day, so it’s important to get our thinking right about business. To make things a bit easier, I’ve taken some of the top business news of the year measured by upvotes from the subreddit r/business. It’s of course hard to tell what an upvote means, but let’s assume it means that other people think we should pay attention to and learn from some piece of news. So let’s see how it goes.

So, here are the pieces of news: Mylan, the makers of EpiPen, is being asked to pay a $465 million dollar fine; Getty Images is being sued for $1 billion; marketing company SteelHouse gives their employees $2000 for vacation; Unilever buys Dollar Shave Club for $1 billion; Zynga is worth less than its office building ($2 billion including $1.5 billion in cash vs. $580 million).

Whew. That’s a whole mix of names and numbers. I think you’ve probably heard of all of the names except for SteelHouse. So how do we make sense of all of this? What do these pieces of news mean for our world? Should we all quit our jobs and work at SteelHouse? Is NBC going out of business? Is Zynga a failure?

Name of company Valuation Cash on hand Number in article
Mylan 2.515 * 10^10 1.2 * 10^9 4.65 * 10^8
Getty Images 3.3*10^9 2.7*10^7 1.0*10^9
SteelHouse 4.9*10^7 ? 2*103*160 employees= 3.2*10^5
Unilever 1.28*10^11 3*10^9 1.0*10^9
Dollar Shave Club 1.0*10^9 ? 1.0*10^9
Zynga 2.0*10^9 1.5 * 109 5.8*10^8
You 1.4 * 10^5 5.0*10^3 NA


Let’s get into it. First, apples-to-apple comparisons, to make this easy. We need to establish some common basis of comparisons. For valuation, I’ve chosen market cap or acquisition price, and for cash on hand, I’ve chosen cash and cash equivalents. A seasoned investor would have some difficulties with this, but it’s good enough for our purposes. For you, I’ve chosen that you have a net worth of $140,000, including your home and car, and cash in the bank of $5000. This may seem ludicrously low or high to you, but I think it works.

Now we can begin to get a sense of the articles. First off, Mylan was asked to pay $465 million in a fine. That’s about 50% of their cash on hand, or 2% of their valuation. This would be the equivalent of you paying a $2500 fine by either number. It would be rough, and definitely make you rethink some of your plans. You wouldn’t end up on the streets, though. Instead you might have to give up vacations for a couple years until you can rebuild your cash to a point you find comfortable.

Getty Images is being sued for $1 billion. That’s 3700% of their cash on hand, or 33% of their valuation. This would be the equivalent of you being fined $185,000 if we go by your cash, or $46,000 if we go by your valuation. If you were fined $185,000, you’d have to declare bankruptcy. Even if you sold everything you owned, it wouldn’t cover it. If you were fined $46,000, you’d almost certainly have to sell your house, or get an enormous loan against it. First option is certain financial ruin, second is probable financial ruin.

SteelHouse gives each of its employees $2,000 for vacation, and I found online that they have 160 employees. That’s a total of $300,000 they’ve spent on this program. They raised $49 million dollars in funding in their last round, so let’s say they’re worth $100 million. I don’t have a great reason for this assumption, because I don’t know how much of the company was valued at $49 million, but I’ll make it anyways. I have no way of knowing how much cash they have. Given these uncertain numbers, their vacation program has cost them 0.3% of their valuation. That’s the equivalent of you spending $400, based on your valuation. That’s an amount that you shouldn’t throw every weekend, but its disappearance would not make any difference in the trajectory of your life.

Unilever spent $1 billion acquiring Dollar Shave Club. That’s 33% of their cash on hand, or 0.7% of their valuation. That’s the equivalent of $1650 going by your cash, or $980 going by your valuation. Pretty similar to the Mylan story: that’d be your big purchase of the year, and maybe of 2 years, but that likely wouldn’t matter beyond that.

Finally, Zynga is worth less than its office building, when we don’t include its cash on hand. Its office building is worth $580 million, while its valuation excluding cash is $500 million. That’s not really a proper financial calculation, because we have to consider its debt as well, so I’m not going to go by their calculation. Instead, we’ll compare $580 million, to $2 billion valuation, to $1.5 billion cash. So, its office building is 39% of its cash on hand, and the office building is 29% of its valuation. That’s like a property you own being worth either $1950, if we go by cash, or $40,600, if we go by valuation. Obviously, there’s a big difference there. A property you own worth $1950 is something you’re probably looking to sell, while a property you own worth $40,600 is something you’d want to pass down to your kids. It’s not crazy to own a property worth either amount given your financial circumstances, but you would think about it differently.

Now that we’ve gotten our comparisons done, let’s rewrite these headlines:

“Mylan is forced to pay a fine that will make it rethink near-term plans, but will not be devastating otherwise”

“Getty Images is being sued for an amount that would almost certainly end it if the lawsuit worked”

“SteelHouse has put a small amount of money towards a somewhat silly vacation scheme”

“Unilever has likely made its big purchase of the year”

“Zynga has way too much cash, given the size of the rest of Zynga. They should do something about that.”

This isn’t the only way to think about these amounts or these news headlines, but it is a fair way. Otherwise, we’re left with all these numbers that are impossible to think about. Our minds glaze over, and journalists can sneak in and tell us stories without us examining them critically.

This question about how to think about companies is an interesting one. While I’ve approached it here just as an intelligent way to think about the news, other people think about it for their daily bread. In my next essay, I’ll be discussing two different ways of valuing companies: the Grahamanian model, and the startup model. Stay tuned!

Hard science, soft science, and pseudoscience

One of the strange parts of our world is how many different types of knowledge swirl about us at any given moment. On your computer screen, a marvel of engineering brought about by, among other things, our understanding of quantum physics, you can read about how quantum physics isn’t real. You feel letters on a keyboard, you see glowing pixels on a screen, you understand (vaguely) how those pixels are made to glow, and then someone on your screen tells you that the whole setup is impossible.

How do we put together these disparate pieces of knowledge? A large part of it is intuitive, which is why you are able to put together your various sensory experiences of your computer into an object you interact with. Kant was the first to recognize this. But there’s also this top layer: you probably vaguely understand how quantum physics makes your computer possible. You see on the screen someone telling you, convincingly, that quantum physics is ridiculous. You personally don’t know enough about quantum physics to argue with the guy on the computer screen. But, most of us would still disagree with him, regardless of our knowledge.

The reason we would disagree with this guy ultimately boils down to authority. Unlike our philosophers, we can’t trace the deductive or inductive arguments of quantum physics back to its source. Even if we did read the papers and the arguments, the mathematics of them are difficult enough to be impossible for an ordinary person. We have to rely on others to tell us what the arguments are and how they work. Our strongest piece of physical evidence for quantum physics is the fact that the computer does work, and that does seem incontrovertible, but the connection to quantum physics again relies on authority. We rely on the New York Times, which presumably relies on the physicists themselves, to tell us what is true or not.

The New York Times gets its authority from the physicists it interviews, who get their authority from the fact that they can make things work. If any piece of that link were broken, like if physicists denounced the New York Times for misrepresenting statements, or if leading computer manufacturers denying the physicists had anything to do with their products, we’d be much harder pressed to defend quantum physics against our convincing arguer.

Another way to think about this is our degree of certainty. For now, I want to focus only on the degree of certainty we have in the physicists themselves, so I won’t examine the New York Times too closely. As I mentioned, these physicists have an advantage, in that their findings can be seen in the real world. They can lend this advantage to their fellow physicists by endorsing them, explicitly or implicitly. Thus, physicists like Stephen Hawking, who work on topics which will never present physical evidence to the layman, like black holes, also tend to be trusted by the layman.

As we get further away from both the physicists and their endorsements, trust breaks down. Or, to be more precise, from the fields in which physicists’ endorsements count, which tends to just be physics. Global warming, for instance, is a theory fairly well supported by inductive and deductive arguments, if you actually look at them. It’s also fairly well supported by scientific authorities. However, after a long and vigorous misinformation campaign, it’s not very well supported by the public, because it lacks physical evidence for the layman. Indeed, the layman doesn’t even know what evidence to look for. If January 20th is a hot day in Boston, that doesn’t prove global warming, and it doesn’t disprove it if it’s a cold day.

I studied Geosciences in university, which means that I understand the arguments themselves for climate change. Most people, however, will not study the arguments, and wouldn’t be able to understand the arguments if they studied them. Their acceptance of global warming, like their acceptance of quantum physics, has to be based on what authorities say. But there are various prominent politicians and businessmen who deny global warming, and these are traditional authorities. So, if the layman is to accept global warming as a fact, as I believe they should, they have to follow the specific authorities that I believe they should.

These specific authorities are scientific authorities, like the National Science Foundation (NSF) and NASA. They’re not “fringe”; they represent the scientific mainstream. But I realize that, even still, they aren’t easy authorities for everyone to follow. I am certain that both the NSF and NASA have believed incorrect things before. I can even name one off the top of my head: NASA claimed that they discovered a lifeform that used arsenic as one of its building blocks (instead of phosphorus), before rapidly backpedaling once they found out their “discovery” was just scientific error.

So, I can’t encourage people to trust in the NSF and NASA based on their infallibility. Instead, I have to encourage people to trust in the NSF and NASA based on their dedication to the scientific method. First, when the NSF and NASA have been wrong, they have admitted they were wrong upon being shown appropriate evidence. This fits in well with Popper’s idea of falsifiability as a key idea in science.

The second reason is that these are both organizations which believe in a truth and strive honestly to achieve it. This is evident in how the organizations are set up, and their accomplishments thus far, especially NASA’s. So, once again, it’s necessary to rely on the physical evidence of science, except this time instead of directly relying on it, we indirectly rely on it as evidence of a general drive towards scientific truth. And striving honestly towards truth, by the way, has to do with everything we’ve talked about so far in this series of essays.

Generally speaking, I believe this is how we should evaluate authorities on matters we don’t know about. In the case of those that speak against global warming, opponents tend not to accept or even understand any evidence that would be contrary to their views. They will not admit that they were wrong, and they do not use others’ work in an attempt to find truth.

And that brings me to the title of this essay. The definitions of hard, soft, and pseudoscience are difficult to get everyone to agree on. For me, however, hard science is distinguished from soft science in how easy it is to disprove it, and how the field tends to react when major theories are disproved. In economics, for instance, it’s very hard to falsify most theories, as deliberately simplified models cannot be disproved by examples from the real world. Furthermore, it’s rare that economic theories actually die, even if the weight of evidence seems firmly against them. Free trade as a method for countries to develop their economy, for instance, is something which does seem to have clear evidence against it, but still has an enormous amount of institutional support.

Soft sciences are distinguished from pseudoscience, though, in their honest pursuit of truth. Economists consider the available evidence and mention it in their papers, even if it’s only to explain why evidence that seems to go against their theory in reality does not. Pseudoscience does not honestly pursue truth. Evidence that goes against pseudoscientific theories is ignored, or, if mentioned, is not explained fairly. Practitioners of alternative medicine, for instance, neither read nor cite the vast traditional medical literature, even though the accomplishments of traditional medicine are evident.

So, when you’re trying to come up with an opinion of some “knowledge” that someone has presented to you, try first to go to their arguments. If you lack the expertise to understand their arguments, go to the authorities, and see who seems to argue in a way that acknowledges the whole story, and is open to be proven wrong. That is the authority you should trust, and that is how you should come up with your opinion on the matter.

Probability, Bayesian theory, and causation

Let’s say that one day you go into the doctor’s office. He asks you what’s wrong, and you say that you have a fever and a headache.

“Hm… ,” he says. “Is the headache particularly sharp, and towards the back of your head? And your fever, is it exactly 101.3 degrees, and it’s been that way for a couple days?”

“Yes,” you say, “exactly that, for both of those things.”

“I read about this in the medical literature,” the doctor responds, “I think you have a brain parasite from Nicaragua. 99% of Nicaraguan brain parasite sufferers have that headache and that fever, and it’s the most reliable way of diagnosing Nicaraguan brain parasites. We need to start treatment immediately.”

This would be an alarming thing to hear from your doctor, and you’d likely want to start treatment immediately. That is, unless, you’d been reading my philosophy essays, and were wondering if this is really sufficient justification to start treatment for brain parasites. The answer is that it’s not, and I’ll tell you why.

The doctor read in the medical literature that 99% of Nicaraguan brian parasite sufferers have those symptoms, and assumed that 99% percent of people with those symptoms have brain parasites. Unfortunately, that’s not a fair assumption. It’s a big world, and there are many diseases. Presumably, at least one of them could cause those symptoms without it having to be a Nicaraguan brain parasite. In fact, you should at least check if the person has been to Nicaragua, because, if they haven’t, the chance that the patient has a brain parasite from Nicaragua is presumably zero. In this case, correlation does not mean causation.

But let’s take this a step further. Let’s say the doctor isn’t quite so hasty. Instead, he says:

“Have you been to Nicaragua?”

“Yes,” you respond.

“Oh, that’s quite bad. Of the 100 people in Nicaragua with the brain parasite, 99 of them had your exact symptoms. Only 1% of Nicaragua’s population at large has these symptoms. That means it’s a specific, reliable test, which both tells us reliably when people have the parasite and when they do not. We need to start treatment immediately.”

Is that better? Well, slightly, but still not by much. In fact, the doctor has now given us a mathematical way of proving why it’s not a good idea to trust his judgment. Not everyone in Nicaragua has a chance of having the parasite. In fact, most do not. We’re told that 100 people had the parasite, and Nicaragua has a population of 6 million. 1% of 6 million is 60,000. 100 is 0.2% of 60,000. That means if you picked a person at random who had those symptoms from Nicaragua, you only have a 0.2% chance of them being a sufferer of a brain parasite. It is almost certain you don’t have a brain parasite.

This method of reasoning is called Bayesian probability, after Thomas Bayes, the guy who (sort-of) discovered it. In brief, it’s about taking prior probabilities into account when discussing current probabilities. When the doctor read about the power of the test, he imagined the test being given to 100 people, each of whom might have a parasite. Then the test would identify with 99% accuracy who had the parasite and who didn’t. But the doctor didn’t consider the prior probability of a person having the parasite in the first place. This was probably in part due to how hard it is for us to think intuitively about big numbers, which I’ve discussed previously.

This might seem like a strange and silly example to you, but it’s actually pretty important. Breast cancer screenings have this trouble all the time. When we say that mammograms are 90% accurate at early detection, we normally mean that if 100 women have breast cancer, on average mammograms can early detect 90 of them. But we should be asking how often a woman doesn’t have breast cancer and gets a “false positive” from a mammogram.

So far, so good. Bayesian proponents (of which there are many who are surprisingly fervent) claim Bayesian statistics should also be used to evaluate evidence itself. So, for instance, there was a murder case in Britain in the 1990’s. In this case, a mother was accused of murdering her children, because both infants had died in their sleep. Roy Meadows, a pediatrician, calculated that if only 1 in 8500 infants died in their sleep, then the chance of it happening twice was 1 in 8500 squared, or 1 in 73,000,000.

Bayesians were outraged. They said that that was true for just the current probability, but the prior probabilities that had to be considered were: the chance a mother would murder her children (unlikely), the chance of it being completely random that some infants die and others live (very unlikely), and the chance that one environmental or health condition caused the death of two children who lived in the same house (very likely). In other words, Bayes’s theorem wasn’t just used to evaluate the tests, but the evidence. Nobody knew the probabilities of mothers murdering their children, the randomness of infant deaths, or environmental conditions causing two infants’ deaths, but Bayesians refused to let them just be ignored.

You can see their logic. Meadows claimed that almost no children die of natural causes, so the mother shouldn’t be believed when she said two of her children died of natural causes. The equivalent is that almost nobody is born on January 1, so any friend of yours who claims to be born on January 1 is a liar. There’s some prior probability that needs to be taken into account of how likely it is that people would lie about their birthday.

So far, so good. But Bayesians are confident people, and they take these ideas further. To them, you can set up an entire system of science like this. You take a hypothesis, see what the hypothesis entails, and start gathering evidence. If the evidence fits into the hypothesis, it helps confirm the hypothesis. If it doesn’t fit into the hypothesis, it helps disprove the hypothesis. And, of course, you have to consider your prior probabilities.

So let’s go back to the brain parasites. Bayesians gather their evidence, meaning your symptoms, the frequency of brain parasites in Nicaragua, and the frequency of your symptoms among sufferers of brain parasites. If you have a headache and fever, the Bayesians are slightly more confident that you have brain parasites, because they’ve considered the prior probability and know that many people with those symptoms do not have brain parasites. If you don’t have a headache and fever, the Bayesians are very confident that you don’t have brain parasites, because someone with brain parasites should have those symptoms and it was unlikely for you to have brain parasites in the first place.

The Bayesians claim that this way they solve the problem of induction. They see themselves as like Popper, careful not to ever say causation is assured, but they believe their method is more rigorous. Forcing people to assign probabilities and likelihoods to evidence means that you actually get a number for how likely a theory is, and the Bayesians see this as a distinct advantage.

Let’s evaluate their method. Popper, for one, hated this idea, although he was an ornery guy in general. He thought that it didn’t matter if you were somewhat or completely sure that you’ve proved causation, your method of induction is still flawed. You are still assuming that two things happening simultaneously is a reason to believe they have a necessary connection, which Hume told us is never justified.

They also, ironically, suffer from the same problem as Popper, in that this is a practically difficult way to do science. Knowing the probabilities of everything is impossible (like Einstein knowing the probability that gravity would bend light), and forcing yourself to assign probabilities when you don’t know them is a recipe for disaster. Bacon warned us of humanity’s tendency to leap from evidence to conclusion, and here the Bayesians tell us that we need to start forming conclusions about the likelihood of things before science begins.

However, even considering these criticisms, it still seems like if there’s a place for Bayesian statistics in science. After all, the Nicaraguan brain parasite example was convincing, as was the example of the mother’s conviction. At the very least, it serves as a proper framework for statistics, and for evaluating the likelihood of events. If we have the sort of evidence that the Bayesian framework requires, it’s useful. If we don’t, we shouldn’t force it. And, as the mother’s conviction example shows, we can also use Bayesian statistics to show that other sorts of statistics are inappropriate as well.

However, one advantage of our era of computers is that probabilities aren’t as hard to come by as they used to be. In my field of Geosciences, Monte Carlo simulations were often used to come up with probabilities. So, for instance, a scientist would come up with an equation that he believed described some system, like the chance of rain. Then, using a computer, he’d put in all the possible values for the system into the equation, like temperatures ranging from 0 degrees to 100 degrees. He could then come up with a probability for the weather. If, for instance, the weather has to be 45 degrees for it to rain, and, if it’s 45 degrees, there’s a 50% chance of rain, then all he needs is for someone to tell him the probability of 45 degree weather for him to say the probability of rain. And he got these probabilities just by feeding a bunch of random numbers into his computer, and seeing how often and in what cases rain occurred.

If you’re computer savvy and interested in predicting things, check out Monte Carlo simulations, and try to figure out how to apply them to Bayesian statistics. The math is pretty easy to do through programs like Excel, but the tricky part is knowing what it means and what it doesn’t. If you’re not computer savvy, though, you can use Bayesian statistics to come up with your own ideas of what’s likely, and avoid being tricked by the statistics of others. After all, as Mark Twain said, “There are three types of lies: lies, damned lies, and statistics.”

The limits of argumentation

As I write this essay, it is still undecided whether Hillary Clinton or Donald Trump will be President [Editor’s note: turns out it was Trump]. Regardless of who wins, though, it is obvious that argumentation and debate will suffer. I won’t be so bold as to say that national discourse has never been worse, or that these debates were the worst of all time, because that’s a tough statement to make. I will say, however, it is really, really bad right now.

In order to illustrate this, I’ve taken and edited a segment from the third presidential debate. I’ll put it up for you, walk you through it step by step, and then comment on exactly why it’s such a poor example of how to argue.


WALLACE: “First of all, where do you want to see the court take the country? And secondly, what’s your view on how the Constitution should be interpreted? Do the founders’ words mean what they say or is it a living document to be applied flexibly according to changing circumstances?”

CLINTON: “..I feel strongly that the Supreme Court needs to stand on the side of the American people, not on the side of the powerful corporations and the wealthy. For me, that means that we need a Supreme Court that will stand up on behalf of women’s rights, on behalf of the rights of the LGBT community, that will stand up and say no to Citizens United, a decision that has undermined the election system in our country because of the way it permits dark, unaccountable money to come into our electoral system.

I have major disagreements with my opponent about these issues and others that will be before the Supreme Court. But I feel that at this point in our country’s history, it is important that we not reverse marriage equality, that we not reverse Roe v. Wade, that we stand up against Citizens United, we stand up for the rights of people in the workplace, that we stand up and basically say: The Supreme Court should represent all of us.”

WALLACE: “Mr. Trump, same question. Where do you want to see the court take the country? And how do you believe the Constitution should be interpreted?”

TRUMP: “We need a Supreme Court that in my opinion is going to uphold the Second Amendment, and all amendments, but the Second Amendment, which is under absolute siege. I believe if my opponent should win this race, which I truly don’t think will happen, we will have a Second Amendment which will be a very, very small replica of what it is right now. But I feel that it’s absolutely important that we uphold, because of the fact that it is under such trauma.

I don’t think we should have justices appointed that decide what they want to hear. It’s all about the Constitution of — of — and so important, the Constitution the way it was meant to be. And those are the people that I will appoint.”

Let’s take this step-by-step. Wallace, the moderator, asks a question pertinent to this election. Given that the next president will have a lot of say over the Supreme Court, and given that Supreme Court Justices are appointed for life, it’s important to know by what criteria the candidates would choose a Justice.

Wallace limits them to two options, which isn’t necessarily the best idea. First, he asks them “what direction they want the court to take the country”. In doing so, he assumes that they are choosing Justices based on what they think the Justices will do, rather than who the Justices are. In other words, he assumes this is a results-based judgment, rather than a value-based judgment. This bothers me especially because judges are supposed to be impartial, and so, presumably, a good judge’s opinions should only be dictated by the facts. If a judge’s opinions can be predicted by the candidate, then I’m not sure they are a good judge.

Second, he asks if the candidates believe in a “living” or “originalist” interpretation. The simplest way of putting this is if the judge should interpret the Constitution based on the founders’ intentions, or if there are other factors which should go into it. This is a more complicated question than it sounds, because it’s not simple to determine the founders’ intentions (the infamously confusingly-worded Second Amendment is a perfect example), and it’s not clear what constitutes changing times. For example, neither Democrats nor Republicans existed in 1789, but the idea of political parties would not be foreign to the founders. Have times changed or not?

Also, again, there is no room in Wallace’s question to defer to the Justices. The Supreme Court Justices are supposed to be the utmost authority on the Constitution, which means that they should be the ones deciding on the interpretation of the Constitution. Wallace is representing “living” vs “originalist” as an opinion question, which it shouldn’t be.

But that’s enough with Wallace’s question. Overall, it’s not terrible, and a candidate could presumably raise the points I talked about, especially the idea of judging Justices based on their virtues, rather than their expected rulings, and then deferring to their experience.

The candidates do not. In fact, they do nothing of the sort. Both use this question, as they use all questions, as a chance to raise talking points.

Clinton uses the first part of the question as a chance to list the opinions she has. These are not controversial opinions: people over corporations, women’s rights (unspecific, of course), LGBT rights (unspecific), and no to Citizens United (which, at last, is a controversial opinion, although not in her base).

The second part of the question she, again, uses as a chance to list the opinions she has: marriage equality, abortion, workplace equality. These are more controversial opinions. Then she concludes by saying, “The Supreme Court should represent all of us,” which is not remotely what the Supreme Court is for. In fact, the judicial branch is not representative, unlike the executive or legislative branch. This was a very deliberate choice on the part of the Founding Fathers.

This isn’t new to anyone, though. Everyone knows that nobody answers the question in the debate, and that the moderators’ questions are mostly for the moderators to try their best to impress America with how intelligent they are. What is more interesting is what precisely the candidates are attempting to do, if they are not trying to answer the questions. They are dog-whistling.

An actual dog-whistle is a whistle that emits at a frequency only dogs can hear. If you blow one, a human won’t hear anything, but a dog will hear a painful, high-pitched sound. The dog instantly pays attention. These answers are designed to do the same thing. Clinton says certain words which are designed to make her followers pay attention, and make them believe that she believes in their paradigm.

A paradigm is the way in which you see the world, or at least a certain segment of the world. For instance, in high school physics, you learn a Newtonian paradigm, in which every combination of objects and movement is put into a “forces diagram”. Aristotle, as you might know from a previous essay of mine, may be considered the inventor of the scientific paradigm. At this point in our democracy, with the proliferation of news sites and self-contained bubbles, political affiliations also have their own paradigms. When a liberal or conservative hears certain news, they can fit this news into what they already “know” about the world. These paradigms are mental shortcuts, in other words.

Women’s rights, marriage equality, Citizen’s United, Roe vs. Wade. Clinton uses these terms and quickly makes her opinion known on them. She’s not trying to convince anyone. She is merely reminding her supporters of whatever groundwork has already been laid in their minds in building these paradigms. In the case of Citizen’s United, she is attempting to bring followers of another paradigm, the corruption of money in politics paradigm (represented best by her erstwhile opponent, Bernie Sanders), into her fold.

There’s no reason why all of these issues have to be tied together. In fact, it’s easy to imagine someone who supports only one of these, or two of these, or three of these. But they are tied together in the current Democratic party, and so Clinton makes an appeal to her followers to also tie them together in their mind. If they do, they will have a united opinion on social issues, and be a potent force in politics.

Let’s turn to Trump. Trump is sloppier than Clinton in these sorts of debates, and so his dog-whistling isn’t nearly so sophisticated. He really only gets two across: the Second Amendment, and an originalist interpretation of the Constitution.

Again, Trump has no need to explain in detail what he thinks about the Second Amendment, or convince anyone of his point of view. The current conservative view of the Second Amendment is that it supports unlimited gun ownership by any citizen, and that fits in really nicely with the generally distrustful viewpoint of the Republican Party. Gun ownership is the ultimate in independence, in freedom from unnecessary intervention, and in distrust of the unknown. The idea that the Second Amendment is “under siege” fits into the distrustful narrative as well, because it promotes a mentality of people trying to take what’s rightfully yours.

The originalist interpretation of the Constitution works as a dog-whistle only because of a historical accident. Trump’s campaign is built on nostalgia: “Make America Great Again”. It’s a very certain sort of nostalgia, for an era in America unquestionably dominated the world and white men unquestionably dominated America. This era never really existed in the way Trump imagines, but that doesn’t matter. Clinton’s campaign is not based on moving forward into the future, but Trump’s campaign is based on the past, which is normally what conservatism is based on (it’s in the name, after all). Originalism serves as nostalgia.

So Trump, with his combined Second Amendment and originalist dog-whistles, appeals to his followers to remind them of the conservative paradigm, and convince them that he is in line with it. Again, the two do not have to be together. In fact, they probably shouldn’t be, given that the conservative interpretation of the Second Amendment is not in line with the founders’ intentions. But they are.

I’m disappointed with these debates because I believe they are emblematic of the state of discourse in our country as a whole. It is very, very difficult to argue if two people share different paradigms. It’d be like arguing art with a colorblind person. You two are experiencing things in fundamentally ways, and you are understanding things in fundamentally different ways. Your only hope would be to try to understand things from each other’s perspectives, which is hard enough if you make a legitimate effort. If you don’t, like Clinton and Trump, then it will not happen.

This situation is made worse by the fact that our paradigms don’t just color how we understand the world around us. They also color what facts we accept about the world around us. Before the Internet, the Wall Street Journal and the New York Times had different paradigms, represented in their opinion sections and in the way their reporters would interpret news. But they accepted the same facts and news almost all of the time. Now, websites like BeitBrart (I refuse to help their name-recognition) deliberately promote different and untrue facts, which are accepted by conservatives because they fit in comfortably with their paradigm. The intersection between paradigms has to be facts. If it’s not, it’s like arguing art with a colorblind person, except you’re arguing about two different pieces. You won’t even know why you’re disagreeing.

Clinton and Trump here are attempting to represent and embody two different paradigms. The most generous interpretation is that they therefore represent and embody two different ways of looking at the world, and two different systems of opinion. The more realistic interpretation is that they also represent and embody two different sets of facts, as shown by how many times they call each other liars during the debate.

Given that that’s the case, the only way left for them to argue, and for the audience to judge their arguments, is arguing about each other personally. There, at least, there is common ground. We can agree about who seems better prepared for the debate, and we can also agree on who seems a better person, if we can manage to agree on facts about the candidates. The country is actually more unified on individual ethical standards than ever before, strangely enough. Epithets like racist or sexist would have been embraced proudly by previous Presidential candidates, like George Wallace or Andrew Jackson. Now they are denied vehemently.

Does this agreement show hope for the country as a whole? Maybe. In order for debates and the national discourse to progress, we need to agree on a common set of facts based on evidence and we need to acknowledge our separate paradigms. If conservatives or liberals insist on only one way to interpret the world, they will be continually baffled by their neighbor’s refusal to do so. Furthermore, we need to critically examine our own beliefs, and make sure we know what they are based on. We don’t need to be Descartes, and dive deep down into the foundations of our knowledge, but if supporting the Second Amendment is important to us, for instance, we need to make sure we know why we think that. Then, we can apply our standards of knowledge taught to us by philosophers, and examine our assumptions, deductions, and inferences.

I’m not sure what will happen until then. I fear another blow-up, like the riots of 1968, or even an attempted revolution. Trump’s paradigm is especially frightening to me, as his facts and understanding could easily lead to atrocities. For instance, if you take him literally when he says that most Mexicans are murderers and rapists, you would want to deport Mexicans, forcibly if necessary. You would want another Trail of Tears.

By the way, if you’re reading this, and agreeing with me, one thing you might consider is Socratic questioning, like this example here. Ask your friends why they believe what they believe, or even strangers. Ask them about their definitions, and the basis for their facts and opinions. You don’t have to disagree. Just make them think. People today are friendlier than they were in Socrates’s time, and, unless your friends are particularly vicious, they will probably not put you on trial.

Karl Popper and falsificationism

C.D. Broad, a 20th century British philosopher, once said, “Induction is the glory of science and the scandal of philosophy.” Induction, as we recall, is essentially reasoning from patterns. Given that something’s happened in the past, we think it will happen in the future. So, for instance, you believe that the sun will rise tomorrow because it’s risen in the past. If you lived on some planet where that wasn’t true, like a non-rotating planet, you would not have that same belief. Deduction, by contrast, is reasoning by definitions. Aristotle’s syllogisms are famous examples of these, but there are more complex examples, including much of math. For instance, no matter what planet you lived on, you’d still believe in the Pythagorean theorem.

Induction is the glory of science because it’s what most of science is based on. Our earliest science was a primitive sort of induction, in which our reasoning from patterns was careless. That’s how the Aztecs ended up believing that they needed to sacrifice people to keep the universe from collapsing, because they saw the pattern that they sacrificed people and the universe didn’t collapse, so assumed a necessary connection. Later on, philosophers like Francis Bacon helped us get more careful with reasoning from patterns, warning us about “idols” which can prevent us from properly reasoning. Science then split from philosophy and developed its own standards of proof, including double-blind studies and statistical analysis.

However, no matter how far science advances, it always relies on induction. Every prediction we make about the world relies on the idea that there are necessary connections. The law of gravity, for instance, supposes that every object obeys the law of gravity, and we assume that because that’s been true so far. However, we don’t have any proof. It’s entirely possible that beachballs, when painted neon yellow, do not obey the law of gravity. We assume they do because that would be a strange and inexplicable exception to our pattern of gravity. However, if tomorrow we find out that neon yellow beachballs do not obey the law of gravity, it’s the law that will have to change, not the beachballs.

Induction is the scandal of philosophy because, in the end, it’s impossible to prove outside of induction. Let’s go back through Hume’s critique so we have a good idea of why this is. Say you took a sip of tea, and burned your tongue. You say, “the tea burned my tongue”, or “the hot tea caused the burn on my tongue.”

Hume disagrees with you. He tells you that you only believe that because the two happened almost simultaneously, but there’s no necessary connection. You respond that every time you’ve taken a sip of hot tea, you’ve burned your tongue, therefore there is a necessary connection. He disagrees, and says that he doesn’t believe you can assume that.

You respond that everything works that way. You’ve formed all of your beliefs by patterns like hot tea burning your tongue, and this is why you’ve been able to avoid repeating other stupid mistakes, like that time you stuck a fork in the toaster. Hume responds like this: “So, you’re reasoning from the pattern that you’ve successfully formed beliefs in the past by reasoning from patterns? Sounds pretty circular to me.”

That’s the essential critique. Now, we can dress it up in fancy ways, and add words like probably, or statistically significant, and it seems that those methods work better than just guessing from patterns. We can even add theories and equations, which try to explain why certain patterns hold up better than others in predicting the future. But, in the end, the only proof of those methods being better is that they’ve been better in the past. In other words, we can only prove based on the pattern, or prove via induction.

Hume’s response to his own critique was that this is why philosophy should only be done for fun. Scientists, on the other hand, have mostly ignored Hume’s critique, and it’s very rarely taught in science classes, even when students learn the scientific method. Philosophers, of course, love it, and have taught it to their own students for centuries. But nobody ever really tried to solve it, or thought of it as something to be solved.

Until a philosopher came along: an Austrian Jew who escaped the Holocaust to become a philosophy professor in England. No, not Ludwig Wittgenstein, although that description fits him perfectly. It was Karl Popper, who, ironically, considered Wittgenstein his mortal enemy. Popper took the critique of induction very seriously, and was bothered that scientists had ignored it. After all, it seemed to attack the foundations of what they believed.

Now Popper was a big believer in science. During his lifetime, Freud and Einstein both rose to prominence. Both were taken equally seriously as innovators of a new sort of science; Freud as the innovator of psychoanalysis, and Einstein as an innovator in physics. But Popper thought this was unjustified. In Popper’s eyes, Freud’s theories were much less trustworthy than Einstein’s. The question was how he could prove that.

What Popper came up with was the theory of falsificationism. In short, a good scientific theory should result in unlikely predictions. The test of a scientific theory is if the predictions are confirmed or not. Einstein’s theories, for instance, were confirmed by the observations of Sir Arthur Eddington. This was especially nice because Einstein’s predictions weren’t just unlikely, but actually went against the predictions of the dominant theory in physics at the time, Newton’s physics.

Freud’s theories, by contrast, were not falsifiable. As Popper saw it, any action could be explained by Freud’s theories. If a man killed someone, Freud could explain it. If that same man then saved someone’s life, Freud could explain that. Therefore, no matter what a man did, Freud was never wrong, which meant his theories were not scientific. It’s important to note that Popper didn’t think they were necessarily useless, but he did think that meant they weren’t science.

Now, there are good parts and bad parts to Popper’s theory of science, here. The good part is that we avoid the problems of the Aztecs. If we focus on proving a theory false, instead of proving it right, we’d never end up in a situation where we think that human sacrifice is necessary to keep the universe from collapsing. Presumably, the first time we don’t sacrifice a human and the universe doesn’t collapse, we realize the fault in our theory (assuming we’re bold enough to do so).

We can also avoid a lot of the idols of Bacon. Many of them revolve around the fact that we often want certain hypotheses to be true, and we’re willing to twist the evidence to “prove” ourselves right. If we’re focusing on proving ourselves wrong, instead of proving ourselves right, we will hopefully avoid twisting the evidence or the experiment.

However, Popper’s original purpose was to avoid Hume’s critique of induction, and to that extent, he fails. Hume’s critique relies on never knowing if a connection is necessary, or just coincidence. However, if we try to prove a theory wrong, like Popper wants us to, we can never know if we really proved it wrong, or if an unknown factor made it look like it was wrong. For instance, we might drop a balloon, watch it float up, and think we disproved gravity, never realizing that we just didn’t know about buoyancy.

There’s also the problem of coming up with theories in the first place. There’s an infinite number of possible ways to explain anything, and Popper can cross some of them out, but they can’t tell us which ones to choose to test, given a limited time frame. For those we need something else, and it’s probably going to be a theory of induction.

Finally, not all of science can be tested so easily as Popper wants. Physics was an easy example for Popper to use, but it took a couple years for Einstein to be proven right by Arthur Eddington. If we had waited to called Darwin’s evolution science, then there’s no telling how long we would have had to wait. We needed to employ different methods of proof, involving the explanatory power of the theory and the number of assumptions needed. Popper never really considered this.

Popper provides us with a new and useful way of thinking about science, and makes a valiant and somewhat successful attempt to reunify science and philosophy. He doesn’t come up with the grand theory of science that he thinks he does, but he provides us with an excellent example of how science should be in certain circumstances.

Other philosophers, most notably Thomas Kuhn, talk about how science should be in other circumstances. More importantly, they describe how science is actually done, which is something that Popper neglects completely. Popper, much like Aristotle, assumed that the world comported itself to his theories, and saw no need to falsify this assumption.

What is the next leap forward in the philosophy of science? Well, to my mind, it has to something about integrating enormous amounts of data. I was trained in geosciences, and the current problem in that field is that there’s too much data and a million different ways to read it. It’s a problem that someone like Bacon would have loved to have, but it’s still a problem, and the field regularly gets roiled by competing data sets and drastically different conclusions. Who knows, perhaps the next philosopher of science will be you!

Wittgenstein and the limits of language

There’s a point in everyone’s life when they suddenly realize that other people have a mind of their own. You always knew it, but you suddenly think to yourself, “That person has his own thoughts and desires.” Then, of course, the next obvious question: “What does he think of me?” This next question is the cause of a lot of angst during teenage years.

If you’re philosophically inclined, at some point you also realize that the link between your language, your thoughts, and their thoughts isn’t as obvious as it seemed to be. When you described something as “blue”, you had always assumed they knew what you meant. But it’s possible, you realize, that they see a different “blue” than you do. Perhaps when you see blue, they see green, and you two have been using the same word to describe completely different experiences without ever knowing.

This can even be a reality for some people. Every colorblind person can tell you a story about the first time they realized they were colorblind. Perhaps it was when they realized that blue and purple were actually separate colors, or when they realized the reason they were stuck on that videogame puzzle was that it relied on telling the difference between green and red. Immediately after they realize this, though, colorblind people have the same thought as the rest of us: how could I have never figured this out before?

It all comes down to the same thing. Language has limits, and colorblind people run smack up against one of them. In fact, once you start thinking about it, it’s astonishing how effective language is. If I combine these letters in the right way, I can cause an image to form in your head: hweti eelptanh. Ready? “White elephant”. Now, I can reasonably assume that you have formed an image in your head that’s pretty similar to the image that I wanted you to form, assuming that nothing crazy is going on, like our color example from before. It’s powerful stuff.

More importantly for our purposes, language is also what I’ve been using to walk you through my arguments, and it’s what Bacon, Descartes, and the rest of our merry bunch used to walk me through their arguments. Or, more precisely for most of them, to walk translators through their arguments, who then transmitted the arguments to me in a way I could understand. This essay series and, indeed, education as a whole, relies on the assumption that language has this capability. It relies on the idea that the understanding someone has in their own mind can be replicated in someone else’s mind through language.

This is a more extreme proposition than we give it credit for. Hume and Kant have both shown us how our understanding can be mistaken, or based on unconscious intuitions. So, before we even get started with transmitting our understanding, our understanding is not guaranteed. Then, when we transmit it, it’s obvious that the understanding can get twisted. Bacon gave us sneaky ways that language can skew our understanding through his “Idols of the Marketplace”, but we don’t need to be as subtle as that. Everyone has misunderstood something that someone else has said.

Presumably, though, if the other person tries really hard, and our language is good enough, we can accurately transmit this information. That is, unless there’s anything else in our language like our color example, which is something that can’t be transmitted. It sure would be nice if some philosopher had discussed the limits to language, and what can and can’t be transmitted…

Luckily, there is! Yet another German (well, Austrian), a 20th century philosopher named Ludwig Wittgenstein. Now Wittgenstein is one of the most interesting, strange philosophers in the history of Western philosophy. Born to one of the wealthiest families in Europe, throughout his life he was a schoolteacher, gardener, philosophy professor, hospital orderly, and laboratory assistant…in order! By which I mean after he became a philosophy professor at Oxford, he quit his job to become a hospital orderly! If this sounds interesting, you can check out his entire biography here.

Wittgenstein is also complicated because he published two major works, one during his lifetime, and one after, that completely contradict each other. So, whenever you discuss Wittgenstein’s beliefs, you always have to clarify whether you mean the earlier Wittgenstein or the later. For us, we’ll be discussing the earlier Wittgenstein, as the later Wittgenstein’s essential belief is that no philosophical statements can be made about language whatsoever, which makes it difficult to write an essay about language philosophy.

The earlier Wittgenstein, on the other hand, is a bit difficult to understand. I’ll be taking a certain interpretation of him, but you should probably know that there are other interpretations. The ambiguity in Wittgenstein’s writing wasn’t helped by the fact that, whenever anyone would ask him about his philosophy, he’d turn to the wall and start humming classical music. As I said, he was a strange guy.

Anyhow, back to the issue at hand. According to Wittgenstein, when we speak to someone else, we’re really trying to convey a certain set of facts, or statements which are definitively true or false, but don’t have to be true or false (so a definition, like “a bachelor is an unmarried man”, doesn’t count as a fact). Each fact is made up of smaller facts, which Wittgenstein called “atomic facts”. So, for instance, if I say “Socrates was a man with white hair,” I’m trying to convey a set of facts about hair, whiteness, men, and Socrates, and combining them together. Wittgenstein’s not concerned about whether or not the facts are real or represent the world correctly. The important part is that I’m trying to convey a true or false idea for you.

But, of course, that doesn’t describe all the ways in which we use language. As later Wittgenstein puts it, there are many different language games. We joke around, sing, talk business, and flirt with our fellow humans. Earlier Wittgenstein tells us, however, that if these statements aren’t attempting to be true or false, then they can’t be relied on to accurately transmit ideas. Or, as he puts it in his own, inimitable way, “Whereof we cannot speak, thereof we must be silent.”

This is a bold claim. It’s an interesting one, but there’s some wiggle room here, which is the reason that later Wittgenstein gets frustrated and blows his whole theory up. It’s certainly true that language often accurately transmits ideas when it’s attempting to be factual, and not otherwise. For instance, if I say “the dog is taller than I am”, then you can imagine the dog’s height, assuming you know mine. But if I say, “the dog is so pretty”, it’s much more difficult for you to imagine what I’m talking about.

However, our “color” example from before would seem to contradict Wittgenstein’s idea. I could say “the apple is red”, and try to convey a fact about the world to you, but fail because you are red-green colorblind, and so you imagine it differently. Or, I could say “Spiderman is obese in the medical sense,” and not convey a true or false statement to you, but you can still imagine fat Spiderman, even if Spiderman, a fictional character, can’t look like anything.

So, let’s modify it a bit. My modified claim is this: language works to convey meaning only so far as we assume that the other person has a similar enough experience that they can understand what we mean. If we’re not assuming that, or if we assume wrongly, language can’t cause ideas to pop up in someone else’s head. This is more vague and weak than Wittgenstein’s theory, but it’s easier for me to defend my sandcastle than it was for him to defend his stone castle.

Socrates, when he persuaded us with his examples, assumed we had a similar enough experience to be persuaded by them. This wasn’t true of Aristotle’s deductions, but that’s because he didn’t need it to be. He didn’t need his deductions to convey meaning, only to make logical sense. Kant, on the other hand, when he made his arguments about space and time, needed us to have the experiences that he had. He needed us to have an impression of our mind similar to his impression of his mind, so that his discoveries about his mind would seem convincing.

This is what Wittgenstein was missing. “Spiderman is obese” makes sense to people who know who Spiderman is, and have experience with the character in a comic or movie. “This apple is red” would have made sense, but we weren’t justified in thinking that everyone had the same experience of apples that we’ve had.

So, for you, when you’re making an argument, or attempting to persuade someone of your idea, think about whether you can assume that their experience is similar enough to your own. If it is, then your language will make ideas pop up in their head. If it’s not, it won’t, or it’ll make the wrong ideas pop up in their head, ideas you don’t want them to have. Even if you are just making a logical argument, like Aristotle, you need to make sure that your audience has the same experience and idea of logic that you do.

But how do you think about other’s experiences? Well, with the help of all the philosophers we’ve talked about who discuss how we experience the world, and the way that we process it and try to make sense of it. Descartes, Hume, Bacon, Kant, all of them can help you.

And above all, when you’re arguing or discussing with someone, make sure that you’re arguing about the ideas, not about the language. Then you’re no longer talking on the same grounds. If you need to, double check that the idea in their head is the same as the idea in yours. Or, if you’re trying to understand someone else’s views, first make sure that you understand their ideas, and the idea that they’re trying to convey to you. As Socrates shows us in his Dialogues, they might not even know themselves. Then a disagreement is really pointless.

But enough about language. Modern philosophy is filled with discussions of language and truth and facts, but that can grow tiring. What about the real world, and, you know, science? We’ve left poor Francis Bacon drifting in the wind, his induction savaged by Hume then forgotten by time. It’s time to leave aside these insubstantial arguments and get back to the meat of the matter. What do we know about this world, how do we know it, and how can we make sure we know it? Well, the first question is left to science, the second already addressed, but the third seems meaningless in the light of Hume’s critique of induction. As you can probably guess, not everyone agrees. Karl Popper is on the case, determined to save science from Hume’s savage attack. Will he succeed? Tune in next time, and find out!


Immanuel Kant and intuition

Here’s a thought experiment for you. Look across the room, wherever you are. Think about how long it’d take you to walk to the opposite wall. Maybe 10 seconds? That’s probably roughly how you’d measure the distance, by thinking about how long it’d take you to walk. Now imagine if you got up, started walking, and reached the wall in 1 second. You didn’t walk faster, or change your stride, you took one step and found yourself at the opposite wall.

It’d be reality-breaking. It’s quite hard to imagine, too, if you’re trying to think what it’d really be like. It’s possible to imagine overestimating the time it would take to drive to Los Angeles, but overestimating the time it’d take to walk to the opposite wall is a different story altogether. Either your perception of distance, or of time, would have to have been radically off. Distance and time are very fundamental ways to experience the world, and to be wrong about them is to be wrong about the foundations of our connection to reality.

In fact, according to 18th century German philosopher Immanuel Kant, distance and time are the fundamental ways in which we experience the world. Kant said that we receive perceptions of the world outside through our senses, and it comes in as a rush of disorganized colors, lights, and sounds. Our brain, before we realize it, organizes these perceptions in terms of distance and time. So, when we look out at the world, we think there’s such a thing as distance and time, and we don’t realize it’s just the way our brain has organized our perceptions.

This is a pretty radical idea. Before now, we’ve looked at a couple different ways of reasoning, and justifications for each. We were told how to deliberately perform deduction by Aristotle, how to deliberately perform induction by Bacon, how to be a skeptic of deduction by Descartes, and how to be a total skeptic of induction by Hume. But there’s always been the assumption that we can examine our thinking. Kant tells us, however, that that’s not always the case. There is “preprocessing” of impressions that is done before we know we’ve received any impression which can’t be undone or examined. It’s like looking at tomato sauce and trying to imagine the tomatoes that went into it. It’s possible to make inferences, but impossible to really know about the processing or the original ingredients from the tomato sauce.

But Kant’s not done. First of all, he crosses out many of the philosophical problems that plagued philosophers before him. Free will, the origin of the universe, and the end of the universe only puzzle us because we’re trying to apply our intuitive understanding of distance and time to problems they don’t apply to. So Kant thinks all the time we’ve spent arguing about them has really just been time wasted. Our intuition about space and time can’t take us to the truth, any more than Aristotle’s logic could take us to the truth about gravity. The difference is that Kant tells us that our intuitions can’t even take us to a satisfying logical conclusion, which is why we can’t stop arguing about these philosophical problems.

Next, Kant takes on Hume. As we remember, Hume laid out a devastating critique of induction. Hume proved we can never assume that something causes something else, or even assume that there’s any reason to assume. To use the words of the great philosopher Bertrand Russell, Hume tells us that when a lunatic claims he is a fried egg, he must be disagreed with solely on the grounds that he is in a minority.

Kant agrees with Hume, but then asks why, if causation is so illogical, it’s still so tempting. Kant answers his own question, as philosophers tend to do. He tells us that causation is just another intuition, like space and time. The only difference is that causation is consciously applied, after we receive impressions, while space and time are unconsciously applied, while we’re receiving impressions.

Hume could examine and criticize causation because he was aware he was applying this intuition. He could act like a third person, seeing himself come to the logical conclusion that hot tea burns his tongue, or that hot tea causes burns on his tongue. He could also imagine the two events being coincidental, which is how he starts his critique of causation. But even the great David Hume can’t examine himself applying distance and time to his perceptions.

A philosophical friend of mine once criticized Kant’s philosophy by saying that Kant leaves no room for argument or discussion. At times like this, it seems somewhat true, because we can’t discuss something that Kant tells us we can’t really understand. Kant claims we’re liable to get stuck in a fruitless discussion, like all the philosophers before Kant bickering about free will.

So instead, let’s look at some of the evolutions of Kant’s ideas. First, the idea of intuitions proves to be a powerful one. While Bacon predicted the ways in which our intuitive thinking could lead us astray, Kant opens up this new idea of an intuitive toolbox, and of tools we apply unconsciously and consciously.

Freud, for instance, a countryman of Kant’s (sort of, because Kant’s country no longer exists by Freud’s time) 200 years later creates theories of unconscious and subconscious motivations. In short, ways in which we feel about the world are influenced or controlled by motivations we don’t know exist. While Freud’s exact theories don’t prove to be supported by science, his general idea is, and we now know that conditioning can lead to this sort of preprocessing. Someone who has been traumatized, for instance, will begin to react to a perceived threat that someone who hasn’t been will not. More interestingly, they will react before they are aware they have reacted. Their intuition preprocesses the perception and applies a fight-or-flight feeling to it, leaving them baffled as to why they are reacting so poorly to something so innocuous.

On the other end of the spectrum, Einstein, another sort-of countryman of Kant’s, puts a scientific basis behind Kant’s ideas that distance and time are fundamental intuitions. Einstein proves that time is dependent on speed, and that time literally goes slower when we go faster. Space and distance can be warped as well with gravity. Unfortunately, you need pretty complicated math to really understand this. As Kant warned us, our intuition isn’t enough.

In these past essays, we’ve covered a lot about how to know things, how to prove things, and the limits of knowledge and proofs. One thing we haven’t covered is, in retrospect, pretty surprising. We’ve never examined the tool that we’re using to cover these things. That is, we haven’t looked at language itself, and we’ve just relied on it to serve our needs, as we normally assume we can. But, as the saying goes, an assumption is a dangerous thing]. For the next time, let’s look at language itself, with Ludwig Wittgenstein as our trusty guide.