Find out what you don’t know, and protect it from half-baked explanations

I’ve thought about these ideas for a long time, and they’ve only been strengthened by reading biographies and articles like Ambidexterity and Cognitive Closure. In this article, I’ll try to untangle this mess of ideas, and also try and provide a refutation of The Case Against Education by Bryan Caplan.

Why are the greats that great?

What is the major difference between scientific revolutionaries like say Newton, Einstein and da Vinci, and researchers that populate various universities around the world, trying to write publishable papers (and also aspiring researchers like me who are at the bottom of the food chain)? Well an easy answer would be “Einstein probably had 25,000 more IQ points than you, and that’s why he did all of those wonderful things that you can’t”. Fine. We can happily accept this argument of “he’s just much, much smarter” and come to peace with our relative mediocrity. However, this goes against my general experience. I have met a lot of very, very high IQ people. People who won multiple gold medals at the International Math Olympiad with perfect scores, people who aced multiple Olympiads and also topped their cohort at Cambridge math, etc. You get the drift. Why aren’t these people discovering new scientific theories and revolutionizing human understanding? Can there really be no Newton amongst them? Was Newton that much smarter than all of them?

I first came across the following idea in Malcolm Gladwell’s Outliers: high IQ people are really good at finding answers, when they know that there’s an answer to be found. But they’re not markably better at asking questions or finding gaps in their understanding. Let’s take an example. Imagine that you were born in a time before Newton’s laws were discovered. You’re asked the following question: “Imagine that you have a smooth surface with no friction. All real-world surfaces have some friction, hence you have to take surfaces with lower and lower friction, and take some sort of limit. If an object is slid on it, will it ever stop unless an external force comes and stops it?” You can re-discover Newton’s First Law in one afternoon without any prior knowledge of it, and feel very smug. Now imagine that you’re instead asked “Clearly all objects that move come to a stop. If you kick a ball on the field, it will stop after traveling some length. What is stopping the object?” The most intuitive answer, which corresponds both to experience and ancient Greek beliefs, is that every object has a propensity to come to its “natural state”, which is a state of rest. Hence, it is the nature of objects itself that is making them come to a stop. In some sense, discovering Newton’s law was not the hard part. It was knowing that there was a law to be discovered at all that made discovering it so difficult. You had to suspend belief in your own experience, and consider a hypothetical smooth surface with no friction. In other words, you had to be led in the right direction with the right questions. Someone had to ask “What if my assumption about objects naturally coming to a halt is really an assumption about the friction exerted by surfaces, and that I can weaken this assumption?”

The same could be said about Einstein. Discovering Relativity was not as difficult as knowing that there was something there was something to be discovered. Of course, Einstein was lucky in the sense that Morley-Michelsen’s experiment had only recently shown that there was “something funny going on with light”, and he just needed to assess the implications of that in order to come up with Special Relativity. Leonardo da Vinci, of course, was famously curious, and it was having these questions in the first place that caused him to discover so many scientific and artistic facts (including, apparently, Newton’s laws before Newton). This brings us to the fact that although a high IQ may be useful in finding answers to questions, it doesn’t help one discover new and important questions to ask. In other words, although it helps us fill in gaps in our knowledge, it doesn’t help us discover those gaps. And discovering those gaps is most of the battle. But what helps in discovering those gaps?

Curiosity, ambidexterity or schizophrenia?

An easy answer is curiosity. You have to be curious about the world around you in order to ask the important questions. However, that is not the complete picture. For instance, I am sometimes curious and ask myself how exactly did trees outside of my window evolve to be so tall? An answer that instantly comes to mind is that trees need to catch sunlight, and taller trees caught more sunlight. Hence, as trees that caught more sunlight probably had a greater chance of survival and procreation, trees have evolved to be tall. I am satisfied with this explanation, and move on. However, if I force myself to think more deeply, I notice that the trees are conical in shape. Hence, although growing taller did make them get more sunlight, they didn’t necessarily prevent shorter trees from also getting a lot of sunlight. So why did trees evolve to become so tall? Clearly a lot of resources must have been expended to become taller. A possible answer is that tall conical trees grew on mountains that were covered in shadows for large parts of the day, and only tall trees could catch sunlight for most of the day. This again is too simplistic an explanation, and there are still more questions to ask. What if tall conical trees originated in sunny mountain valleys, but failed to originate in other shadowed plains? Clearly my hypothesis will be wrong, and I will have to look at alternate explanations.

In general, if I ever ask questions at all, I stop after the first answer. My mind thinks of an explanation, and accepts it without trying to poke holes into it. However, when I write things down, I can reflect upon my explanation much more easily and maybe see some holes. However, this process ends within a couple of iterations, and I move on even though I might not be completely satisfied with my answer. Who are these freaks who keep on questioning their assumptions and hypotheses until they arrive upon earth-shattering facts, and why can’t I be like them?

Scott Alexander, in his article on predictive processing, argues that people like me, who are satisfied with half-baked approximations of the facts, have very high priors. We assume certain things about the world, and stick to them, shielding them from attack until we absolutely have to discard them. We don’t deal well with uncertainty, and prefer inaccurate but convincing untruths over difficult-to-find but accurate truths. Of course there may be an energy-theoretic argument for this: poking holes in your own arguments takes energy, and being satisfied with your own half-truths helps in conserving useful mental energy. I can think of an evolutionary argument for why most humans have this feature. On the other hand, people with schizophrenia or ambidexterity have very low priors, which means that they don’t shield their assumptions about the world from attack (as much), and are open to external inputs changing their priors. They are much more tolerant of uncertainty, and won’t accept anything less than the absolute truth (that which can explain all known observations). In other words, people like me never try to uncover gaps in our knowledge, and rush to fill them with half-truths when they are inevitably exposed. Revolutionary scientists and schizophrenics, on the other hand, uncover the gaps in their understanding with ease, and then try to hold out on filling these gaps until they find an explanation that is completely convincing.

Is the only way to scientific greatness self-induced schizophrenia or ambidexterity? I really hope not. Perhaps if we can try really hard to question our beliefs and hypotheses, to actively seek data that contradicts our half-baked explanations, there is still some hope. Of course writing things out would help. Knowing that certain gaps exist in our knowledge is the first, and most important step. We should spend a considerable amount of effort in exposing these gaps, and not being satisfied with untruths.

The case against The Case Against Education

This leads me to my criticism of Caplan’s “The Case Against Education“. Caplan argues that because students soon forget everything that they learn in school, and that skills in one field are rarely transferable to other fields (I think he also makes the implicit argument that IQ is the most important determinant of professional success), we should stop investing this much money into schools and colleges, and should instead focus on developing marketable skills within children. This goes against my experience of going to school and college.

I have taken a lot of courses whose contents I have mostly forgotten. These include courses that are completely irrelevant to my field of interest, like history, geography and environmental science, and also courses that are aligned with my field of interest, like mathematical physics and geometry. Although I’ve forgotten most of the material from these courses, they did succeed in creating place-holders or gaps of knowledge in my memory. For instance, although I might forget a theorem that I can use in a certain situation, I do know that there does exist such a theorem. I can now look it up and find out more details. Similarly, although I might have forgotten most of the contents of the Constitution, I do know that it does contain something about secularism and freedom of speech. I can now look up credible sources to find out more. Hence, although my formal education has failed in getting me to remember all that I’ve been taught, it has succeeded in something almost as important: creating place-holders for knowledge in my brain, that I can easily fill by a simple internet search.

Would I have been able to learn all of this material (and consequently create place-holders for knowledge) if I had been self-taught, or perhaps been taught at home by my parents. Not if I was extraordinarily brilliant or curious, or perhaps have had parents that were ready to devote considerable amounts of time and effort to educate me in a plethora of fields. What is more likely is that I would have had little or no training in most things. I do believe that there are certain aspects of schooling that are harmful to students, and I have suffered a great deal because of the nature of my schooling. However, there are certain beneficial aspects of it that have been overlooked in certain discourse.

Is it really as simple as we’re making it out to be?

Now let us see what are some failings of my basic argument: it is much easier to make conjectures in mathematics (Fermat’s Last Theorem, Twin Prime Conjecture, etc) than prove them. Hence, exposing “gaps in our understanding” is not half the battle in this case. The same could be said of finding out a unified theory of Physics: we know that a gap exists in our understanding of the universe. It is filling this gap that is proving to be difficult. Hence, my basic hypothesis would have to be re-phrased to also address these cases.

Thanks for reading!

Dostoevsky is a two-trick pony

Like most other people who enjoy self-abuse through reading thousand-page novels, I’ve had the experience of reading Dostoevsky and marveling at his ability to capture “reality”. I’ve read “Crime and Punishment” in the past, and am now reading “The Brothers Karamazov”. Needless to say, there have been many parts that have completely floored me. I used to think that Dostoevsky was perhaps a “god amongst men”, having the ability to capture emotions and human behaviour in a way that is far beyond our abilities. However, in light of my previous two articles on goal conflict and the dynamic model of human personality, I feel that Dostoevsky uses only two tricks again and again to create unbelievably real scenarios, and that we too may perhaps be able to use those tricks to enhance our writing.

The trick that Dostoevsky uses most often is the following: his characters behave in one particular way, and then behave in a completely opposite way the next moment. And for some reason, this only makes them more believable. For instance, the poor Captain in The Brothers Karamazov is elated when he is offered money by Alyosha. He dreams aloud about how he will use that money to pay for medical treatment for his family, and take his son on a long-promised vacation. However, it is at that moment that he chooses to throw the money on the ground, stamp on it in disgust, and let Alyosha know exactly what he think of his charity. What’s more surprising is that Alyosha later says that now that the Captain has rejected his money once, if he is again offered the same money the next day, he will happily accept it. And we know that Alyosha is correct.

This can be seen through the lens of the passive goal guidance system dealing with conflicting goals- his goal of providing for his family vs his goal of preserving/signaling his honor. When the Captain fantasizes in detail about how this money will solve all his problems, his passive goal guidance system mistakes imagination for reality, and assumes that all his financial problems are already solved. This causes disengagement with this goal, and engagement with the conflicting goal of saving his honor and not accepting charity from the brother of his enemy. Moreover, when he makes a big show of how his honor matters to him much more than any of his financial problems, his goal of proving to the world that his honor cannot be bought is also fulfilled, and now the conflicting goal of providing for his family again re-surfaces. Hence, Alyosha correctly predicts that if the Captain is offered that money again, he will readily take it.

This jumping between conflicting goals is something we also see in the most other characters in the book. For example, Grushenka is torn between taking revenge on the man who betrayed her, or running into his arms when he comes calling again. However, it is only after fantasizing in great detail, about how she will take her revenge by insulting him and turning down his offer to go with him, that she decides to run into his arms. This may be interpreted as her being torn between two conflicting goals- her goal of taking revenge for earlier wrongdoing, vs her goal of being with a man she still loves. After she imagines taking revenge on him in great detail, her passive goal guidance system assumes that this goal has been fulfilled, and disengages with it. This causes her other goal to surface- that of running into his arms, and this is exactly what she does.

Another trick that Dostoevsky uses is that his characters readily abandon the good and solid things in their life, and value only those things that carry an element of risk or uncertainty. For instance, Mitya has a beautiful, rich and virtuous fiance, Katerina Ivanovna, who is ready to forgive all of his infidelity and be with him. However, he abandons her and chooses to pursue Grushenka, who is an undependable and promiscuous escort to a rich landlord in town, and also has a relationship with his own father. This brings us back to the paper on the dynamic theory of personality, which told us that our desire for something/someone increases with the uncertainty involved in obtaining the object/person (it is maximum when our chances of obtaining that object/person are 50-50). Hence, Katerina Ivanovna, despite all her qualities, was too much of a sure thing for Mitya to desire. He was self-destructively pulled towards a woman who was much more ambivalent towards him, and with whom his future was much more uncertain.

To Dostoevsky’s credit, the concepts of goal conflict and the dynamic theory of personality are manifestly true for the human nature. Hence, it is through the use of these two concepts that he is able to create hyperreal characters. Tolstoy uses the concept of goal conflict as well. For instance, when Natasha is finally pursued by her childhood love in War and Peace, she doesn’t reciprocate, but falls for someone entirely different. However, I find Tolstoy’s treatment to be much more subtle than Dostoevsky’s. Although Tolstoy and Dostoevsky both occupy a position in world literature that has hardly been challenged in the last couple of centuries, I find Tolstoy to be much more of a literary genius than Dostoevsky. I can perhaps explain this more in a future post.

The futility of contrarian arguments

Since I was very young, I used to want to tell people exactly what I think about something. Most of these views would be contrarian, and I would assume that I am adding value to the conversation by presenting an alternate view. Needless to say, I often had trouble making friends in middle school, high school or college. I would thrive on trying to spot holes in people’s arguments, presenting a contrarian view with some “evidence”, etc. I would share inflammatory posts on social media trying to “present an alternative view”, politicize issues, and then take exception when people couldn’t see why my opinion was correct.

This has slowly changed over time. I realized how I was only needlessly antagonizing people by trying to be contrarian all the time. However, I truly began to understand the issues involved when I read “Trapped Priors as a basic problem of rationality” by Scott Alexander. Let me try and explain some salient points of this phenomenal article.

People argue about all kinds of issues in person or online (of course smarter people mostly stay away from such things). We now have unprecedented access to all kinds of opinions both online and offline. However, it is extremely rare to see anyone change their opinion. With a greater access to information, we are only seeing people further entrench themselves in their opinions. One example of this is that Trump supporters only became more entrenched in their support for Trump after the media flurry of anti-Trump articles early on in his presidency. One explanation of this is confirmation bias: if I already have a particular stand on an issue, I will have a tendency to accumulate information that supports my stand, and cheaply dismiss information that contradicts it. Although Eliezer Yudkowsky thinks that this is a defect of the brain, I’ve also seen a research paper that says that this is actually a feature and not a defect, and helps us form a coherent world view (which was historically important for survival).

Hence, whenever I argue with someone about an issue, I am mostly trying to defend my opinion by sharing cherry-picked facts that support my stand, and omitting facts that don’t. Moreover, when the person I’m arguing with shares their own opinions and facts, I try desperately to find a way to poke holes into their arguments so that I can “win” the argument and claim “intellectual superiority”. Let us assume that I’m successful in doing so: that I’m able to poke holes in my opponents’ arguments, and that they’re not successful in doing so. Does that mean that I’ve managed to convert them to my opinion?

You’ve obviously guessed that the answer is a resounding “No”. But why is that? Is it just out of spite that my opponent would not like to accept my stand as the only correct one? I don’t think so. An opinion is a manifestation of many conscious and unconscious inferences about the world, and being able to contradict one inference does not mean that all inferences have been contradicted. For instance, if my opponent supports Republican politics, they might argue that Republican policies have generally been better for the layman’s income than Democrat policies. In response to this, I might bring up a research paper that claims that this is only a myth, and that Democrat policies are in fact slightly better for the layman’s income than Republican policies. Does that mean that my opponent will understand that they’re wrong, and hence should morally accept my stand? No, because their support for Republicans rests on many other factors like how Republicans seem to be doing a better job of developing their area than the previous Democrat candidate, how a lot of Democrat policies seem ineffective and counter-productive, how Republicans don’t carry the “holier than thou” attitude, etc. These opinions still stand unchallenged. Moreover, because research papers contradict each other all the time and often reflect vested interests, bringing up one research paper that disproves their point doesn’t really prove anything. It only shows that I was pretentious enough to bring up a research paper in a casual conversation about politics, probably as a means of signaling awareness.

Essentially, if I try and poke holes in the arguments that my conversation partner is making, I’m not really trying to understand the deeper reasons for their stand. Curiously, this was something I first came across in Tolstoy’s “Anna Karenina”, and have since come across in many places like Yudkowsky’s Sequences on Rationality and Scott Alexander’s blog. We only antagonize people further by trying to argue with them, and not attempting to understand their deeper values. What is the solution to this? Should we just agree to disagree on everything that we don’t agree on already? Do people’s opinions never change?

I don’t know. However, I do know that our only responsibility is to listen carefully to other’s opinions, strengthen them instead of trying to poke holes into them, and then see how we should change our opinions in order to fit the facts that the other person has presented. Of course it is entirely possible that the other person has said only incorrect things, and that we should not change our opinion at all. However, what is much more likely is that we will try to cheaply dismiss their opinions despite the value inherent in them, and look for reasons to stick to our previous opinions. We should avoid that temptation.

I grew up thinking that my main aim was to be right. Very, very slowly, it has (partly) changed to understanding how and where I’m wrong, and possibly changing myself/my opinions in order to form a more accurate view of the world. Writing this essay has hopefully been a step in that direction. In Yudkowsky’s words, our main aim should be to create a “map that reflects the territory“.

Dealing with conflicting goals

Today I read an amazing paper titled “Behavioural Consistency and Inconsistency in the Resolution of Goal Conflict” by Laran and Janiszewski. It was published in the Journal of Consumer Research in 2009, and is one of the most highly cited papers in the field. I think this paper really helped me understand my daily behaviour, and can hopefully help to rectify whatever it is that I do everyday.

Goal Conflict

So what really is goal conflict? Well, most of us have at least two conflicting goals- say eating delicious fatty food vs eating healthy/staying fit. Clearly, chasing after any one of these goals will undercut the other (unless you’re one of those salad-loving freaks). So which of these goals do we end up pursuing?

There are two theories of goal management: active goal guidance (AGG) and passive goal guidance (PGG). The active goal guidance theory states that humans rationally measure and calibrate how close or far they are from their goals, and then make decisions based on that. For instance, after a week full of 4 mile runs, you might calculate that you’ve done enough to deserve a pizza meal tonight. However, humans aren’t always calculating and calibrating. Most of our impulses and desires that shape the bulk of our behaviour stem from something deeper and much more irrational. If there’s a piece of cake lying in your refrigerator that you’ve been avoiding for a day or two because you haven’t been working out at all, there will come a moment when you’ll go stick your head into the refrigerator and just inhale the cake before you know what hit you. This move to eat the cake wasn’t really a result of a conscious, rational calculation. So what’s going on? Hint: it’s the other goal guidance system.

Passive goal guidance is an imprecise goal management system that is always running in the back of our minds when we’re not actively thinking. And it is a pretty shoddy system. How so? Let us assume that you’ve decided to start working out in order to lose weight. It’s your first day, and you go for a run. Ideally, you should continue this for at least a couple of months, eat healthy, and start eating unhealthy food only when you’ve reached your target weight. However, the passive goal guidance system tells you that you’ve done enough towards your fitness goals….on the first day itself! Hence, you disengage with your goal of becoming fitter, and engage with your other goal: to eat delicious fattening foods. All of this is a result of the fact that the passive goal guidance system cannot distinguish between whether you’ve actually achieved your goals or just taken a small step towards them. Hence, we keep bouncing between conflicting goals, never actually attaining either of them.

This is something that resonates very deeply with me. I have multiple goals: to become a good mathematical researcher, to learn about a lot of different and disparate fields, read good fiction and non-fiction, etc. These goals are in conflict because I only have limited time and energy to allot to these goals. Hence, working towards any of them means not working towards the other goals. I can of course use the active goal guidance system to rank these goals in order of importance, and then allot time to them judiciously in order to maximize my personal utility function. However, what ends up happening in practice is that whenever I make even a small amount of progress in one of my goals, I abandon it to pursue something else. Context switching often nullifies whatever little progress I might have made towards any of my goals, and I end up not achieving much towards any of my goals. The most effective strategy that I’ve ever applied to myself is that of not abandoning a task until I finish it completely. Although I couldn’t continue that strategy beyond the first week or so, mostly because of the level of discipline required to carry it out, this strategy effectively eliminated the problem that the passive goal guidance system creates. Hence, my personal experience suggests that the authors are correct: the passive goal system seems to be responsible for a lot of the context switching and procrastination I see in my daily life.

Another important fact to note is that goals are often unconsciously created inside us. For instance, watching an advertisement for the latest cola product might create an unconscious desire inside us to buy it. This kind of unconscious goal creation is called priming, and is also controlled by the passive goal guidance system. Priming plays a very important role in this paper, which will be discussed below.

Goal Management in the presence of conflicting goals

The following quote sums up most of the arguments made above:

Goal management models seek to explain goal pursuit and disengagement in the presence of goal conflict. Fishbach and Dhar (2005, 2008; see also Fishbach, Dhar, and Zhang 2006) propose that the determinant of consistency versus inconsistency in two consecutive behavioral decisions is the extent to which an initial behavioral decision signals goal progress or commitment. They posit that when people perceive an initial act as goal progress, they become less likely to pursue the same goal, and they end up pursuing opposing goals when making the second decision. When people perceive an initial act as commitment to a goal, they become more likely to pursue the same goal when making the second decision. Therefore, people make an inference of how far they are from achieving a certain goal and, depending on whether there is an inference of progress or commitment, choose their subsequent behaviors.

Another interesting aspect to note is that when we have to trade off goals vs resources, we are mostly consistent in our choices. For instance, when faced with either indulging in expensive shopping or saving money (resource), we either consistently buy a lot of expensive things, saving little money, or very rarely buy expensive things, saving a lot of money in the process. We strike some sort of optimum balance, and stick to it. However, when dealing with conflicting goals, we are not consistent in our choices, indiscriminately hopping between eating junk food and trying to have a healthy lifestyle. We rarely ever intend to strike an optimal balance between the two goals: we first religiously stick to one goal, and then completely abandon it to pursue a conflicting goal. This suggests that when deciding what to do with our resources, we let our active guidance system guide us. However, when we’re merely comparing two conflicting goals, eventually our passive goal guidance system takes over, often making contradictory and self-defeating choices for us.

Some features of the passive goal guidance

The propositions of this system of goal guidance are:

  1. As goal activation increases (decreases), means that are relevant to the goal will gain (lose) value. What is goal activation, you ask? Imagine that you read a biography of Elon Musk. You are heavily prepped to do fantastic things and change the world. This is goal activation (and obviously not fulfillment). The means through which you may attain these goals, perhaps your computer and books, gain importance for you, while other unrelated means, like your Xbox, might at least temporarily reduce in value.
  2. When you take a step towards your goal, it might be misinterpreted as goal completion, leading to disengagement with that goal. This has been explained above.
  3. Goal activation for one goal will result in the inhibition of other goals. For instance, as explained above, when your “create something great” goal has been activated, your “play with your Xbox for long hours” goal may take a backseat.
  4. Goal achievement will result in disengagement with that goal, leading to a rebound of previously inhibited goals. For instance, after you’ve achieved your goal of having a great job, your inhibited goal of “spend more time with family” will come to the forefront, and you’ll try to make amends on that front.

    Remember that our false belief of goal completion, promoted and abused by our unconscious mind and our passive goal guidance system, is a major reason why we jump to conflicting goals and fail to accomplish what we’re capable of.

Experiments and discussion

Five different experiments are described in the paper, which I will not discuss in much detail. However, I will discus the results obtained from those experiments, and how they support the main claims of this paper.

Initial steps taken towards goal completion are hugely significant. For example, if you are trying to avoid eating unhealthy food at a party, saying “no” to it early on hugely boosts your chances of not eating that food at all. However, if you decide to partake of a small amount of that food so that you may somehow quench your desire to eat that food, that’ll turn out to be a disastrous strategy, as you’ll soon succumb to eating increasingly large amounts of that unhealthy food. Hence, if you’re going on a diet, throwing out all the unhealthy food in your house is a much better strategy than trying to restrict yourself to “one cookie a day”. This is something that I’ve struggled with myself. “I’ll only check facebook once or twice everyday” has turned out to be a spectacularly bad strategy. What has turned out to be a better strategy is deactivating my profile and saying “no” to all my future selves with less self control.

Active goal guidance is activated when you compare your goal progress with others or your past self, while it fails to be activated when no comparison is being made. For instance, after a run you may compare yourself to a fitter family member and find yourself lacking. You will then persist in your goals for longer. However, if you don’t form any such comparisons, your passive goal guidance system is likely to tell you that you’ve achieved your fitness goals and that you deserve to eat all your favorite junk food.

“Small indulgences” that don’t result in goal completion can prime you for other related goals. For instance, if you’re satiated only after eating a full bowl of chocolates but are allowed to eat only one chocolate, you are more likely to indulge in all kinds of pleasurable activities like eating more chocolates, buying expensive merchandise online, drinking expensive wine, etc for a short while. However, indulging in your guilty pleasures until you’re satiated will result in goal completion, and then you can move on to other conflicting goals. For instance, if you finish a whole bowl of chocolates, you’re more likely to be more interested in healthy food, living a simpler life without expensive indulgences, etc. In some sense, it is better to say “no” right away to an indulgence, or to completely immerse yourself in it until you’re satiated, than to partake only a little of it (which will prime you for bigger indulgences and worse crashes).

Another interesting observation in one of the experiments was that goals eventually fade away with time. For instance, if a TV advertisement primes you to go buy its product, and you restrain yourself from doing so for some time, eventually you’ll disengage with that goal and move on to other things. Hence, if you ever feel a very strong impulse to do something, just restrain yourself for long enough until that goal fades away.

Conclusion

Like everybody else I spend way too much time on my phone, keep jumping between unrelated tasks, procrastinate after completing small subtasks of my main task, find it difficult to complete most tasks, etc. This paper gave me a scientific basis for why this happens, and also an idea for what I can do to reduce such distractions. I can perhaps write a post a month or two later, detailing the extent to which this paper changed my productivity and life.

I also feel that procrastination is the act of jumping between conflicting goals, and am unsure why the authors do not address it in this paper.

References

  1. Behavioural consistency and inconsistency in the resolution of goal conflict, by Laran and Janiszewski

Decoding human personality

I recently had the opportunity to read a fascinating paper titled “A model for personality at three levels” by Revelle and Condon. I was expecting the article to be rather dry, from which I would need to extract useful facts after a lot of effort. However, it proved to be fantastically “wise”, with deeply penetrating insights into human personality. Written more like a Tolstoy novel than a research paper, it is hands down the most amazing thing I’ve read in recent times.

Levels of individual differences

The authors state that people change with time at an intra-person level, inter-person level and inter-group level. Intra-person refers to the changes within a person, like a sudden surge in anger or hunger, inter-person refers to a relative change between individuals, like a faster decrease in awkwardness in one person than in another which causes that person to speak first in a social situation, and inter-group refers to group level differences, like some majors such as Physics being negatively correlated with agreeableness.

Why are we concerned with trait changes with respect to time, and not stable characteristics? For instance, why are we not concerned with a person’s stable level of anger, as opposed to a “rate of increase of anger”? The authors contend that it is the rate of change of a characteristic, and not the stable value of a characteristic that determines human behaviour. Why is that? Consider the following situation: you’re walking on the road, and you suddenly see an acquaintance from college. There will be a part of you that will want to go talk to them, while the anxious part of you will want to feign ignorance and move on. Whichever tendency rises fastest at that moment will lead to the consequent action. Hence, it is the rate of change of a desire/trait inside a person that determines their actions.

As an aside, consider the following quote:

When others evaluate our reputation, they are evaluating our behavior in critical situations and how it changes across situations. When we think of our identity, we interpret our behavior as the result of our affects and our cognitions.

This is a supremely deep quotation that set me thinking. Yes, people do seem to evaluate the true worth of others only in “critical situations”. If you didn’t help your friend in their time of need, you will forever be thought of as unreliable and “not really a friend”. However, we interpret our behavior as a result of our thoughts and desires (that are not accessible to others), and hence tend to give ourselves far more leeway than others do. Although this quote is not directly related to the rest of the paper, I thought it was a great quote and deserved to be discussed on its own merit.

The authors’ main contention in this paper is that the behaviour of individuals depends majorly on intra-person dynamics, and which tendencies of theirs are reinforced by social approval and encouragement, etc. Of course, this reinforcement only works when an individual is receptive to external opinion and feedback from past performances. For instance, one of the major reasons why I switched to Mathematics in grad school was that I didn’t enjoy engineering, and had previously performed very well in Mathematics in high school. Hence, my past performances reinforced my affinity for Mathematics, which led me to change my whole life trajectory. On the other hand, I was never very receptive to reinforcement by social acceptability, which has hampered my social behaviour for many years. Hence, receptivity to reinforcement is perhaps a factor that the authors have left out from the paper.

Different levels can be different

Consider alcohol consumption. Alcohol consumption is negatively correlated with cognitive abilities at an intra-person level, while being positively correlated with cognitive abilities at an inter-person level. How do we reconcile these two facts? These are completely different and non-contradictory facts. We are just saying that smart people tend to drink more alcohol than not so smart people (sorry teetotaling Hitler), and smart people get less smart when they drink a lot of alcohol.

This example illustrates the fact that correlations between two traits can be completely different at an intra-person and inter-person level. However, individual intra-person traits are still indicative of inter-person traits.

Dynamics within individuals

Consider the following quote:

Dynamic models imply more than the mere observation that people differ over time, for this could just be random fluctuations around a mean level.

This is a realization that I had myself a couple of weeks back! People don’t generally stay the same their whole life, their behaviour fluctuating around a stable mean. People change slowly but irrevocably with time (their mean behaviour changes with time). And it is around this changing mean that behaviour fluctuates.

We will now describe the mathematical model of dynamic personality developed by the authors. Let T be the desire or action tendency related to a particular action, F be the external conditions causing that desire, and c the “amount of action” taken to quench that desire. Then the rate of change of desire is dT=F-cT. In other words, your desire to do something will keep increasing with time unless you act upon it, which will then cause it to decrease sharply. If you desire to eat a chocolate cake, your desire to do so will keep increasing until you take a bite, which will cause it to decrease momentarily. Moreover, the decrease is proportional the desire itself!

When does dT become 0? This happens when F=cT, or in other words when you’ve done enough to quench that desire. Note that desire may not actually become 0 when dT=0. Hence, when the action c is removed, the desire again starts growing, until more action is taken. Effectively, enough action c has to be taken until T=0.

What is F, exactly? It is expressed as p_s(1-p_s)N_{ach}, where p_s is the probability of success and N_{ach} is the “need for achievement” (in other words, ambition). In other words, our desire to do something doesn’t necessarily increase with the ease with which it can be done. Desire is maximal when the probability of success is equal to the probability of failure. What does that mean? If I’m a naturally gifted athlete, and it is apparent that I am almost certain to win the Olympiad gold, I will not really want to pursue athletics that badly. However, if I’m a great musician but my chances of making it to the big stage are 50-50, I will be tempted to gamble on my life and see if I have what it takes to make it. This hit home for various reasons. As humans, we are not really maximizing our chances of having a comfortable life. We are always looking to gamble on our lives, and see if we have what it takes. That is why we are so open to new challenges and get excited by uncertainty (under the right circumstances). Although I probably have to think through some of the deeper implications of this, this equation did blow my mind!

There is an analogous model for negative tendencies or anxiety. Let N be the anxiety related to a particular task, I the inhibitory forces (factors that are causing anxiety), and r the cost of resisting action. Then we have dN=I-rN. In other words, anxiety about a pending homework assignment keeps rising (if the cost of doing homework is too low) or falling (if the cost of doing homework is too high) until it stabilizes. There is no action that you can perform which will make doing your homework less of an anxious experience.

What does this teach us? We all have tasks that we find distasteful but have to do from time to time. And waiting longer to do them is not going to make them any less distasteful (might only increase anxiety). Hence, whenever you have a task that you have to do, do it first. And then when it is over, you can get back to enjoying your life.

Again I is defined as p_s(1-p_s)N_{af}, where p_s is the probability of success and N_{af} is the need to avoid failure. We don’t really get anxious about things that have a low probability of success (asking someone out who will almost surely say no) or a high probability of success (asking someone out who will surely say yes). Our anxiety is the highest in situations which have a 50-50 chance of success (asking someone out just above our league).

Essentially, whether we do something or not depends on p_s(1-p_s)(N_{ach}-N_{af}). Things that have a 50-50 chance of success induce both the highest excitement and the highest anxiety in us. This says something very deep about human nature, which I’ll leave up to you to interpret.

The authors note that in this model of the human personality, there is no fixed or control point. There is no fixed value that positive or negative tendencies return to. In some sense, both desires and anxieties are unbounded.

Some more aspects of the model

You might want to eat pizza and go for a run at the same time. However, what you end up doing depends on what you want to do more badly. Hence, although desires for various actions grow in parallel, the actions themselves can only happen one after the other, and the action with a higher desire function is performed first. For instance, a newt copulates underwater, but comes up to the surface for oxygen. If oxygen content is increased, the chances of the newt being able to breathe successfully increase. Hence, the desire to go up on top decreases (remember that F=p_s(1-p_s)N_{ach}), and the newt copulates underwater for longer.

Another aspect of this is that because attention is limited, it can only be allotted to various tasks sequentially. Hence, one must not text while driving, watch tv while doing a cognitively intense task, etc.

Inter-person dynamics

The first observation that the authors make is that rates of change are more important than absolute levels. For instance, a person whose anxiety suddenly spikes is more likely to behave irrationally than someone who is always anxious.

Secondly, they state that the average levels of what one person does is different from the maximum levels that that person is capable of. For instance, if you work for 5 hours a day, that does not preclude the fact that you may be capable of 20 hours of work in a day when motivated.

Thirdly, the authors state that most cognitive abilities correlate with other cognitive abilities. A straight A student is probably also good at music and debating and all kinds of other activities that you may think of. This is perhaps the reason why top Business schools prefer that their candidates demonstrate excellence in multiple fields- a demonstrated expertise in multiple areas is of the best signals of overall cognitive abilities.

Cognitive abilities are sometimes described as the speed at which a solution to a problem is found/speed of processing. It has also been seen that the amount of crystallized knowledge is positively correlated with processing speed. Hence, smart people are more likely to be repositories of knowledge…at least in school (this may change later in life, when smart people may no longer be incentivized to accumulate irrelevant knowledge).

Now consider the following quote:

If temperament is what you usually do, and ability is what you can do, interests are what you like to do and how you spend your time.

One way to interpret this line is that abilities or the capacity for a certain kind of work does not necessarily form one’s temperament. Moreover, temperament may be different from interests. For instance, although you may have an angry temperament (reinforced perhaps by a lack of opposition from others), being angry might not be how you like spending your time. The authors also note that interests can be classified into people vs things and facts vs ideas.

Now let us consider the following quote:

That what one can do (ability or competence) is not necessarily shown by what one does has been known since at least Tolman and Honzik (1930) who studied the effect of reward on maze performance. With the same number of learning trials, non-rewarded rats take far longer to run a maze than when given a reward.

One way of interpreting this line is that if you want to assess a person’s maximal capabilities, attach a reward to the successful completion of a task. Only then will that person really exert themselves in attaining that goal.

Group differences as the consequence of individual choices

Imagine that there are two students- A and B. A feels slightly more anxious during public speaking than B. Hence, B generally does a better job at speeches than A. A receives negative reinforcement through audience disinterest or discouragement, while B receives positive reinforcement in the form of adulation. These effects compound, and A may join a profession that does not involve much public interaction while B may become a lawyer or a politician. Hence, individual differences compound and result in individuals joining different groups.

The natural question to ask at this point is what personality traits lead a student to choose a particular major, say Physics? In a large scale study, it was found that having high cognitive abilities and being less agreeable was highly correlated with choosing Physics (sorry Physicists). This is just a small example of how individual differences lead to group differences.

Conclusion

The authors conclude by clarifying that individual differences don’t deterministically lead to group differences. It is not as simple as that. Although individuals are blessed with their own abilities and desires, individuals change as a result of reinforcement from society or other external factors. “Personality needs to be conceived at multiple temporal durations.”

The authors also emphasize the importance of using softwares and formal methods to analyze data and create more accurate mathematical models.

References

  1. A model for personality at three levels, by Revelle and Condon

Status signaling

Of the subtle things in life, status signaling is one of the subtlest. It is mostly a delicate art refined in the forging fires of one-upmanship (and perhaps envy), but it can become quite unpleasant in the hands of brutes.

So what is status signaling? Well if I am earning a lot more than you, or went to a better school than you, or perhaps can stake some claim to professional success, I will need to signal that in some way, lest you think you’re better than me.

There are a couple of things about status that one needs to carefully consider before we get into status signaling. Status in a varied society is not a completely ordered set. There is not just one parameter by which you can compare the statuses of two people. For example, a rich person is not necessarily better than an MIT professor. However, an MIT professor in a particular field may be considered to be “better” than a professor at a lesser known school who works in the same field.

This, too, is not the complete picture. Some communities do give greater importance to some parameters over others. For instance, my own community in India gives much more importance to wealth than academic achievement. Hence, status may be thought of as a weighted sum of all parameters, and these weights vary amongst communities and cultures.

Secondly, status signaling is mostly a defensive strategy, and not an offensive strategy. If my social group already accords my a high social status for my unique talents and personality, and I do not feel that another person has been accorded a higher social status than me, I will mostly be comfortable, and will not feel the need to compete with another person for status. However, if my group accords me a lower status than someone else, or if someone slights me and hence challenges me to a status duel, I will feel a need to somehow pull myself up (or maybe pull them down) in order to gain higher status than them.

As an adult, I spend a lot of my time watching other people signaling their status, or perhaps doing some signaling of my own. Such signaling is mostly subtle. If I went to reputed school for my education, I cannot just bring it up right away. I’ll need to find the right opportunity to bring it up: maybe when the conversation meanders to the topic of education, I can casually talk about how I had a terrible time in college. This of course would lead to “where did you go for college?”, in which instance I would bring it up. Some people may casually slip in where they went for vacation, what senior administrative managerial position they occupy in their company, how they once beat the odds and got that very coveted job, etc.

The delicate art of signaling, however, can get less subtle in the hands of brutes. I once saw two math professors compete for status by comparing the scores they got on some standardized exams two decades ago. I’ve seen multiple students from reputed institutes in India bring up their entrance exam ranks in their very first conversations with me (I’ve seen profs do that too). I’m sure that brutes may also signal wealth and professional achievement in a similar manner, although I do not have real life examples of these sort.

What is perhaps even more interesting is not status signaling, but status slapping. What is status slapping, you ask? I’d thought of a similar concept myself a couple of months back, but then I came across Eliezer Yudkowsky’s comment on slatestarcodex that perhaps expressed it much better. If society thinks that you have a status S, and you try to convey that your status is S+2, people around you will push you down/make fun of you, etc until you begin to accept that your status is indeed S. Consider the following example: someone in my college cohort did very well on a standardized exam in India. Everyone was suitably impressed. Then she said something to the effect of “I think I have the potential to become a tech genius like Ironman”. Predictably, this led to a lot of derision until that person accepted their lower “true” status.

Another aspect of status slapping/signaling is status by association. If you signal that you have higher status than me, I can remind you that there are people who have even higher status than you, so that you don’t feel all high and mighty. For instance, if you casually let slip that you went to Berkeley for grad school, I might bring up the fact that I had a close friend who went to Harvard last year. Hence, I know people that “are better than you”. Why is this a successful form of status slapping? Even though I know someone who went to Harvard, I myself remain where I am! Why does bringing up this overachieving friend reduce our mutual status gap? Maybe status can also be earned by association. Although going to Harvard/being rich/being athletic may give me direct social status, being a spouse/friend/family member of someone with high status may also accord some secondary status to me. This feels weird in our supposedly individualistic societies. However, it exposes the fact that we still possess monkey brains, and status by association is a real thing.

It is important to note that status signaling is a lost battle for most people. If you are rich/well educated/etc, you are likely to socialize with people who enjoy the same status as you, if not more. For instance, if I went to Yale and joined a law firm right after, I am likely to work with other well educated college graduates, perhaps some from Harvard, who may go on to do much better than me. Hence, status signaling will mostly be a lost battle for me in the long run. However, status signaling does not affect those people (as much) who accept their status. If I accept my position in the status hierarchy and do not try to compete with another for status, I will mostly be left alone. Everyone I know who refuses to status signal actually has a much better time socializing, fitting in, etc.

So what should be the best strategy for life, as far as social signaling is concerned? Should I just stop signaling status? Sure. That plan should work for most people…..until the next jerk who comes along and signals their superior status. Faced with such a situation, we may feel an irresistible urge to either challenge them to a status duel, or perhaps pull off a “status by association” move. I would perhaps give some zen advice on how not to care about status, and that we’re all stupid mortals anyway, etc…but this would run directly opposed to our fundamentally simian brains that have been hardwired to indulge in status challenges for literally millions of years. Can we subdue our monkey brains? Can we not care when someone of a supposedly higher social status slights us? Can we just keep learning and growing forever, exploring our fascinating universe and ourselves, and not competing for status? Thus ends my status signaling post on status signaling.

On virtue signaling

Let me detail some of the many ways in which I am basically a hypocrite.

I am a “liberal”. Does that mean that I fight for women’s rights? Do I work tirelessly for the downtrodden? Well I got into an argument with a family member once in which I said that men and women deserved equal professional opportunities. I sometimes post articles on social media when I’m particularly “disgusted” by women not getting a chance. A lot of people see it for what it is- a stab at the social applause meter.

However, let us imagine a better version of me. Imagine a (slightly better-looking) version of me who went on long marches advocating for equal pay for men and women, gave money to the poor, etc. I would probably do all of that, and then expect some social capital out of it. Although I’d be discreet enough to not bring it up myself in conversations, people would eventually “come to know”, and then they’d think of me as passionate about equality of justice, etc. Come to think of it, even this slightly better version of me is not that great.

Well let’s imagine an even better version of me (who’s even better looking). I maintain strict anonymity, but work really hard to serve the marginalized sections of society. I go on long marches (perhaps wearing a Guy Fawkes mask), and distribute a lot of my income to the poor who I think have better use for it than I do. Although people don’t really know my true identity, I feel pretty good at the end of the day. A true savior. The closest thing to Batman a baniya from Kolkata can get. Is this guy “good”?

Moloch is probably the word/feeling that most closely resembles what this much better version of me is trying to do. “I will defend the rights of all the marginalized sections of society. And I will do so because I matter, and my chosen actions are exactly what are needed to correct these evils.” The sheer arrogance of it. What if going on marches and redistributing my income are not what is needed to cure these problems?

If I really wanted to help, I would have made an effort to discover what actually helps. And what actually helps is often counter-intuitive, and can only be found after experimentation. For instance, Abhijit Banerjee discovered that it was very difficult to convince people in India to have their kids take the Polio vaccine. However, when they started a policy of giving away 2 kgs of rice free after administering the vaccine, this relatively inexpensive intervention convinced hundreds of thousands of people to buy much more expensive tickets, and brave hours and hours of journeying to travel to the polio administration centers, kids in tow.

What actually helps in improving the general condition of people is not redistributing wealth to the poor, but allowing the rich to hoard it (Capitalism)! This is perhaps one of the most counter-intuitive ideas in history. If you allow the rich to hoard wealth, they pump in so much money into a country’s infrastructure that the poor are bettered as a consequence. Who’d have thunk!

Did this much-much better version of me care about finding out what actually helps? No. He was mostly invested in an ineffective cocktail of arguing with family members, sharing updates on social media, and perhaps going on marches and redistributing income.

Most people don’t really want to do anything! Whether it be helping others, doing meaningful research, basically anything at all! They mostly want to be thought of as doing something meaningful or important. If I can shut down a “male chauvinist” uncle at a family gathering, I will have emancipated all women alive. Because it is I, the emancipator of women, that stands at the crossroads of history, on whose actions the future arc of humanity depends. What if I am basically a stupid and inconsequential bystander, who is too stupid to realize this?

What if asking that uncle to shut up is not the best intervention? In fact, I will only have further alienated him. Instead of taking the higher moral ground at family functions, what if I look up ways that actually work to emancipate women? What if I help my mother cook? What if I *really listen* to her with full attention, as she tells me stories of her past? What if I devote more time and effort to what the women in my life want me to do to help them? What if I stop assuming that the only way that women can be made equal to men is if they’re doing the same jobs, because I, me, myself think that way? What if it is not my f@#king place to make that choice for them? What if I just shut up, listen and learn?

Mimetic desire

One of my favorite things in life is to discover counterintuitive truths. Hence, I was happy to recently learn about mimetic desire, as discovered by the French polymath René Gerard.

Let me reproduce some real zingers from the article:

 We borrow our desires from others. Far from being autonomous, our desire for a certain object is always provoked by the desire of another person—the model—for this same object. This means that the relationship between the subject and the object is not direct: there is always a triangular relationship of subject, model, and object. Through the object, one is drawn to the model, whom Girard calls the mediator: it is in fact the model who is sought.

Mediation is external when the mediator of the desire is socially beyond the reach of the subject or, for example, a fictional character, as in the case of Amadis de Gaula and Don Quixote. The hero lives a kind of folly that nonetheless remains optimistic. Mediation is internal when the mediator is at the same level as the subject. The mediator then transforms into a rival and an obstacle to the acquisition of the object, whose value increases as the rivalry grows. This is the universe of the novels of StendhalFlaubertProust and Dostoevsky, which are particularly studied in this book.

What Paris wanted, then, was not Helen, but to be a great king like Agamemnon. A person who desires seeks to be like the subject he imitates, through the medium of object that is possessed by the person he imitates.

This was, and remains, a pessimistic view of human life, as it posits a paradox in the very act of seeking to unify and have peace, since the erasure of differences between people through mimicry is what creates conflict, not the differentiation itself.

It is the last quote in particular that seems like a fundamentally counter-intuitive fact about the world. I was always taught that it is our differences that create strife and conflict, and that if we could reduce the differences amongst ourselves, we could be peaceful. However, it now seems clear on reflection that conflicts are the harshest between people that are similar or close to each other. This is because, at least according to mimetic theory, all conflict is a desire to become someone- someone who you envy. You desire what they have. And hence you’re ready to maim them, insult them, somehow occupy the exalted position that they currently command.

If we could accept our differences, and accept that we are unique and incomparable with everyone else, we would perhaps have peace. It is this acceptance of our differences that would reduce conflict. However, we do not consider ourselves to be fundamentally incomparable with people that we envy or despise. We want to become them, and think that we indeed can become them. And it is in this act of aspirational mimicry that we shatter any hope of peace or contentment.

The true cost of an education

I come to bury Caesar, not to praise him

Main takeaway: Moloch has killed your curiosity (and hence the cat still lives)

People that are much more intelligent and successful than me have written about the perils of standardized education. Some of them are Paul Graham, Scott Alexander and Bryan Caplan. I would highly recommend that anyone reading this post read these linked articles. There are some of the most eye-opening arguments that I’ve heard on the subject of education.

Some highlights from Paul Graham’s article are given below:

I

Like most other children, I would want constantly want to ask adults questions about the world. However, after trying to humor me for some time, my family would ask me to stop, and my teachers would ask me to “first understand what’s written in the book, and then ask all of these unrelated questions”. I soon learned to shut up and read the textbook.

This approach mostly worked OK for me throughout school. Now I’m a PhD student, and I realize that I have no questions to ask. I train all day to try and become better at answering questions that others ask me. “Can you solve this problem?” I learn theorems and techniques so that I may be able to attack most problems that I see.

I recently came across this blogpost by Zvi Moshwowitz. Zvi is a member of the rationalist community who writes very, very long posts performing detailed statistical analysis of the government’s COVID response. And he posts these extremely detailed statistical analyses every 2-3 days.

My first response on seeing this post was disgust. I was disgusted by the fact that this person, somewhere in the world, spends hours and hours everyday performing statistical analyses, running numbers through some number-crunching software, and then compiling them to post on a blog that he cannot possibly earn much financial or social credit from. Why does he do it? No one asked him to!

When I saw this article a couple of days back, I wanted to write a blogpost right away. And I wanted to call it “The difference between successful people and….me”. This is become it is unfathomable for me that I would put in this much work into anything that wouldn’t go towards my career or oiling my social network. I had absolutely no curiosity about the world around me. There was no question that I’d come up with for a long time, that I wanted to answer. I only aspired to answer questions that other had, so that I could signal some combination of intelligence and hard work, in order to perhaps gain employment, etc.

Over the last couple of days, this made me angry. I have now spent most of my life trying to jump through hoops. I let the people around me kill my curiosity. And as I near the age of 30, my only aspiration is to get better at jumping through these hoops in order to gain employment, or perhaps earn social credit. But what about the education system killed my curiosity? I can only speak about it from an Indian perspective, although these problems seem true for the US (at least as per Paul Graham’s article) as well.

II

An education is mostly supposed to broaden your horizons, and teach you facts about the world that an adult is expected to know. However, in practice, it becomes a system that can be gamed. Although I have learned a lot by going to class in school and college, I ultimately knew that doing well in that course only partially correlated with “loving the subject”. Some combination of intelligence + solving the questions at the back of the chapter + access to notes from class + access to past years’ exams would be required. I don’t really think that any of these factors in themselves are bad. Of course trying to solve problems from the back of the book or from past years’ exams would only strengthen my understanding of the concept. The abuse arose from the fact that it was only through solving the problems at the back of the book or from past years’ exams that one could prepare for the exam or signal an understanding of the subject. Let’s explore this a little bit.

Suppose I’m taking a class in college. I know that if I am able to solve all the questions at the back of the textbook and from past exams, I can realistically be assured of an A/B+. Because humans mostly do the least amount of work required (perhaps because there is an evolutionary advantage to conserving energy that has been hard-coded into our brains), my study strategy would soon evolve into solving just those questions, and not caring as much about being curious about the subject and asking my own questions/reading other unrelated material. Although sometimes there are questions on exams that aim to test whether students have a “deeper” understanding of the material, trial and error soon teaches us what’s the least we have to do to “get by”.

Because curiosity is not really a skill that is required or moulded in school or college, it soon dies out. But what can our education systems really do? Is there really a way to encourage and mould students’ curiosities? Having taught a number of classes myself over the years, I find that this is a difficult problem to solve. I’ve often toyed with the idea of asking students to just “write a detailed summary of the whole course, and any particular problems you like, why you like them, etc”, and then just grade it like a subjective essay rather than an objective assessment. This perhaps sounds silly because science and mathematics are expected to be about “objectivity” and “accuracy”, and do not really lend themselves to long essays filled with “feelings” and such. Although I mostly agree with this, I think that this stance is partly overstated. I would have a much higher opinion of a student’s mathematical understanding if they could write a detailed essay on the development of calculus, why it was shocking for scientists of that time, and what some of their favorite calculus problems are, than one who can differentiate and integrate the problems given on the exam.

Moreover, a lot of hard mathematical problems are just about trying enough things until something clicks. Even if I have to give students numerical problems to solve on an exam, I would perhaps give them most, if not all points if they could write down ten ways that they tried solving the problem but couldn’t. At least in this way, students will be encouraged to try.

III

Education has been a contentious issue for a very long time. Surely a lot of people must have thought of the same points as me. Why haven’t they been implemented yet? Perhaps part of the answer is that it is much easier to test and grade students on numerical problems than on opinion-driven essays. For instance, grading an exam paper consisting only of numerical problems would not take more than 5-7 minutes. However, grading an essay would take perhaps twice as long, because you would often have to address the points made by the student and critically evaluate them. I definitely find this to be true for myself when I grade my students’ exams. There are lots of other ways in which the education of students is harmed by lazy administration, but that perhaps calls for a different post.

I’ve actively tried to be more curious over the last couple of days. This has led me to address fundamental doubts about subjects that I’ve already studied, and has even made research more pleasurable for me. I hope to pass this on to my students who I meet 4 times a week.

Don’t optimize for the world.