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!

Published by ayushkhaitan3437

Hello! My name is Ayush Khaitan, and I'm a graduate student in Mathematics. I am always excited about talking to people about their research. Please please set up a meeting with me if you feel that I might have an interesting perspective to offer-

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