This blog is about Simon, a young gifted mathematician and programmer, who had to move from Amsterdam to Antwerp to be able to study at the level that fits his talent, i.e. homeschool. Visit https://simontiger.com
If you’re interested in why #covid-19 tracing apps are important and the most privacy-friendly way to implement them, please read this interactive essay by Nicky Case and play with the colorful simulations of all our possible futures. For Simon, this has been the entrance into the Nicky Case @ncasenmare universe (first recommended by 3Blue1Brown). Simon has been gulping down the playable essays on human networks and the spread of complex ideas, self-synchronization in nature, the shape of society and several other burning themes (like coming out and anxiety) and watching Nicky Case’s talks, like this one. Nicky is a self-made indie artist, programmer and writer making very edgy, very 21st century multimedia products that are both profound in content and have an engaging/interactive interface. It’s as if reading an informative piece is turned into a game. And that’s exactly what Nicky stands for: learning through play and messing about. Maybe that’s why Simon has embraced his works so eagerly, Nicky has proven to be one of those perfect matches for our self-directed learning style.
Thanks to the lock-down, Simon’s got new friends. For a little over a month now, he has been part of exciting daily discussions, challenging coding sessions and just playing together with his new gang (warning: playing always involves math). We’ve never seen him like this before, so drawn to socializing with his peers, even taking the lead in some meetings and initiating streams.
And then we realized: this is how social Simon is once he meets his tribe and can communicate in his language, at his level. Most of his new friends are in their late teens and early twenties. Most of them didn’t use to hang out together before the crisis, probably busy with college, commuting, etc. The extraordinary circumstances around covid-19 has freed up some extra online time for many talented young people, creating better chances to meet like-minded peers across the world. Finally, Simon has a group of friends he can really relate to, share what he is working on, ask for constructive help. And even though he is the youngest in the group, he is being treated as an equal. It’s beautiful to overhear his conversations and the laughter he shares with the guys (even though sometimes I wish he wasn’t listening to a physics lecture simultaneously, his speakers producing a whole cacophony of sound effects, but he likes it that way and seems to be able to process two incoming feeds at once).
Last week, Simon took part in a World Science Scholars workshop by Dr. Ruth Gotian, an internationally recognized mentorship expert. The workshop was about, you guessed it, how to go about finding a mentor. One of the things that struck me most in Dr. Gotian’s presentation was her mentioning the importance of ‘communities of practice’. I looked it up on Etienne Wenger’s site (the educational theorist who actually came up with the term in the 1990s):
A community of practice is a group of people who share a concern or a passion for something they do, and learn how to do it better as they interact regularly. This definition reflects the fundamentally social nature of human learning. It is very broad. It applies to a street gang, whose members learn how to survive in a hostile world, as well as a group of engineers who learn how to design better devices or a group of civil servants who seek to improve service to citizens. their interactions produce resources that affect their practice (whether they engage in actual practice together or separately).
It is through the process of sharing information and experiences with the group that members learn from each other, and have an opportunity to develop personally and professionally, Wenger wrote in 1991. But communities of practice isn’t a new thing. In fact, it’s the oldest way to acquire and imperfect one’s skills. John Dewey relied on this phenomenon in his principle of learning through occupation.
It has been almost spooky to observe this milestone in Simon’s development and learn the sociological term for it the same month, as if some cosmic puzzle has clicked together.
Of course, it would be a misrepresentation to say nothing of the internal conflict the new social reality unveiled in my mothering heart as I struggled to accept that Simon started skipping Stephen Wolfram’s livestreams in favour of coding together with his new friends. 👬Yet even those little episodes of friction we experienced have eventually led to us understand Simon better. We sat down for what turned into a very eye-opening talk, which involved Simon asking me to take down the framed Domain of Science posters we’d recently put up above his desktop and pointing to those infographics depicted on the posters that represented the areas of his greatest interest.
Simon simply guided us through the Doughnut of Knowledge, Map of Physics, Map of Computer Science and Map of Mathematics posters as if were on tour inside his head. And he made it clear to us that he seriously preferred pure mathematics, theoretical computer science and computer architecture and programming to applied mathematics (anything applied, really) and even computational physics, even though he genuinely enjoyed cosmology and Wolfram’s books.
“Mom, you always think that what you’re interested in is also what I’m interested in”, he told me openheartedly. It was at that moment it hit me he had grown up enough to gain a clearer vision of his path (or rather, his web). That I no longer needed to absolutely expose him to a broadest possible plethora of the arts and sciences within the doughnut of knowledge, but that from now on, I can trust him even more as he ventures upon his first independent steps in the direction he has chosen for himself, leaning back on me when necessary.
So far, in just one month, Simon has led a live covid-19 simulation stream, programming in JS as he got live feedback from his friends, cooperated on a 3D rendering engine in turtle (🤯), co-created Twitch overlays, participated in over a hundred Clashes of Code (compelling coding battles) and multiple code katas (programming exercises with a bow to the to the Japanese concept of kata in the martial arts).
Last month, ten young programmers including Simon formed a separate “Secret Editors’ Club Riding Every Train” group on Discord, uniting some “nice and active” people who met on The Coding Train channel (they also included Dan Shiffman in the group). Simon really enjoys long voice chats with the other secret editors, going down the rabbit holes of math proofs and computer algorithms. Last Tuesday, he was ecstatic recounting his 3-hour call with his new peer Maxim during which Simon managed to convince Maxim that 0.999… equals 1 by “presenting a written proof that involved Calculus”:
We even talked about infinity along the way, aleph null and stuff. There was a part where he almost won, because of the proof I showed him when we talked about infinities. I was almost stumped.
The guys have now inspired Simon to take part in the Spring Challenge 2020 on CodingGame.com, a whole new adventure. To us, the lockdown experience has felt like an extra oxygen valve gone open in our world, another wall gone down, another door swung open, all allowing Simon to breathe, move and see a new horizon.
What has been your silver lining during this COVID-19 crisis so far, in terms of self-directed learning? Simon is happy that Grant Sanderson, Stephen Wolfram and Brian Greene all have more time now to make frequent streams and tutorials. In fact, he can’t even follow all of them live as they often overlap!
Luckily, years of homeschooling have allowed us to develop a very flexible approach to daily routine, enabling us to embrace learning opportunities from across the Atlantic, that mostly present themselves in the evening hours. Our learning is circular, cyclical, not linear (we learn around the clock and Simon often returns to the topics he has already covered before but at a new level).
Brian Greene publishes daily videos called “Your Daily Equations” on the World Science Festival channel, and viewers can “order” which equation they want to discuss next. He also does a weekly live Q&A.
It’s funny how both Wolfram and Greene are Simon’s professors as part of the World Science Scholars program, but he seems to have gotten a better chance to engage with them personally now that we’re all stuck at home (through the live chat and comments) than during the official World Science Scholars sessions!
December was all about computer science and machine learning. Simon endlessly watched Welch Labs fantastic but freakishly challenging series Learning to See and even showed me all the 15 episodes, patiently explaining every concept as we went along (like underfitting and overfitting, recall, precision and accuracy, bias and variance). Below is the table of contents he made of the series:
While watching the series, he also calculated the solutions to some of the problems that Welch Labs presented, like the question about the number of possible rules (= grains of sand) for a simple ML problem if memorisation is applied. His answer was that the grains of sand would cover all land on earth:
Simon loved the historical/philosophical part of the course, too. Especially the juxtaposition of memorising vs. learning, the importance of learning to make assumptions, futility of bias-free learning, and the beautiful quotes from Richard Feynman!
I have since then found another Feynman quote that fits Simon’s learning style perfectly (and I believe is the recipe to anyone’s successful learning as opposed to teaching to the test): “Study hard what interests you the most in the most undisciplined, irreverent and original manner possible.” We have discussed the possibilities of continuing at the university again. I have also asked Simon how he sees himself applying his knowledge down the road, trying to understand what academic or career goals he may have set for himself, if any. Does he have a picture of himself in five years from now, where does he want to be by then? He got very upset, just like when asked to sum himself up in one sentence for an interview last spring. “Mom, I’m just having fun!”
Simon’s just finished auditing a class at the University of Antwerp. His first experience at the university came via a road less traveled. But then again, one may argue that we all walk the road less traveled because there’s no “normal pathway” that fits everyone.
Last spring, I shared a few videos of Simon studying at home and a couple of university professors in his MathsJam club mentioned he would probably enjoy a course in Complex Analysis (Calculus with complex numbers). I grabbed that opportunity and asked whether they would actually allow him to sit in the lectures.
Simon audited the course for one full semester (September to December), with me accompanying him to all the lectures to make sure he didn’t disturb anyone with his “youthful enthusiasm”. Before we arrived at the first lecture, I’d made it clear to Simon that we absolutely must remain silent in class. I wasn’t sure he would manage to control himself, for the main reason that had never managed to do so before, not even at the theatre. But then again, maybe at the theatre he sensed that the condition wasn’t as crucial. On our first day, I knew the professor was nervous about Simon possibly disturbing the class, I was nervous myself and I couldn’t believe how nervous Simon suddenly was. There was one thought nagging me: Have I spoiled it by my stern warning about keeping quiet?
Simon kept incredibly quiet. He didn’t even dare introduce himself. I had never seen him this way before. The professor was relieved, even elated. On my part, I was shocked by the high level of the course and whether Simon was too tense to tune in. The course turned out to be for college seniors; in Simon’s case, possibly a year or two too early. With Simon you never know. He always learns top down, and when I say “top” I mean Mount Everest top. “We try a couple more lectures and then see if it’s too much for you”, I told Simon.
The second and the third time, he was still quite nervous, but later he let go of most of that tension. Several times he got very bored, two hours felt like a long time for him to sit quietly. Still he said he didn’t intend to quit. And once, at the end of October, at the moment when I positively lost it and didn’t have any clue about what the professor was talking about anymore, he whispered in my ear: “Now it’s actually getting interesting!” During the break, he summed up the general idea about the zeta function and the professor said he understood it correctly.
I don’t like asking Simon how much he understands every time. I don’t think it’s a fair question to ask. We didn’t attend the practice section of the course because it didn’t match Simon’s schedule (the practice lesson started early in the morning and was impossible to combine with Simon’s late night classes from New York). Auditing a class doesn’t involve any compulsory attendance, Simon won’t be doing the exam. During the last several sessions, he was relaxed about being able to control the volume of his voice and sit quietly when necessary. It was at the uni that I heard him whisper for the first time! At the last lecture, he was treated to his favourite topic, the zeta function.
My general conclusion is that auditing a course has been a nice way to get exposed to what studying at the uni is like, even though we may have picked the wrong course in terms of difficulty level or in terms of what interests Simon at the moment (contrary to last spring, when he was all about calculus and complex numbers, he is currently investing most of his time into logic, computer science and computer electronics). He definitely still misses a lot of fundamental knowledge, especially in integral calculus, but by now I’m familiar with his learning style and know that he will come back to what he hasn’t dealt with properly when the time is ripe, at the new turn of the spiral, so to speak.
I know attending classes won’t be Simon’s primary source of knowledge as he learns best through self-study (mainly videos and books), but such experiences are definitely going to mean something both in terms of personal growth and mathematical thinking. “Do you want to audit a more fundamental calculus or integral calculus class here at the uni?” I asked him the other day. “No, of course not! I can just learn that on Brilliant!” he answered. “A course on sequences perhaps, as suggested by one of the professors?” – “No, I don’t want to”, – Simon replied.
Maybe we’ll be back at the uni at a later stage, with more practical discussion involved instead of passive listening, and in a subject/at a level he feels less timid to actively contribute to that discussion. What would also help is if there was a more official way to follow university courses for bright young minds like Simon. At the moment, it’s only possible as a personal favour or if I sign myself in and take Simon along, which contributed to Simon’s timidity and being afraid to feel present.
We’ll just be taking it one step at a time, grateful for the freedom that we have. My very special thanks go to Simon’s math professor who has a kind and courageous heart. He has also signed his newly published book for Simon:
Is there an equation for intelligence? Yes. It’s F = T ∇ Sτ.
Prior to a World Science Scholars live session on November 25, Simon had been asked to watch this TED talk given by a prominent computer scientist and entrepreneur, Alex Wissner-Gross, on intelligent behavior and how it arises. Upon watching the talk, Simon and I have discovered that the main idea presented by Wissner-Gross can serve as a beautiful scientific backbone to self-directed learning and explain why standardized and coercive instruction contradicts the very essence of intelligence and learning.
What you’re seeing is probably the closest equivalent to an E = mc² for intelligence that I’ve seen. So what you’re seeing here is a statement of correspondence that intelligence is a force, F, that acts so as to maximize future freedom of action. It acts to maximize future freedom of action, or keep options open, with some strength T, with the diversity of possible accessible futures, S, up to some future time horizon, tau. In short, intelligence doesn’t like to get trapped.Intelligence tries to maximize future freedom of action and keep options open. And so, given this one equation, it’s natural to ask, so what can you do with this? Does it predict artificial intelligence?
Recent research in cosmology has suggested that universes that produce more disorder, or “entropy,” over their lifetimes should tend to have more favorable conditions for the existence of intelligent beings such as ourselves. But what if that tentative cosmological connection between entropy and intelligence hints at a deeper relationship? What if intelligent behavior doesn’t just correlate with the production of long-term entropy, but actually emerges directly from it?
As an example, Wissner-Gross went on to demonstrate a software engine called Entropica, designed to maximize the production of long-term entropy of any system that it finds itself in. Entropica was able to pass multiple animal intelligence tests, play human games, and even earn money trading stocks, all without being instructed to do so. Note that Entropica wasn’t given learning goals, it simply decided to learn to balance a ball on a pole (just like a child decides to stand upright), decided to use “tools”, decided to apply cooperatvive ability in a model experiment (just like animals sometimes pull two cords simultaneously to release food), taught itself to play games, network orchestration (keeping up connections in a network), solve logistical problems with the use of a map. Finally, Entropica spontaneously discovered and executed a buy-low, sell-high strategy on a simulated range traded stock, successfully growing assets. It learned risk management.
The urge to take control of all possible futures is a more fundamental principle than that of intelligence, that general intelligence may in fact emerge directly from this sort of control-grabbing, rather than vice versa.
In other words, if you give the agent control, it becomes more intelligent.
“How does it seek goals? How does the ability to seek goals follow from this sort of framework? And the answer is, the ability to seek goals will follow directly from this in the following sense: just like you would travel through a tunnel, a bottleneck in your future path space, in order to achieve many other diverse objectives later on, or just like you would invest in a financial security, reducing your short-term liquidity in order to increase your wealth over the long term, goal seeking emerges directly from a long-term drive to increase future freedom of action”.
The main concept we can pass on to the new generation to help them build artificial intelligences or to help them understand human intelligence, according to Alex Wissner-Gross is the following: “Intelligence should be viewed as a physical process that tries to maximize future freedom of action and avoid constraints in its own future. Intelligence is a physical process that resists future confinement”.
Simon’s reaction to Alex Wissner-Gross’s TED Talk was: “But this means school only makes you less intelligent!” (in the sense that school reduces your chances at seeking goals yourself, introduces constraints on your future development).
During the actual live session, neuroscientist Suzana Herculano-Houzel, famous for inventing a method to count the exact number of neurones in the human brain and comparative studies of various species, defined intelligence as behavioral and cognitive flexibility. Flexibility as a choice to do something else than what would happen inevitably, no longer being limited to purely responding to stimuli. Flexibility in decisions that allow you to stay flexible. Generically speaking, the more flexibility the more intelligence.
Animals with a cerebral cortex gained a past and a future, Professor Herculano-Houzel explained. Learning is one of the results of flexible cognition. Here learning is understood as solving problems. Hence making predictions and decisions is all about maximizing future flexibility, which in turn allows for more intelligence and learning. This is very important guideline for educational administrations, governments and policy makers: allowing for flexibility. There is a problem with defining intelligence as producing desired outcomes, Herculano-Houzel pointed out while answering one of the questions from students.
Replying Simon’s question about whether we can measure intelligence in any way and what the future of intelligence tests could be like, Professor Herculano-Houzel said she really liked Simon’s definition of IQ testing as a “glorified dimensionality reduction”. Simon doesn’t believe anything multidimensional fits on a bell curve and can possibly have a normal distribution.
Professor Herculano-Houzel’s answer:
Reducing a world of capacities and abilities into one number, you can ask “What does that number mean?” I think you’d find it interesting to read about the history of the IQ test, how it was developed and what for, and how it got coopted, distorted into something else entirely. It’s a whole other story. To answer your question directly, can we measure intelligence? First of all, do you have a definition for intelligence? Which is why I’m interested in pursuing this new definition of intelligence as flexibility. If that is an operational definition, then yes, we can measure flexibility. How do we measure flexibility?
Professor went on to demonstrate several videos of researches giving lemurs and dogs pieces of food partially covered by a plastic cylinder. The animals would have to figure it out on their own how to get to the treat.
You see, the animal is not very flexible, trying again and again, acting exactly as before. And the dog that has figured it out already made its behavior flexible. It can be measured how long it takes for an animal to figure out that it has to be flexible, which you could call problem solving. Yes, I think there are ways to measure that and it all begins with a clear definition of what you want to measure.
As a side note, Professor Herculano-Houzel also mentioned in her course and in her live session that she had discovered that a higher number of neurons in different species was correlated with longevity. Gaining flexibility and a longer life, it’s like having the cake and eating it! We are only starting to explore defining intelligence, and it’s clear that the biophysical capability (how many neurons one has) is only a starting point. It is through our experiences of the world that we gain our ability and flexibility, that is what learning is all about, Professor concluded.
“When the overall behavior is complex, it becomes impossible to characterize it in any complete way by just a few numbers”, Stephen Wolfram writes in A New Kind of Science. Simon: “This is like the essence of my life!”
Today, Simon returned to a problem he first encountered at a MathsJam in summer: “Pick random numbers between 0 and 1, until the sum exceeds 1. What is the expected number of numbers you’ll pick?” Back in June, Simon already knew the answer was e, but his attempt to prove it didn’t quite work back then. Today, he managed to prove his answer!
The same proof in a more concise way:
At MathsJam last night, Simon was really eager to show his proof to Rudi Penne, a professor from the University of Antwerp who was sitting next to Simon last time he gave it a go back in June. Rudi kept Simon’s notes and told me he really admired the way Simon’s reasoning spans borders between subjects (the way Simon can start with combinatorics and jump to geometry), something that many students nurtured within the structured subject system are incapable of doing, Rudi said. Who needs borders?
Later the same evening, Simon had a blast demonstrating the proof to a similar problem to a larger grateful and patient audience, including Professor David Eelbode. The first proof was Simon’s own, the second problem (puzzle with a shrinking bullseye) and proof came from Grant Sanderson (3Blue1Brown) on Numberphile.
“Don’t allow any constraints to dull his excitement and motivation!” Rudi told me as Simon was waiting for us to leave. “That’s a huge responsibility you’ve got there, in front of the world”.
Such a pleasant play date last week with another eager learner. Simon shared his GeoGebra skills and some geometrical paper tricks, among other things. It’s heartwarming to see Simon blossom socially, he is growingly attentive to younger kids and generally engaging with people of various ages, as long as they show interest in anything Simon has an understanding of.