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
In a complete binary tree, every node has two children (except for the bottom nodes that don’t have any children at all). This means one mind-blowing thing: that the bottom row always has more nodes than the number of nodes in the entire rest of the tree! Example: if there’s one node at the top of the tree, two nodes in the second row, four nodes in the third row and eight nodes in the bottom row, the bottom row has more nodes (8) than the remaining part of tree (7). I’ve been thinking about this, and I applied this to the real world:
The average number of children a parent has in the world is 2.23 (I’ve used an arithmetic mean, which is oversimplistic, should have probably used the harmonic mean). Does this mean that currently, the number of children exceeds the number of parents? The definition of “children” I’m using are people who don’t have children, so the last row of nodes so to speak. By “parents” I’m counting all generations. If you just want to talk about now, the parents living now, then you have to trim the top rows (the already dead generations). If the average number of children is 2 or more, are there going to be more children in the world than parents?
Well, in this model, I’m ignoring crossover. This means we should consider every node in our tree for 2 people*. So, now, if the average number of children is 4 or more, there’re going to be more children than parents. So, what I said earlier was wrong. The average number of people doesn’t exceed 4, so there aren’t more children than parents. But the number of children today may still exceed the number of parent generations still alive.
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.
In reaction to Yuval Noah Harari’s book Homo Deus (the part about humans evolving to break out of the organic realm and possibly breaking out of planet Earth):
When you cross the street there’s always a risk that an accident will happen that has a non-zero probability. If you live infinitely long, anything that has a non-zero probability can happen infinitely many times in your life. For example, if the event we are talking about is an accident, the first time it will happen in your life, you’re already dead. So when you cross the street and want to live infinitely long there’s a risk that an accident will happen and you die. So we come to the conclusion, that if you want to live infinitely long it’s not worth crossing the street. But there’s always a risk that you die, so if you live infinitely long, it’s not actually worth living. So we’ve got a little bit of a problem here. Unless you come to the more extreme idea of detaching yourself from the physical world all together. And I’m not talking about the sort of thing that you don’t have a body, but somehow still exist in the physical world. I mean literally that you live in a pure mathematical world. Because in mathematics, you can have things that have zero probability of happening. You can have something definitely happening and you can also have something that is definitely not happening.
However, there’s another thing. How does mathematics actually work? There are these things called axioms and it’s sort of built up from that. What if we even do away from those axioms? Then we can actually do anything in that mathematical world. And what I mean by anything is really anything that you can from any set of axioms that you can come up with. There’s a little bit of a problem with that, you can come to contradictions, it’s a little bit risky. We are really talking about the ultimate multiverse, we’re talking about quite controversial stuff here. The only way anyone can come up with this is by pushing to the extremes.
In reaction to Yuval Noah Harari’s book Homo Deus (the paragraph about the a-mortals anxious about dying in an accident):
With individual intelligences, you can have the car that’s driving down the street not knowing that you are going to be crossing the street at that point in time and then poof! You got yourself an accident. With collective intelligence though, that doesn’t happen. Because the whole definition of knowing something or not knowing something breaks down. The members of collective intelligence don’t have the notion of knowing something. It’s only the “central intelligence” that the members are hooked up to that has the notion of knowing something. Which means that you can have the central intelligence deciding that a car driving down the street does not create an accident with the person crossing the street.
In this video, Simon (a 9 year old mathematician and programmer) shares his views on what absolutely needs to change in the educational system and why self-directed learning works better than traditional schooling. Simon’s main points are:
At school, you’re forced to master a few subjects at the same “average” level while when learning at your own pace you tend to follow your talent and passion and learn some subjects at a much higher level. Simon depicts this difference as two bar chart diagrams. On the schooled diagram, there’re fewer subjects/areas of exploration and they are all at about the same level. On the self-directed learning diagram, the bars resemble a diverse metropolis with multiple buildings of varying height (or a garden with many sorts of flowers). Simon also explains that standardized tests and IQ tests expect a child to have developed evenly in all areas, while it may be more natural for a child to be much more developed in a few specific areas depending on her interests, and that there therefore such a thing as a total score simply shouldn’t exist.
Simon’s second point is that the internet, with its online educational opportunities, is going to kill traditional schooling. Simon himself is a perfect example of someone who learns a lot more on than off line.
Simon regrets that the way a student’s proficiency is evaluated today is mainly based on testing the speed at which the student can apply trained strategies as opposed to looking at the student’s original problem solving ability in an untimed setting.
Simon’s final argument against the traditional school system is that it doesn’t allow for failure. Failure is being discouraged and stigmatized, a bad grade can have serious consequences. That is very counterproductive, says Simon, because failure is an important part of the learning process. You don’t learn from your successes (when you have simply used what you already knew), you learn from your failures (because you start to look into why you’ve failed and that makes you a little smarter every time).
Simon has been teaching himself since he was a toddler. He is especially fond of math and sciences, doing university level math and researching serious questions about quantum mechanics and general relativity. He is also fluent in several programming languages. We have had to move from Simon’s native Amsterdam to Belgium to be able to homeschool, because homeschooling is nearly illegal in The Netherlands. Simon is an adamant advocate of educational freedom. There is a growing body of evidence that forced learning is not only ineffective and damaging to the intrinsic motivation, but may also be psychologically detrimental.
This is the fourth video in Simon’s short series Infinities Driving You Mad. In this episode, Simon attempts to start to comprehend indescribable numbers. To Simon’s knowledge, no one has ever made a video about indescribable numbers on YouTube before. Simon is planning to record a follow-up to this video, something like part 4b.
“I can see that your son has native speaker skills, but we still cannot give him a passing grade”, the English examinator told me in an apologetic tone of voice. She and her colleague at the Brussels examination committee had just finished their assessment of Simon’s oral English and brought Simon, a whole storm of emotions on his face, back to me in the waiting room.
“We were just wondering, does he speak Dutch? We weren’t sure he understood the tasks and they were written in Dutch”, — the examinator was sympathetic of Simon’s young age as most of the other kids taking the same test were about 6 years his seniors. As it turned out, the first task was to describe several photos of “criminals” (one of them with many piercings), the second task involved choosing two things that Simon would like to do from a list of recreational activities (the list included an escape room and a Stonehenge trip). “I just didn’t know what to say!” Simon was catching his breath in between the sobs. “It’s an impossible question, because I had to choose two things I like from a list where there wasn’t anything I liked!” The examinator suggested Simon could have said why he disliked those things. “If you don’t find something interesting you just don’t find it interesting, it’s a given fact! You can’t explain it!” – he told her in English.
Another fact is that Simon wouldn’t be able to perform these tasks in any of the three languages he speaks. Not because his vocabulary or grammar don’t stretch that far. I often hear him construct amazingly intact sentences, which I immediately record, like this one recently: “This is incredible! We’ve found a connection between a discrete problem, of what’s the smallest number that divides all of the numbers in a given sequence, to a continuous problem, of what is the fundamental frequency of a combination of sine waves. In other words, we found a discrete solution to a continuous problem!” Simon loves deep philosophical or scientific questions, but often cannot answer open questions lacking substance. He doesn’t care if you ask him to describe someone’s looks on a picture, it’s not important to him. He doesn’t know how to pick two things he likes from a list of things he doesn’t like. It’s just the way his mathematical brain is wired.
“Can I send you one of the many videos on Simon’s YouTube channel as an alternative proof of his excellent oral English skills?” I asked, still shocked at the absurdity of the situation. “Because I dare to say Simon speaks English better than any other student you have examined today”. The examinator agreed that I was probably right in my judgement but couldn’t accept anything else but a completed exam task.
Although distressed about what Simon had to go through, I can’t help feeling content with today’s scoop. What can provide a more obvious proof that exams don’t do a good job measuring one’s skill than this example of a 9 year old who gives hour-long science lessons on YouTube, speaks at grown-up creative coding meet-ups and is often mistaken for a native speaker, but doesn’t pass his oral English exam because he’s being asked questions that don’t interest him?
It wasn’t Simon who failed today, it was the exam that failed to measure his English. And this raises a whole lot of questions. Why is this system of measurements, that clearly doesn’t work for everyone, has become decisive in how our society views someone’s ability? And what is the use of spending so much money and nervous cells on something that doesn’t work?
Wouldn’t it be more fair towards both the students and anyone who honestly wants to know their level to actually look at what they can do with their knowledge in real life (their actual projects, videos of their social engagement) instead of the fake setting at the exam? Wouldn’t it be wiser to observe a student’s gradual progress in a given area, instead of stressing the students out and giving them the impression that it’s all about the examinator checking off that box and they can forget what they have learned the next day, because all that matters in our society is the passing grade?
“I’m so neutral about this”, Simon told me (in English) when he was lying in bed the same evening. “Because on the one hand, I kind of feel bad. And on the other hand, it’s so beautiful how we sort of accidentally taught them how exams can show false negatives or false positives. Because the exam showed a false negative. Even the examinators know it’s a false negative”.