A Mathematical Model for Human Intelligence

Curiosity and Intelligence

People have been trying to figure out what intelligence is, and how it differs from person to person, for centuries. Much has been written on the subject, and some of this work has helped people. Unfortunately, much harm has been done as well. Consider, for example, the harm that has been done by those who have had such work tainted by racism, sexism, or some other form of “us and them” thinking. This model is an attempt to eliminate such extraneous factors, and focus on the essence of intelligence. It is necessary to start, therefore, with a clean slate (to the extent possible), and then try to figure out how intelligence works, which must begin with an analysis of what it is.

If two people have the same age — five years old, say — and a battery of tests have been thrown at them to see how much they know (the amount of knowledge at that age), on a wide variety of subjects, person A (represented by the blue curve) may be found to know more, at that age, than person B (represented by the red curve). At that age, one could argue that person A is smarter than person B. Young ages are found on the left side of the graph above, and the two people get older, over their lifespans, as the curves move toward the right side of the graph.

What causes person A to know more than person B, at that age? There can be numerous factors in play, but few will be determined by any conscious choices these two people made over their first five years of life. Person B, for example, might have been affected by toxic substances in utero, while person A had no such disadvantage. On the other hand, person A might simply have been encouraged by his or her parents to learn things, while person B suffered from parental neglect. At age five, schools are not yet likely to have had as much of an impact as other factors.

An important part of this model is the recognition that people change over time. Our circumstances change. Illnesses may come and go. Families move. Wars happen. Suppose that, during the next year, person B is lucky enough to get to enroll in a high-quality school, some distance from the area where these two people live. Person B, simply because he or she is human, does possess curiosity, and curiosity is the key to this model. Despite person B‘s slow start with learning, being in an environment where learning is encouraged works. This person begins to acquire knowledge at a faster rate. On the graph, this is represented by the red curve’s slope increasing. This person is now gaining knowledge at a much faster rate than before.

In the meantime, what is happening with person A? There could be many reasons why the slope of the blue curve decreases, and this decrease simply indicates that knowledge, for this person, is now being gained at a slower rate than before. It is tempting to leap to the assumption that person A is now going to a “bad” school, with teachers who, at best, encourage rote memorization, rather than actual understanding of anything. Could this explain the change in slope? Yes, it could, but so could many other factors. It is undeniable that teachers have an influence on learning, but teacher quality (however it is determined, which is no easy task) is only one factor among many. Encouraging the “blame the teacher” game is not the goal of this model; there are already plenty of others doing that.

Perhaps person A became ill, suffered a high fever, and sustained brain damage as a result. Perhaps he or she is suddenly orphaned, therefore losing a previous, positive influence. There are many other possible factors which could explain this child’s sudden decrease of slope of the blue “learning curve” shown above; our species has shown a talent for inventing horrible things to do to, well, our species. Among the worst of the nightmare scenarios is that, while person B is learning things, at a distant school, the area where person A still lives is plunged into civil war, and/or a genocide-attempt is launched against the ethnic group which person A belongs to, as the result of nothing more than an accident of birth, and the bigotry of others. Later in life, on the graph above, the two curves intersect; beyond that point, person B knows more than person A, despite person B‘s slow start.  To give credit, or blame, to either of these people for this reversal would clearly be, at best, a severely incomplete approach.

At some point, of course, some people take the initiative to begin learning things on their own, becoming autodidacts, with high-slope learning curves. In other words, some people assume personal responsibility for their own learning. Most people do not. Few would be willing to pass such judgment on a child who is five or six years old, but what about a college student? What about a high school senior? What about children who have just turned thirteen years old? For that matter, what about someone my age, which is, as of this writing, 48? It seems that, the older a person is, the more likely we are to apply this “personal responsibility for learning” idea. Especially with adults, the human tendency to apply this idea to individuals may have beneficial results. That does not, however, guarantee that this idea is actually correct.

I must stop analyzing the graph above for now, because the best person for me to examine, at this point, in detail, is not on the graph above. He is, however the person I know better than anyone else: myself. I’ve been me now for over 48 years, and have been “doing math problems for fun” (as my blog’s header-cartoon puts it) for as long as I can remember. This is unusual, but, if I’m honest, I have to admit that there are inescapable and severe limits on the degree to which I can make a valid claim that I deserve credit for any of this. I did not select my parents, nor did I ask either of them to give me stacks of books about mathematics, as well as the mathematical sciences. They simply noticed that, when still young, I was curious about certain things, and provided me with resources I could use to start learning, early, at a rapid rate . . . and then I made this a habit, for, to me, learning is fun, if (and only if) the learning is in a field I find interesting. I had absolutely nothing to do with creating this situation. My parents had the money to buy those math books; not all children are as fortunate in this respect. Later still, I had the opportunity to attend an excellent high school, with an award-winning teacher of both chemistry and physics. To put it bluntly, I lucked out. As Sam Harris, the neuroscientist, has written, “You cannot make your own luck.”

At no point in my life have I managed to learn how to create my own luck, although I have certainly tried, so I have now reached the point where I must admit that, in this respect, Sam Harris is correct. For example, I am in college, again, working on a second master’s degree, but this would not be the case without many key factors simply falling into place. I didn’t create the Internet, and my coursework is being done on-line. I did not choose to be born in a nation with federal student loan programs, and such student loans are paying my tuition. I did not create the university I am attending, nor did I place professors there whose knowledge exceeds my own, regarding many things, thus creating a situation where I can learn from them. I did not choose to have Asperger’s Syndrome, especially not in a form which has given me many advantages, given that my “special interests” lie in mathematics and the mathematical sciences, which are the primary subjects I have taught, throughout my career as a high school teacher. The fact that I wish to be honest compels me to admit that I cannot take credit for any of this — not even the fact that I wish to be honest. I simply observed that lies create bad situations, especially when they are discovered, and so I began to try to avoid the negative consequences of lying, by breaking myself of that unhelpful habit. 

The best we can do, in my opinion, is try to figure out what is really going on in various situations, and discern which factors help people learn at a faster rate, then try to increase the number of people influenced by these helpful factors, rather than harmful ones. To return to the graph above, we will improve the quality of life, for everyone, if we can figure out ways to increase the slope of people’s learning-curves. That slope could be called the learning coefficient, and it is simply the degree to which a person’s knowledge is changing over time, at any given point along that person’s learning-curve. This learning coefficient can change for anyone, at any age, for numerous reasons, a few of which were already described above. Learning coefficients therefore vary from person to person, and also within each person, at different times in an individual’s lifetime. This frequently-heard term “lifelong learning” translates, on such graphs, to keeping learning coefficients high throughout our lives. The blue and red curves on the graph above change slope only early in life, but such changes can, of course, occur at other ages, as well.

It is helpful to understand what factors can affect learning coefficients. Such factors include people’s families, health, schools and teachers, curiosity, opportunities (or lack thereof), wealth and income, government laws and policies, war and/or peace, and, of course, luck, often in the form of accidents of birth. Genetic factors, also, will be placed on this list by many people. I am not comfortable with such DNA-based arguments, and am not including them on this list, for that reason, but I am also willing to admit that this may be an error on my part. This is, of course, a partial list; anyone reading this is welcome to suggest other possible factors, as comments on this post. 

3 thoughts on “A Mathematical Model for Human Intelligence

  1. Reading your experiences, influenced by Asperger’s syndrome, makes me remember the very similar experiences from my life. I also became curios about the mathematics from early childhood. I also do the math for fun. And my “learning coefficient” increases over the time. However, at junior school I was probably considered a moron. Today nobody believes me if I tell him about these early years.

    Regarding the “learning curve” I personally consider that each person, who has some interest to life, has more or less the same interest to learn and speed of learning. However personal situation, and other factors greatly influence what exactly a person learns. The different learning paradigms include, but are not limited to:
    – Learn fundamental knowledge, that doesn’t change over the time;
    – Study quickly changing information, which means learning much every day with no increasing of knowledge over the time;
    – Learn foreign languages, which is a difficult and time consuming task, but in result it duplicates, and not increases, someone’s knowledge;
    – Learn social habits, including emotional intelligence.
    The intelligence tests for these categories of persons will greatly differ while all them involve about the same amount of information.

    Another point is the speed at which some person looses his knowledge – his forgetfulness. And different persons forget different amount of their knowledge. Many people can remember less then 10% of all the information they knew at school. When they become parents and their children are at the school, they realize they forget much part of what is studied there.

    Liked by 1 person

    • You raise some good points. To address one of them: if one is losing knowledge over time (on net), then that part of the person’s learning curve would have a negative slope. As I wrote the post, I thought about including negative slope in learning curves, but Alzheimer’s disease was what I thought of that could cause negative slope in a person’s learning curve. I know realize why I chose not to include anything on this subject in the original post, and it’s quite simple: Alzheimer’s scares me intensely.

      I make no claim to be rational, nor objective, on the subject of Alzheimer’s (or similar conditions), so I don’t blog about such subjects. Mentioning such topics in a comment is, of course, different.

      Regarding short-term, typical forgetting of what people hear in a classroom: such knowledge was, in my opinion, never really learned, and so would never affect the learning curve in the first place. Learning is, to me, an entirely different process from rote memorization.

      As far as learning on different subjects, I did lump them altogether for the sake of keeping this model simple (for now). One could, of course, show (for one person), a mathematical learning curve, a linguistic learning curve, or even a primarily-kinesthetic, completely sports-related learning curve. That would be the next logical step in the development of this model: establishing definitions for the various types of learning curves, based on different types of learning, or learning in different subject areas. I make no claim that this model is complete, and would love to see someone else improve upon it. It isn’t copyrighted, and if you read my “Note Regarding Copyright,” on any page of this blog, you’ll see why.

      Thank you for your input!


  2. The theory of multiple intelligences differentiates human intelligence into specific ‘modalities’, rather than seeing intelligence as dominated by a single general ability. Howard Gardner proposed this model in his 1983 book Frames of Mind: The Theory of Multiple Intelligences. According to the theory, an intelligence ‘modality’ must fulfill eight criteria:

    potential for brain isolation by brain damage
    place in evolutionary history
    presence of core operations
    susceptibility to encoding (symbolic expression)
    a distinct developmental progression
    the existence of savants, prodigies and other exceptional people
    support from experimental psychology
    support from psychometric findings

    Gardner proposed eight abilities that he held to meet these criteria:


    However, speaking about Asperger’s, I have to say it is linked to the Central Nervous System, for better or worse. According to Gardner’s model, an Asperger’s maybe could or would not possess bodily-kinesthetic, interpersonal, or even intrapersonal skills, although the other ones would/could fit in (visual-spatial, logical-mathematical, verbal-linguistic, or even naturalistic). I screwed up the mathematical part in my IQ test when I was 14 years old, but because of my visual-spatial and verbal-linguistic skills I beat the Wechsler Adult Intelligence Scale the psychologist thrust upon me.

    Although the learning coefficient can change for anyone, at any age, for numerous reasons, even high functioning autism keeps individuals still at a disadvantage. Autism spectrum disorders are generally thought to be caused by deficits in brain development, that at least some aspects of the disorder — including how touch is perceived, anxiety, and social abnormalities — are linked to defects in another area of the nervous system, the peripheral nerves found throughout the limbs, digits, and other parts of the body that communicate sensory information to the brain. (https://www.sciencedaily.com/releases/2016/06/160609134247.htm)

    What’s amazing is what’s being said about the cerebellum. I include myself on this one:
    while it is best known for its role in movement, the cerebellum is also important in higher cognitive functions such as language and attention, both of which are affected in autism. People with autism often have language delays and pay unusually intense attention to objects or topics that interest them. (https://www.sciencedaily.com/releases/2017/11/171102131330.htm).

    I think these studies apply even to high functioning autism, in the sense that I myself take on new information atypically. It’s not through the typical sensory channels everyone else uses. I’m tuned to the environment differently. I am more sensitive to sounds, lights, vibrations, therefore ‘patterns’ form in my mind. This is why I feel I absorb more information than a ‘typical’ person does.


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s