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. 

The Inverted Popularity of This Aspie’s Phobias and Philias, Part I: An Explanation

phobias and philias

The image above contains three colors: white, black, and red. The words appear in red because I see it as a color denoting positive or negative intensity, and phobias and philias are both certainly intense. To “see red,” I have learned, does not usually mean what it would mean if I said it myself. Consistent with Asperger’s Syndrome, which I have, I tend to be almost completely literal in the words I use, while the non-Aspie majority often uses words in confusing (to me) non-literal ways. Over the years, I have figured out that this phrase means, when non-Aspies say it,  that they are extremely angry. (I, however, would only say “I see red” if I was actually seeing light with the wavelength-range, ~620 to ~740 nm, which our species has labeled, in English, as “red.”) On the other hand, red roses and Valentine’s Day hearts are popularly used to symbolize romantic love, which is an intensely positive emotion, while extreme anger is extremely negative. White and black, the other colors above, in much of the world, are commonly associated with, respectively, positive and negative things. I, on the other hand, view these colors the opposite way: I have avoided sunlight for much of my life, and continue to do so (to the point where I need to take supplements of vitamin D), while also reveling in darkness, in much the same way that I revel in my “Aspieness.” Right now, it is daytime here, and I am writing this inside, in a dark room, with the only artificial light reaching me coming from computer screens.

It is a common misconception that Aspies (an informal term many people with Asperger’s use for ourselves) are non-emotional. After all, two well-known fictional characters from different incarnations of Star Trek, Spock and Data, are based, in my opinion, on Aspie stereotypes. Stereotypes, I have observed, are usually based on some real phenomenon, and in this case, that phenomenon is that many Aspies experience emotions in radically different ways from the non-Aspie majority — so differently that we are sometimes perceived by non-Aspies to be emotionless, although that is not the case. This causes a considerable amount of tension, and no small amount of outright hostility, between the community of Aspies and the non-Aspie majority. When I write on the subject of Asperger’s Syndrome, I try to do so with the goal of explaining and understanding our differences, in order to reduce Aspie/non-Aspie misunderstanding, which is both common and unhelpful — in both directions. This is the reason I use the factual, non-hostile term “non-Aspie,” in place of the unhelpful and perjorative term “neurotypical” (a word in common use within the Aspie community), one of three unfortunate words discussed in this post.

Explaining my choices of colors in the image above was a prelude to a personal, mathematical analysis of the inverted popularity of my own phobias and philias. I have long observed that I have an intense, inexplicable affinity (in many cases, reaching the level of a “philia,” an often-misunderstood word and suffix, for reasons I will discuss below) for things which the majority, in my part of the world (the American South) hates and/or fears. Examples include spiders, cats, the number thirteen (and all other prime numbers), mathematics in general, geometry in particular (strangely, even many people who like mathematics still dislike the subfield of geometry), being different from those around me, darkness, the color black, night, the physical sciences, evolution (which happens, like it or not), enclosed spaces, heights, flying on airplanes, women, and Muslims. I have also struggled with phobias, working (with professional help) on eliminating them, one by one, but they tend to be less common. Examples of targets for my current and past phobias include light, especially sunlight, to the point where I actually have to take vitamin D supplements; as well as voice calls on cell phones (human voices coming out of small boxes freak me out); death; the life sciences; insurance; sports (and related events, such as pep rallies); loud noises; efforts to control me; and, since my mother died, last November 16, the 16th day of any month, especially at, and after, six months after her death.

I’m a teacher, and it’s the 16th of July, and I simply do not have the option of falling apart on the 16th of every month when school starts again next month, at a new school, with new students, for, as the saying goes, the students will arrive — whether I’m ready or not. That’s no way to start a school year.

I have much to be optimistic about, for I will be teaching in a different building, but on a much-improved schedule, with far fewer different subjects to prepare for each day than I had last year. When I fell asleep last night, after completing four full days of training to teach Pre-AP Physical Science for the first time, starting next month, some part of me knew that mental health improvement — before the 16th hit again, today — was essential. Was that something about which I was consciously thinking? No. I apparently rewrote my mental software (again) last night, an ability I have worked on developing for over thirty-five years. When this brain-software-debugging process first became evident, a few years back, it was happening in my sleep, just as happened again last night, and it took some time for me to figure out exactly what was going on, and how my ability to rapidly adapt to change had improved. 

In Part II of this post, I will analyze, mathematically, the inverted popularity of my phobias, compared to the most common phobias, ranked by incidence among the population. First, however, it is necessary for me to explain what I mean — and do not mean — by the word “philia.” There is a serious problem with this word, in English, when it appears as a suffix, and that is due to an unfortunate linguistic error: the incorrect application of a Greek idea, and word, to the horrific, disgusting, and criminal behavior of child molesters, as well as those who have sex with corpses. The ancient Greeks, as is well-known, used four different words for different kinds of love, and “philia” (φιλία) referred specifically to fraternal, or “brotherly,” love. This was not a word the ancient Greeks used for any type of sexual act. The words “pedophilia” and “necrophilia” are, for this reason, historical anomalies. Both terms are misnomers, meaning, simply, that they are messed-up words, and their existence creates the potential for misunderstanding. A philia, properly understood, is simply the opposite of a phobia. Phobias are better-understood, of course, and require no detailed explanation. 

One example of what I mean by my own philias should suffice. I have, for many years, had an abnormally strong fascination with spiders. I like them — a lot — so much so, in fact, that I actually have a tattoo of a spider, and often wear a spider necklace, to express how much I like this one biological order, the largest within the class of arachnids. Despite my strong affinity for spiders, however, I have zero sexual interest in them. It is accurate to call me an arachnophiliac, which is the opposite of an arachnophobe.

It is now near 9 pm on Saturday, November 16, and Friday night’s sleep therapy gave me the energy to work on the needed improvements to my mental health during the day today, by using reflective writing as a therapeutic technique. I also have a new appreciation for sleep, which will come soon. Part II will be posted soon, but it will not be written until after I have enjoyed a full night of sleep — starting, hopefully, in a few minutes. Goodnight, and thank you for reading Part I.

[Update, July 17: Part II is now posted here.]

Asperger’s Syndrome and “Emotional Vision”


The source of the term “emotional vision” is the same as the source of the image above: this New York Times article. This blog-post is my response, so I recommend reading the NYT article before you continue.

The story was written by, and about, a man on the autistic spectrum, and, if you’re on the autistic spectrum and get published in that newspaper, you’re high-functioning. High-fuctioning autism (HFA) and Asperger’s syndrome were “merged” in the United States in 2013, shortly before I started figuring out that I am, myself, an Aspie. By the time I discussed the idea with my doctor, it was too late to get an “official” diagnosis. (Yes, that does mean no diagnosis for me, but that’s simply the way things happened, and I’m fine with that.)

Many in the Asperger’s community have a form of emotional blindness — an inability to “read” the emotions of others — and that described me accurately until, well, this week, when I awakened my own emotions, and also gained the ability to understand emotions of some other people. Which people? Only the ones I know well, generally by having contact with them for at least a year. Shortening this time is high on my mental “to-do” list.

In the article linked above, the author voluntarily had his emotional light-switch “turned on” in an experimental treatment designed by other people. That, I believe, is the key difference between his case and mine, for I made the decision to turn mine on myself, wrote the “mental software” behind it myself, and am testing it at every opportunity, in accordance with the way I think. This ability to reprogram my own brain’s software isn’t magic, nor a super-human ability power, but simply a project I have been working on, for, well, over thirty years.

The author of the article above has many regrets about accepting the experimental medical treatment he had to turn his emotions “on.” This treatment involved letting doctors mess around with his brain. My own doctor knows me well, and therefore does not try to force any sort of treatment on me, for he knows that my biggest compulsion involves an intense need to be free from control by other people. Not all Aspies have compulsions, but some of us do, and I am one of them.

Something most Aspies do have are “special interests,” as they are called, but they vary widely. My special interest is mathematics. I learned to speak, read, and write so that I could express my own mathematical ideas. My parents provided me with books about mathematics, one they realized the intensity of my need, driven by curiosity, to absorb mathematical ideas which were new, at the time, to me. I have never stopped wanting more.

My interest in science came later, but not much later, due to that same curiosity. Once I learned how linked the physical sciences and mathematics are, this was inevitable. The more mathematical a given subject was, the faster I could learn it. Without mathematics involved, however, learning was a chore, and deciphering the mysteries of human behavior has been, for this reason, very difficult. Why did people do such bizarre and confusing things? For a long time, I had no idea, and wasn’t willing to do the hard work of figuring it out, either. I puzzled other people, and they puzzled me right back. I made little progress, on this front, for many years.

Why did understanding anything about emotions come so much later in life, for me? That’s an easy question to answer: emotions are more complicated than anything else I have learned, in the sense that emotions are extremely difficult to understand, or express, mathematically. To do this in a way that would work well, I had to rewrite my “software” myself, and that took a lot of hard work, time, and thought. This is entirely unlike the case of the man who told his story in the New York Times, who was thrown into an emotional nightmare by an experimental treatment he willingly received, but did not design. He has my sympathy, and I hope his life gets better in the future. 

On “Thinking Outside the Box” (Thanks, Mom)

outside the box

The secret to “thinking outside the box” is to never have your thinking put in a box in the first place. Thanks, Mom. This would not have happened to me without you making it happen, and I only just now figured this out.

Please do not play pranks on the Aspies

Please do not play pranks on the Aspies

This is not a joke. I don’t know any way to make it plainer than this.

“Aspie,” in case you do not know, is a word commonly used within the community of people with Asperger’s Syndrome, as an informal term for ourselves. I don’t claim to speak for all Aspies, of course, but this is certainly true for me, as well as many others I know. Some Aspies have figured out how to recognize and understand pranks, but I am not one of them. I don’t understand why people play pranks . . . at all.

What do you mean, you “can’t wait?” Obviously, you can!


you can wait

Why do people so often, and completely incorrectly, say they “can’t wait” for things? No one ever says this, it seems, unless they already are waiting for whatever they are talking about.

A seasonal example: “I can’t wait for Christmas!”

When I hear this, I generally point out to people that they are already waiting, and therefore, obviously, they can do so.

What is it with this? Why do so many people say this thing that clearly makes no sense at all?

On Teaching Students with Asperger’s Syndrome

teaching Aspies

Teaching students with Asperger’s Syndrome is a challenge. As a teacher who also has Asperger’s, I have some suggestions for how to do this, and wish to share them.

  1. Keep the administrators at your school informed about what you are doing.
  2. Know the laws regarding these matters, and follow them carefully. Laws regarding confidentiality are particularly important.
  3. Identify the special interest(s) of the student (these special interests are universally present with Asperger’s; they also appear, sometimes, with students on other parts of the autism spectrum). Do not expect this/these special interest(s) to match that of anyone else, however — people with Asperger’s are extremely different from each other, just as all human beings are. As is the case with my own special interests in mathematics and the “mathy” sciences, it’s pretty much impossible to get students with Asperger’s to abandon their special interest — and I know this because I, quite literally, cannot do much of anything without first translating it, internally, into mathematical terms — due to my own case of Asperger’s. Identifying the special interest of a student with Asperger’s requires exactly one thing: paying attention. The students themselves will make it easy to identify their special interest; it’s the activity that they want to do . . . pretty much all the time.
  4. Find out, by carefully reading it, if the student’s official Section 504 document, or Special Education IEP, permits item #5 on this list to be used. If it doesn’t, you may need to suggest a revision to the appropriate document. (Note: these are the terms used in the USA; they will be different in other countries.)
  5. Of things done in class which will be graded, if the relevant document permits it, alter them in such a way as to allow the student to use his or her special interest to express understanding of the concepts and ideas, in your class, which need to be taught and learned. This is, of course, the most difficult step, but I cannot overemphasize its importance.
  6. Use parental contact to make certain the parent(s) know about, and agree with, the proposed accommodations/modifications. (504 students get accommodations, while special education students receive modifications. Following both 504 plans, and Special Education IEPs, is not optional for teachers — it is an absolute legal requirement, by federal law, and the penalties for failure to do so are severe. It is also, of course, the ethical thing to do.)
  7. Do not make the mistake of punishing any student for behavior related to a documented condition of any kind, including Asperger’s Syndrome.