In this post I lay out the first part of a mathematical foundation for comparing the capabilities of machines to humans. Let me begin by saying that this is not intended to be a formal mathematical proof that machines can be more intelligent than humans and I have made no attempt at approaching this with any sort of mathematical rigor. The purpose of this exposition is simply to focus attention on existing mathematical concepts in the context of the ongoing discussion regarding the metrics we use to measure intelligence. I will later use this to show why I believe that machines will far outperform us no matter what criteria we might adopt in measuring intelligence. The notion of intelligence and intelligent behavior is an abstract one and any attempts to quantify intelligence in humans, animals, or other entities, both living and non-living, have fallen short of expectations. Mathematics has long served as a powerful tool for abstraction of concepts and manipulating those concepts in useful ways. We have successfully deployed it as a tool for working in domains where we have a solid understanding, such as chemistry, as well as domains in which we are still discovering their true nature, such as quantum field theory. Perhaps the most famous example of using mathematics to represent concepts which were not yet understood is Einstein’s theories of relativity. In this case we see an example of using mathematics to build a theory which predicts the existence of phenomenon which were formerly unknown. By representing the world around us in abstract mathematical terms, we are able to use the rich set of mathematical tools at our disposal to test hypotheses and follow them to their logical conclusion.

Representing Humans with Vectors

I begin by introducing a simple concept used in many domains throughout business and science alike – the vector. The notion of a vector is a simple one. It is a series of numbers which comprise a collection treated as a single unit. Although vectors can be manipulated and compared completely independently of any association with real world values, for our purposes we will assume that they represent something ‘real’. Choosing a simple example will make it easier to introduce some concepts typical of vectors . For example, we might use a vector to represent a person with respect to two values such as height and weight. Most of us are familiar with graphing values on an X/Y axis so we will use this to review some simple concepts before moving on to more useful examples. Let’s start with a vector to represent a person (P₁) with a height of 72 inches and weight of 220 pounds and another person(P₂) with a height of 76 inches and weight of 180 pounds, represented as:

P₁ = [72, 220]  P₂ = [76, 180]

These two vectors represent our simplistic categorization as depicted in 2-dimensional space shown below.

We can easily see that P₁ is greater in terms of weight and that P₂ is greater in height. We can also see that taking the two attributes combined (if we assume that height and weight are equally important) P₁ is slightly greater than P₂ because the line representing P₁ is longer than the line representing P₂ . This measurement is known as the magnitude of a vector. We say that the magnitude of P₁ is greater than the magnitude of P₂ .

Now let’s take this comparison and apply it to a situation. Assume we are trying to choose between two athletes for a sports team such as a basketball or football team. In our simple world we only have two data points: height and weight. Also, we will assume that height and weight are equally important on our fantasy sports team. We can see that athlete 1 is heavier at 220 pounds but athlete 2 is taller at 76 inches. But as shown above, athlete 1 has a greater overall magnitude when we combine the two values. So by these very simple criteria athlete 1 is our clear choice. But even in our simple version of the world we would soon realize that there is more to selecting the better athlete than just comparing height and weight, so we begin looking for better criteria to improve our selection process. Perhaps one of the most obvious is strength. There are many ways to measure strength, but since we want to keep our examples simple, we will assume there is some test that gives us a strength index that goes from 0 to 500, where 500 is the strongest human alive. We find that our first athlete has a strength index of 300 and the other a strength index of 450. Our numerical representation of our two athletes now looks like this:

P₁ = [72, 220, 300]  P₂ = [76, 180, 450]

Visually we can represent this in 3-dimensional space as shown below.

Just as in our 2-dimensional example, we can evaluate each vector in terms of magnitude, or overall length of the line shown from the origin to the endpoint. Given this added dimension in the two vectors, we can clearly see that the second athlete, P₂ , is our better choice.

There are many details and considerations to take into account such as the number of data points beyond our simple 3-dimensional example and how to account for the fact that not all data points should be treated equally. For example, we may want to construct a model where strength is three times as important as weight. I will address these issues and more later on. But for now we have enough to put forth the basis for evaluating the level of intelligence in a human or a machine.

If we take our simple example and translate into the domain of intelligence we might choose to evaluate the three criteria previously suggested in Criteria for Intelligence: speed intelligence, collective intelligence, and quality intelligence. As in the above example where we simplified the notion of the athlete’s strength into a strength index, we will assume we have come up with a method of representing these three rather complex aspects of intelligence with a numerical index. For brevity’s sake, I will refer to these three values which correspond to the three different types of intelligence as speed, breadth, and quality. At this point we haven’t established any real meaning for these measurements so the values are arbitrary. Using the same notation as before to represent these values as a vector for two different people  we have:

P₁ = [72, 220, 300]  P₂ = [76, 180, 450]

Once again we can represent the intelligence of our two subjects in 3-dimensional space as shown below:

By the given criteria  our second subject P₂ is clearly the more intelligent overall as shown by the magnitude of the vector.

The above examples show, albeit in a very simplistic manner, a very straightforward and well-defined way of evaluating multiple characteristics to evaluate and compare two or more subjects using the mathematical structure known as a vector. This concept, along with other mathematical tools which I will introduce later on, have been used for hundreds of years in diverse fields ranging from mathematics and physics to business and finance. I intend to apply these resources in the ongoing effort to explore and ultimately give some clarity to the many questions that fall under the general heading of “What is Intelligence?”, as well as to further my case for the ultimate superiority of machines in the not so distant future.

Nick Bostrom has described collective intelligence as “joint problem‐solving capacity”. There are many examples of problems that have been solved by more than a single individual and which probably could not have been solved by a single person. When a new drug is discovered and developed it is not done by a single person but by a team of people working toward a collective goal, sometimes in collaboration with other teams performing similar, related research. Projects such as the Human Genome Project was a massive research project that was only able to achieve its goal by the collective research of twenty teams across six countries over thirteen years. In our daily lives we often tackle jobs at work that require the cooperation of several people or large teams to solve problems and implement solutions. We are surrounded by evidence that our problem-solving potential is increased by combining the skills of multiple individuals. This increase in potential is realized for two reasons.

First, we have an additive effect. Think of a simple example where our goal is to identify all red objects in a warehouse. A single individual may be able to pick out and identify each object in 1 second. Therefore they can sort through 60 objects per minute, 3600 objects in an hour. But if we can enlist ten people to take on this task, assuming they are all able to sort through objects at the same rate, we can sort through 36,000 in an hour. In other words, we can accomplish the task ten times as fast! This is of course a rather simple uninteresting example, but this is the way many tasks are accomplished. The pyramids of Egypt were built this way, and many intellectual tasks are as well.

The second reason we can accomplish difficult tasks more easily with a group is due to the breadth of skills required to complete the task. We live in an increasingly complex world and the problems we are faced with are constantly increasing in complexity as well. As the tasks become more complex, solving them requires an ever-increasing array of problem-solving skills. While this phenomenon has been evolving for a long time, the first good example with a significant impact is found in Henry Ford’s approach to building the automobile. Rather than relying on a single individual or even a few individuals with the requisite skills to build a car, he broke the overall tasks into smaller tasks requiring a degree of skill that could be developed in the workers in a short amount of time. Today we see an even greater divergence of skills required in our world. Think of the number of people that contribute to the treatment of a patient during their stay at a hospital. Even going to a store to buy something sometimes involves several people. We ask where we can find what we are looking for and are directed to the correct department. Once there we can’t find what we are looking for and have to ask someone who works in that department. We then ask them about some feature of the product, they don’t know the answer and go find someone who knows more about that product. Finally, we go to the register where we pay for what we have selected. While this may not seem like something that exemplifies the height of human intelligence, it demonstrates how much of what we undertake relies on the knowledge of multiple individuals. This is a simple form of collective intelligence.

There is one more aspect of collective intelligence that is important to recognize. In the past we see examples of a brilliant individual such as Thomas Edison or Alexander Bell who were great innovators. But even these great names relied on the discoveries and knowledge gained from others, some their contemporaries and some from the past. More than any other species, humans have the ability to learn from their predecessors. This ability is referred to by Michael Tomasello and Steven Mithen as cultural learning. In its simplest form it can be seen as a child learning not to cross the street without looking for oncoming cars. It can be seen as the fundamentals of reading, writing, and arithmetic that we learn in our first few years of school. It is seen in our education where we learn the necessary skills for our careers. Although we don’t often think of it, when we attend a year of school to learn a trade or skill we are learning skills and knowledge that took hundreds of years to accumulate. The long term effect of this type of knowledge transfer is incredible. It is what has allowed us to develop rocket ships that fly to the moon and understand the complex system we know as the human body. It is what has led to the development of  the computers that are so advanced, so fast, so intelligent – that they may soon surpass us in intelligence in its every form.

Before delving further into how machines might become intelligent it is helpful to define, or at least describe, what is meant by intelligence when referring to machines. I. J. Good, who worked with Alan Turing during World War II and is credited as the originator of the oft cited term ‘singularity’, described what he called an “ultraintelligent” machine as “a machine that can far surpass all the intellectual activities of any man however clever”. Good and many others tend to focus not just on what constitutes intelligence but specifically on how and when we will know that machines will surpass the capabilities of human. Regardless of whether we are defining intelligence or ultraintelligence, the type of the criteria should be the same. This gives us the basis for one possible criterion for intelligence, viz. the ability to perform a task or activity as well as an average person. This gives us the basis for one possible criterion for intelligence, viz. the ability to perform a task or activity as well as an average person.

In Superintelligence: Paths, Dangers, Strategies Nick Bostrom states : “…we use the term ‘superintelligence’ to refer to intellects that greatly outperform the best current human minds across many very general cognitive domains.” He goes on to suggest that it is helpful to decompose this notion into three categories of superintelligence: speed superintelligence, collective superintelligence, and quality superintelligence.

Speed superintelligence is defined as “a system that can do all that a human intellect can do, but much faster.”

Collective superintelligence is defined as “a system composed of a large number of smaller intellects such that the system’s overall performance across many very general domains vastly outstrips that of any current cognitive system.”

Quality superintelligence is defined as “a system that is at least as fast as a human mind and vastly qualitatively smarter.”

Each of these three definitions covers a different form of intelligence and is in fact the product of a different type of system. The next three posts will cover these in more detail.

This century will bring the advent of true intelligence in machines. But exactly what does that mean? We often speak of someone as intelligent. We might speak of an intelligent decision or an intelligent statement. But what exactly does it mean for an action, or an utterance, or a person to be intelligent? I have set a goal to come up with some sort of criteria for determining what counts as an intelligent action and to determine if and when the label of “intelligent” is conferred upon a person or a machine. Loosely defined, we think of someone who is intelligent as being capable of doing something correctly. If a person seems to have all the right answers, we might say they are intelligent. If a person can solve math problems well, we think they are intelligent. But these two things are certainly within the grasp of many computers today and have been for some time. So computers certainly are intelligent at least in this simple, perhaps trivial sense. But when heated discussions arise about whether a computer is truly intelligent we are looking for something deeper, more profound. Most of us would still differentiate between the type of intelligence described above and the intelligence that a human possesses. We even have a special term for it: artificial intelligence.

So what exactly do we mean by artificial intelligence as opposed to real or genuine intelligence? To look at the question another way, precisely what criteria must a machine, a computer, satisfy for us to assert that it possesses intelligence that is not artificial – that it possesses genuine intelligence? The term artificial has several definitions, but fall into two categories. The first infers that something is artificial if it was produced to seem as if it were natural. The second infers that something is not real. The first type of definition is a de facto definition which eliminates anything that is not natural. In other words, if it is not part of the animal or plant kingdom it is by definition artificial. While I cannot say that this definition is wrong, it doesn’t seem to be of much use for the purposes of this discussion. We already knew that machines do not occur in nature. They are made by humans. What I want to focus on is the question of what constitutes real or genuine intelligence.

So back to the original question: what is real intelligence? What criteria constitute intelligence on the same level as we humans? I have seen many definitions of intelligence, and I don’t want to limit myself at this time to any one or even limit myself to a specific list of definitions.

At an intuitive level, it seems that intelligence is closely tied to knowledge. The more knowledge one has and the more one is able to make use of that knowledge, the more intelligent one is. My computer may be able to give me the correct solution for an equation, but can it tell me why it is important to know that solution? Can I trust my computer to let me know what equation I should be trying to solve? We are seeing great progress in these types of “artificial intelligence” and as these very smart and very useful machines become more and more a part of our lives, it will become more and more difficult to find differences between what we humans can do and what machines can do. Seen in this light, it is only a matter of time until the term artificial intelligence becomes arcane and machines will be accepted as truly intelligent. Going forward, I will explore how and in what way advances in the intelligence of machines will take place.