Problem Solving (Insight)

The Inferential System probably was formed during evolution to produce perceptions. But it produces most (if not all) of the thoughts that appear in human consciousness. This includes insights in problem solving.

This essay continues the theme of learning more about the Inferential System. The following essay has some practical advice for improving your creativity and problem-solving ability.


Archimedes, an ancient Greek, had a problem -- was the king's new crown made of pure gold, or did it have lead inside? Archimedes could answer this question without destroying the crown if he could figure out the volume of the crown. But while there are mathematical formulas for the volume of shapes like cubes and spheres, there are no mathematical formulas for the volume of odd-shaped objects like crowns.

One day, Archimedes stepped into his bath. The solution to his problem flashed into his brain – he could put the crown in water, use his formulas to determine the volume of the water both with and without the crown, then subtract the two to find the volume of the crown.

This is what is called an insight in problem solving – the solution to the problem occurs suddenly in conscious. In another classic example, Poincare was on vacation and stepping on a bus, when he suddenly realized that the transformations used to define the Fuchsian functions (whatever they are) were identical to those of non-Euclidean geometry. This thought just appeared in Poincare's thinking – he wasn't trying to produce that thought; he wasn't even trying to think about this topic.

In a now-classic study, Wallas asked famous mathematicians and scientists how they made their famous discoveries. Almost all of them reported using insight. In hindsight, this makes sense. If a discovery could be made by consciously-known methods, it would be an ordinary discovery, not a great discovery. Almost by definition, the great discoveries had to be produced by the mysterious process of insight. Anyway, the point is that insights are a very important method of solving problems in math and science.

Two Methods of Solving Problems

Sometimes you know a straightforward method of solving a problem. Then you can use this method without much thought (other than remembering the method). For example, there is a method for multiplying two numbers, which can be used to solve the problem of multiplying 38 times 62. Using a known method isn't fun, and it isn't creative. It is done with conscious thought, not the Inferential System.

That isn't what happened to Archimedes or Poincare. Archimedes didn't know of any way to solve his problem. Yet he kept thinking about it and trying to solve it. Then the answer just appeared suddenly in his consciousness. This essay is about insight problem solving.

From the Inferential System

Several pieces of evidence confirm that insights in problem solving are produced by the Inferential System. First, your conscious experience is that the insights just appear in consciousness by themselves, with no accompanying thoughts about the process of forming the insight. Of course, if you have an insight while working on a problem, you might have the illusion of knowing what thoughts preceded the insight. Or maybe you really do know. But you don't always know.

The inability to be aware of the process of forming the insight was demonstrated long ago in an experiment by Maier. The subjects had to solve a problem using insight. While they were thinking, Maier subtly gave them a clue -- he swung an object on a string. (The correct answer involved swinging a weight on a rope). The clue helped subjects solve the problem. But many subjects were unaware of the clue. When asked how they solved the problem, they offered a variety of stories. Some of these stories were long and creative. But most of the subjects did not mention the clue.

A few subjects did mention the clue. But Maier presented another clue that actually didn't help subjects solve the problem. If they then solved the problem, subjects were as likely to mention this useless clue as they were to mention the useful clue. So subjects could mention a clue if they noticed it, but they didn't know which clues were actually useful. This again shows that subjects were not conscious of their brain's production of the insight.

Modern experiments confirms the role of the Inferential System in producing insights. First, when information is presented so quickly that subjects cannot perceive it, it can nonetheless hasten problem solving (Bowden, 1997). This is because the Inferential System receives the cue and uses it to hasten the insight. Second, people who are good at solving difficult perceptual tasks are also good at solving insight problems (Schooler & Melcher, 1994), suggesting that both are solved by the same part of the brain (the Inferential System). Third, subjects in an experiment could not predict when they would solve a problem using insight. This again suggests a lack of awareness of the process of producing the insight, and that it just appears in consciousness.

The Process of Insight

Wallas found four stages to insights. The first stage is working on the problem without solving it. During the first stage, you are learning about the problem and building mental models which you will later use to solve the problem. For example, when the Rubik's cube first came out, I worked about 8 hours on it without solving it. Failure? Technically yes, but I was learning how the Rubik's cube worked. The Inferential System might seem magical, but it needs information and experience to do its "magic".

The second stage of insight problem solving is incubation. During incubation, you are not making any active efforts to solve the problem. This stage is optional, depending on the difficulty of the problem you are trying to solve -- sometimes of course you simply start working on a problem and then have an insight without any incubation period.

The value of incubation could be simply giving your unconscious a chance to rest, or giving it a chance to make a fresh start on solving the problem. It is also possible that you are "mulling over" the problem during incubation -- maybe your unconscious is thinking about the problem, or maybe you are consciously thinking about the problem in a loose, unstructured way.

The best activities for incubation are those that give your brain a chance to rest and/or mull over a problem. Essentially, if you can daydream during an activity, it is probably good for incubation. For example, walking and resting are good incubation activities; watching television is not.

The third stage is finally producing the insight. This most often occurs when you return to the problem, though as the stories of Archimedes and Poincare illustrate, the insight can also occur when you are not actively working on the problem. When this occurs, I would guess that some cue in the environment triggers the solution. For example, the rising water in the bathtub was the trigger for Archimedes' solution.

To complete the list, the fourth stage of insight problem solving is verifying that the insight is correct. For problem-solving, this is usually straightforward.

Inputting a Goal

There is a neat "trick" to using your Inferential System to solve a problem. The Inferential System does nothing more -- and nothing less -- than construct a conclusion consistent with its inputs. In problem solving, the facts of the problem are given to the Inferential System. Additionally, consciousness inputs the to-be-accomplished goal. Then the Inferential System constructs a conclusion consistent with the facts of the problem and the goal.

The trick is this. Any conclusion that is consistent with the goal will also be a method of achieving the goal. Suppose you are given the problem of getting from Los Angeles to New York City by boat. Your Inferential System will produce something consistent with using a boat, starting in Los Angeles and arriving in New York. And anything consistent with all of these is a solution. Your Inferential System won't produce the answer of driving, because that is inconsistent with going by boat. Your Inferential System won't produce the answer of taking the Mississippi River, because that is inconsistent with starting in Los Angeles and ending in New York. Instead, your Inferential System produces something like sailing around the tip of South America.

The presence of conscious input to the Inferential System is included in the figure below. The input of consciousness into the Inferential System could also have been inferred from studies of perception -- people tend to perceive what they consciously expect.

The conscious input to the Inferential System also explains the conscious control of attention. If you attend to red -- consciously think about red -- your Inferential System produces perceptions and other thoughts consistent with red. Put simply, the perception of a black dog is not consistent with "red". Similarly, if you consciously attend to a location, your Inferential System is constrained to producing perceptions about that location.


This essay made two important points about the Inferential System. First, the Inferential System can be very powerful and creative. It is the source of most major scientific and mathematical breakthroughs.

Second, consciousness can also input information into the Inferential System. This gives consciousness some control over the Inferential System. This is important control, but it also means that consciousness can impair functioning of the Inferential System if it doesn't know what it is doing.