Metalevels in thinking

Kalevi Kolttonen <>
23 September, 2018


In order to keep this article short, I assume that you are familiar with the concept meta. If you are uncertain, it is probably wise to read the Wikipedia entry for Meta first. Personally, I have always found meta easy to comprehend and apply, in particular in the context of metatheories, metalanguages and to a lesser extent in metaprogramming.

For instance, understanding Alfred Tarski's truth definition is not possible without understanding the distinction between:

Tarski's friend Willard Van Orman Quine always took the distinction very seriously and rightly so. In his text book Mathematical Logic he provided a memorable example of the crucial importance of the object language and metalanguage distinction. I do not remember his exact wording, and I do not have access to the book now, but Quine's example was like this:

Boston has 500 000 inhabitants. 'Boston' has six letters.

His example is excellent. The first sentence refers to the city Boston and the people who live there. In the second sentence, Quine has used single quotation marks to refer to the word 'Boston'. As you can see, it makes a huge difference which language we use. It is also important to make the distinction evident by always using correct symbolism (in Quine's case, quotation).

Quine's illustration seems self-evident and everybody understands it, because Boston and cities are familiar objects. Words are also familiar to us. But in the realm of abstract entities such as numbers, the distinction is harder to spot for some people. There can be considerable confusion in that area. Let me take one example of incorrect thinking that is caused by the lack of understanding the object language and metalanguage distinction.

I have seen some puzzles that are claimed to be "mathematical", but their supposedly mathematical, numeric problems in reality deal with symbols and not numbers at all! It is pure confusion, but unfortunately I am unable to remember any of those puzzles, so I cannot provide examples. But the good news is that I remember an equivalent example: Some researchers have claimed that concatenation applied to numbers is a mathematical operation comparable to a familiar function such as addition.

Let's start with what we know to be true:

1 + 1 = 2

This is a concrete instance of addition function: Adding 1 and 1 yields sum 2. It is well-defined and this is a function that accepts numbers as input and outputs a number as well.

Now the confused researchers claim that concatenation can be applied to numbers too, yielding numbers as a result. It is not important which symbol they pick to represent concatenation operation, so let's just use '|' here. For example, they say we can have operations like these:

1 | 1 = 11
2 | 2 = 22
3 | 3 = 33
123 | 456 = 123456
444 | 171 = 444171

Further, they make claims like "Using concatenation, we can construct numbers such that...". But like I said earlier, they fail to see that their concatenation operator applies to symbols only and not numbers. I can prove that easily by using hexadecimal notation and simple logic.

This is the same truth about numbers expressed using different notation:

In base 10, we have: 10 + 10 = 20
In base 16, we have: a + a = 14

Now let's use concatenation as proposed. In base 10 they claim we have:

10 | 10 = 1010

So far so good. Let's now convert the input (10 and 10) into hexadecimal notation (i.e. base 16) and apply concatenation:

a | a = aa

Now let's compare the results in base 10. The first operation yielded 1010. What is aa expressed in base 10? It is 170. So we see that concatenation operator yields different results, depending on the base we use: 1010 is not equal to 170!

But we know for sure that addition and all other functions that operate on numbers never depend on the base we choose to use. After the results are converted to uniform base, proper mathematical operations always yield the same results. This is not controversial, it is a basic fact.

So we have proved that concatenation is not a mathematical operation on a par with, say, addition. Concatenation function accepts symbols as input and yields a symbol as a result.

What have we learned? That the distinction between abstraction levels is very important. We always need to think clearly and avoid errors.


About 15 years ago my colleague asked me whether I was capable of metathoughts. In other words, he meant thinking about thinking in the general sense, not in the narrower context of, say, metaprogramming or metalanguages. I replied that of course I was, and added that thinking about thinking is one of the most important capabilities if you want to do any deeper lever analysis about your mental processes.

To my surprise he said that according to a study he had read, only 10% of all population is capable of metathoughts. I am not sure about the number, it could have been even less, maybe 8%. And he never referred me to the actual research paper, so I cannot be sure whether the numbers are accurate at all. But whatever the case, the number is totally irrelevant to our topic today. On the other hand, if the result is true, then metathoughts are not accessible to many.

I added that not only was I capable of metathoughts, I had realized that it is possible to have metametathoughts, metametametathoughts, and so on. In my own thought experiments I had done ascension to perhaps ten levels, so the model of my mental process was like this, described in words:

  1. Thinking about thinking
  2. Thinking about thinking about thinking
  3. Thinking about thinking about thinking about thinking
  4. Thinking about thinking about thinking about thinking about thinking
  5. Thinking about thinking about thinking about thinking about thinking about thinking
  6. Thinking about thinking about thinking about thinking about thinking about thinking about thinking
  7. Thinking about thinking about thinking about thinking about thinking about thinking about thinking about thinking
  8. Thinking about thinking about thinking about thinking about thinking about thinking about thinking about thinking about thinking
  9. Thinking about thinking about thinking about thinking about thinking about thinking about thinking about thinking about thinking about thinking
  10. Thinking about thinking about thinking about thinking about thinking about thinking about thinking about thinking about thinking about thinking about thinking

I told my colleague about this and informed him that although unlimited ascension to metalevels of thinking seemed possible to the human mind, in my own experiments I found that only the metalevels 1 and 2 had been useful. Further ascension was possible for me and took place, but I failed to see any benefits of being in the metalevel 3 and above. However, I added that I did not know why this was so.


I occasionally returned to this problem, but never took the time to think about it for that long. It all seemed mysterious until about a week ago I quite effortlessly came up with a very plausible explanation. I have absolutely no idea why that happened, because I was not at all obsessed with this problem. It was just a sudden realization of how this could be solved. Now the solution to the mystery seems simple and self-evident, and I am even a little bit embarrassed to talk about it. I am sure others have solved this before, but I still feel the need to document this in my own way.

Let's see a diagram of the levels of thinking:

metalevels of thinking

In the model above, Level 0 is ordinary everyday thinking in our everyday world. It includes thoughts such as:

Level 0 thinking does not contain any metathoughts. All objects of our thoughts are non-thoughts, but they could be abstract or concrete.

Level 1 is the first ascension to metathoughts. If we continue with the same examples, we could have:

Let's take more examples. On Level 0 we could be trying to solve a programming or system administration problem of some kind. We could think: "How do I get this component to do this operation?". Now a useful ascension to metathoughts on Level 1 could be: "I have been thinking about this component and its workings. But maybe I have been thinking about a wrong way to solve the problem. Could I step back, ignore this component, and use something else instead of this component?".

On Level 2 we could think: "I was thinking about my thinking of solving this problem with that component at first. Then I thought about that and of stepping outside the box. I see my ascension to Level 1 was good, therefore I should probably visit Level 1 more often!"

But the problem with ascensions to the upper levels, is, as I said earlier, often pretty useless. To handle most situations, it seems to me that Level 1 is sufficient. The mystery was why this is so.

My proposed solution is based purely on the number of the objects of our thoughts. Again, let's illustrate this with examples.

Level 0 deals with the everyday life and the objects of our thoughts are non-thoughts. Our world contains a huge number of objects, concrete and abstract (e.g. cars, humans, computers, food, animals, plants, emotions). So Level 0 is useful and familiar, because it enables us to survive and live in our world. Our mind contains a vast amount of thoughts that are connected to Level 0.

Level 1 is the first ascension level to the world of metathoughts. On Level 1, we are thinking about our thoughts that took place on Level 0. Thinking, analytic, introspective persons often enter Level 1. There are many benefits of this ascension, because Level 0 contains many thoughts that are accessible and useful objects from Level 1 perspective.

We are getting closer to solving the metathought mystery. On Level 2, the objects are thoughts that we had on Level 1. At this point it should be clear that we are talking about a simple hierarchy here, nothing more, but as we keep ascending to the higher metalevels of thinking, we always enter a more unfamiliar terroritory. If we ascend to say, Level 10, we can be pretty sure that Level 9 was almost empty, i.e. it had no objects except for the one that was "created" when our mind entered Level 9.

That is why higher level ascension is useless, because there is usually nothing to think about for ordinary people. Not many objects are accessible, and thus we have little chance of learning anything by doing the ascension. The purpose of ascension is, I guess, to gain some new insights, but in the higher levels it is not possible.

By using induction principle, I believe this observation holds to all sufficiently high metalevels of thinking, all the way up to countable infinity. So if Level 10 is now useless to us, it makes perfect sense to assume that Level 10000000 is currently useless for us too, because the hierarchy is simple and closed in its structure. Each and every Level N accesses Level N-1 and that is all there is to it.

To sum it up, the lower levels (e.g. 0, 1, 2) are useful to us, because they are populated with many objects! Starting from Level 1, all objects are thoughts. In our everyday world we can survive and solve most of our problems using only the lower levels. This is just a fact of life on this earth, but could be otherwise in different worlds or universes.

Because our minds live in the lower levels only, the higher levels never get populated. I am not sure what the so-called meditation gurus in India think of, but it is possible that they enter the higher metalevels of thinking and find them useful, because they spend their time in those levels - thus they have accessible content to process.

After all this, I am still not absolutely sure that my proposed explanation is correct. But it is the only one that I can now think of. It is unclear why the number of accessible objects would affect the usefulness of metalevel thinking. One possibility could be that having more accessible content allows us to make better generalizations based on the previous level's content. But when Level N-1 contains only one object, it is perhaps harder to gain additional insights when accessing it from Level N.