Vagueness and Being Human

The inherently creative nature of human beings means every exact description of what it means to be a human being is incomplete. As a result, any concept of the human that is attempting to be general must be vague. The human being can always develop, so it can develop out of the scope of any given description. So either  we get a partial exact description or one that seems to be general but where the concepts involved have incomplete definitions (vagueness being a form of incompleteness, on the level of the definitions of the concepts used).

Fixing ourselves or others into roles that are strictly and formally defined, is thus dehumanizing.

We are partial, we are vague. That is what makes us human.

The Core of Philosophy

In a way (that I am going to explore in some articles on Creativistic Philosophy), one could say that computability theory (which could be called “formalizability theory”), as one can find it in the works of Post, Kleene, Turing, Church and Gödel, forms the very core of philosophy. From here, one can investigate why philosophy still exists, why it will not go away and what is the nature of the analytic/continental divide and the science/humanities divide.

Project Sketch

Sketch of the line of argumentation, to be developed in a sequence of articles. The plan is to write each article in such a way that it appears to be almost trivial. The argument is broken up into very small steps that can be understood without special knowledge of mathematics or computer science. The line of thought should be presented in a form that shows it is actually simple and trivial (which it is).

Programs as finite texts over finite alphabets. Each program only contains a finite amount of information.

Programming languages – Interpreters – Special purpose languages – Universal programming languages – Turing machines and other mathematical “programming languages”

Computable functions. Programs computing functions. Functions as (infinite) lists of input-output pairs. Programs of computable functions as compressed representations of such lists. Regularity in such lists expressed by the programs.

Representation of arbitrary data as natural numbers. Representation of Programs by natural numbers. Gödel numbers. Results valid for functions (and programs) of natural numbers are valid for functions (and programs) of arbitrary data.


Programs computing total functions of natural numbers are not Turing-enumerable. Proof of this by the diagonal method. Constructive nature of this proof. So every algorithm producing programs computing total functions is incomplete. The diagonalization method can always be used to produce another computable function and the program computing it, but although this operation is Turing-computable itself, integrating it into an algorithm yields an incomplete program again. So it must be applied “from the outside”, not under the control of the algorithm itself.

Side-step: Turing-enumerability of programs of a programming language (programming languages are decidable). Halting-problem for Turing machines. Impossibility to prove equivalence of arbitrary programs with an algorithm. Impossibility to prove correctness of arbitrary software by an algorithm. Programming is always risky and error-prone.

Set of Programs producing programs computing total functions is again not Turing-enumerable. Sketch of Proof. Productive sets and productive functions. The set of such programs is a productive set. Trying to integrate the productive function into the algorithm does yield an incomplete program. So again, the extension process must be applied from the outside, not under the control of the algorithm itself.

Definition of creative systems. Creative systems cannot be algorithms.

Because of the possibility of Gödelization (mapping of data onto natural numbers) all these results are valid for programs processing arbitrary types of data.

Any kind of knowledge can be viewed as programs calculating total functions or programs producing such programs. Declarative knowledge can be viewed as programs formulated in a special purpose programming language and interpreted by some procedures that act as the interpreter. Applying such knowledge can be viewed as the production and subsequent execution of programs. All these programs halt after some time, so they can be viewed as programs computing total functions.

Creativity (adding new programs to a set of programs that is not Turing-enumerable) is the core of general intelligence. A generally intelligent system cannot be an algorithm but must be a creative system. Any algorithm (even an algorithm producing programs) is limited. It contains a limited amount of knowledge that has a limited reach. General intelligence requires a mechanism to extend the set of programs (the knowledge) but this cannot be part of the system as far as it can be viewed as an algorithm.

Algorithms and formal theories are equivalent notions. There cannot be formal theories of creative systems. If science is about describing systems with fixed laws, creative systems are outside its scope. They are inside the scope of a wider area of “Wissenschaft”, however.

Artificial intelligence may be possible but truly intelligent systems cannot be algorithms. They must contain an extension mechanism not under the control of their algorithmic part.

It is interesting to note that the basic results from computability theory where already known in the 1950s and 1960s (and even earlier) when the traditional AI paradigm was created. The traditional AI paradigm ignored these insights. This is the reason it developed into a dead track. All contemporary “AI” systems can be described as algorithms. Where they contain learning mechanisms, these are limited. It would be interesting to work out the history of early AI to see how this happened. Why where the results of people like Gödel, Turing, Kleene etc. ignored by AI, instead of turning them into the core of the discipline and defining the aim of the discipline as developing creative systems, i.e. systems that can go beyond algorithms? Has this been worked out by any historian of science already?

Thoughts about Intelligence and Creativity

Some unordered notes (to be worked out further) on some general principles and limits of intelligence.

Reality has more features that we can perceive. What we perceive is more than what we understand. And our understanding has several levels, from perceiving shapes to conceptual interpretation and deep analysis. On each level, we can capture only a fraction of the information of the level before it. (See also

The primary sense data are processed quickly, by neuronal systems having a high degree of parallelism. However, the level of analysis is rather shallow. To process large amounts of data quickly, you have to have an algorithm, a fixed way of processing the data. Such an algorithm can only recognize a limited range of structures. An algorithm limits the ways in which the bits of data are combined. An algorithm is a restriction. It prevents universality. The data could be combined in so many ways that you would get what is known as a combinatorial explosion if you would not limit it somehow. The system, having only a limited processing capacity, would be overwhelmed by the hyper-astronomically growing number of possibilities. Therefore a system processing a large amount of data must restrict the way it combines the data. As a result, it can process large amounts of data quickly but will be blind to a lot of the regularity that is contained in the data and could theoretically be discovered.

In order to discover such hidden features, you cannot process large amounts of information at once because this would lead to a combinatorial explosion. You would, instead, have to process small amounts of information at any given time, trying to find some pattern. Only when you discover a pattern, you can try to scan large amounts of data for it, essentially applying a newly found algorithm to the data. But that algorithm will in turn be blind to other regularity the data might contain. Each algorithm you may use to analyze data is incomplete, because it has to limit the way data is combined, or it will not be efficient, leading to combinatorial explosions again.

Intelligence could be defined as the ability to find new instances of regularity in data, regularity that was not known before. It can therefore be defined as the ability to construct new knowledge (new algorithms). This is only possible, in principle, by analyzing small amounts of data at any given time. Any algorithm you may use to analyze larger amounts of data will be limited and may be missing some of the structure that is there (i.e. it will restrict the generality of the intelligence). (See also and

This limit to intelligence should be valid for single human beings but also for groups of human beings, like scientific communities or cultures. It would also hold for any artificial intelligent system. Such systems cannot be made arbitrarily intelligent. One could try to do so by putting many small intelligent systems in parallel (something like an artificial intelligent community) but since such systems would not be limited by any algorithm (or formal theory), they could develop into totally different directions, disagree with each other and suffer from misunderstandings if one would try to connect them together. If you connect them in a way that limits the possibility of misunderstandings in their communication or that stops them from disagreeing or from developing into totally different directions, you end up with a parallel algorithm again that can harmoniously analyze large amounts of data but is limited in what it can do.

You either get shallow processing of large amounts of data or deep analysis of small amounts of data with the potential of new discoveries, but you cannot have both at once. As a result, there is a limit to how intelligent a system can become.

There is no limit to what can be discovered by an intelligent system: if a structure is present in a set of data, it can be found if the system doing the analysis is not an algorithm (i.e. a system describable in terms of a finite formal theory – an algorithmic system, on the other hand, will necessarily be systematically blind to some structures). On the other hand, an artificial superintelligence is not possible. Processes of intelligent data analysis in such a system might be faster than they are in a human being, but they will not be much more sophisticated. Higher sophistication by adding of smart algorithms leads to limitations, i.e. to systematic blind spots. Higher sophistication by attempting to process more data at a time leads to combinatorial explosions which cannot be managed by whatever additional speed or processing power one would add. (See also and also

For shallow analysis you need algorithms. Speed in terms of amount of data (bits) processed per time (seconds) may be high, but the depth of processing is limited. If the goal of cognition is to find regularity (and thus compress data), the algorithmic system will not find all regularity that is there. It cannot compress data optimally in all instances. Such a system will have blind spots.

Finding all regularity may be viewed as the ability to find the smallest self-expanding program that can produce the data (i.e. an optimal compression of the data). If an algorithm analyzes a stream of data, i.e. it parses the data, and the stream of data is longer than the algorithm itself, the algorithm may be seen as a compression of the data. If the compression is loss-free, i.e. the algorithm can reproduce the original data then the data must contain some redundancy if it is longer than the algorithm. The data will then not exhaust the information carrying capacity of the information channel. Therefore, it must be possible to add some information to that channel that is not parsed by the given algorithm. Hence the algorithm must be incomplete since there is data it cannot parse. It systematically has a blind spot.

Therefore, an intelligent system able to find arbitrary regularity cannot itself be an algorithm. Instead it must be a system that can produce new knowledge (and thus does not have a fixed representation as a finite text, and does not have a Goedel number). It must be changing over time, incorporating information that enters it from the analyzed information stream. This information reprograms the system, so it changes the way the system works. The system cannot have a fixed way in which it is working because then it would be an algorithm and would have a blind spot.

The possibility that the system self-destructs (becomes mad) cannot be excluded. That is a risk involved in intelligence/creativity.

Sophisticated knowledge has a high efficiency but a low universality. It is special and will “miss” many features of the data it processes (i.e. it has blind spots). On the other hand, it is efficient, which means that it allows large amounts of data to be processed. The processing of large amounts of data in a short time means that only a limited subset of the properties of that data can be considered, making analysis shallow.

Simple knowledge, on the other hand, has a high universality but a low efficiency. It allows for new features of data to be discovered. It therefore has the potential of a deep analysis that does not miss properties, but it has a low efficiency and can only process small amounts of data at a time, since applying it to large sets of data leads to combinatorial explosions.

The simple knowledge is what is called “reflection basis” in K. Ammon’s dissertation. (see Ammon, Kurt: “The Automatic Development of Concepts and Methods“, Doctoral Dissertation, University of Hamburg, 1987).

New knowledge forms by incorporating information from data into the knowledge base. This might occasionally happen through the application of sophisticated knowledge but most of the time is the result of applying simple knowledge to small amounts of data, leading to the discovery novel (from the system’s point of view) properties of the data. As a result, new more sophisticated knowledge forms. This knowledge is special and more efficient.

The small amounts of data that are processed by simple knowledge might be input data from the input stream, but might also be chunks of knowledge that are experimentally plugged together in different ways and then experimentally applied to the input stream (perhaps in its entirety). This might occasionally lead to sudden changes of perception (e.g. changing from two-dimensional vision to three-dimensional vision). Successful (i.e. efficient) structures are then retained. This is a way of incorporating information from the environment into the system.

The total universality of a creative system lies in the emptiness of its core (i.e. there is no fixed, i.e. unchangeable, special knowledge restricting what it can do).

The trade-of between efficiency and generality is a special case of (or another way of expressing) the trade of between explicitness/exactness and generality described in A result of it is that there is a fundamental limit to how intelligent a system can become.

Sophisticated knowledge can be used to filter out expected components from the data stream, leaving the surprising parts that can then be analyzed by less sophisticated knowledge. The end result might be more comprehensive sophisticated knowledge where the previously surprising data becomes expected data.

(A lot of this is already contained in K. Ammon’s dissertation in one form or another).

Does Meditation Reduce Creativity?

Just a question. I have no answer to this and answering it would require some serious scientific research:

Meditation is, first and foremost, training of attention. As far as I understand, regular practice of meditation can, to a great extent, improve attention by promoting the ability to reduce the uncontrolled straying of thoughts. My question is: does this have a negative impact on creativity? What one learns to suppress through meditation seems to be what is called the “default network” of the brain.

 My personal experience in the “default mode” of the mind is that my thoughts are wandering around. It looks like I am analyzing all kinds of problems. At the same time, my impression is that in this default state, I am having my best ideas. So it seems to me that this is the “creative mode” of the mind.

While in a state of concentration, I can work on a specific problem and apply known methods; however, my creativity, i.e. the ability to generate new ideas, to move out of the scope of known methods of thinking, seems to be highest in the state in which the mind is wandering uncontolled. when brain activity is strongest in the default network.

As far as I understand, it seems to be this default activity that is reduced in meditation (if I am not wrong on this). So could it be that people who practice a lot of meditation gain a highly improved ability of concentration, but at the same time loose some creativity? If the highly improved attention gained through practicing meditation where only advantagous, our brains would likely have developed in such a way that attention would be better from the start. This, however, has not happened. Our thoughts are straying around without controll. The reason might simply be that this uncontrolled straying is necessary for developing new ideas and that the resulting creativity was selected for. The way our thinking and perception is working, with less than optimal attention and thoughts being distracted and wandering might be a compromise between the advantages and disadvantages of attention and concentration on one side and creativity and innovativeness on the other.

So  I suggest that researchers working on meditation and its effects try to design experiments investigating a possible (negative) effect of meditation on creativity.