In his theory of creativity, Dr., Kurt Ammon has defined creativity as the ability of a system to calculate functions that are not “Turing-computable”.
What does this strange definition of creativity mean?
“Turing-computable” means that you can write a program to calculate a function in a very simple programming language called “Turing machines”. Such programs are called algorithms. Functions that can be calculated by Turing machines are called “Turing-computable”. For several decades, most computer scientists believed that anything that can be calculated by a machine can be expressed as a Turing machine. However, some people did not share this view. One of them is Kurt Ammon.
Ammon is a mathematician. When doing mathematical proofs, he noticed that there is no general method by which you can calculate the proof for a given mathematical statement (or “theorem” in mathematics jargon). This and other activities in mathematics require creativity. You have to invent something. There are patterns and you sometimes do it the same way you did in in another case before, but there are again and again situations where you have to do it differently. There is no pattern or method that always fits.
Interestingly, there are mathematical proofs showing that say that this must be so. Such proofs show that for certain activities in mathematics, no calculation method is possible that covers every case. There are “incompleteness proofs” – the first one Kurt Gödel’s proof of his famous “incompleteness theorem” – that say that for certain types of tasks, no complete methods are possible. There are no complete formal theories and no “algorithms” to do the calculations in all cases. In Ammon’s terminology, each of these limited methods forms an “analytical space” and there is no way to unify all analytical spaces into a single one covering all cases.
But astonishingly, humans are still doing such tasks. The ability that enables them to do so is what Ammon calls creativity. It is the ability to get out of any defined scheme and invent new ways. If he is right, the human mind cannot be described completely by any one formal theory because humans do things for which mathematicians can prove that no complete formal theory exists.
If the human mind is creative in this way, there would be no fixed laws of thinking. Our thought processes are able to get out of any fixed scheme that could be described as “the laws of thinking” or “the laws of cognition”. If our thinking processes where limited by a set of “laws”, by an algorithm that describes once and for all how thinking works, we would be unable to do mathematics and other activities for which it is possible to proof that creativity in the sense of Ammon’s definition is required (like, for example, programming computers).
However, there are no mysterious processes needed to describe creativity. Ammon thinks that it is possible to create computer programs that are creative in the sense of his definition. And in his article An Eﬀective Procedure for Computing “Uncomputable” Functions he has shown that such computer programs are indeed possible.
Such programs have the strange property that they cannot be described completely by any formal theory. What does that mean?
Think of a formal theory as a box. There are some things that fit into the box and others that don’t. A formal theory is similar to such a box in a way. A formal theory describes a set of statements. The ones that fit into the “box” are “derivable” inside that formal theory.
“Normal” computer programs “fit” into such a box. Programs for which such a “box” exists are called algorithms (it is possible to describe them as “Turing machines”). In a formal theory about such a program, we can derive statements of the form “if we apply the program to the inputs x, it will produce the outputs y”. This way, it is possible to describe completely what the program does, with a finite amount of information.
Now, Ammon’s Proof shows it is possible to write programs for which such a “box” does not exist. Whatever box you put it into, there are inputs that let it move out of the box. Think of a small box (a formal theory describing what the program is doing) that describes everything the program has done so far. Then there is a way to give it an input so that it will move out of that box. You can find a still larger box that will contain the smaller one and also the process of moving out of the smaller, inner box (i.e. the process not covered by the “inner” theory), than you still don’t describe the program completely. Instead, there is a process that will move it out of that larger “outer” box as well, and so on.
Think of a program that you install on your computer. You download it as a file. It is a finite amount of information. If it is an algorithm, this finite amount of information determines completely what that program can do. Not so for a “creative” program. Here, the program you get initially does not determine completely what will happen. Some inputs may change and reprogram the program, and it will grow (literally) bit by bit.
Ammon’s proof shows that this may happen in such a way that no single formal theory can completely describe what the program does. Whenever you describe exactly what the program has done so far and so you find a box into which everything fits that the program has ever done, there is a way out of that box, no matter how large you make that box.
Since what the program will do depends on previous inputs, not only on the initial program, such a program has a “history” or a “biography” when it runs and develops. You can think of it as reprogramming itself, depending on its input. You may think of it as a learning system.
An algorithm, on the other hand, is a program without a history. It will always react in the same way. Even if it can store some information, as long as its behavior can be completely described by a formal theory, it is an algorithm.
In Ammon’s view, algorithms are always limited. They cannot develop and they can only solve a limited set of problems. For a system to be truly intelligent, on the other hand, creativity is required.
The possibility of computer programs that are creative in the sense of Ammon’s definition of creativity hints at the possibility that human creativity and intelligence may be scientifically explained. This explanation, however, has a very strange property. A creative system in Ammon’s sense cannot be described completely by any single formal theory. What happens inside it can always be described and explained exactly in hindsight, and theories about it can always be created that include those processes, but every such theory is incomplete.
While science normally deals with systems that are described completely by some formalism, here we have systems for which this is impossible in principle. So simple epistemological approaches (like the “HO-Scheme”, see http://plato.stanford.edu/entries/hempel/ , section 4.) are not sufficient.
And here, I think, lies the deeper reason for the science/humanities divide. In historical and cultural sciences, the impossibility of complete description and the historical relativity of all descriptions is a normal and accepted state of affairs. For creative systems, in fact nothing else is possible. We see here a new kind of reductionism. Culture, history and psychology are, in a way, reduced to or explained by material processes, but there are no fixed laws because the human mind is self-reprogrammable without limits and any law, algorithm, “representation language” or whatever structure it contains might be changed, extended or replaced by something else. In the same way, there are no fixed laws describing cultures, societies or historical processes. As a result, the physical reality humans and their societies and cultures are part of has more properties than any single formal theory can capture.
We can always leave the box!
(the picture is from