Creativity / Incompleteness / Mythology / Philosophy / Science



“The limits of my language mean the limits of my world.” – Ludwig Wittgenstein, Tractatus Logico-Philosophicus, 5.6.

There is a gap in our language and I want to try to close it by suggesting a new philosophical term. What I am looking for is a term for an entity that exists but cannot be described completely by any single formal theory (or algorithm), something for which a complete and at the same time exact description is impossible in principle or at least for all practical purposes, e.g. for reasons of complexity.

It seems to me that human beings as well as their societies and cultures are such entities, but I think many physical systems are such entities as well. Actually, the word “system” does not really fit here because it has a connotation of something systematic, i.e. something that can be captured completely and exactly by some theory. Scientists use the term “system” as a very general term for practically everything and just assume that everything is a “system”. There seems to be a tacit (and I suppose: wrong) assumption in science that everything is formalizable, and this assumption seems to be hidden right inside the term “system”.

It is, however, not the aim of this article to investigate the question if such entities really exist in physical reality (they definitely do within mathematics, so at least there is no contradiction in the concept). What I want to do in this post is just to propose a word so that it becomes a little bit easier to think and talk about such things, a word that can then be used in discussions about what things are (or are not) such entities, why they are (or are not) or such things even exist.

Instead of using a circuitous circumlocution like “something for which every formal theory of it is incomplete” and instead of using a negative term derived from another word by means of “non-“, “in-“, “un-“ or “a-“, like “non-formalizable”, “informal” etc., we should have a simple positive term.

Antique mythology has often provided the material for new words and terms in science and philosophy, so looking for a fitting new philosophical term, I came upon Proteus. My previous post on this blog was an attempt to play with this figure a little bit and attach a few philosophical thoughts to it in order to see if it fits the bill.[1]

Proteus, an ancient god of the seas, has the ability of changing shape. He does not even have any primary, “real”, definite shape. The idea of such a good might have emerged from the experience of sailors with the unpredictability of the weather and the seas. Modern science has made some progress in the area of weather forecasts, but there is reason to believe that an exact long term forecast of the weather will always remain impossible, so here we have a prime example from science.

In an article about Proteus, Wikipedia tells us:

“He can foretell the future, but […] will change his shape to avoid having to.[…] From this feature of Proteus comes the adjective protean, with the general meaning of “versatile”, “mutable”, “capable of assuming many forms”. “Protean” has positive connotations of flexibility, versatility and adaptability.”[2].

The German counterpart of that article informs us that stoic philosophy and its allegorical interpretation of myths used the Proteus figure and its ability to transform itself to illustrate the creative changes of the elements in the cosmos.[3]

So I want to suggest deriving the term we are looking for from the name “Proteus”. I suggest the word “proteon” to mean something (an object or a process) that cannot be described by any single formal theory. This means that each formal (i.e. exact) description of a proteon it is incomplete, although the possibility to extend or improve any given description of it might exist. There might be different reasons why something cannot be described completely but I would like to keep the meaning of the term a little bit vague here, so there might be several different types of proteons that have this property for different reasons. The term may also be used for cases where a complete formal description might be possible theoretically (“in principle”) but where it is practically impossible, e.g. for reasons of complexity. A proteon is not a system. Within the terminology proposed here, the term “system” may be viewed as the opposite of the term “proteon”. Moreover, in the context of this terminology, I would use the (already existing) adjective “protean” to refer to something that is a proteon. I suggest using the plural “proteons” but I would not mind if somebody prefers to speak of “protea”[4].

I don’t know if this proposal is going to catch on. Language is a part of culture and I suppose both culture and language are proteons, so I cannot predict with certainty what is going to happen. When it comes to proteons, seeing and noticing is more important than deriving, since they are full of surprises. Let’s wait and see.

(The picture, showing a Greek vase painting of some ships, is from It hints at the nautical context in which the ideas surrounding the deity of Proteus probably emerged.)


[1] The form of that piece, the more or less philosophical dialogue between mythological figures, was inspired by a book written by Cesare Pavese. I have tried my hand on this form a few times before, see


[3]“Die Philosophie der Stoa und ihre allegorische Mythendeutung nutzte die Proteus-Gestalt und deren Verwandlungsfähigkeit zur Veranschaulichung der schöpferischen Veränderungen der Elemente im Kosmos“, see The wikipedia article referes to “Georg Pfligersdorffer:Studien zu Poseidonios. Wien 1959, S. 120ff“ as a reference for this and states that even Francis Bacon picks up this interpretation referring to “Sapientia veterum c. 13”. I have not checked these claims but it does not matter for our purpose here.

[4] Although this is not a Greek word – it is a pseudo-Greek neologism -. Moreover, there is no rule forcing us to import grammatical forms from a source language along with a borrowed term. So even if “proteon” was a Greek word, we could still use a plural “proteons”, just as we use “kayaks” as the plural of “kayak”, although the Inuktitut word “qajaq” from which it has been borrowed, has the plural form “qajait” (as far as I can tell from my very limited knowledge of that language).

17 thoughts on “Proteons?

  1. An interesting proposal Nannus. But aside from mathematical objects, I wonder if everything isn’t ultimately a proteon. It seems like anything we understand, we understand only in terms of its components, and those components we only understand in terms of their components. Eventually, we always arrive at primitives we don’t yet understand.

    Or maybe we can consider a system to be a subset of a proteon, the portion of a thing whose dynamics we can currently describe. Proteon might refer to the systematic and the currently unknown (or unknowable) aspects. (Maybe I’m just repeating what you said.)

    • I think many real world entities have more properties than can be derived inside any single formal theory about them. Such a formal theory describes a system that is, one could say, emulated by the underlying proteon. The system is a part or an aspect that can be described exacty, but there will be a rest or “residuum” (as I have called it somewhere else). Scientists try to isolate physical systems from the environment to get something that can be described in terms of some systematic theory. Part of the art of the experimentalis consists in achieving this isolation. But the resulting descriptions will normally be incomplete with respect to the real physical entitiy. For example, if you think of some table top experiment, you could take a hammer and smash it. Being smashed is a normal process for the physical, protean entity, but it destroyes the emulated system. Of course, you can extend the theory to include the hammer etc. but will it ever become complete.
      Moreover, it is possible that the description of a theory in terms of a set of equations involves non-computable functions (non-turing-computable (I guess the three-body or n-body problem is such a case) or practically non-computable for reasons of computational complexity – protein folding seems to be such a case). In such cases, you can only solve special cases. By adding more mathematical knowledge (methods for calculation, algorithms etc.), you may extend what you can calculate, but it is never going to be complete. You will have to use approximations (numerical methods, grid simulations etc.), but in case of non-linear systems (like the turbulences of weather systems) these will yield totally wrong results.
      In classical physics, you would end up with proteons because you are dealing with real numbers (the continuum) and there are more real numbers than can be expressed in any formal language for the representation of numbers. I a quantum mechanical description, on the other hand, the information content of a finite size system is finite, but you cannot predict what the system is going to do. For example, you cannot produce a formal theory predicting when a radioactive nucleaus is going to decay.
      The reasons why complete formal descriptions are impossible might be different and there is some work involved in analyzing the different types of proteons, but actually I see proteons everywhere, even in science, so I think science is self-limiting by pretending to be about formalizable and computable systems only.

      • Hi Joh, I am not actually proposing any metaphysical hypothesis here. If panpsychism was right, reality would definitely consists of proteons, but I don’t subscribe to that hypothesis. One could argue that every physical process may also be described as a process of information processing and that what we call the psyche is the informational aspect of the processes in the brain, but I don’t see what would be gained to call all those processes of information interchange psychic.
        I think that very normal physical entities, like, for example, storms, are proteons. Even something as simple as a few (more than two) planets moving around each other can probably not be calculated. One can put up a system of equations, but there is no general way to solve those equations in all instances. In computational terms, such a system of equations would be an incomplete description. To predict or describe what the planets or stars are doing, you would need additional mathematical knowledge from time to time, i.e. an extension of the theory. by additional axioms or rules of inference. The physical proteon behaves according to those equations, but it does not calculate them. We, when we calculate them, would need an unlimited amout of mathematical knowledge. So a proteon might pop up inside very normal physics.
        In other cases, it is simply complexity. Remember what I once wrote about the impossibility of describing the cloud of gas and dust that gave rise to the solar system (or rather, the solar proteon, since the solar system is just an idealization of the real thing): One would thing that such a cloud is a simple thing, but…

  2. Creative philosophy indeed. I’m loving it!

    I’m curious to see where you plan to go with the term. It seems like new terms or revisions of older terms take on their own territory, just by being created, at least that’s the way it seems to me. Consider the word “monad” which Leibniz used to describe a soul (in his Monadology.) That word used to mean something like an “atom” in the original sense, an indivisible unit. I don’t think monads were exclusively human souls, which may be why he came up with the term. There was no monad-over-body involved in his philosophy. He was attempting to create a new way of thinking, and despite the Aristotelian style of his writing, there’s a kind of poetry in his ideas. I sense you’re planning something similar? 🙂

    • Initially, I am going to reformulate some things I have written before using the new terminology. I think some things will come out simpler and easier to understand. I think one should always try to work on the language, until you arrive at terms in which the respective thoughts become easy, even trivial (
      I don’t know if this term will catch on with others, but I am going to use it a lot. I might write a couple of short “case studies” to look into different kinds of proteons and “claim the territory”. I also want to explor connections to other parts and schools of philosophy.
      Leibniz was indeed a very interesting man. I have recently moved a bit nearer to him, since I am now staying in the city of Hannover, where he spent a lot of his time. The university here is called “Leibniz Univerität”, the library is named after him and they actually have an archive there with his herritage. They are still not finished editing all his writings. You see Leibniz a lot here, last year there was a big campaign and his picture was on busses and tram trains and many other places. The other way I have moved nearer to Leibniz is by finding out a few weeks ago that I am related to his friend Otto Mencke ( – he is my first cousin 9 generations removed) who founded and edited the first scientific journal in Germany, in which Leibniz published a lot.
      At Leibniz’ time, it was still possible to just speculate (and the Monadology is an example of that), but Leibniz and Mencke are examples of a generation that set the course for modern science and the use of formal theories inside it (think of Newtonian mechanics as the paradigmatic example of a formal theory of some aspects of reality). Subsequent generations became a bit too optimistic about how far one can go with formalization. In the 1920s and 1930s, the limits of formal theories became visible inside mathematics (think of Gödel, Post, etc.) but somehow, this was ignored by the scientists, perhaps because science had been defined as the project of understanding reality in terms of complete, exact, i.e. formal descriptions. I think the concept of science must be extended.
      Over the next weeks, I will have to do some work on another project first, before getting back to Proteons. I will also continue working on my other project of exploring some strands of German phlosophy of the 1920s. And there are more projects in the pipeline (including a translation project, and the comparision of our Frege-Translation with the previous ones,…). Ars longa, vita brevis 🙂

      • The fact that you see Leibniz’s image on busses and trams there, that’s just amazing. Most people here would have no idea who that was. I sometimes see “Leibniz” cookies, but I don’t know if those are actually named after him. (Students brought those to a Leibniz class once, but you don’t see them all over the place.) Very cool that you have such a history, and ties to his friend.

        I’m looking forward to reading your thoughts on Proteons, and hearing what you find in comparing our translation to those that have been published. Hopefully our translation didn’t deviate too far from the ones out there. That would make me question things. Let me know what you find!

        • The connection to the Leibniz cookies is that they are from Hannover. The company producing them is here in Hannover. When they introduced them (1891), there was a fashion of naming products after famous people, so they named them after him (
          Personally I eat a low-carb diet, so I don’t eat them that often, but from time to time… 🙂

          • Interesting! So they’re actually named after him. How about that.

            They say that low-carb is good. I can tell you from experience that no-carb is horrible. I tried to do the Atkins diet with my husband once, but I found that too extreme. Somewhere around the two week point I got really spaced out, probably from low blood sugar. Everyone needs a Leibniz cookie from time to time. 🙂

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