The Formal and the Creative

February 15, 2012 § Leave a comment

If there is such a category as the antipodic at all,

it certainly applies to the pair of the formal and the creative, at least as long as we consult the field of propositions1 that is labeled as the “Western Culture.” As a consequence, in many cultures, and even among mathematicians, these qualities tend to be conceived as completely separated.

We think that this rating is based on a serious myopia, one that is quite common throughout rationalism, especially if that comes as a flavor of idealism. In a small series of essays—it is too much material for a single one—we will investigate the relation between these qualities, or concepts, of the formal and the creative. Today, we just will briefly introduce some markers.

The Basic Context

The relevance of this endeavor is pretty obvious. On the one hand we have the part of creativity. If machine-based episteme implies the necessity to create new models, new hypothesis and new theories we should not only get clear about the necessary mechanisms and the sufficient conditions for its “appearance.” In other chapters we already mentioned complexity and evolutionary processes as the primary, if not the only candidates for such mechanisms. These domains are related to the transition from the material to the immaterial, and surely, as such they are indispensable for any complete theory about creativity. Yet, we also have to take into consideration the space of the symbolic, i.e. of the immaterial, of information and knowledge, which we can’t find in the domains of complexity and evolution, at least not without distorting them too much. There is a significant aspect of creativity that is situated completely in the realm of the symbolic (to which we propose to include diagrams as well). In other words, there is an aspect of creativity that is related to language, to story-telling, understood as weaving (combining) a web of symbols and concepts, that often is associative in its own respect, whether in literature, mathematics, reading and writing, or regarding the DNA.

On the other hand, we have the quality of the formal, or when labelled as a domain of activity, formalization. The domain of the formal is fully within the realm of the symbolic. And of course, the formal is frequently conceived as the cornerstone, if not essence, of mathematics. Before the beginning of the 20ieth century, or around its onset, the formal was almost a synonym to mathematics. At that time, the general movement to more and more abstract structures in mathematics, i.e. things like group theory, or number theory, lead to the enterprise to search for the foundations of mathematics, often epitomized as the Hilbert Program. As a consequence, kind of a “war” broke out, bearing two parties, the intuitionists and the formalists, and the famous foundational crisis started, which is lasting till today. Gödel then proofed that even in mathematics we can not know perfectly. Nevertheless, for most people mathematics is seen as the domain where reason and rationalism is most developed. Yet, despite mathematicians are indeed ingenious (as many other people), mathematics itself is conceived as safe, that is static and non-creative. Mathematics is about symbols under analytic closure. Ideally, there are no “white territories” in mathematics, at least for the members of the formalist party.

The mostly digital machines finally pose a particular problem. The question is whether a deterministic machine, i.e. a machine for which a complete analytic description can exist, is able to develop creativity.

This question has been devised many times in the history of philosophy and thinking, albeit in different forms. Leibniz imagined a mathesis universalis and characteristica universalis as well. In the 20ieth century, Carnap tried to proof the possibility of a formal language that could serve as the ideal language for science [1]. Both failed, Carnap much more disastrously than Leibniz. Leibniz also thought about the transition from the realm of the mechanic to the realm of human thought, by means of his ars combinatoria, which he had imagined to create any possible thought. We definitely will return to Leibniz and his ideas later.

A (summarizing) Glimpse

How will we proceed, and what will we find?

First we will introduce and discuss some the methodological pillars for our reasoning about the (almost “dialectic”) relation between creativity and formalization; among those the most important ones are the following:

  • – the status of “elements” for theorizing;
  • – the concept of dimensions and space;
  • – relations
  • – the domain of comparison;
  • – symmetries as a tool;
  • – virtuality.

Secondly, we will ask about the structure of the terms “formal” and “creative” while they are in use, especially however, we are interested in their foundational status. We will find, that both, formalization and creativity belong to a very particular class of language games. Notably, these terms turn out to be singular terms, that are at the same time not names. They are singular because their foundation as well as the possibility to experience them are self-referential. (ceterum censeo: a result that is not possible if we’d stick to the ontological style by asking “What is creativity…”)

The experience of the concepts associated to them can’t be externalized. We can not talk about language without language, nor can we think “language” without practicing it. Thus, they also can’t be justified by external references, which is a quite remarkable property.

In the end we hopefully will have made clear that creativity in the symbolic space is not achievable without formalization. They are even co-generative.

Introductory Remarks

Let us start with creativity. Creativity has always been considered as something fateful. Until the beginnings of psychology as a science by William James, smart people have been smart by the working of fate, or some gods. Famous, and for centuries unchallenged, the passage in Plato’s Theaitetos [2], where Sokrates explains his role in maieutics by mentioning that the creation of novel things is task of the gods. The genius as well as concept of intuition could be regarded just a rather close relatives of that. Only since the 1950ies, probably not by pure chance, people started to recognize creativity as a subject in its own right [3]. Yet, somehow it is not really satisfying to explain creativity by calling it “divergent” or “lateral” thinking [4]. Nothing is going to be explained by replacing one term by another. Nowadays, and mostly in the domain of design research, conditions for creativity are often understood in terms of collaborations. People even resort to infamous swarm intelligence, which is near a declaration of bankruptcy.

Any of these approaches are just replacing some terms with some other terms, trying to conjure some improvement in understanding. Most of the “explanations” indeed look rather like rain dancing than a valuable analysis. Recently a large philosophical congress in Berlin, with more than 1200 inscribed participants, and two books comprising around 2000 pages focused on the subject largely in the same vein and without much results [5]. We are definitely neither interested in any kind of metaphysical base-jumping by referring directly or indirectly to intuition, and the accompanying angels in the background, nor in phenomenological, sociological or superficial psychological approaches, tying to get support by some funny anecdotes.

The question really is, what are we talking about, and how, when referring to the concept of creativity. Only because this question is neither posed nor answered, we are finding so much esoterics around this topic. Creativity surely exceeds problem solving, although sometimes it occurs righteous when solving a problem. It may be observed in calm story telling, in cataclysmic performances of artists, or in language.

Actually, our impression is that creativity is nothing that sometimes “is there”, and sometimes not. In language it is present all the time, much like it is the case for analogical thinking. The question is which of those phenomena we call “creative,” coarsely spoken, which degree of intensity regarding novelty and usefulness of that novelty we allow to get assigned a particular saliency. Somehow, constraints seem to play an important role, as well as the capability to release it, or apply it, at will. Then, however, creativity must be a technique, or at least based on tools which we could learn to use. It is, however, pretty clear that we have to distinguish between the assignment of the saliency (“this or that person has been creative”) and the phenomenon and its underlying mechanisms. The assignment of the term is introducing a discreteness that is not present on the level of the mechanism, hence we never will understand about what we are talking about if we take just the parlance as the source and the measure.

The phenomenon of language provides a nice bridge to the realm of the formal. Today, probably mainly due to the influence of computer sciences, natural languages are distinguished from artificial languages, which often are also called formal languages. It is widely accepted, that formalization either is based on formal languages or that the former creates an instance of the latter. The concept of formal language is important in mathematics, computer science and science at large. Instantiated as programming languages, formal languages are of an enormous relevance for human society; one could even say that these languages themselves establish some kind of a media.

Yet, the labeling of the discerned phenomena as “natural” and “formal” always strikes me. It is remarkable that human languages are so often also called “natural” languages. Somehow, human language appears so outstanding to humans that they call their language in a funny paradoxical move a “natural” thing, as if this language-thing would have been originated outside human culture. Today, as we know about many instances of cultural phenomena in animals, the strong dichotomy between culture and nature blurred considerably. A particular driver of this is provided by the spreading insight that we as humans are also animals: our bodies contain a brain. Thus, we and our culture also build upon this amazing morphological structure, continuously so. We as humans are just the embodiment of the dichotomy between nature and culture, and nothing could express the confusion about this issue more than the notion of “natural language.” A bit shamefaced we call the expressions of whales and dolphins “singing”, despite we know that they communicate rather complicated matters. We are just unable to understand anything of it. The main reason for that presumable being that we do not share anything regarding their Lebensform, and other references than the Lebensform are not relevant for languages.

Language, whether natural or formal, is supposed to be used to express things. Already here we now have been arriving in deep troubles as the previous sentence is anything than innocent. First, speaking about things is not a trivial affair. A thing is a difficult thing. Taking etymology into consideration, we see that things are the results of negotiations. As a “result,” in turn, “things” are reductions, albeit in the realm of the abstract. The next difficulty is invoked by the idea that we can “express” things in a formal language. There has been a large debate on the expressive capabilities of formal languages, mainly induced by Carnap [1], and carried further by Quine [6], Sneed [7], Stegmüller [8], Spohn [9], and Moulines [10], among others, up to today.

In our opinion, the claim of the expressibility of formal language, and hence the proposed usage of formal languages as a way to express scientific models and theories, is based on probably more than just one deep and drastic misunderstanding. We will elucidate this throughout this series; other arguments has been devised for instance by Putnam in his small treatise about the “meaning of meaning” [11], where he famously argued that “analyticity is an inexplicable noise” without any possibility for a meaningful usage. That’s also a first hint that analyticity is not about the the same thing(s) as formalization.

Robert Brandom puts the act of expressing within social contexts into the center of his philosophy [12], constructing a well-differentiated perspective upon the relation between principles in the usage of language and its structure. Following Brandom, we could say that formal language can not be expressive almost by its own definition: the mutual social act of requesting an interpretation is missing there, as well as any propositional content. If there is no propositional content, nothing could be expressed. Yet, propositional content comes into existence only by a series of events where the interactees in a social situation ascribe it mutually to each other and are also willing to accept that assignment.

Formal languages consist of exactly defined labeled sets, where each set and its label represents a rewriting rule. In other words, formal languages are finite state machines; they are always expressible as a compiler for a programming language. Programming languages organize the arrangement of rewriting rules, they are however not an entity capable for semantics. We could easily conclude that formal languages are not languages at all.

A last remark about formalization as a technique. Formalization is undeniably based on the use, or better, the assignment of symbols to particular, deliberately chosen contexts, actions, recipes or processes. Think of proofs of certain results in mathematics where the symbolized idea later refers to the idea and its proof. Such, they may act as kind of abbreviations, or they will denote abstractions. They also may support the visibility of the core of otherwise lengthy reasonings. Sometimes, as for instance in mathematics, formalization requires several components, e.g. the item or subject, the accompanying operators or transformations (take that as “usage”), and the reference to some axiomatics or a explicit description of the conditions and the affected items. The same style is applied in physics. Yet, this complete structure is not necessary for an action to count as a formalization. We propose to conceive of formalization as the selection of elements (will be introduced soon) that consecutively are symbolized. Actually, it is not necessary to write down a “formula” about something in order to formalize that something. It is also not necessary, so we are convinced, to apply a particular logic when establishing the formalization through abstraction. It is just the symbolic compression that allows to achieve further results which would remain inaccessible otherwise. Or briefly put, to give a complicated thing a symbolic form that lives within a system of other forms.

Finally, there is just one thing we always should keep in mind. Using, introducing or referring to a formalization irrevocably implies an instantiation when we are going to apply it, to bring it back to more “concrete” contexts. Thus, formalization is deeply linked to the Deleuzean figure of thought of the “Differential.” [13]

part 2: Strong Singularity of certain Terms

Part 3: A Pragmatic Start for a Beautiful Pair

Part 4: Elementarization and Expressibility


1. Here we refer to Michel Foucault’s concept of the “field of propositions” / “field of proposals”, which he developed in the book “The Archaeology of Knowledge.”

  • [1] Rudolf Carnap, Logische Syntax der Sprache, Wien 1934 [2. Aufl. 1968].
  • [2] Platon, Theaitetos.
  • [3] Guilford,  Creativity , 1950
  • [4] DeBono about lateral thinking
  • [5] Günther Abel (ed.), Kreativität. Kolloquiumsband XX. Kongress der Deutschen Philosophie. Meiner Verlag, Hamburg 2007.
  • [6] Quine, Two Dogmas of Empiricism.
  • [7] Sneed
  • [8] Stegmüller
  • [9] Spohn
  • [10] Moulines
  • [11] Hilary Putnam, The Meaning of Meaning
  • [12] Robert Brandom, Making it Explicit.
  • [13] Gilles Deleuze, Difference and Repetition.

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