Towards an ontology of design: lessons from C–K design theory and Forcing

For instance, there is an active quest for new theoretical physics based on String theory that could replace the standard model of particles.

There is also documented material on its practical applications in several industrial contexts (Elmquist and Segrestin 2007; Ben Mahmoud-Jouini et al. 2006; Hatchuel and Weil 2003; Hatchuel et al. 2004, 2006; Gillier et al. 2010; Elmquist and Le Masson 2009).

It can also be formulated as: “The class of objects x, for which a group of properties p1, p2, pk holds in K is non empty”.

The literature about C–K theory discusses two ways to treat this issue: the class of “non-rubber tyres for ordinary cars” can be seen as a special kind of set, called C-set, for which the existence of elements is K-undecidable (Hatchuel and Weil 2009). This is the core idea of the theory and the most challenging aspect of its modelling. Clearly assuming “elements” of this C-set will be contradictory with the status of the concept, or we would have to speak of elements without any possibility to define them or to construct them. This is in contradiction to the classic elementarist approach of sets (see Jech, Dehornoy). It means that the propositions “C-set is empty” or “a C-set is non-empty” is K-undecided and only after design is done, we will be able to decide this question. Technically, Hatchuel and Weil suggest that C-sets could be axiomatized within ZF if we reject the axiom of choice and the axiom of regularity, as these axioms assume necessarily the existence of elements. More generally, in space C, where the new object is designed, the membership relation of Set theory has a meaning only when the existence of elements is proved. Hendriks and Kazakci (Hendriks and Kazakçi 2011) have studied an alternative formulation of C–K theory only based on first-order logic. They make no reference to C-sets, and they reach similar findings about the structure of design reasoning.

It may be surprising that the inclusion relation becomes possible: it becomes possible only when the existence is proved.

The idea of Design as Chimera forming can be traced back to Yoshikawa’s GDT (Yoshikawa 1985) (see the Frobird, p. 177) although the authors did not use the term chimera and the theoretical properties of such operations were not fully described in the paper.

He has been awarded a Fields medal for this work.

Properly speaking, ZF has infinitely many axioms: its axiomatization consists of six axioms and two axiom schemas (comprehension and replacement), which are infinite collections of axioms of similar form. We thank an anonymous reviewer for this remark.

Such models of things are also present in design theories (see for instance the “entity set” in GDT (Yoshikawa 1981)).

The two propositions of this type that gave birth to the forcing method are well known in set theory. The first one is “every set of nonempty sets has a choice function”; the second one is the existence of infinite cardinals that are intermediate between the cardinal of the integers and the cardinal of the reals also called the continuum hypothesis.

Complete presentations of Forcing can be easily found in standard textbooks in advanced set theory (Kunen 1980; Jech 2002; Cohen 1966).

Filters are standard structures in Set theory. A filter F is a set of conditions of Q with the following properties: nonempty; nestedness (if p < q and p in F then q is in F) and compatibility (if p, q are in F, then there is s in F such that s < p and s < q).

G is not in M as soon as Q follows the splitting condition: for every condition p, there are two conditions q and q’ that refine p but are incompatible (there is no constraint that refines q and q’). Demonstration (see (Jech 2002), exercise 14.6, p. 223): Suppose that G is in M and consider D = Q\G. For any p in Q, the splitting condition implies that there q and q’ that refine p and are incompatible; so one of the two is not in G, hence is in D. Hence, any condition of Q is refined by an element of D. Hence, D is dense. So G is not generic.

To give a hint on this strange property and its demonstration: Cohen follows, as he explains himself, the reasoning of Cantor diagonalization. He shows that the “new” real is different from any real g written in base 2 by showing that there is at least one condition in G that differentiates G and this real (this corresponds to the fact that G intersects Dg, the set of conditions that are not included in g).

Forcing is a mathematical tool that can design new sets using infinite series of conditions. In real design, series of conditions are not always infinite.

There are several forms of extensions in mathematics that cannot be even mentioned in this paper. Our claim is that Forcing, to our knowledge, presents the highest generality in its assumptions and scope.

In a simulation study of C–K reasoning (Kazakçi et al. 2010) voids could be modelled, since knowledge was assumed to have a graph structure.

One can also use the image of a “hole”. The metaphor of “holes” has been suggested by Udo Lindemann during a presentation about “Creativity in engineering” (SIG Design theory Workshop February 2011). It is a good image of the undecidable propositions, or concepts in C–K theory, that trigger a design process. Udo Lindemann showed that such “holes” can be detected with engineering methods when they are used to find Design ways that were not yet explored.

Proof: for any dense subset D of C, there is a refinement of C _k that is in D. But since C _k is also in K, any refinement of C _k is in K and cannot be in C. Hence C _k is in D.

The output of Forcing is not one unique new Set G, but a whole extended model N of ZF. The building of the extension model combines subsets of the old ground Model M and the new set G. Thus, new names have to be carefully redistributed so that an element M with name a gets a new name a’ when considered as an element of the new set. As a consequence of these preserving rules, the extension Set is well formed and obeys ZF axioms (Jech 2000).