This paper presents results of an ongoing interdisciplinary study to develop a computational theory of creativity for engineering design. Human design activities are surveyed, and popular computer-aided design methodologies are examined. It is argued that semiotics has the potential to merge and unite various design approaches into one fundamental theory that is naturally interpretable and so comprehensible in terms of computer use. Reviewing related work in philosophy, psychology, and cognitive science provides a general and encompassing vision of the creativity phenomenon. Basic notions of algebraic semiotics are given and explained in terms of design. This is to define a model of the design creative process, which is seen as a process of semiosis, where concepts and their attributes represented as signs organized into systems are evolved, blended, and analyzed, resulting in the development of new concepts. The model allows us to formally describe and investigate essential properties of the design process, namely its dynamics and non-determinism inherent in creative thinking. A stable pattern of creative thought - analogical and metaphorical reasoning - is specified to demonstrate the expressive power of the modeling approach; illustrative examples are given. The developed theory is applied to clarify the nature of emergence in design: it is shown that while emergent properties of a product may influence its creative value, emergence can simply be seen as a by-product of the creative process. Concluding remarks summarize the research, point to some unresolved issues, and outline directions for future work.