The Geometry of Information Retrieval
The Geometry of Information Retrieval
The Algorithmic Beauty of Sea Shells
The Algorithmic Beauty of Sea Shells
How can nature help us compute?
SOFSEM'06 Proceedings of the 32nd conference on Current Trends in Theory and Practice of Computer Science
Algorithmic randomness, quantum physics, and incompleteness
MCU'04 Proceedings of the 4th international conference on Machines, Computations, and Universality
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It would be nice if science answered all questions about our universe. In the past, mathematics has not just provided the language in which to frame suitable scientific answers, but was also able to give us clear indications of its own limitations. The former was able to deliver results via an ad hoc interface between theory and experiment. But to characterise the power of the scientific approach, one needs a parallel higher-order understanding of how the working scientist uses mathematics, and the development of an informative body of theory to clarify and expand this understanding. We argue that this depends on us selecting mathematical models which take account of the ‘thingness' of reality, and puts the mathematics in a correspondingly rich information-theoretic context. The task is to restore the role of embodied computation and its hierarchically arising attributes. The reward is an extension of our understanding of the power and limitations of mathematics, in the mathematical context, to that of the real world. Out of this viewpoint emerges a widely applicable framework, with not only epistemological, but also ontological consequences --- one which uses Turing invariance and its putative breakdowns to confirm what we observe in the universe, to give a theoretical status to the dichotomy between quantum and relativistic domains, and which removes the need for many-worlds and related ideas. In particular, it is a view which confirms that of many quantum theorists --- that it is the quantum world that is ‘normal', and our classical level of reality which is strange and harder to explain. And which complements fascinating work of Cristian Calude and his collaborators on the mathematical characteristics of quantum randomness, and the relationship of ‘strong determinism' to computability in nature.