Physics, Topology, Logic and Computation: A Rosetta Stone #categorytheory
arXiv:0903.0340 [quant-ph]
In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology: namely, a linear operator behaves very much like a "cobordism". Similar diagrams can be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of interest in quantum cryptography and quantum computation, it became clear that there is extensive network of analogies between physics, topology, logic and computation. In this expository paper, we make some of these analogies precise using the concept of "closed symmetric monoidal category". We assume no prior knowledge of category theory, proof theory or computer science.
#categorytheory, #physics, #topology
[1012.5647] An informal introduction to topos theory
Author: Tom Leinster
This short expository text is for readers who are confident in basic category theory but know little or nothing about toposes. It is based on some impromptu talks given to a small group of category theorists.
Subjects: Category Theory (math.CT); Algebraic Geometry (math.AG); Logic (math.LO)
[1503.02776] The Emergence of Life as a First Order Phase Transition
We demonstrate a phase transition from non-life to life, defined as non-replicating and replicating systems respectively, and characterize some of its dynamical properties. The transition is first order and demonstrates many characteristics one might expect from a newly emergent biosphere. During the phase transition the system experiences an explosive growth in diversity, with restructuring of both the extant replicator population and the environment. The observed dynamics have a natural information-theoretic interpretation, where the probability for the transition to occur depends on the mutual information shared between replicators and environment. Through the transition, the system undergoes a series of symmetry breaking transitions whereby the information content of replicators becomes increasingly distinct from that of their environment. Thus, the replicators that nucleate the transition in the non-life phase are often not those which are ultimately selected in the life phase. We discuss the implications of these results for understanding the emergence of life, and natural selection more broadly.
#arXiv, #itbio
Interestingly, Erwin Schrödinger's seminal paper "What is Life?" came out in 1944, almost contemporaneously with the creation of information theory in 1948. Unbeknownst to most Claude Shannon's (the researcher who created the field of information theory and whose Masters thesis literally launched the digital revolution) Ph.D. thesis (1940) was entitled: "An Algebra for Theoretical Genetics," so he could certainly be said to be the god-father of the entire area.
Since then, certainly dozens of researchers have looked at the potential of using information theory to create a better hard core definition of what life really is (certainly the place where any self-respecting Platonist would begin.)
One of the best definitions and frameworks I've seen thus far has to be that of Christoph Adami. To start, and depending on your level of sophistication, take a look at his recent arXiv paper (Information-theoretic considerations concerning the origin of life - http:/
For further references, I maintain a nice list of resources at Information Theory and Biology Resources (http:/
If you really want to blow the top off of your definitions, you might also consider taking a much broader look at the universe (a distant reading, so-to-speak) and read through David Christian's conceptualization (Maps of Time: An Introduction to Big History: David Christian, William H. McNeill http:/