Monday, 20 May 2013

Problems with knowledge



What are the fundamentals of knowledge? What does it mean to say that one thing knows another thing? Although human knowledge is one of our fundamental defining characteristics we have only scant clues concerning the nature of knowledge.


The academic knowledge industry has largely settled on a definition of knowledge only slightly modified since it was first proposed by Plato: knowledge is justified true belief. In my view this definition suffers from a possible implication that only humans may have knowledge. However if we loosen up the meanings of ‘belief’ to include non-human expectations and narrow the meaning of ‘justified’ to ‘justified by the evidence’ I can accept this definition.


More technically information theory offers the concept of mutual information. If we know one thing, X, with a certain degree of certainty, and X is correlated with another probabilistic event Y then if we know X we also are more certain about Y. The added certainty is mutual information. For instance if we know the sky is cloudy we know it is more likely to be raining than if we did not know the state of the sky. Knowing the sky is cloudy provides information about the likelihood of rain.





Figure 1: Individual (H(X),H(Y)), joint (H(X,Y)), and conditional entropies for a pair of correlated subsystems X,Y with mutual information I(X; Y).



From a human perspective there seems to be a maximum amount of mutual information we can share with any other entity. Even those who we love most dearly largely remain mysteries. In many respects our inner selves are quite isolated; we are unable to effectively communicate very much of our wondrous complexity.


One view of this problem is that to know something requires a mental model of it. The degree to which this model reproduces the known thing is the degree to which we know it. However the world around us is much more complex than our individual human brain; it includes numerous other human brains as well as numerous other complex entities. Our mental models can be at best only rough sketches of the true reality.


Recent neural research shows that mental models consist of a hypothesis space which divides the possibilities into an exhaustive and mutually exclusive list of hypotheses and assigns probabilities to each. As new information is received the probabilities are updated in accordance with the support their respective hypotheses have received from the new information or evidence. This Bayesian brain school of neuroscience has succeeded in explaining many of the brain’s mysteries.


Perhaps unsurprisingly it turns out that our brains are most adept at gathering knowledge concerning other members of our species. In a way this is to be expected as we are best suited to know other things that are similar to ourselves but the mechanics may be surprising. 


The brain contains families of ‘mirror’ neurons. When for example we observe someone smiling a family of neurons in our own brain fire. This is the same family of neurons that fire when we smile and are correlated with the emotions we have when we smile. Thus we gain understanding of the other person’s mental state; on the basis of this information we judge some hypotheses concerning the other person as more likely and others as less.


This brilliant mechanism in some ways enables us to know others as we know ourselves. There are however ambiguities. Even our self-knowledge is incomplete and a smile by itself is somewhat ambiguous; it could mean many things.

We are not alone in our problem of knowing. Another view of this conundrum is at the level of fundamental physics. Here we see that information can be transferred between entities in only four ways, through the four quantum forces of nature. For example when we see someone smile the information is transferred from the reflected light to our visual system via a quantum interaction in our retina.


Quantum information transfer is hugely constrained. Only a minute portion of the information that would fully describe a quantum system may be transferred to entities in its environment. Quantum systems too are isolated.

Quantum systems may be modeled in a hypothesis space or state space which is the set of density matrices on N-dimensional complex Hilbert space (ref.: Information-theoretic postulates for quantum theory). This technical description conveys some of the mind boggling complexities of quantum systems; the density matrices are hugely complex and the Hilbert space may approach infinite dimensionality.

 


Information transfer between quantum systems and their environment takes place through communication channels, the nature of which are beginning to be unraveled. It appears that information regarding only a very small subset of the quantum complexity may be transferred through these channels and that subset consists of a simple probability distribution, it is the same probability distribution which quantum theory provides for the outcome of measurements on quantum systems (ref.: No-local-broadcasting theorem for quantum correlations).


We may now see quantum theory in a new light. Our physical reality is based on quantum systems; however quantum systems are weird and spooky compared to the logical and familiar classical world we inhabit. This situation exists because we (and all other classical entities) may only experience a very small subset of the quantum world and again this knowledge is in the form of probability distributions. These probability distributions form our best model and the best knowledge we may have of the quantum reality.


A further interesting point is that quantum communication channels can be viewed as selection mechanisms; only certain aspects of the quantum system can survive the transfer to the system’s environment. In this sense quantum information transfer may be seen as a Darwinian process as described by the theory of quantum Darwinism (ref.: Objectivity Through State Broadcasting: The Origins Of Quantum Darwinism).



Once we note that life, neural models and cultural knowledge such as science may also be best understood as evolving via Darwinian processes, we are led to the stunning conclusion that reality is at base a nested hierarchical system of knowledge accumulation mediated by Darwinian processes.


While knowledge is by its nature always incomplete and we may not be fully known to any other entity we may gain some solace from understanding that the rest of reality is in the same boat with us. As our knowledge is based upon forms of quantum, biological, neural and cultural knowledge we should understand that the reason we may see further than other natural entities is because we are standing on the shoulders of giants.

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