From Other Angles
On this page, I attempt to analogize the representational ability of our brain from a linear algebra and a causal perspective. Then ask some further questions.
Starting from Linear Algebra: transformation of basis
Part of linear algebra talks about changing a vector x with bases B to a new basis C
(x)B --> (x)C.
And we could apply a series of computations to get the change-of-coordinates vector
P(C<--B)
which gives a formula
(x)C = P(C<--B)*(x)B
If we think of our cognitive processes as mapping the reality into our mental space, then would there be a corresponding "change-of-coordinates" computation in between the two states?
For a system to be linear, it has to preserve some kind of rules related to addition and scalar-multiplication. That is to say, a space full of parallel and evenly spaced grid lines still needs to preserve this property after applying a linear set of rules.
Then the question becomes, does our brain follow a "linear" set of rules?
Suppose that we do, then our ability to integrate different sensory information into a whole "experience" might be interpreted as our neural activities going under a complex set of adding and multiplying. Following the logic, it seems plausible for us to find the computation of each stage if we could find all the transitional stages in between the start and the end product.
However, considering the complexity of our brain and the large amount of steps in between, the "linear algebra" approach might still be a hard and time-consuming approach to take.
Moreover, we need to consider the automated nature of our brain. Beside from having the capability to perform computations, our brain also has the ability to sort computations in the correct sequential order. Though both "perform computations" and "sort computations" could be due to the same hardwired neuro networks (the biological aspect), the later adds a layer of dynamics (the psychological aspect) into our brain. It is worth our effort to combine the two and come up with more holistic ideas.
Going back to the linear algebra analogy, is the relationship between the hardwired neuro-network and the dynamics more like:
1. a linear system and a much more complex linear system
or
2. a linear system and a non-linear system that is quite different in nature
Further questions on causation and neuroscience
When someone ask me: "What did you do today"? I often answer in a sequential order:"I've been doing ... this morning, and in the afternon, I ..."
Or when someone ask of my favorite color and why, I am likely to say:"I like blue because it is a cool and calm color to look at."
Why is our memory sequential or associational? It is a kind of reflection for some properties of the physical world, or is it completly due to the properties of the brain itself?
Some Properties of Causation:
1. Assymetry: heat caused ice-cream to melt, not the other way.
2. Regularity: heat always causes ice-cream to melt. It is regularly happening based on our experience
3. Simultaneity: the cause and effect are linked temorally in some way. Though the time may vary, effect would always happen if the cause has happened.
4. Non-locality: the distance between the cause and the effect could be far away.
A reason: effectiveness
Our ability to both sustain stability and allow rapid transitions to novel percepts is required to survived in this complex world. However, our brain only has limited capacity, and could not take in everything at the same time. Therefore, it needs some way to fold and store the information. Consctruting information in a causal way seems to be quite effective as we could quickly predict what to do next when something happens.
An example of the limitation of our brain capacity: the rabbit-duck illusion
In a rabbit-duck illusion, we are able to see either a rabbit or a duck from the same graph by directing our cognition. However, we are unable to view the graph as anything else other than a rabbit or a duck. Overall, it seems that our representational power is tightly linked to what we percieve, but we still have certain freedom in labelling what we saw.
Inversly, though we cognitively know that the graph could be a rabbit or a duck, we could only see either one at one stare. In other words, there is an upper limit to our speed of switching between mental states.