Why do we need to care category theory and quantum cognition?

In my latest Consciousness Dialogue with Alex Maier [Link], @alexvmaier we chatted a bit about this issue.

Why am I motivated to learn and apply category theory on the data in consciousness research? Why am I interested in quantum cognition by Busemeyer, Bruza, Trueblood, @EmmanuelPothos , ….?

In our discussion, I pointed out an empirical need for a theory that can explain enigmatic findings in similarity experiments. In most similarity experiments, we (implicitly) assume that inverse of similarity (=dissimilarity) is like a distance. A metric (or distance) needs to satisfy the following three axioms. https://en.wikipedia.org/wiki/Metric_(mathematics)

1. Minimality: d(a,b) = 0 <=> a = b
2. Symmetry: d(a,b) = d(b,a)
3. Triangle inequality: d(a,b) <= d(a,c) + d(c,b)

In the video, we chatted a bit on the violation of “2. symmetry”. We are preparing several drafts on this. If you are interested, please see Tversky 1977, Pothos et al 2013 Psych Review.

There is one thing that I couldn’t mention in the video. And in fact, this may be more important. We need a “theory” that is “independent” of the data. This is most eloquently described Steve Phillips’s paper in 2010 (PLoS Comp)

“The Ptolemean (geocentric) theory’s additional assumption (called “epicycles”) is ad hoc. It is unconnected with the rest of the theory and motivated only by the need to fit the data—the assumption could not be confirmed independently of confirming the theory.”

Category theory is a mathematical theory constructed to study “structures”.

Quantum cognition is a mathematical theory that is based on noncommutative probability theory.

These theories are built independent of the data that we want to explain (e.g., the properties of consciousness / qualia). We don’t know if it’s applicable to the data or not.

This is different from testing “theories” that were constructed from the data to be explained. Such “theories” is unfortunately immune to “ad hoc” explanation. Even the theorists themselves do not notice how the theory has changed over the time “to fit” it with the data….

On the process theory and qualia

On Picturing Quantum Processes by Coecke & Kissinger

David Bohm and David Peat wrote in 1987 “We haven’t actually paid much attention to thought as a process. We have engaged in thoughts, but we have only paid attention to the content, not to the process”

I’m guessing that the same may be true for qualia / consciousness.

Here “process” means what is discussed in “process theory”. Process is anything that has zero or more inputs and zero or more outputs. Process theories seem quite useful in the context of quantum theory (and it has a link to category theory).

A function takes one or more inputs and outputs one. Functions are just one example of processes.

A more general notion is a relation. A relation links many inputs to many outputs. It can be considered as a stochastic output. Relations are much closer to what neurons are and more likely to be useful to understand how the brain works. (But most neuroscientists including myself probably don’t know much about relations…)

Ch3 of the Picturing Quantum Processes explains it very well “3.3 Functions and Relations as Processes”.

And processes are even more general. It allows even NO inputs or NO outputs. And this is the reason I think processes are much better concepts than functions or relations when we think about qualia and consciousness.

Thinking about consciousness as something that is defined / induced / evoked by stimulus is misleading. Something that causes motor output is also misleading. Consciousness exists in dreaming and hallucination. Consciousness exists in locked-in or even more severe brain damaged patients. Phenomenal consciousness may or may not be reportable. But if phenomenal consciousness is considered as a process, it doesn’t matter.