Levels of structure and consciousness

In this section we begin to describe how the structures of mathematics connect to conscious experience. Central to this notion and to the structure of our brain is the idea of feedback. There is a general progression of information processing that starts with data from our senses and proceeds to processes that perform ever higher levels of integration. A low level process might detect a sharp change in color at one point in the visual field. A higher level process might integrate these into a line or edge. A higher level process might recognize a door which may be what we experience consciously. We are not aware of the lines that make up the door unless something shifts our attention from the scene as a whole to such details. The process of recognizing a door is not strictly one way. There are feedback loops where higher level processes generate inputs to lower level processes. One speculative theory suggests that the brain, at each level of processing, is continually making predictions of what to expect[25]. Feedback to lower level processes is generated from those predictions. If the expectations are not met then signals to higher level processes are generated. Whatever the detailed structure of the brain, feedback plays a central role. The ordinal numbers in mathematics can be thought of as characterizing the subtlety of feedback in a mathematical system. They are the tool I use to begin to connect mathematical structure with conscious experience.

The ordinal numbers described in Section 5.6 characterize the power of feedback, iteration or self reflection that a system is capable of. There is an enormous richness of possible mathematical structures that can be defined at higher levels of the ordinal hierarchy. Exploring ordinals is not the primary focus of mathematics. Ascending to higher levels of structure is not the primary focus of evolution. However ordinals characterize the power of a mathematical system and the limit of sophistication of a physical structure. This suggests they may do the same for conscious experience embodying such structures.

There is no precise mapping of ordinals to physical systems or biological structures, but biological structures for modeling external and internal state can be assigned ordinals that characterize them. The goal is to develop a feel for the connection between these mathematical structures and biological structures.

Table 9.1 gives some simple examples. The first entry is for a fixed response for fixed input. There is no iteration and thus no sense of the potentially infinite. Thus the limit ordinal for these structures is the first non finite ordinal $\omega$ . A limit ordinal encompasses all smaller (in this case finite) ordinals, but is not reachable by the structures that define these smaller ordinals. Next is the amplification of an input. The intensity of the response is determined by the intensity of the input. For example if a creature is running to escape a predator it will try to travel proportionately faster than its attacker. The limit ordinal for this is $\omega\times\omega$ . The next two entries involve stringing together amplifiers. The response is proportional to the product of two inputs or more inputs. Finally we consider a variable number of memories with the ability to mutually reinforce each other. If the number of these has no fixed limit then the ordinal that characterizes this is $\omega^\omega$ .

It is only simple toy structures like the above that we can easily relate to ordinals. There is a certain artificiality to this process, but it gives a feel for the relationship between feedback mechanisms and mathematical structure. More complex thought processes have ordinal limits, but it is more difficult to determine these.

The most complex structures in the human brain are not for dealing with simple external stimulus. They are for dealing with our fellow creatures. Subtle forms of iteration and self reflection evolve because evolution creates an environment in which they are valuable.

Table 9.1: Ordinals that characterize biological structures

feedback structure	ordinal
fixed response for input	$\omega$
single amplifier	$\omega\times\omega$
double amplifier	$\omega\times\omega\times\omega$
series of n amplifiers	$\omega^{n+1}$
variable number of distinct memories
with ability to connect and mutually reinforce	$\omega^\omega$