About mathematical cognition in TGD Universe

When one ponders about consciousness, one sooner or later realizes that the emergence of mathematical consciousness has meant an enormous evolutionary leap. Something completely exceptional seems to have occurred when the apple fell down about 382 years ago. Of course, also the emergence of our species was this kind of event? And there are my heros of science. This raises questions. What really happens in this kind of giant leaps of mathematical consciousness or cognitive consciousness in general. What does being or becoming conscious of a mathematical concept mean? Could one see this kind of event as an emergence of a new reflective level of consciousness? One cannot answer these questions unless one can identify the physical correlates of cognition, in particular mathematical cognition. In the sequel these questions are considered in the TGD framework in which the number theoretic and geometric views of physics are dual to each other.

This raises questions. What really happens in this kind of giant leaps of mathematical consciousness or cognitive consciousness in general. Is our species even in principle able to answer this kind of questions?

What does being or becoming conscious of a mathematical concept mean? Could one see this kind of event as an emergence of a new reflective level of consciousness? But how to describe this kind of hierarchy of levels of consciousness? What kind of phenomenon, bringing in mind phase transition, took place when humankind became conscious of differential and integral calculus, number theory, algebra, logic? Or did already the emergence of our civilization lead to this even?

One can imagine a more modest goal. What could be the physical correlates of these kinds of cognition. The easy solution of the problem would be that deterministic computations are conscious and one can formally regard any time deterministic time evolution as a computer program. This hypothesis does not however explain anything and is untestable.

Even the understanding of how the basic notions and algorithms are realized consciously at the level of cognitive consciousness seems very difficult in the framework of the recent day physics which has hitherto refused to say anything about conscious experience. Is the existing view of physics enough to meet the challenge?

These challenges look formidable but one can try! Maybe one could say at least something about cognition and mathematical cognition?

1. What are the physical correlates of cognition? Cognition is discrete and finite and cognition represents. Could one identify cognitive representations as a discretization of the sensory world. TGD leads to a number theoretic vision about physics dual to the geometric vision and provides a theory of cognition.

   p-Adic topologies seems to be very natural candidates for the topology of cognition. p-Adic number fields fusing together with reals to what is called adele. One can also defined entire hierarchy of adeles induced by algebraic extensions of rationals. There is also a second adele-like structure defined by the union of the p-adic number fields. Two p-adic number fields are glued together at interfaces formed by numbers, which have an expansion in powers of an integer divisible by both primes.

2. Concepts, in particular mathematical concepts, are a key element of cognition. What could be the quantum description of concepts and their emergence. Here standard quantum theory suggests an answer. Classically the field of concept is the set of its instances. In quantum theory wave functions in the set could define the instances of the concept.
3. What gives a concious meaning to the concept? Category theoretical thinking suggests that "arrows" as entanglements between concept and other concepts provide the meaning as state function reduction selecting one particular instance of the rule representd by entanglement. In physics this means quantum measurements.
4. What are the quantum physical correlates for Boolean logic? The Fock states define a Boolean algebra and in TGD framework these states span an infinite-D state space. In zero energy ontology (ZEO) \cite{allb/ZEO} \cite{btar/zeoquestions} of TGD this leads to a natural realization of Boolean algebras and zero energy state defines a quantum version of Boolean map \cite{allb/intsysc}.
5. We learn in school mathematics as associations such as "1+2"$\rightarrow$ "3" and the recent successes of GPT have demonstrated how powerful tool associations are. I have consider the possible quantum aspects of AI and GPT in \cite{btart/GPT,tgdcomp}) Could associative rules be represented by using quantum entanglement? This could reduce the development of mathematical understanding to the emergence of entangled states representing correlations as rules.
6. What conscious computation could mean? This requires a theory of consciousness and here TGD inspired theory of consciousness provides an approach. Could state function reduction reducing the entanglement defining associative rule give rise to a conscious experience associated with the association? Could one also imagine some kind of quantum hardware of mathematical consciousness perhaps representing the basic arithmetical operations?

Returning to the motivating question about the emergence of new reflective levels of consciousness and cognition, humans are distinguished from the other species by a highly evolved social organizations and in the TGD Universe the emergence of higher levels of consciousness assignable to social structures could be a central element. Single human viz. society would be like a single neuron viz. the entire brain. Here the hierarchy of magnetic bodies (MBs) is highly suggestive. The emergence of revolutionary ideas certainly requires a highly developed society with a large and complex MB. MBs involve an onion-like hierarchy of extensions of rationals and their dimension measuring their complexity serves as a kind of IQ. Could the emergence of a new level in this hierarchy give rise to these kinds of revolutionary events?

See the article About mathematical cognition in the TGD Universe .

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.