Recently a humanoid robot known as Sophia has gained a lot of attention in net (see the article by Ben Goertzel, Eddie Monroe, Julia Moss, David Hanson and Gino Yu titled with title " Loving AI: Humanoid Robots as Agents of Human Consciousness Expansion (summary of early research progress)" .
This led to ask the question about the distinctions of Natural and Artificial Intelligence and about how to model Natural Intelligence. One might think that idiot savants could help answering this kind of question but so it turned out to be!
Mathematical genii and idiot savants seem to have something in common
It is hard to understand the miraculous arithmetical abilities of both some mathematical genii and idiot savants lacking completely conceptual thinking and conscious information processing based on algorithms. I have discussed the number theoretical feats here.
Not all individual capable of memory and arithmetic feats are idiot savants. These mathematical feats are not those of idiot savant and involve high level mathematical conceptualization. How Indian self-taught number-theoretical genius Ramajunan discovered his formulas remains still a mystery suggesting totally different kind of information processing. Ramanujan himself told that he got his formulas from his personal God.
Ramajunan's feats lose some of their mystery if higher level selves are involved. I have considered a possible explanation based on ZEO, which allows to consider the possibility that quantum computation type processing could be carried out in both time directions alternately. The mental image representing the computation would experience several deaths following by re-incarnations with opposite direction of clock time (the time direction in which the size of CD increases). The process requiring very long time in the usual positive energy ontology would take only short time when measured as the total shift for the tip of either boundary of CD - the duration of computations at opposite boundary would much longer!
Sacks tells about idiot savant twins with intelligence quotient of 60 having amazing numerical abilities despite that they could not understand even the simplest mathematical concepts. For instance, twins "saw" that the number of matches scattered along floor was 111 and also "saw" the decomposition of integer to factors and primality. A mechanism explaining this based on the formation of wholes by quantum entanglement is proposed here. The model does not however involve any details.
Flux tube networks as basic structures
One can build a more detailed model for what the twins did by assuming that information processing is based on 2-dimensional discrete structures formed by neurons (one can also consider 3-D structures consisting of 2-D layers and the cortex indeed has this kind of cylindrical structures consisting of 6 layers). For simplicity one can assume large enough plane region forming a square lattice and defined by neuron layer in brain. The information processing should involve minimal amount of linguistic features.
Flux tube dynamics
- A natural geometric representation of number N is as a set of active points (neurons) of a 2-D lattice. Neuron is active it is connected by a flux tube to at least one other neuron. The connection is formed/strengthened by nerve pulse activity creating small neuro-transmitter induced bridges between neurons. Quite generally, information molecules would serve the same purpose (see this and this).
Active neurons would form a collection of connected sets of the plane region in question. Any set of this kind with given number N of active neurons would give an equivalent representation of number N. At quantum level the N neurons could form union of K connected sub-networks consisting Nk neurons with ∑ Nk=N.
- There is a large number of representations distinguished by the detailed topology of the network and a particular union of sub-networks would carry much more information than the mere numbers Nk and N code. Even telling, which neurons are active (Boolean information) is only part of the story.
The subsets of Nk points would have large number of representations since the shape of these objects could vary. A natural interpretation would be in terms of objects of a picture. This kind of representation would naturally result in terms of virtual sensory input from brain to retina and possibly also other sensory organs and lead to a decomposition of the perceptive field to objects.
The representation would thus contain both geometric information - interpretation as image - and number theoretic information provided by the decomposition. The K subsets would correspond to one particular element of a partition algebra generalizing Boolean algebra for which one has partition to set and its complement (see this).
- The number N provides the minimum amount of information about the situation and can be regarded as a representation of number. One can imagine two extremes for the representations of N.
- The first extreme corresponds to K linear structures. This would correspond to linear linguistic representation mode characteristic for information processing used in classical computers. One could consider interpretation as K words of language providing names for say objects of an image. The extreme is just one linear structure representing single word. Cognition could use this kind of representations.
- Second extreme corresponds to single square lattice like structure with each neuron connected to the say 4 nearest neighbors. This lattice has one incomplete layer: string with some neurons missing. This kind of representation would be optimal for representation of images representing single object.
For N active neurons one can consider a representation as a pile of linear strings containing pk neurons, where p is prime. If N is divisible by pk: N= Mpk one obtains a M× pk lattice. If not one can have M× pk lattice connected to a subset of neurons along string with pk neurons. One would have representation of the notion of divisibility by given power of prime as a rectangle! If N is prime this representation does not exist!
The classical topological dynamics for the flux tube system induced by nerve pulse activity building temporary bridges between neurons would allow phase transitions changing the number of sub-networks, the numbers of neurons in them, and the topology of individual networks. This topological dynamics would generalize Boolean dynamics of computer programs.
This dynamics would also allow to realize basic arithmetics. In the case of summation the initial state of the network would be a collection of K disjoint networks with Nk elements and in final state single connected set with N=∑ Nk elements. The simplest representation is as a pile of K strings with Nk elements. Product M× N could be reduced to a sum of M sets with N element: this could be represented as a pile of M linear strings.
- Flux tube networks as sets of all active neurons can be also identified as elements of Boolean algebra defined by the subsets of discretize planar or even 3-D regions (layer of neurons). This would allow to project flux tube networks and their dynamics to Boolean algebra and their dynamics. In this projection the topology of the flux tube network does not matter much: it is enough that each neurons is connected to some neuron (bit 1). One might therefore think of (a highly non-unique) lifting of computer programs to nerve pulse patterns activating corresponding subsets of neurons. If the dynamics of flux tube network determined by space-time dynamics is consistent with the Boolean projection, topological flux tube dynamics induced by space-time dynamics would define computer program.
- At the next step one could take into account the number of connected sub-networks: this suggests a generalization of Boolean algebra to partition algebras so that one does not consider only subset and its complement but decomposition into n subsets which one can think as having different colors (see this). This leads to a generalization of Boolean (2-adic) logic to p-adic logic, and a possible generalization of computer programs as Boolean dynamical evolutions.
- At the third step also the detailed topology of each connected sub-network is taken into account and brings in further structure. Even higher-dimensional structures could be represented as discretized versions by allowing representation of higher-dimensional simplexes as connected sub-networks. Here many-sheeted space-time suggests a possible manner to add artificial dimensions.
Number theoretical feats of twins and flux tube dynamics
Flux tube dynamics suggests a mechanism for how the twins managed to see the number of the matches scattered on the floor and also how they managed to see the decomposition of number into primes or prime powers. Sacks indeed tells that the eyes of the twins were rolling wildly during their feats. What is required is that the visual perception of the matches on the floor was subject to dynamics allowing to deform the topology of the associated network. Suppose that some preferred network topology or network topologies allowed to recognize the number of matches and tell it using language (therefore also linear language is involved). The natural assumption is that the favored network topology is connected.
The two extremes in which the network is connected are favored modes for this representation.
Option II provides also number theoretical information allowing a model for the feats of the twins.
- Option I corresponds to any linear string giving a linguistic representation as the number neurons (which would be activated by seeing the matches scattered on the floor). A large number of equivalent representations is possible. This representation might be optimal for associating to N its name. The verbal expression of the name could be completely automatic association without any conceptual content. The different representations carry also geometric information about the shape of the string: melody in music could be this kind of curve whereas words of speech would be represented by straight lines.
- Option II corresponds to a maximally connected lattice like structure formed as pile of strings with pk neurons for a given prime: N= M1× pk+M2, 0≤ Mi < pk. The highest string in the pile misses some neurons. This representation would be maximally connected. It contains more information than that about the value of N.
For the storage of memories the 2-D (or possibly 3-D representation) is non-economical and the use of 1-D representation replacing images with their names is much more economic. For information processing such as decomposition into primes, the 2-D or even 3-D representation are much more powerful.
- As far the checking the primeness of N is considered, one can assume k=1. For the primes pi dividing N one would find a representation of N as a rectangle. If N is prime, one finds no rectangles of this kind (or finds only the degenerate 1× p rectangle). This serves a geometric signature of primeness. Twins would have tried to find all piles of strings with p neurons, p=2,3,5,... A slower procedure checkes for divisibility by n=2,3,4,....
- The decomposition into prime factors would proceed in the similar manner by starting from p=2 and proceeding to larger primes p=3,5,7,.... When a prime factor pi is found only single vertical string from the pile is been taken and the process is repeated for this string but considering only primes p>pi. The process would have been completely visual and would not involve any verbal thinking.
See the chapter Artificial Intelligence, Natural Intelligence, and TGD or the article with the same title.