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Structures, Learning and Ergosystems: Chapters 1-4, 6 Structures, Learning and Ergosystems: Chapters 1-4, 6. Misha Gromov December 30, 2011 Contents 1 Ergobrain and Evolution. 2 1.1 Ergo, Ego and Goal Free Learning. 2 2 1.2 Peacock Tail, Maxwell Equation and E = mc . 10 1.3 Eyes, Wings, Energy, Information and Icosahedra. 14 1.4 Animals Play, Animals Learn, Animals Dance, Animals Talk. 21 1.5 Learning to Talk. 26 1.6 Interesting, Amusing, Amazing, Funny and Beautiful. 30 1.7 Mathematics Supernovas. 31 1.8 Ergoworld, Ergothinking and Mathematics. 34 1.9 What Kind of Mathematics? . 42 2 Structures within and next to Mathematics. 44 2.1 Symmetry in Motion. 44 2.2 Ambiguity, Galois Symmetry and Category Theory. 50 2.3 Problem with Numbers and other Mathematical Notions. 56 2.4 What is Equality? . 59 2.5 Equivalences, Partitions, Graphs, Trees, and Categories. 62 2.6 Spaces, Meters and Geometries. 72 2.7 Continuity, Robustness and Fixed Points. 74 2.8 Similarity, Clustering and Compression of Information. 74 2.9 Edges, Arrows, Orders, Sums and other Relations. 80 2.10 Spaces of Structures. 82 2.11 Statistics, Probability, Improbability, Prediction and Entropy. 85 2.12 Mathematics of a Bug on a Leaf. 91 2.13 Automata and Turing Machines. 94 3 Structures in Life. 98 3.1 Connectivity, Trees and Evolution. 98 3.2 Trees, Sequences and Mutations. 98 3.3 Polypeptides and Proteins. 98 3.4 Protein Spaces and Protein Counting. 98 3.5 Proteins Energy Landscapes. 98 3.6 Sweet Learning by Bacteria. 98 3.7 Tongue, Nose and the Space of Smells. 98 3.8 One+One for Hearing, Three+One for Vision . 98 1 3.9 Dimensions of Motion. 98 3.10 Universality in Biology. 98 4 Syntactic Structures. 98 4.1 Information, Redundancy, Understanding and Prediction. 98 4.2 Libraries and Dictionaries. 100 4.3 Short Range Correlations, Segments and Words. 107 5 Classification of Words, Tags and Inter-word Correlations. 111 6 Mundane Stories. 111 6.1 Self Referential Loop. 112 6.2 Turing Test. 115 6.3 Bug in a Bug . 123 6.4 Taxonomy Problem. 130 6.5 Babel Chamber. 135 6.6 Who Made Who? . 145 6.7 Whose is Your Self? . 149 7 Bibliography. 154 Abstract We introduce a concept of an ergosystem which functions by building its "internal structure\ out of the "raw structures\ in the incoming flows of signals. 1 Ergobrain and Evolution. We shall argue in this section that the essential mental processes in humans and higher animals can be represented by an abstract universal scheme and can be studied on the basis of few (relatively) simple principles. 1.1 Ergo, Ego and Goal Free Learning. The brain of a human or an animal, for ever confined to the scull, has no direct access to the outside world. The "outside world" for the brain is nothing else but a muddy flow of electric/chemical signals it receives from receptor cells and the endocrine system(s). The brain responds with similar signals directed mostly to the muscles of the body. (Some signals go to the brain itself and some to the endocrine glands in the body.) Here is a vivid description by Charles Sherrington [80]: "The eye sends, as we saw, into the cell-and-fibre forest of the brain through- out the waking day continual rhythmic streams of tiny, individually evanes- cent, electrical potentials. This throbbing streaming crowd of electrified shift- ing points in the spongework of the brain bears no obvious semblance in space pattern, and even in temporal relation resembles but a little remotely the tiny two-dimensional upside-down picture of the outside world which the eyeball paints on the beginnings of its nerve fibers to the brain. But that little picture sets up an electrical storm. And that electrical storm so set up is one which effects a whole population of brain cells. Electrical charges having in themselves 2 not the faintest elements of the visual - having, for instance, nothing of "dis- tance," "right-side-upness," nor "vertical," nor "horizontal," nor "color," nor "brightness," nor "shadow," nor "roundness," nor "squareness," nor "contour," nor "transparency," nor "opacity," nor "near," nor "far," nor visual anything - yet conjure up all these. A shower of little electrical leaks conjures up for me, when I look, the landscape; the castle on the height, or, when I look at him approaching, my friend's face, and how distant he is from me they tell me. Taking their word for it, I go forward and my other senses confirm that he is there". Ergo. In sum, the transformation of signals by the brain can be depicted by an arrow incoming signals ↝ outcoming signals; which, metaphorically, corresponds to "ergo\ in the Cartesian "Cogito, ergo sum\. We call the (hypothetical, formal) structure implementing this transforma- tion an ergobrain or an ergosystem. We think of this as a dynamic entity con- tinuously being built by the brain from the moment of birth of a human or an animal by the process of (goal free) structure learning. Very roughly, an ergosystem is something like a network of "ergo-ideas\ which grows in the course of learning by incorporating and "metabolizing\ new "ideas\ from the incoming flows of signals. But from ergobrain's own perspective, the "real world" emerges via the reversed arrow symbolizing the transformation from the signals the ergobrain generates to their "echos" { the responses to these signals. The ergobrain learns by analyzing/structuralizing the "information" carried by these echos { "patches of ripples" on the external flows of signals resulting from ergobrain's "actions", where some "echos" originate within the ergobrain itself. 3 The simplest instance of learning performed by the ergobrain is conditioning: the process of acquisition of conditional reflexes. We are interested, however, in more "complicated\ mental processes, e.g. acquisition of native languages by children. Such processes performed by the ergobrain, be they simple or compli- cated, are unconscious but they leave some perceptible traces in our conscious mind (we shall explain this later on). We take an avenue of mathematical, albeit informal and/or unrigorous, rea- soning and try to argue "on brain's terms\ with a minimal reference to the "meaning\ which our "mind\ assigns to these signals. (In practice, this is as difficult as not assigning, albeit unconsciously, any purpose to evolution. In Haldane's words: "Teleology is like a mistress to a biologist: he cannot live without her but he's unwilling to be seen with her in public".) We do not hypothesize on how this arrow is implemented by the synaptic network of the brain, although we try to keep track of restrictions on possible structures of "↝" which are due to limitations of network architectures. And we constantly keep in mind possible algorithmic realizations of ergosystems. Neuro-brain. This signifies a (semi-mathematical) representation (model) of the physiology of the brain, including "streaming crowd of electrified shifting points in the spongework of the brain'\ along with a (the ???) slower biochem- istry of middle-term and long-term memory. An ergobrain can be thought of as a (kind of) dynamical reduction (quotient) of a neuro-brain, while an ergosystem is something similar to an ergobrain but which is not, a priori, related to a neuro-brain. Ego-mind. This refers to (manifestations of) mental processes (more or less) directly serving and/or reflecting the survival/reproduction needs of an organism. Ego-mind is pragmatic and it is goal oriented, where the goals are set up by the evolutionary genetic (pre)history of an (animal or human) organism and they are modulated by interactions of an organism (or of its mind) with the outside world. Most (but not all) of what we consciously perceive is produced by the ego- mind. (The Freudian type subconscious and most of what we see in dreams are byproducts of the ego-mind.) In these terms, the ergobrain includes the totality of mental processes which are not covered by the ego-mind as well as some "ego-parts", such that the mechanisms of conditional and unconditional reflexes which were, apparently, directly targeted by the evolution selection process. The egomind and its ego-reasoning (common sense) are adapted to the every day life, they stay on guard of our survival and passing on our (and of close kin { ego can be altruistic) genes. "Myself" is the central reference point in the visible part of our mind as much as the sun is central for accounting for the observed planetary orbits. But as the centrality of the sun is secondary to such concepts as ”differential equation" and "Hamiltonian system", the centrality of ego is secondary to the (unknown to us) "laws" running general ergosystems. Besides, our ergomind was not designed by evolution for a structural mod- eling of the world. The self glorifying ego-vocabulary: "I think\, "I am\, "important" "intelligent\, "understanding", "conscious", "in- tuitive", "insightful", "creative", \motivated\, "purposeful", "rational\, "adap- tive\, ”efficient\, "resourceful", "rewarding\, \useful\, etc. is not adequate for structural modeling of most essential mental processes, such as learning; we shall do our best to keep our distance from these "concepts" 4 except for a metaphorical usage of the words. Goal Free Learning. The ergobrain, as we shall argue, plays the key role in the learning processes, where our notion of learning is far from the con- cept of "adaptation\ and it does not essentially depend on any external "re- ward/punishment\ reinforcement. Our ergobrain models are "self propelled learners\ which need no additional external or internal reinforcement stimuli, similar to some current robotic models, e.g. as in [72]. Ergobrains and ergosys- tems are structures (evolutionary or artificially) designed for recognizing, selecting, analyzing, interacting with and building structures; (ergo)learning is a spontaneous structure building and rebuilding process.
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