A partial index of discussion notes is in http://www.cs.bham.ac.uk/research/projects/cogaff/misc/AREADME.html
We need to develop a kind of spatial formalism that is not necessarily logical, but should plausibly be something that could be implemented in a variety of animal brains (at least mammals, reptiles, birds, carnivorous fish, maybe octopuses). It doesn't (initially) need most of the concepts of Euclidean geometry: point, line, plane, parallel, length, area, volume, etc. Though it requires a host of (more or less fuzzy) special cases. Maybe some ideas in this draft, incomplete, specification of p-geometry will be useful: http://www.cs.bham.ac.uk/research/projects/cogaff/misc/p-geometry.html
It doesn't (initially) need most of the concepts of Euclidean geometry: point, line, plane, parallel, length, area, volume, etc. Though it requires a host of (more or less fuzzy) special cases. It requires time so that movements are possible (as in p-geometry,) though not exact speeds, exact trajectories, etc. Partial orderings of speeds of change (X is moving away faster than Y, gap G1 is closing faster than G2) are needed. These may be absolute (G1 will close completely before G2) or relative (G1 will shrink to about half its size before G2). It does need objects with surfaces and visual appearances (projections to some sort of retina), and something like a notion of reaching or moving towards, which changes the appearances and reachability. It doesn't need equality (of anything) though a fuzzy "more-like" relationship between triples of distances, angles, areas, volumes, gap-sizes, object-widths, temporal intervals (e.g. observed or expected). X is more like Y than like Z X is more like Y than Z is It needs a notion of a perceiver in the space, and probably needs a distinction between processes that the perceiver can initiate, speed up, slow down, terminate, reverse, etc. and processes that it can merely observe. It needs to be able somehow to develop, or to have available from the start, modal notions of possibility and impossibility/necessity. E.g. X could be closer to Y X cannot pass through gap G and condititional possibility/impossibility if X does not change direction it will eventually bump into Y if that happens either X will stop moving or Y will start moving or X will change direction (e.g. bounce). if X changes its direction of motion to head more to the right (or more away from the heading to Y) then X can continue without bumping into Y and many more, some of it making use of notions of different kinds of stuff of which things are made. I have some thoughts about this that are scattered in online presentations here http://www.cs.bham.ac.uk/research/projects/cogaff/talks/#babystuff Ontologies for baby animals and robots From "baby stuff" to the world of adult science: Developmental AI from a Kantian viewpoint http://www.cs.bham.ac.uk/research/projects/cogaff/talks/#toddler Why (and how) did biological evolution produce mathematicians? A New Approach to Philosophy of Mathematics: Design a young explorer, able to discover "toddler theorems" and others are in more or less fragmentary online discussion notes, e.g. http://www.cs.bham.ac.uk/research/projects/cogaff/misc/simplicity-ontology.html And the polyflap project: http://www.cs.bham.ac.uk/research/projects/cogaff/misc/polyflaps A hard project: bringing all these ideas together to see whether someone cleverer than I am can work out how to use them in the design of a baby robot (perhaps simulated) that learns about its environment. After it develops a fair amount of behavioural competence it should be able to reorganise what it has learnt into a sort of deductive theory that allows it to work out what will happen in novel situations instead of having to go on learning empirically. (A way of using robotics to support Kant's philosophy of mathematics.)TO BE CONTINTUED/REVISED/ETC
M. Aiello, I.Pratt-Hartmann and J.van Benthem What is Spatial Logic? in Handbook of Spatial Logics http://www.springer.com/philosophy/epistemology+and+philosophy+of+science/book/978-1-4020-5586-7 Springer. Price GBP 341.00 The editors' introduction is available online: http://www.cs.man.ac.uk/~ipratt/papers/mereogeometry/APHvB-chapter.ps Torsten Hahmann Model-Theoretic Analysis of Asher And Vieu's Mereotopology Master of Science Thesis, Graduate Department of Computer Science University of Toronto http://www.cs.toronto.edu/~torsten/Hahmann_MScThesis.pdf
Maintained by
Aaron Sloman
School of Computer Science
The University of Birmingham