Hidden Depths of Triangle Qualia University of Birmingham

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When will the first baby robot grow up to be a mathematician. Introduction ways to change or not change features of a triangle. An Evolutionary Conjecture, Two disclaimers concerning impossibility necessity. 1 Local vs global impossibility,2 What CURRENT computer systems cannot do. Doing without a global length metric,Offline vs online intelligence. Flaws in enactivism and theories of embodied cognition. On seeing triangles again,The Median Stretch Theorem. NOTE How can areas be compared, Using containment instead of area Added 27 May 2013.
Is this geometry or topology 27 May 2013, Using the Side Stretch Theorem to prove the Median Stretch Theorem. Modified 27 May 2013, The roles of continuity and discontinuity in SST and SCT. Virtual discontinuities 28 May 2013,Another way of modifying a triangle. Some observations on the above examples,The role of meta cognition. Perception of affordances,The perpendicular stretch theorem.
A problem with the proof using Figure P a,Toward robot mathematicians discovering geometry. Chemical computation,Comparison with logical proofs. A partial list of references to be expanded,Acknowledgements. Related documents,Maintained By, A Preface was added 7 May 2013 removed 12 May 2013 now a separate document. Biology Mathematics Philosophy and Evolution of Information Processing. http www cs bham ac uk research projects cogaff misc bio math phil html. BACK TO CONTENTS, Introduction ways to change or not change features of a triangle.
Aspects of mathematical consciousness of space, This document may appear to some to be a mathematics tutorial introducing ways of doing. Euclidean geometry It may have that function but my main aim is to draw attention to products of. biological evolution that must have existed before Euclidean geometry was developed and. organised in Euclid s Elements over two thousand years ago. An online English version of Euclid s Elements is here. http aleph0 clarku edu djoyce java elements elements html. That was arguably the most important and most influential book ever written ignoring highly. influential books with mythical or false contents Unfortunately this seems to have dropped out of. modern education with very sad results, I ll give a few examples of apparently very simple but actually very complex human spatial. reasoning capabilities concerned with perception of triangles I think these competences are deeply. connected with the abilities of human toddlers and other animals to perceive what James Gibson. called affordances though I don t think he ever understood the full generality and depth of those. animal competences I suspect we shall not understand the functions and mechanisms of. perception of affordances until we know a lot more about the evolutionary transitions that produced. different kinds of affordance related competences in evolutionary ancestors of humans one of the. aims of the Meta Morphogenesis project Sloman 2013. A core aspect of intelligence of many animals is the ability to perceive what is possible in a. situation while certain features of the environment remain unchanged For example it is possible for. you to walk through a gap in a wall as long as the width of the gap does not shrink to less than the. narrowest width across your body left to right or front to back Another core aspect is perception of. what is impossible, Perceiving possibilities involves acquiring information about structures and processes that do not. exist but could have existed and might exist in future and grasping some of the constraints on. those possibilities and also some of the consequences if they are realised. This is not to be confused with discovering probabilities possibilities are obviously more. basic The differences between learning about impossibilities i e constraints on what s. possible and learning about probabilities seem to have been ignored by most researchers. studying probabilistic learning mechanisms e g Bayesian mechanisms Deep learning that. draw conclusions from statistical evidence cannot provide information that something is. impossible However only one instance of a type of configuration or process is required to. prove that it is possible, The examples I ll present look very simple but have hidden depths as a result of which there is as. far as I know nothing in AI that is even close to modelling those animal competences and nothing. in neuroscience that I know of that addresses the problem of explaining how such competences. could be implemented in brains How could a neural net encode the information that something. e g separation of interlocking rings made of impermeable material without temporarily making a. gap in one of the rings is impossible, I am not claiming that computer based machines cannot model these discovery process as.
Roger Penrose sometimes does only that the current ways of thinking in AI Computer science. Neuroscience Cognitive Science Philosophy of mind and Philosophy of mathematics need to be. extended to allow for perception of possibilities impossibilities and necessary connections. I ll be happy to be informed of working models or even outline designs for implementing such. abilities in robots, For reasons that should become clearer below this could be dubbed the problem of accounting for. mathematical qualia or contents of mathematical geometric consciousness their evolution. their cognitive functions and the mechanisms that implement them. I have some ideas about the layers of meta cognitive and meta meta cognitive mechanisms. that are involved in these processes which I think are related to Annette Karmiloff Smith s. ideas about Representational Redescription 1992 but I shall not expand on those ideas. here the purpose of this document is to present the problem. For more on this see the Meta Morphogenesis project Sloman 2013 and these examples of. perception of impossibilities, http www cs bham ac uk research projects cogaff misc impossible html. An Evolutionary Conjecture, My aim here is to provide examples supporting the following conjecture. The discoveries organised and presented in Euclid s Elements depended to a. considerable extent on products of biological evolution that humans share with several. other species of animals that can perceive understand reason about construct and. make use of structures and processes in the environment competences that are also. present in pre verbal humans e g toddlers, Human toddlers and some other animals seem to be able to make such discoveries. but they lack the meta cognitive competences that enable older humans to inspect and. reason about those competences and the discoveries they give rise to. I suspect that important subsets of those competences evolved independently in several. evolutionary lineages including some nest building birds elephants and primates because. they all inhabit a 3 D environment in which they are able to perceive understand produce. maintain or prevent various kinds of spatial structures and processes Some of those competences. are also present in very young even pre verbal children But the competences have largely been. ignored or misunderstood by researchers in developmental psychology animal cognition. philosophy of mathematics and more recently AI and robotics Thinkers who have noticed the gaps. sometimes argue that computer based systems will always have such gaps e g Roger Penrose. That is not my aim though there is an open question about how research motives should be. related to particular sorts of experiments, Research on tool use in young children and other animals often has misguided motivations.
based on some ill defined notion of tool and the assumption that there is a well defined. sub category of actions that are examples of tool use. The label Tool use should perhaps be replaced with matter manipulation a label well suited. for a wide range of examples I suspect research on tool use in young children and other. animals tends to have an arbitrary focus and should be replaced by research on. matter manipulation including uses of matter to manipulate matter. But that s really a topic for another occasion, Very often these spatial reasoning competences are confused with very different competences. such as abilities to learn empirical generalisations from experience and to reason probabilistically. In contrast this discussion is concerned with abilities to discover what is possible and constraints. on possibilities i e necessities, These abilities were also noticed by Immanuel Kant who I suspect would have been actively. attempting to use Artificial Intelligence modelling techniques to do philosophy had he been alive. In young humans the mathematical competences discussed here normally become evident in the. context of formal education and as a result it is sometimes suggested mistakenly that social. processes not only play a role in communicating the competences or the results of using them but. also determine which forms of reasoning are valid a muddle I ll ignore here apart from. commenting that early forms of these competences seem to be evident in pre school children and. other animals though experimental tests are often inconclusive we need a new deep theory. more than we need new empirical data, The capabilities illustrated here are to the best of my knowledge not yet replicated in any AI. system though some machines e g some graphics engines used in computer games may. appear to have superficially similar capabilities if their limitations discussed below are not. Two disclaimers concerning impossibility necessity. 1 Local vs global impossibility,Added 9 Apr 2017, The examples presented here concern local or relative impossibility not the sorts of global. impossibility found in pictures by Reutersvard Penrose Escher and others discussed here. http www cs bham ac uk research projects cogaff misc impossible html. Some Possibly new considerations regarding impossible objects. That discusses depictions and descriptions of various types of spatial configuration that could not. possibly exist because instances would violate some mathematical constraint such as that further. than is transitive and irreflexive, In contrast the examples here all concern structures and processes that are possible but which.
cannot be altered in a way that produces a change satisfying some description E g if X increases. then it is impossible for Y to decrease However those impossible changes are not ones for which. compelling depictions are presented unlike the impossible objects that have compelling depictions. 2 What CURRENT computer systems cannot do, I am not claiming that computers cannot do things done by humans and other animals merely that. novel forms of representation and reasoning and possibly new information processing. architectures will be required developing a claim I first made in Sloman 1971 though I did not. then expect it would take so long to replicate these animal capabilities That is partly because I did. not then understand the full implications of the claims especially the connection with some of J J. Gibson s ideas about the functions of perception in animals discussed below and the distinction. between online intelligence and offline intelligence also discussed below which challenges. some claims made recently about embodied cognition and enactivism claims that I regard as. deeply confused because they focus on only a subset of competences associated with being. embodied and inhabiting space and time, It is possible that computers will turn out to have a limitation that isn t present in brains but so far. there is no evidence that that is the case An investigation of that possibility is a major feature of. the Meta Morphogenesis project of which this document is a part Sloman 2013. NOTE The ideas presented here overlap somewhat with ideas of Jean Mandler on early. conceptual development in children and her use of the notion of an image schema. representation though she seems not to have noticed the need to account for competences. shared with other animals Studying humans and trying to model or replicate their. competences while ignoring other species and the precocial altricial spectrum in animal. development can lead to serious misconceptions, I am grateful to Frank Guerin for reminding me of Mandler s work accessible at. http www cogsci ucsd edu jean, Another colleague recently drew my attention to this paper. http psych stanford edu jlm pdfs Shepard08CogSciStepToRationality pdf. Roger N Shepard, The Step to Rationality The Efficacy of Thought Experiments in Science Ethics and Free.
In Cognitive Science Vol 32 2008, It s one thing to notice the importance of these concepts and modes of reasoning Finding a. good characterisation and developing a good explanatory model are very different more difficult. Note added 3 Jan 2013, This document is also closely related to my 1962 DPhil Thesis attempting to explain and defend. Immanuel Kant s claim 1781 that mathematical knowledge includes propositions that are. necessarily true i e it s impossible for them to be false but are not provable using only definitions. and logic i e they are not analytic they are synthetic necessary truths. The thesis is available online in the form of scanned in PDF files kindly provided by the university. of Oxford library enhanced in 2016 by re typed searchable and less bulky files. Hidden Depths of Triangle Qualia Especially their areas Theorems About Triangles and Implications for Biological Evolution and AI The Median Stretch Side Stretch and Triangle Area Theorems Old and new proofs NOTE Some of the contents of this document and autobiographical background are also presented in a

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