From aaron Tue Jul 25 02:39:47 BST 1995 Newsgroups: comp.ai.philosophy,sci.logic References: <3u3vkr$bn7@agate.berkeley.edu> <3u48ri$o9v@netnews.upenn.edu> <3u4mju$38@agate.berkeley.edu> <3u4tre$38@agate.berkeley.edu> <3u5t9v$oba@netnews.upenn.edu> Subject: Re: Putnam reviews Penrose. weemba@sagi.wistar.upenn.edu (Matthew P Wiener) writes: > Date: 14 Jul 1995 14:00:31 GMT > Organization: The Wistar Institute of Anatomy and Biology > Everything in a program is formal, and everything about its output is ^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^ > thus susceptible to the conclusions of formal logic. That is all. Not so. It is perfectly possible to use something formal (a computer program) to model or implement something informal, e.g. an incomplete or partially erroneous method of reasoning used by human beings. Then not everything in the program is formal, and not everything in its output is susceptible to the conclusions of formal logic (in any sense I can attach to that phrase). There's a whole strand of AI research which is not concerned with producing some super-intelligent artificial system so much as with trying to model aspects of the human mind. For example there has been work by John Seely Brown and others on modelling the reasoning processes in young children who make mistakes in doing simple subtractions, and work by Richard Young on modelling the erroneous efforts of children to assemble a pile of bricks into ascending order of height, and work by Phil Johnson-Laired attempting to model the erroneous `logical' deductions people make. CYC is probably another example, at least in intention, and there are many more in many labs. E.g. I am trying to model what happens to human beings when their higher level decision-making and reasoning processes get overloaded by external interrupts that come too frequently to be handled by resource-limited processes. (Is that formal??) In all these cases there is a lower level, an implementation level, that is formal in the sense of being expressed in a program, but also one or more higher level virtual machines that are quite different. So far, I am not aware of anyone who has seriously tried to model from the bottom up the kind of grasp of mathematical concepts that a young child has when learning about numbers, nor the processes by which that grasp develops. E.g. what happens when a child learns to recite number names? What happens when a child learns to identify, in sequence, the various members of a set of objects (by successively pointing at them, stepping on them, or merely attending to them)? What happens when a child learns to combine and synchronise these two activities -- e.g. counting steps whilst walking upstairs? What's the difference between the child who grasps that if the two activities get out of phase, or if an object is omitted, something is wrong, and the child who merely does it (and perhaps always gets it right)? What happens when a child goes from being able to recite number names in synchrony with identifying objects, to being able to answer questions or requests like How many apples are there? Please pass me three buttons. What changes when a child who can recite number names but cannot answer questions like "what comes before eight?" develops the ability to answer such questions? (My five year old son could answer the question if the kitchen clock was visible, but not otherwise, till I taught him the algorithm: "count silently to the number specified then say the previous number". He could do this because he could easily remember the number just preceding the last one counted to. Later he learnt a different method which was unconscious, and much faster, I know not what. Perhaps using a trained neural net?) What happens when a child goes from memorising an initial sequence of number names to grasping a generative principle that can produce an unending sequence of number names? What happens when a child goes from - treating as merely empirical the generalisation that counting items in two different orders produces the same result, to - grasping (implicitly) that it's not an empirical generalisation, but something deeper? What has such a child seen? And how? What happens when a child grasps that besides the number names it is useful to think of numbers themselves as distinct from their names? (I.e. what happens when the child's ontology is enriched with this unending sequence of abstract entities?) What happens when a child goes from - using numbers to characterise other things to - treating numbers as objects that can be counted, grouped in sets, related, etc.? I.e. how does the child's ontology for numbers grow to include properties, relations, functions applied to numbers? What happens when a child goes from grasping the natural numbers, which can be used to answer the question "How many?" to grasping that they can be embedded in a richer, doubly infinite set including positive and negative integers, so that instead of 7 - 5 being undefined it now has a value, -2? What happens when a child discovers that there are multiple interpretations of number names, e.g. answers to `how much?' as well as `how many?', or discovers that there's a "vector" interpretation of numbers, as representing translations, so that negative numbers have opposite translations to positive numbers? What happens when the child grasps that concepts like addition and multiplication, which were originally understood as operations on sets of sets of objects, can now be given a host of new interpretations, e.g. adding translations to produce a bigger translation, multiplying translations to produce ... what? What happens when the learner first understands mathematical induction? Is it really just a matter of learning to apply a logical axiom schema or is there something different going on, including, for example, a grasp of the analogy with a row of dominoes balanced on edge, where knocking down the first causes them all, successively, to be toppled? Why does pointing out such analogies help some learners? More generally, why do people find helpful the mapping between - numbers and operations on numbers, and - spatial structures and operations on them? (Cartesian coordinates are but one example?) What happens when the learner first grasps definitions of concepts and relations that hold between sets too large to be explicitly represented? (E.g. any even numbered set of objects no matter how large can be arranged in two rows matched one to one)? What change in the child's mind makes possible thinking about things too complex to be perceived? What's the relationship between abilities to grasp various kinds of necessities, e.g. - the necessity that adding an even and an odd number yields an odd number, - the necessity that if one end of a rigid centrally pivoted lever goes down the other goes up, - the necessity that if one gear wheel rotates in one direction another meshed with it rotates in the opposite direction? (If both have fixed axles.) Do all these insights in the learner depend on the same mental mechanisms, or are there different mechanisms at work in grasping these different kinds of necessity? How is all this related to the child's ability to understand how things work, as opposed to merely knowing that they work? What happens when the ideas of properties and relations of sets of numbers, or other objects, are first applied to infinite sets, e.g. in grasping that the set of positive integers has the same cardinality as a subset, e.g. the set of even integers? What happens when a learner first acquires the ability to reason formally about mathematics, e.g. using an explicit formal system with an explicit recursive definition of a proof? How exactly does this relate to the abilities a child had previously? What happens when a learner first grasps the idea of class of structures characterised by a formal definition which can be arbitrarily varied (groups, rings, fields, etc.) as contrasted with facts perceived to hold true of structures that already exist? What happens when some learners begin to distrust their apparent ability to think about infinite sets and end up with finitist, or constructivist or intuitionist philosophies of mathematics? (And why do the not all end up with the same philosophy?) =================================================================== There's an awful lot of pontification that goes on, by Penrose, his followers and his detractors, about what humans or computers can or cannot do -- when we don't yet know very much about any of these things. It's time to stop the endless bickering and try carefully to work out decent theories about what sorts of mechanisms are capable of supporting the sort of conceptual development exhibited by nascent human mathematicians. We may then have a clearer idea of what the end state of a fully fledged adult mathematician is, and what kinds of mechanisms can or cannot support that state. One thing that's clear is that no amount of introspecting will enable an adult mathematician directly to grasp the properties of all the mechanisms underlying all these capabilities and their growth from infancy to adulthood. I've given, above, a partial, incomplete specification of what needs to be explained. This specification needs to be extended and refined: there are very many details I've left out. (It could have been much longer.) Of course, some people believe the brain (of a mathematician) is not a mechanism - this research programme will have no interest for them. Some believe the brain of a mathematician is just a logic engine - this programme will have no interest for them. Some believe the brain of a mathematician has powers that cannot be explained by any mechanism. I, for one, don't believe in magic. Aaron