All Roads Lead to Rome and if they don't, make'em While I agree that the idea of reasoning about a problem in multiple ways is an important one, I'm concerned about talk that sounds like there are several rather distinct reasoning systems involved that get traded off depending on how well they work for certain problems. Any system that depends on a limited number of reasoning systems will face a number of hard problems, and I believe some of these problems may be showstoppers. 1) How do problem representations get converted between the different reasoning systems? 2) What happens when no reasoning system seems adequate? 3) How can we know which and how many reasoning systems there should be? It seems likely that different reasoning systems will need different sorts of representations to compute on. Thus, if we are to think about the same problem in several systems, we need to convert it to various representations. If the systems do not already share a common expressive language it seems like a hard problem to come up with a reliable conversion mechanism. On the other hand, forcing every system to compute on the same representation would likely make some systems extremely inefficient or even impossible, as I believe the neural network and logic traditions in AI show. The availability of several reasoning systems solves some of the problems of reasoning flexibility that machines have and humans do not. However, I do not believe it matches the generativity we see in human problem solving. Especially during childhood, but also later in life, human beings seem to spawn new reasoning systems, sometimes on-the-fly. During childhood these may be ones we take to be fundamental, like spatial and temporal reasoning (see Piaget's studies on the Child's Conception of Time), later they may be others like inventing a new game (an example due to Jackendoff) where many notions we reason about like 'points' have their meaning only relative to this newly spawned theory of playing the game. Again, I do not believe that simply having several reasoning systems explains the generativity humans show when they do not have an appropriate way to reason, and does not solve the question of how many and which system there should be. So how can we explain the flexibility of human reasoning? I believe that the answer is that human beings do tend to learn to represent new domains in terms of the representations old ones can deal with - the prime example of a fallback system being spatial reasoning. We make spatial analogies to almost every other realm - be it temporal, social or economic, and studies like Lera Boroditsky's show that at least in some cases (and, I suspect, perhaps in all others) these are not just analogies in that we talk about the realms in the same way - we actually reuse our spatial reasoning system to reason about time, and thus change our temporal thinking when we change our spatial thinking. Ray Jackendoff has taken this idea extremely seriously, and I think rightly so. He has performed detailed analyses of which spatial notions like origin, endpoint, path and manner of traveling can be and are applied in other realms. For example, both the temporal and the possession realm are one-dimensional. In the temporal realm, you can only travel in one direction, whereas we do not consider the path in the possession realm, but rather just origin and endpoint. He finds large amounts of evidence in linguistic data for these mappings, and can predict the types of concepts we should be considering in new realms due to these mappings. Here is one solution to the problems raised above, then: When we need a new way to reason, we find an old system and invent a mapping from the new problem domain to the old system, often using language as guidance. This is not necessarily a complete problem conversion mapping, but rather a bootstrapping mechanism to start attacking a new domain that may have properties that were not part of the old system. Note that some of this mapping is bound to be underdetermined, like in the case of the time to space mapping where English maps onto the horizontal forward axis, whereas some Chinese languages map onto the vertical axis. Sometimes, perhaps, no such mapping can be made - and it is not clear to me, then, that we do come up with good ways to reason about these problems. The real flexibility we show thus lies in spawning a new shell around an old reasoning system and learning about the limitations and extensions that a new domain imposes onto the old system - like time not moving backwards and possession moving instantly from origin to endpoint. However, having all the other reasoning systems around that use a similar representation mechanism also allows us to consider things we might not think of otherwise - like notions of time moving backwards or perhaps in several parallel lines. Obviously, at the same time as providing flexibility, this way of generating solution systems also restricts us - and perhaps this leads back to some of the problems kids have with Math when they learn it. If they do not discover and are not taught a way to map Math to, say, spatial knowledge, they simply have no way of spawning a useful reasoning system that covers the new problem domain. References: @Article{Lera, Author = {Boroditsky,Lera}, Title = {Metaphoric structuring: understanding time through spatial metaphors}, Journal = {Cognition}, Volume = {75}, Number = {1}, Pages = {1--28}, year = 2000, } @Article{Lera2, Author = {Boroditsky,Lera}, Title = {Does Language Shape Thought? Mandarin and English speakers conceptions of time}, Journal = {Cognitive Psychology}, year = {in press}, } @Book{Jackendoff, Author = {Jackendoff,Ray}, Title = {Foundations of Language: Brain, Meaning, Grammar, Evolution}, Publisher = {Oxford University Press}, year = {2002}, } @Book{Piaget, Author = {Piaget, Jean}, Title = {The Child's Conception of Time}, Publisher = {Ballantine Books}, year = 1971, }