Louise: "But I would still argue that most, if not all, of your examples were reducible to logic".
OK I did not realise that that was your point. The notion of something being "reducible" to logic is quite problematic.
Suppose Hilbert did manage to produce a logical axiomatisation of Euclidean geometry: i.e. a set of logical axioms (plus the standard rules of inference) such that every theorem and every valid inference in Euclidean geometry is mirrored by a logical theorem and a valid proof in the formal system.
This structural correspondence between geometry and a formal system would certainly be interesting and in some cases might be useful, but it would be a *discovery* rather than a *reduction*. As I think Frege argued somewhere, establishing the correspondence would require use of geometrical knowledge.
As one of my slides points out the arithemtisation of geometry by Descartes was one of the most important intellectual achievements of a human mind, which has made many things possible that would have been very difficult or impossible without it.
But it still remains the case that there is such a thing as using geometric capabilities to investigate problems and that is not the same thing as using logical or algebraic capabilities. However, as we agree, it is not easy to implement that collection of capabilities on current computers -- though various fragments have been demonstrated, e.g. in the PhD theses of Mateja Jamnik and Daniel Winterstein (both in Edinburgh).
I suspect we don't yet have a good requirements specification for the kind of virtual machine that is needed. When we do, it may turn out that such a VM cannot be implemented in von Neumann computers, though I don't think there is evidence yet that establishes that.
However, current computers don't have to limit the possibilities for information-processing engines to go into future robots, though there are some rigid and dogmatic people who insist that *only* what can be implemented on von Neumann or Turing-equivalent machines should be used for AI.
If it turns out, for example, that chemical computers are needed then they will be used. (Evolution produced a wide variety of chemistry-based information processing systems, including all the organisms that don't have brains.)
On re-reading your original posting ("Aaron instead argued that you could not treat von Neumann or Turing machines as logical since they were embedded in the real world"), I think it is possible that I missed the point of the question being asked. However, if you build a robot that shares some of its information-processing between internal processing and external mechanisms (e.g. diagrams on paper, etc.) then it is not clear that the whole process is one that can be run on a Turing machine, and so it may just be wrong to say it is a logic machine, even if a logic machine is a part of the total system. That was the point I was trying to make.
Most of the learning that humans do depends crucially on learning from interactions with the environment: the resulting system is not one that could have been derived by some kind of logical inference from the initial state of a new-born infant.
Humans are to some extent able to dispense with external diagrams and to use imagined ones just as well.
The question whether some of the uses of external reasoning aids can be internalised on a VN-based computer, or whether some different kind of engine, perhaps something used by human brains but not yet understood, is a separate question, mentioned above.
Re: The slides for the talk (still developing) are available online
OK I did not realise that that was your point. The notion of something being "reducible" to logic is quite problematic.
Suppose Hilbert did manage to produce a logical axiomatisation of Euclidean geometry: i.e. a set of logical axioms (plus the standard rules of inference) such that every theorem and every valid inference in Euclidean geometry is mirrored by a logical theorem and a valid proof in the formal system.
This structural correspondence between geometry and a formal system would certainly be interesting and in some cases might be useful, but it would be a *discovery* rather than a *reduction*. As I think Frege argued somewhere, establishing the correspondence would require use of geometrical knowledge.
As one of my slides points out the arithemtisation of geometry by Descartes was one of the most important intellectual achievements of a human mind, which has made many things possible that would have been very difficult or impossible without it.
But it still remains the case that there is such a thing as using geometric capabilities to investigate problems and that is not the same thing as using logical or algebraic capabilities. However, as we agree, it is not easy to implement that collection of capabilities on current computers -- though various fragments have been demonstrated, e.g. in the PhD theses of Mateja Jamnik and Daniel Winterstein (both in Edinburgh).
I suspect we don't yet have a good requirements specification for the kind of virtual machine that is needed. When we do, it may turn out that such a VM cannot be implemented in von Neumann computers, though I don't think there is evidence yet that establishes that.
However, current computers don't have to limit the possibilities for information-processing engines to go into future robots, though there are some rigid and dogmatic people who insist that *only* what can be implemented on von Neumann or Turing-equivalent machines should be used for AI.
If it turns out, for example, that chemical computers are needed then they will be used. (Evolution produced a wide variety of chemistry-based information processing systems, including all the organisms that don't have brains.)
On re-reading your original posting ("Aaron instead argued that you could not treat von Neumann or Turing machines as logical since they were embedded in the real world"), I think it is possible that I missed the point of the question being asked. However, if you build a robot that shares some of its information-processing between internal processing and external mechanisms (e.g. diagrams on paper, etc.) then it is not clear that the whole process is one that can be run on a Turing machine, and so it may just be wrong to say it is a logic machine, even if a logic machine is a part of the total system. That was the point I was trying to make.
Most of the learning that humans do depends crucially on learning from interactions with the environment: the resulting system is not one that could have been derived by some kind of logical inference from the initial state of a new-born infant.
Humans are to some extent able to dispense with external diagrams and to use imagined ones just as well.
The question whether some of the uses of external reasoning aids can be internalised on a VN-based computer, or whether some different kind of engine, perhaps something used by human brains but not yet understood, is a separate question, mentioned above.
Aaron
http://www.cs.bham.ac.uk/~axs/ (http://www.cs.bham.ac.uk/~axs/)