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From: shankar@netcom.com (Shankar Ramakrishnan)
Subject: Re: Penrose, Etc.: Why No Such Argument Can Possibly Work
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Date: Wed, 2 Aug 1995 21:18:42 GMT
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In article <3vncm8$ma7@percy.cs.bham.ac.uk> A.Sloman@cs.bham.ac.uk (Aaron Sloman) writes:
>
>
>[Some wild, vague, speculation starts here:]
>My own half-baked hunch is that it involves, at the very least, having
>concurrent layered control systems, where one layer can do things like
>generate number names, sequentially pick out objects, etc. and another
>layer can observe and monitor and form generalisations about what the
>first layer is doing.
>
>But how the monitoring is done, what form of representation is used for
>the generalisations, and what exactly happens when the child first
>explicitly grasps that many operations can go on indefinitely, i.e. the
>natural numbers form an infinite set, all still remain to be explained.
>
>I don't think we'll get far with this if we START by discussing the
>capabilities of a fully fledged professional mathematician. We need to
>understand the developmental and learning processes and mechanisms (i.e.
>high level virtual machines as well as their neural implementations)
>that produce that end state, and I suspect that only then will be
>understand what that end state is.

I agree with you. Formalization and rigidity should come from
generalized common sense, not the other way round. Number and set theories
evolved from using apples and oranges, not from a mathematician with
pencil and paper.
>
>My guess is that many properties of the monitoring layer are there
>not just because they are part of the mechanisms supporting
>knowledge of numbers, but because they are part of the more general
>capability to learn from performing actions.
>
>Often this requires not merely doing something (e.g. not merely carrying
>out a plan) but also observing the interactions between the various
>steps and their consequences. So mathematical abilities arise out of
>solutions to general requirements for intelligence. They are not
>special.
>
>E.g. the ability to discover that addition is commutative is a very
>special case of a more general ability to learn that two actions can be
>performed in either order.

I am currently developing a two-layered reasoning system. There is an
'informal' layer at the top level which understands concepts like
'count', 'set', and a few operations on them. 'Count' is a non-negative
integer, and is akin to the number of oranges one has in one's hands.
The commutativity of the addition of two counts _count1 and _count2 
is an axiom at this level, since this would be the equivalent of an
informal observation if we had _count1 and _count2 oranges in both
hands and switch them. The commutativity of multiplication is not
so apparent, unless the system could visualize a _count1*_count2
array of oranges and rotate it. However, if we introduce a _repeat
operation as an axiom that effectively translates to 'if you have
no oranges to start with, and you add one orange at a time _count1
times, you end up with _count1 oranges', it is possible to obtain
the commutativity of multiplication as a theorem at the informal level. 
At the formal lower level, the informality gets translated. For example,
the _count'th natural number can be defined as n(_count). Then, if
we define the formal 'add' operator to be n(_count1)+n(_count20) =
n(_count1 _+ _count2) (I use underscores before variables and operators
at the informal level), then commutativity of addition of natural numbers 
at the formal level can be easily shown, since n(_count1)+n(_count2)
= n(_count1 _+ _count2) = n(_count2 _+ _count1) = n(_count2)+n(_count1).

The same kind of formalism also applies to some of the elemenatary
axioms of set theory, like D'Morgan's laws. Other kinds of concepts
and operators would deal with strings (reprsentation of numbers in
decimal format, for example). Negative numbers, rationals, and
complex numbers would be treated within the 'formal' level.

Apart from these two, there would also be a 'verbal' layer that
intrinsically understand the meanings of the words 'prove', 'hence', etc.
The flow of information between all these three is not strictly
unilateral, since in human beings, elementary common sense can undergo
changes with the type of formal training they undergo.

Shankar
