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