On 14 Jul 2015, at 03:26, Pierz wrote:
On Tuesday, July 14, 2015 at 2:08:35 AM UTC+10, Brent wrote:
Just ask yourself how you grasp the notion of infinity. It's not by
dividing by zero. It's by using "and then..."
Sure. It's a concept even very young children can understand -
probably almost as easily as zero. "There are no more slices of
pudding" versus The Magic Pudding. Divide by zero is just one way
computers routinely come across infinity.
There's no obstacle in principle to having a computer reason about
the consequences of having an axiom of succession.
That is a facile response and makes me wonder if perhaps you don't
program computers. Please explain how you make a computer "reason
about the consequences of having an axiom of succession." If you can
do that in any satisfactory manner, you will indeed have answered my
question, because that is precisely the problem I can't conceive of
a solution to, and I program computers day in day out. Computers
just iterate until told or forced to stop, they cannot reason about
their own iterative processes. It's easy to use words like the ones
you have above, but it's a bit like saying "there's no obstacle in
principle to an ideal society." Maybe there is, maybe there isn't,
but what would one look like?
I'm not saying that some kind of ability to deal with infinity
couldn't emerge from a very complex machine (it's hard as hard to
show it can't as that it can), but what is at least interesting is
that it seems to require such a complex machine, indeed an AI so
complex that one can't even easily describe its programming, but has
to fall back on emergent properties. Yet infinity and zero are about
equally easy mathematical concepts to grasp - historically both
appeared in Indian mathematics around the same time. It's odd and
possibly telling that the simplest computer can readily deal with
zero, but obscure emergent properties are required for infinity.
It doesn't need to have an infinite memory capacity to do so
(anymore than you do).
I didn't say it did. But I can extrapolate in the abstract. I'm
trying to understand how a computer can make the same extrapolation.
I am not sure I understand. Surely a theorem prover for ZF can do such
extrapolation.
And ZF is a small theory with very few principles, and it is easy to
conceive a machine getting such principle by learning, or chance, or
getting other but similar abstract principle.
On the general problem of grounding symbols, I think they must be
grounded thru interaction with the world and the things symbolized.
Which would make it impossible for the symbol "infinity" to ever be
grounded, and this is indeed my point. You can certainly argue that
a rover grounds the symbols for its location and so on, but it's in
principle impossible for a computer to ground an infinity symbol
through interaction with either the world or infinity itself.
OK. But that is true for us too. What "grounds" us with the infinite
is that we met it through things which we bet will not stop relatively
to us, and which ask us to move at a higher level of abstraction, like
in the passage of understanding enough of I, II, III, IIII, IIIII, ...
to the understanding of {I, II, III, IIII, IIIII, ... }.
Bruno
A computer depends on human beings to interpret and ground the
symbols. Only a robot can ground the symbols itself, e.g. a Mars
rover grounds they symbols for it's location, for the local
terrrain, for the charge in its batteries,...
Brent
On 07/13/15, Pierz wrote:
Here's something that bothers me when I try to think of the brain
too much as a computer. How would I teach a computer the notion of
infinity? In simple terms, how can I represent infinity in a
computer program? All a computer knows about infinity is 'stack
overflow' (or simply integer overflow), an inability to continue a
calculation due to lack of computational resources. Yet obviously
such a state has nothing to do with infinity, and the halting
problem shows that no computer can "grasp" the notion of infinity
through some algorithmic method. I could program in an infinity
symbol and under certain circumstances teach the computer to spew
out this symbol, such as when asked to divide by zero, but that's
like teaching a monkey to press a button that causes a recording of
Hamlet's soliloquy to be played whenever the monkey is asked, "How
do you feel about your life?" The understanding of the connection
between input and output is all external to the system. I'm curious
to know - how could one even in principle write a program that would
produce an output that could plausibly be regarded as a
representation of infinity, generated by the machine? How, for
example, could one get it to "work out" that a number divided by
zero gives an infinite or undefined result, without simply
programming that response? This is just another variant on the
symbol grounding problem of course, but it seems a telling one,
because infinity is a mathematical concept, and in its way almost as
important as zero.
I'm not advancing this exactly as an argument against
computationalism, but I am curious to know if anyone has a better
answer to it than simply something like "it will emerge at higher
levels" or something - which to me just begs the question of how.
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