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. > > 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. > 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. > > > > > > -- > > You received this message because you are subscribed to the Google Groups > "Everything List" group. > > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected] <javascript:>. > > To post to this group, send email to [email protected] > <javascript:>. > > Visit this group at http://groups.google.com/group/everything-list. > > For more options, visit https://groups.google.com/d/optout. > > -- You received this message because you are subscribed to the Google Groups "Everything List" group. 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