Daniel, thanks for your very detailed reply. I shouldn’t have said ‘Self-evident Truths’. The only self evident truth that I know is ‘all men are created equal’, and that’s not true. So I was using it to mean ‘assertions’. Which was a stupid thing to do. I am ever tempted to rhetorical slyness over clarity.
I don’t find myself forced to believe in uncomputable things in the same way that I find myself forced to accept the existence of countable infinite sets . I’ve always been really sceptical about the ‘real numbers’.
People win the lottery every day!
No it wouldn’t. A universe of finite size would work. For example, Conway’s game of life on a torus.
Yes, that would be interesting, especially if the torus were very large. I retract that sentence. I think what’s bugging me is that a TM with a finite tape seems more complicated than a TM with an infinite tape. If the finiteness is large, then there’s more information in the size of the constant than there is in the machine. And in fact I believe that some very simple systems are turing equivalent.
I think my argument is something like ‘If you let countability in at all, then you’ve probably got everything. Over and over again.’
I don’t find myself forced to believe in uncomputable things in the same way that I find myself forced to accept the existence of countable infinite sets .
I’m not saying that uncomputable things exist. I’m just saying that they might.
I think what’s bugging me is that a TM with a finite tape seems more complicated than a TM with an infinite tape.
So? Just because the computer can run forever doesn’t mean that the program never halts or repeats.
Also, Occam’s razor doesn’t work that way. If you add a constant of a specific value, that makes it less likely because the probability has to be split over all possible values, and it’s unlikely to be that specific one. If you’re just suggesting that there is a constant, this does not apply.
Daniel, thanks for your very detailed reply. I shouldn’t have said ‘Self-evident Truths’. The only self evident truth that I know is ‘all men are created equal’, and that’s not true. So I was using it to mean ‘assertions’. Which was a stupid thing to do. I am ever tempted to rhetorical slyness over clarity.
I don’t find myself forced to believe in uncomputable things in the same way that I find myself forced to accept the existence of countable infinite sets . I’ve always been really sceptical about the ‘real numbers’.
People win the lottery every day!
Yes, that would be interesting, especially if the torus were very large. I retract that sentence. I think what’s bugging me is that a TM with a finite tape seems more complicated than a TM with an infinite tape. If the finiteness is large, then there’s more information in the size of the constant than there is in the machine. And in fact I believe that some very simple systems are turing equivalent.
I think my argument is something like ‘If you let countability in at all, then you’ve probably got everything. Over and over again.’
I’m not saying that uncomputable things exist. I’m just saying that they might.
So? Just because the computer can run forever doesn’t mean that the program never halts or repeats.
Also, Occam’s razor doesn’t work that way. If you add a constant of a specific value, that makes it less likely because the probability has to be split over all possible values, and it’s unlikely to be that specific one. If you’re just suggesting that there is a constant, this does not apply.