Then I would agree that intuitions don’t provide evidence for such statements, although it is a little bit confusing to say that there is no correlation between intuitions and truth for this class of statements since people don’t have any intuitions pertaining to members of this class of statements.
To further nitpick, one googol is small as compared to what exactly?
(I’d find it way more natural for “randomly constructed statement” to be selected from statements people actually construct and have intuitive opinions about, given the context of the discussion.)
Maybe I can get my point across using this analogy:
“You would expect a human to be able to remember a random natural number.”
“No—because only an infinitessimal fraction of random numbers fall into the range of having a number of digits that fit into a typical human’s memory.”
“Ah, I thought we are only talking about typical natural numbers in human use, from which we randomly choose one.”
The discussion we’re leading has some similarities with talking about a “random mind”—discussing our little corner of the mindspace versus choosing from the whole of it.
Similarly, “truth” is a property of a vastly, vastly bigger class of statements than just those our intuitions were conditioned on. Of course there are heuristics we can use even on e.g. enormous k-SAT statements, but would we still call that a human intuition? I’m not talking about uncomputability here, merely on the nonapplicability of human intuition for the vast majority of potentially true statements.
(I’d find it way more natural for “randomly constructed statement” to be selected from statements people actually construct and have intuitive opinions about, given the context of the discussion.)
If indeed you only choose from statements that people “have intuitive opinions about”, it would of course follow that people have intuitive opinions about said statements.
The googol amount of clauses was referred to as small if e.g. the number of clauses is randomly chosen as a natural number.
The original post was about analytic philosophy and, in particular, about how much intuitions should be trusted. This is a pretty interesting question when we are investigating hypotheses whose truth or falsity are subject to human intuitions. Therefore I find it natural to interpret the statement you were initially responding to, i.e.
And a claim that intuitions are uncorrelated with truth would be pretty hard to defend.
as refering to statements inside the domain of human intuitions. There is certainly no disagreement between us about the fact that human intuitions isn’t applicable to all statements.
The googol amount of clauses was referred to as small if e.g. the number of clauses is randomly chosen as a natural number.
I understood that you sort of say that googol is small as compared to a (typical?) random natural number, but I am not sure what it exactly means. What is a typical random natural number? Do you have in mind some specific distribution whose mean is greater than googol? What would be a number which weren’t small in comparison to random natural numbers?
(I have a feeling that my responses may appear somewhat obnoxious. If it is so, it’s not intentional.)
Then I would agree that intuitions don’t provide evidence for such statements, although it is a little bit confusing to say that there is no correlation between intuitions and truth for this class of statements since people don’t have any intuitions pertaining to members of this class of statements.
To further nitpick, one googol is small as compared to what exactly?
(I’d find it way more natural for “randomly constructed statement” to be selected from statements people actually construct and have intuitive opinions about, given the context of the discussion.)
Maybe I can get my point across using this analogy:
“You would expect a human to be able to remember a random natural number.”
“No—because only an infinitessimal fraction of random numbers fall into the range of having a number of digits that fit into a typical human’s memory.”
“Ah, I thought we are only talking about typical natural numbers in human use, from which we randomly choose one.”
The discussion we’re leading has some similarities with talking about a “random mind”—discussing our little corner of the mindspace versus choosing from the whole of it.
Similarly, “truth” is a property of a vastly, vastly bigger class of statements than just those our intuitions were conditioned on. Of course there are heuristics we can use even on e.g. enormous k-SAT statements, but would we still call that a human intuition? I’m not talking about uncomputability here, merely on the nonapplicability of human intuition for the vast majority of potentially true statements.
If indeed you only choose from statements that people “have intuitive opinions about”, it would of course follow that people have intuitive opinions about said statements.
The googol amount of clauses was referred to as small if e.g. the number of clauses is randomly chosen as a natural number.
The original post was about analytic philosophy and, in particular, about how much intuitions should be trusted. This is a pretty interesting question when we are investigating hypotheses whose truth or falsity are subject to human intuitions. Therefore I find it natural to interpret the statement you were initially responding to, i.e.
as refering to statements inside the domain of human intuitions. There is certainly no disagreement between us about the fact that human intuitions isn’t applicable to all statements.
I understood that you sort of say that googol is small as compared to a (typical?) random natural number, but I am not sure what it exactly means. What is a typical random natural number? Do you have in mind some specific distribution whose mean is greater than googol? What would be a number which weren’t small in comparison to random natural numbers?
(I have a feeling that my responses may appear somewhat obnoxious. If it is so, it’s not intentional.)