I can’t tell you where you took a wrong turn, because I don’t know whether you did. But I can tell you where you lost me—i.e., where I stopped seeing how each statement was an inference drawn from its predecessors plus uncontroversial things.
The first place was when you said “This corresponds to the notion that intuitionism captures the concept of truth.” How does is correspond to that? “This” is the idea that tthe territory has no errors in it, whereas the map has errors, and I don’t see how you get from that to anything involving intuitionism.
… Oh, wait, maybe I do? Are you thinking of intuitionism as somehow lacking negation, so that you can only ever say things are true and never say they’re false? Your “summary” paragraph seems to suggest this. That doesn’t seem like it agrees with my understanding of intuitionism, but I may be missing something.
The second time you lost me was when you said “If The Map is included in The Territory [...] that neatly dovetails with the idea [...] that classical mathematics is a proper subset of intuitionistic mathematics”. Isn’t that exactly backwards? Intuitionistic mathematics is the subset of classical mathematics you can reach without appealing to the law of the excluded middle.
Finally, your “summary” paragraph asserts once again the correspondence you’re describing, but I don’t really see where you’ve argued for it. (This may be best viewed as just a restatement of my earlier puzzlements.)
Regarding errors: It’s not that intuitionism never turns up errors. It’s that the classical approach incorporates the concept of error within the formal system itself. This is mentioned in the link I gave. There are two senses here:
Falsehood is more tightly interwoven in the formal system when following the classical approach.
Errors are more integral to the process of comparing maps to territories than the description of territories in themselves.
It is possible that these two senses are not directly comparable. My question is: How meaningful is the difference between these two senses?
Regarding subsets: It is true that intuitionism is often regarded as the constructive subset of classical mathematics, but intuitionists argue that classical mathematics is the proper subset of intuitionistic mathematics where proof by contradiction is valid. I’m basically paraphrasing intuitionistic mathematicians here.
This (i.e. subsets thing) is not intended as an irrefutable argument. It is only intended to extend the correspondence. After all, if either classical or intuitionistic approaches can be used as a foundation for all of mathematics, then it stands to reason that the other will appear as a proper subset from the foundational perspective of either.
Edit: This doesn’t add any new information, but let me give an example for the sake of vividness. Suppose you have a proposition like, “There is a red cube.” Next, you learn that this proposition leads to a contradiction. You could say one of two things:
This proves there is no red cube.
This means the context in which that proposition occurs is erroneous.
Does it make sense to say that 1 is the strategy of correcting a map and 2 is the strategy of rejecting a description as inaccurate without seeking to correct something?
I can’t tell you where you took a wrong turn, because I don’t know whether you did. But I can tell you where you lost me—i.e., where I stopped seeing how each statement was an inference drawn from its predecessors plus uncontroversial things.
The first place was when you said “This corresponds to the notion that intuitionism captures the concept of truth.” How does is correspond to that? “This” is the idea that tthe territory has no errors in it, whereas the map has errors, and I don’t see how you get from that to anything involving intuitionism.
… Oh, wait, maybe I do? Are you thinking of intuitionism as somehow lacking negation, so that you can only ever say things are true and never say they’re false? Your “summary” paragraph seems to suggest this. That doesn’t seem like it agrees with my understanding of intuitionism, but I may be missing something.
The second time you lost me was when you said “If The Map is included in The Territory [...] that neatly dovetails with the idea [...] that classical mathematics is a proper subset of intuitionistic mathematics”. Isn’t that exactly backwards? Intuitionistic mathematics is the subset of classical mathematics you can reach without appealing to the law of the excluded middle.
Finally, your “summary” paragraph asserts once again the correspondence you’re describing, but I don’t really see where you’ve argued for it. (This may be best viewed as just a restatement of my earlier puzzlements.)
Thank you for the response.
Regarding errors: It’s not that intuitionism never turns up errors. It’s that the classical approach incorporates the concept of error within the formal system itself. This is mentioned in the link I gave. There are two senses here:
Falsehood is more tightly interwoven in the formal system when following the classical approach.
Errors are more integral to the process of comparing maps to territories than the description of territories in themselves.
It is possible that these two senses are not directly comparable. My question is: How meaningful is the difference between these two senses?
Regarding subsets: It is true that intuitionism is often regarded as the constructive subset of classical mathematics, but intuitionists argue that classical mathematics is the proper subset of intuitionistic mathematics where proof by contradiction is valid. I’m basically paraphrasing intuitionistic mathematicians here.
This (i.e. subsets thing) is not intended as an irrefutable argument. It is only intended to extend the correspondence. After all, if either classical or intuitionistic approaches can be used as a foundation for all of mathematics, then it stands to reason that the other will appear as a proper subset from the foundational perspective of either.
Edit: This doesn’t add any new information, but let me give an example for the sake of vividness. Suppose you have a proposition like, “There is a red cube.” Next, you learn that this proposition leads to a contradiction. You could say one of two things:
This proves there is no red cube.
This means the context in which that proposition occurs is erroneous.
Does it make sense to say that 1 is the strategy of correcting a map and 2 is the strategy of rejecting a description as inaccurate without seeking to correct something?