Edit: A close reading of Shramko 2012 has resolved my confusion. Thanks, everyone.
I can’t shake the idea that maps should be represented classically and territories should be represented intuitionistically. I’m looking for logical but critical comments on this idea. Here’s my argument:
Territories have entities that are not compared to anything else. If an entity exists in the territory, then it is what it is. Territorial entities, as long as they are consistently defined, are never wrong by definition. By comparison, maps can represent any entity. Being a map, these mapped entities are intended to be compared to the territory of which it is a map. If the territory does not have a corresponding entity, then that mapped entity is false insofar as it is intended as a map.
This means that territories are repositories of pure truth with no speck of falsehood lurking in any corner, whereas maps represent entities that can be true or false depending on the state of the territory. This corresponds to the notion that intuitionism captures the concept of truth. If you add the concept of falsehood or contradiction, then you end up with classical logic or mathematics respectively. First source I can think of: https://www.youtube.com/playlist?list=PLt7hcIEdZLAlY0oUz4VCQnF14C6VPtewG
Furthermore, the distinction between maps and territories seems to be a transcendental one in the Kantian sense of being a synthetic a priori. That is to say, it is an idea that must be universally imposed on the world by any mind that seeks to understand it. Intuitionism has been associated with Kantian philosophy since its inception. If The Map is included in The Territory in some ultimate sense, that neatly dovetails with the idea of intuitionists who argue that classical mathematics is a proper subset of intuitionistic mathematics.
In summary, my thesis states that classical logic is the logic of making a map accurate by comparing it to a territory, which is why the concept of falsehood becomes an integral part of the formal system. In contrast, intuitionistic logic is the logic of describing a territory without seeking to compare it to something else. Intuitionistic type theory turns up type errors, for example, when such a description turns out to be inconsistent in itself.
Also possibly problematic is the dichotomy described by the summary:
classical logic is the logic of making a map accurate by comparing it to a territory, which is why the concept of falsehood becomes an integral part of the formal system. In contrast, intuitionistic logic is the logic of describing a territory without seeking to compare it to something else. Intuitionistic type theory turns up type errors, for example, when such a description turns out to be inconsistent in itself.
seems more appropriate to contrast scientific/Bayesian reasoning, which strives to confirm or refute a model based on how well it conforms to observed reality vs deductive (a priori) reasoning, which looks only at what follows from a set of axioms. However, one can reason deductively using classical or intuitionistic logic, so it is not clear that intuitionistic logic is better suited than classical logic for “describing a territory without seeking to compare it to something else”.
I don’t see how distinguishing between deductive and inductive reasoning is mutually exclusive with the map/description distinction. That is to say, you could have each of the following combinations: deductive map, deductive description, inductive map, and inductive description.
Edit: On second thought, I see what you were saying. Thanks, I will think about it.
I can’t shake the idea that maps should be represented classically and territories should be represented intuitionistically.
But, it seems to me that a map is a representation of a territory. So, your statement “maps should be represented classically and territories should be represented intuitionistically” reduces to “representations of the territory should be intuitionistic, and representations of those intuitionistic representations should be classical”. Is this what you intended, or am I missing something?
Also, I’m not an expert in intuitionistic logic, but this statement from the summary sounds problematic:
classical logic is the logic of making a map accurate by comparing it to a territory, which is why the concept of falsehood becomes an integral part of the formal system
But, the concept of falsehood is integral to both classical and intuitionistic logic. Intuitionistic logic got rid of the principle of the excluded middle but did not get rid of the concept of falsity.
Regarding falsehood: I would say that intuitionistic logic ejects falsehood from its formal system in the specific sense mentioned in my link. I could dig up more references if you want me to. I agree that there are many reasonable interpretations in which it does not do so, but I don’t think those interpretations are relevant to my point. I only intended to argue that proof by contradiction is the strategy of correcting a map as opposed to describing a territory.
Regarding mapping versus description: I agree that my motivations were semantic rather than syntactic. I just wanted to know whether the idea I had made sense to others who know something of intuitionistic logic. I guess I have my answer, but for the sake of clarifying the sense I was going for, here’s the example I posted below:
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?
Regarding mapping versus description: I agree that my motivations were semantic rather than syntactic. I just wanted to know whether the idea I had made sense to others who know something of intuitionistic logic.
Understood. But, the point that I raised is not merely syntactic. On a fundamental level, a description of the territory is a map, so when you attempt to contrast correcting a map vs rejecting a description of a territory, you are really talking about correcting vs. rejecting a map.
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?
Yes, in the case of number 1 you have proved via contradiction that there is no red cube, and in #2 you have concluded that one or more of your assumptions is incorrect (i.e. that your map is incorrect). However, this is not a map vs. territory distinction; in both cases you are really dealing with a map. To make this clear, I would restate as:
1 is the strategy of correcting the map and 2 is the strategy of rejecting the map as inaccurate without seeking to correct it.
So, I guess I don’t really have anything additional to add about intuitionistic logic—my point is that when you talk about a description of the territory vs. a map, you are really talking about the same thing.
Thanks. The next thing I was going to say is that the intuitionistic strategy of neutrality with regard to affirming or negating propositions in worlds until proof comes along roughly (i.e. in a sense to be argued for later) differentiates the classical and intuitionistic approaches like so:
The classical approach is good for having one “world” description that is almost certainly inaccurate. This can be gradually updated, making it represent one map.
The intuitionistic approach is good for having multiple world descriptions that are almost certainly incomplete. Their contours are filled in as more information becomes available and rejected as inaccurate when they lead to contradictions, making each one a holistic representation of a possible territory. (Shoehorning the same approach into classical logic is possible, but you have to create a set of conventions to do so. These conventions are not universal, making the approach less natural.)
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?
Edit: A close reading of Shramko 2012 has resolved my confusion. Thanks, everyone.
I can’t shake the idea that maps should be represented classically and territories should be represented intuitionistically. I’m looking for logical but critical comments on this idea. Here’s my argument:
Territories have entities that are not compared to anything else. If an entity exists in the territory, then it is what it is. Territorial entities, as long as they are consistently defined, are never wrong by definition. By comparison, maps can represent any entity. Being a map, these mapped entities are intended to be compared to the territory of which it is a map. If the territory does not have a corresponding entity, then that mapped entity is false insofar as it is intended as a map.
This means that territories are repositories of pure truth with no speck of falsehood lurking in any corner, whereas maps represent entities that can be true or false depending on the state of the territory. This corresponds to the notion that intuitionism captures the concept of truth. If you add the concept of falsehood or contradiction, then you end up with classical logic or mathematics respectively. First source I can think of: https://www.youtube.com/playlist?list=PLt7hcIEdZLAlY0oUz4VCQnF14C6VPtewG
Furthermore, the distinction between maps and territories seems to be a transcendental one in the Kantian sense of being a synthetic a priori. That is to say, it is an idea that must be universally imposed on the world by any mind that seeks to understand it. Intuitionism has been associated with Kantian philosophy since its inception. If The Map is included in The Territory in some ultimate sense, that neatly dovetails with the idea of intuitionists who argue that classical mathematics is a proper subset of intuitionistic mathematics.
In summary, my thesis states that classical logic is the logic of making a map accurate by comparing it to a territory, which is why the concept of falsehood becomes an integral part of the formal system. In contrast, intuitionistic logic is the logic of describing a territory without seeking to compare it to something else. Intuitionistic type theory turns up type errors, for example, when such a description turns out to be inconsistent in itself.
Where did I take a wrong turn?
Also possibly problematic is the dichotomy described by the summary:
seems more appropriate to contrast scientific/Bayesian reasoning, which strives to confirm or refute a model based on how well it conforms to observed reality vs deductive (a priori) reasoning, which looks only at what follows from a set of axioms. However, one can reason deductively using classical or intuitionistic logic, so it is not clear that intuitionistic logic is better suited than classical logic for “describing a territory without seeking to compare it to something else”.
I don’t see how distinguishing between deductive and inductive reasoning is mutually exclusive with the map/description distinction. That is to say, you could have each of the following combinations: deductive map, deductive description, inductive map, and inductive description.
Edit: On second thought, I see what you were saying. Thanks, I will think about it.
But, it seems to me that a map is a representation of a territory. So, your statement “maps should be represented classically and territories should be represented intuitionistically” reduces to “representations of the territory should be intuitionistic, and representations of those intuitionistic representations should be classical”. Is this what you intended, or am I missing something?
Also, I’m not an expert in intuitionistic logic, but this statement from the summary sounds problematic:
But, the concept of falsehood is integral to both classical and intuitionistic logic. Intuitionistic logic got rid of the principle of the excluded middle but did not get rid of the concept of falsity.
Thanks.
Regarding falsehood: I would say that intuitionistic logic ejects falsehood from its formal system in the specific sense mentioned in my link. I could dig up more references if you want me to. I agree that there are many reasonable interpretations in which it does not do so, but I don’t think those interpretations are relevant to my point. I only intended to argue that proof by contradiction is the strategy of correcting a map as opposed to describing a territory.
Regarding mapping versus description: I agree that my motivations were semantic rather than syntactic. I just wanted to know whether the idea I had made sense to others who know something of intuitionistic logic. I guess I have my answer, but for the sake of clarifying the sense I was going for, here’s the example I posted below:
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?
Understood. But, the point that I raised is not merely syntactic. On a fundamental level, a description of the territory is a map, so when you attempt to contrast correcting a map vs rejecting a description of a territory, you are really talking about correcting vs. rejecting a map.
Yes, in the case of number 1 you have proved via contradiction that there is no red cube, and in #2 you have concluded that one or more of your assumptions is incorrect (i.e. that your map is incorrect). However, this is not a map vs. territory distinction; in both cases you are really dealing with a map. To make this clear, I would restate as:
So, I guess I don’t really have anything additional to add about intuitionistic logic—my point is that when you talk about a description of the territory vs. a map, you are really talking about the same thing.
Thanks. The next thing I was going to say is that the intuitionistic strategy of neutrality with regard to affirming or negating propositions in worlds until proof comes along roughly (i.e. in a sense to be argued for later) differentiates the classical and intuitionistic approaches like so:
The classical approach is good for having one “world” description that is almost certainly inaccurate. This can be gradually updated, making it represent one map.
The intuitionistic approach is good for having multiple world descriptions that are almost certainly incomplete. Their contours are filled in as more information becomes available and rejected as inaccurate when they lead to contradictions, making each one a holistic representation of a possible territory. (Shoehorning the same approach into classical logic is possible, but you have to create a set of conventions to do so. These conventions are not universal, making the approach less natural.)
Something like that anyway, but Shramko 2012 has put a lot more thought into this than I have: http://kdpu.edu.ua/shramko/files/2012_Logic_and_Logical_Philosophy_What_is_a_Genueny_Intuitionistic_Notion_of_Falsity.pdf I defer to expert opinion here.
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?