I think this example brings out how Pearlian causality differs from other causal theories. For instance, in a counterfactual theory of causation, since the negation of a mathematical truth is impossible, we can’t meaningfully think of them as causes.
But in the Pearlian causality it seems that mathematical statements can have causal relations, since we can factor our uncertainty about them, just as we can other statements. I think endoself’s comment argues this well. I would add that this is a good example of how causation can be subjective. Before 1984, the Taniyama-Shimura-Weil conjecture and Fermat’s last theorem existed as conjectures, and some mathematicians presumably knew about both, but as far as I know they had no clue that they were related. Then Frey conjectured and Ribet proved that the TSW conjecture implies FLT. Then mathematician’s uncertainty was such that they would have causal graphs with TSW causing FLT. Now we have a proof of TSW (mostly by Wiles) but any residual uncertainty is still correlated. In the future, maybe there will be many independent proofs of each, and whatever uncertainty is left about them will be (nearly) uncorrelated.
I also think there can be causal relations between mathematical statements and statements about the world. For instance, maybe there is some conjecture of fluid dynamics, which if true would cause us to believe a certain type of wave can occur in certain circumstances. We can make inferences both ways, for instance, if we observe the wave we might increase our credence in the conjecture, and if we prove the conjecture, we might believe the wave can be observed somehow. But it seems that the causal graph would have the conjecture causing the wave. Part of the graph would be:
[Proof of conjecture -> conjecture -> wave <- (fluid dynamics applies to water) ]
[Proof of conjecture ← conjecture → wave ← (fluid dynamics applies to water) ]
Then mathematician’s uncertainty was such that they would have causal graphs with TSW causing FLT.
Well the direction of the arrow would be unspecified. After all, not FLT implies not TSW is equivalent to TSW implies FLT, so there’s a symmetry here. This often happens in causal modelling; many causal discovery algorithms can output that they know an arrow exists, but they are unable to determine its direction.
Also, conjectures are the causes of their proofs rather than vice versa. You can see this as your degrees of belief in the correctness of purported proofs are independent given that the conjecture is true (or false), but dependent when the truth-value of the conjecture is unknown.
Apart from this detail, I agree with your comment and I find it to be similar to the way I think about the causal structure of math.
This is very different from how I think about it. Could you expand a little? What do you mean by “when the truth-value of the conjecture is unknown”? That neither C nor ¬C is in your bound agent’s store of known theorems?
your degrees of belief in the correctness of purported proofs are independent given that the conjecture is true (or false),
Let S1, S2 be purported single-conclusion proofs of a statement C.
If I know C is false, the purported proofs are trivially independent because they’re fully determined incorrect?
Why is S1 independent of S2 given C is true? Are you saying that learning S2⊢C puts C in our theorem bank, and knowing C is true can change our estimation that S1⊢C , but proofs aren’t otherwise mutually informative? If so, what is the effect of learning ⊨C on P(S1⊢C)? And why don’t you consider proofs which, say, only differ after the first n steps to be dependent, even given the truth of their shared conclusion?
What do you mean by “when the truth-value of the conjecture is unknown”? That neither C nor ¬C is in your bound agent’s store of known theorems?
I meant that the agent is in some state of uncertainty. I’m trying to contrast the case where we are more certain of either C or ¬C with that where we have a significant degree of uncertainty.
If I know C is false, the purported proofs are trivially independent because they’re fully determined incorrect?
Yeah, this is just the trivial case.
Why is S1 independent of S2 given C is true?
I was talking about the simple case where there are no other causal links between the two proofs, like common lemmas or empirical observations. Those do change the causal structure by adding extra nodes and arrows, but I was making the simplifying assumption that we don’t have those things.
But in the Pearlian causality it seems that mathematical statements can have causal relations, since we can factor our uncertainty about them, just as we can other statements.
There maybe uncertainty about casual relations and about mathemtical statements, but that does not mean
mathematics is causal.
I think endoself’s comment argues this well. I would add that this is a good example of how causation can be subjective.
The transition probabilites on a causal diagram may be less than 1, but that represents levels of subjective confidence—epistemology—not causality per se. You can’t prove that the universe is indeterministic by
writing out a diagram.
Before 1984, the Taniyama-Shimura-Weil conjecture and Fermat’s last theorem existed as conjectures, and some mathematicians presumably knew about both, but as far as I know they had no clue that they were related. Then Frey conjectured and Ribet proved that the TSW conjecture implies FLT. Then mathematician’s uncertainty was such that they would have causal graphs with TSW causing FLT. Now we have a proof of TSW (mostly by Wiles) but any residual uncertainty is still correlated. In the future, maybe there will be many independent proofs of each, and whatever uncertainty is left about them will be (nearly) uncorrelated.
Yes you can write out a diagram with transitions indicating logical relationships and probabilities representing
subjective confidence. But the nodes aren’t spatio-temporal events, so it isnt a causal diagram. It is another
kind of diagram which happens to have the same structure.
I also think there can be causal relations between mathematical statements and statements about the world.
Causal relations hold between events, not statements.
For instance, maybe there is some conjecture of fluid dynamics, which if true would cause us to believe a certain type of wave can occur in certain circumstances.
What causes us to believe is evidence, not abstract truth.
We can make inferences both ways, for instance, if we observe the wave we might increase our credence in the conjecture, and if we prove the conjecture, we might believe the wave can be observed somehow.
The production of a proof, which is a spatio temproal event, can cause a change in beleif-state, which
is a spatio temporal event, which causes changes in behaviour....mathematical truth is not involved.
Truth without proof causes nothing. If we dont have reason to believe in a conjecture, we don’t act
on it, even if it is true.
Taboo spatio-temporal. Why is it a good idea to give one category of statements the special name ‘events’ and to reason about them differently than you would reason about other events?
“Events” aren’t a kind of statement. However a subset of statements is about events.
The point of separating them out is that this discussion is about causality, and, uncontentiously, causality
links events. If something is a Non-event (or a statement is not about an event), that is a good argument
for not granting it (or what is is about) causal powers.
Yes, I’m trying to get you to reduce the concept rather than take it as primitive. I know what an event is, by I think that the distinction between events and statements is fuzzy, and I think that events are best understood as a subcategory of statements.
Reduce it to whart? You’ve already “tabood” spatio-temporal. I can’t communicate anything to you without some set of common meanings. It;s a cheap trick to complain that someone can’t define something from a basis of shared-nothing, since no one can do that with any term.
I know what an event is, by I think that the distinction between events and statements is fuzzy, and I think that events are best understood as a subcategory of statements.
The diffrerence is screamingly obvious. Statements are verbal communiations. If one asteroid crashes
into another, that is an event but not a statement.. Statements are events, because they happen at particular
palces and times, but most events are not statements. You’ve got it the wrong way round.
I meant ‘statement’ in the abstract sense of what is stated rather than things like when it is stated and who it is stated by. ‘Proposition’ has the meanings that I intend without any others, so it would better convey my meaning here.
Reduce it to whart? You’ve already “tabood” spatio-temporal.
The point of rationalist taboo is to eliminate all the different phrasings we can use to mention a concept without really understanding it and force us to consider how the phenomenon being discussed actually works. Your wording presumes certain intuition about what the physical world is and how it should work by virtue of being “physical”, intuitions that are not usually argued for or even noticed. When you say you can’t explain what an “event” or something “spatio-temporal” is without reference to words that really just restate the concept, that is giving a mysterious answer. Things work a certain way, and we can determine how.
I have no idea what “work” means, please explain...
If you are a native english speaker, you will have enough of an understanding of “event” to appreciate my point. You expect me understand terms like “work” without your going through the process of giving a sematic bedrock definition , beyond the common one.
I think this example brings out how Pearlian causality differs from other causal theories. For instance, in a counterfactual theory of causation, since the negation of a mathematical truth is impossible, we can’t meaningfully think of them as causes.
But in the Pearlian causality it seems that mathematical statements can have causal relations, since we can factor our uncertainty about them, just as we can other statements. I think endoself’s comment argues this well. I would add that this is a good example of how causation can be subjective. Before 1984, the Taniyama-Shimura-Weil conjecture and Fermat’s last theorem existed as conjectures, and some mathematicians presumably knew about both, but as far as I know they had no clue that they were related. Then Frey conjectured and Ribet proved that the TSW conjecture implies FLT. Then mathematician’s uncertainty was such that they would have causal graphs with TSW causing FLT. Now we have a proof of TSW (mostly by Wiles) but any residual uncertainty is still correlated. In the future, maybe there will be many independent proofs of each, and whatever uncertainty is left about them will be (nearly) uncorrelated.
I also think there can be causal relations between mathematical statements and statements about the world. For instance, maybe there is some conjecture of fluid dynamics, which if true would cause us to believe a certain type of wave can occur in certain circumstances. We can make inferences both ways, for instance, if we observe the wave we might increase our credence in the conjecture, and if we prove the conjecture, we might believe the wave can be observed somehow. But it seems that the causal graph would have the conjecture causing the wave. Part of the graph would be:
[Proof of conjecture -> conjecture -> wave<- (fluid dynamics applies to water) ][Proof of conjecture ← conjecture → wave ← (fluid dynamics applies to water) ]
Well the direction of the arrow would be unspecified. After all, not FLT implies not TSW is equivalent to TSW implies FLT, so there’s a symmetry here. This often happens in causal modelling; many causal discovery algorithms can output that they know an arrow exists, but they are unable to determine its direction.
Also, conjectures are the causes of their proofs rather than vice versa. You can see this as your degrees of belief in the correctness of purported proofs are independent given that the conjecture is true (or false), but dependent when the truth-value of the conjecture is unknown.
Apart from this detail, I agree with your comment and I find it to be similar to the way I think about the causal structure of math.
This is very different from how I think about it. Could you expand a little? What do you mean by “when the truth-value of the conjecture is unknown”? That neither C nor ¬C is in your bound agent’s store of known theorems?
Let S1, S2 be purported single-conclusion proofs of a statement C.
If I know C is false, the purported proofs are trivially independent because they’re fully determined incorrect?
Why is S1 independent of S2 given C is true? Are you saying that learning S2⊢C puts C in our theorem bank, and knowing C is true can change our estimation that S1⊢C , but proofs aren’t otherwise mutually informative? If so, what is the effect of learning ⊨C on P(S1⊢C)? And why don’t you consider proofs which, say, only differ after the first n steps to be dependent, even given the truth of their shared conclusion?
I meant that the agent is in some state of uncertainty. I’m trying to contrast the case where we are more certain of either C or ¬C with that where we have a significant degree of uncertainty.
Yeah, this is just the trivial case.
I was talking about the simple case where there are no other causal links between the two proofs, like common lemmas or empirical observations. Those do change the causal structure by adding extra nodes and arrows, but I was making the simplifying assumption that we don’t have those things.
Hmm, you are right. Thanks for the correction!
There maybe uncertainty about casual relations and about mathemtical statements, but that does not mean mathematics is causal.
The transition probabilites on a causal diagram may be less than 1, but that represents levels of subjective confidence—epistemology—not causality per se. You can’t prove that the universe is indeterministic by writing out a diagram.
Yes you can write out a diagram with transitions indicating logical relationships and probabilities representing subjective confidence. But the nodes aren’t spatio-temporal events, so it isnt a causal diagram. It is another kind of diagram which happens to have the same structure.
Causal relations hold between events, not statements.
What causes us to believe is evidence, not abstract truth.
The production of a proof, which is a spatio temproal event, can cause a change in beleif-state, which is a spatio temporal event, which causes changes in behaviour....mathematical truth is not involved. Truth without proof causes nothing. If we dont have reason to believe in a conjecture, we don’t act on it, even if it is true.
Taboo spatio-temporal. Why is it a good idea to give one category of statements the special name ‘events’ and to reason about them differently than you would reason about other events?
Also, what about Newcomb’s problem?
“Events” aren’t a kind of statement. However a subset of statements is about events. The point of separating them out is that this discussion is about causality, and, uncontentiously, causality links events. If something is a Non-event (or a statement is not about an event), that is a good argument for not granting it (or what is is about) causal powers.
What is an event? What properties do events have that statements do not?
Are you a native English speaker?
Yes, I’m trying to get you to reduce the concept rather than take it as primitive. I know what an event is, by I think that the distinction between events and statements is fuzzy, and I think that events are best understood as a subcategory of statements.
Reduce it to whart? You’ve already “tabood” spatio-temporal. I can’t communicate anything to you without some set of common meanings. It;s a cheap trick to complain that someone can’t define something from a basis of shared-nothing, since no one can do that with any term.
The diffrerence is screamingly obvious. Statements are verbal communiations. If one asteroid crashes into another, that is an event but not a statement.. Statements are events, because they happen at particular palces and times, but most events are not statements. You’ve got it the wrong way round.
I meant ‘statement’ in the abstract sense of what is stated rather than things like when it is stated and who it is stated by. ‘Proposition’ has the meanings that I intend without any others, so it would better convey my meaning here.
The point of rationalist taboo is to eliminate all the different phrasings we can use to mention a concept without really understanding it and force us to consider how the phenomenon being discussed actually works. Your wording presumes certain intuition about what the physical world is and how it should work by virtue of being “physical”, intuitions that are not usually argued for or even noticed. When you say you can’t explain what an “event” or something “spatio-temporal” is without reference to words that really just restate the concept, that is giving a mysterious answer. Things work a certain way, and we can determine how.
I have no idea what “work” means, please explain...
If you are a native english speaker, you will have enough of an understanding of “event” to appreciate my point. You expect me understand terms like “work” without your going through the process of giving a sematic bedrock definition , beyond the common one.
Newcomb’s problem is an irrelevant-to-everything Waste Of Money Brains And Time, AFIAC.