That’s what I would expect most mathematical-existence types to think. It’s true, but it’s also the wrong thought.
Wei, do you see it now that I’ve pointed it out? Or does anyone else see it? As problems in philosophy go, it seems like a reasonable practice exercise to see it once I’ve pointed to it but before I’ve explained it.
In logic, any time you have a set of axioms from which it is impossible to derive a contradiction, a model exists about which all the axioms are true. Here, “X exists” means that you can prove, by construction, that an existentially quantified proposition about some model X is true in models of set theory. So all consistent models are defined into “existence”.
A causal process is an unfolded computation. Parts of its structure have relationships that are logically constrained, if not fully determined, by other parts. But like any computation, you can put an infinite variety of inputs on the tape of the Causality Turing machine’s tape, and you’ll get a different causal process. Here, “X exists” means that X is a part of the same causal process that you are a part of. So you have to entangle with your surroundings in order to judge what “exists”.
Eliezer, I still don’t understand Pearl well enough to answer your question. Did anyone else get it?
Right now I’m working on the following related question, and would appreciate any ideas. Some very smart people have worked hard on causality for years, but UDT1 seemingly does fine without an explicit notion of causality. Why is that, or is there a flaw in it that I’m not seeing? Eliezer suggested earlier that causality is a way of cashing out the “mathematical intuition module” in UDT1. I’m still trying to see if that really makes sense. It would be surprising if mathematical intuition is so closely related to causality, which seems to be very different at first glance.
It’s still unclear what you mean. One simple idea is that many formalisms allow to express each other, but some give more natural ways of representing a given problem than others. In some contexts, a given way of stating things may be clearly superior. If you e.g. see math as something happening in heads of mathematicians, or see implication of classical logic as a certain idealization of material implication where nothing changes, one may argue that a given way is more fundamental, closer to what actually happens.
When you ask questions like “do you see it now?”, I doubt there is even a good way of interpreting them as having definite answers, without already knowing what you expect to hear, a lot more context about what kinds of things you are thinking about than is generally available.
That’s what I would expect most mathematical-existence types to think. It’s true, but it’s also the wrong thought.
Wei, do you see it now that I’ve pointed it out? Or does anyone else see it?
I take this to be your point:
Suppose that you want to understand causation better. Your first problem is that your concept of causation is still vague, so you try to develop a formalism to talk about causation more precisely. However, despite the vagueness, your topic is sufficiently well-specified that it’s possible to say false things about it.
In this case, choosing the wrong language (e.g., ZFC) in which to express your formalism can be fatal. This is because a language such as ZFC makes it easy to construct some formalisms but difficult to construct others. It happens to be the case that ZFC makes it much easier to construct wrong formalisms for causation than does, say, the language of networks.
Making matters worse, humans have a tendency to be attracted to impressive-looking formalisms that easily generate unambiguous answers. ZFC-based formalisms can look impressive and generate unambiguous answers. But the answers are likely to be wrong because the formalisms that are natural to construct in ZFC don’t capture the way that causation actually works.
Since you started out with a vague understanding of causation, you’ll be unable to recognize that your formalism has led you astray. And so you wind up worse than you started, convinced of false beliefs rather than merely ignorant. Since understanding causation is so important, this can be a fatal mistake.
--- So, that’s all well and good, but it isn’t relevant to this discussion. Philosophers of mathematics might make a lot of mistakes. And maybe some have made the mistake of trying to use ZFC to talk about physical causation. But few, if any, haven’t “noticed the difference in style between the relation between logical axioms and logical models versus causal laws and causal processes.” That just isn’t among the vast catalogue of their errors.
Is it simply that: causal graphs (can) have locality, and you can perform counterfactual surgery on intermediate nodes and get meaningful results, while logic has no locality and (without the hoped-for theory of impossible possible worlds) you can’t contradict one theorem without the system exploding?
That’s what I would expect most mathematical-existence types to think. It’s true, but it’s also the wrong thought.
Perhaps, but irrelevant, because I’m not what you would call a mathematical-existence type.
ETA: The point is that you can’t be confident about what thought stands behind the sentence “Pearl’s causal networks can be encoded in ZFC” until you have some familiarity with how the speaker thinks. On what basis do you claim that familiarity?
That’s what I would expect most mathematical-existence types to think. It’s true, but it’s also the wrong thought.
Wei, do you see it now that I’ve pointed it out? Or does anyone else see it? As problems in philosophy go, it seems like a reasonable practice exercise to see it once I’ve pointed to it but before I’ve explained it.
Is this it:
In logic, any time you have a set of axioms from which it is impossible to derive a contradiction, a model exists about which all the axioms are true. Here, “X exists” means that you can prove, by construction, that an existentially quantified proposition about some model X is true in models of set theory. So all consistent models are defined into “existence”.
A causal process is an unfolded computation. Parts of its structure have relationships that are logically constrained, if not fully determined, by other parts. But like any computation, you can put an infinite variety of inputs on the tape of the Causality Turing machine’s tape, and you’ll get a different causal process. Here, “X exists” means that X is a part of the same causal process that you are a part of. So you have to entangle with your surroundings in order to judge what “exists”.
Eliezer, I still don’t understand Pearl well enough to answer your question. Did anyone else get it?
Right now I’m working on the following related question, and would appreciate any ideas. Some very smart people have worked hard on causality for years, but UDT1 seemingly does fine without an explicit notion of causality. Why is that, or is there a flaw in it that I’m not seeing? Eliezer suggested earlier that causality is a way of cashing out the “mathematical intuition module” in UDT1. I’m still trying to see if that really makes sense. It would be surprising if mathematical intuition is so closely related to causality, which seems to be very different at first glance.
It’s still unclear what you mean. One simple idea is that many formalisms allow to express each other, but some give more natural ways of representing a given problem than others. In some contexts, a given way of stating things may be clearly superior. If you e.g. see math as something happening in heads of mathematicians, or see implication of classical logic as a certain idealization of material implication where nothing changes, one may argue that a given way is more fundamental, closer to what actually happens.
When you ask questions like “do you see it now?”, I doubt there is even a good way of interpreting them as having definite answers, without already knowing what you expect to hear, a lot more context about what kinds of things you are thinking about than is generally available.
I take this to be your point:
Suppose that you want to understand causation better. Your first problem is that your concept of causation is still vague, so you try to develop a formalism to talk about causation more precisely. However, despite the vagueness, your topic is sufficiently well-specified that it’s possible to say false things about it.
In this case, choosing the wrong language (e.g., ZFC) in which to express your formalism can be fatal. This is because a language such as ZFC makes it easy to construct some formalisms but difficult to construct others. It happens to be the case that ZFC makes it much easier to construct wrong formalisms for causation than does, say, the language of networks.
Making matters worse, humans have a tendency to be attracted to impressive-looking formalisms that easily generate unambiguous answers. ZFC-based formalisms can look impressive and generate unambiguous answers. But the answers are likely to be wrong because the formalisms that are natural to construct in ZFC don’t capture the way that causation actually works.
Since you started out with a vague understanding of causation, you’ll be unable to recognize that your formalism has led you astray. And so you wind up worse than you started, convinced of false beliefs rather than merely ignorant. Since understanding causation is so important, this can be a fatal mistake.
--- So, that’s all well and good, but it isn’t relevant to this discussion. Philosophers of mathematics might make a lot of mistakes. And maybe some have made the mistake of trying to use ZFC to talk about physical causation. But few, if any, haven’t “noticed the difference in style between the relation between logical axioms and logical models versus causal laws and causal processes.” That just isn’t among the vast catalogue of their errors.
Is it simply that: causal graphs (can) have locality, and you can perform counterfactual surgery on intermediate nodes and get meaningful results, while logic has no locality and (without the hoped-for theory of impossible possible worlds) you can’t contradict one theorem without the system exploding?
Perhaps, but irrelevant, because I’m not what you would call a mathematical-existence type.
ETA: The point is that you can’t be confident about what thought stands behind the sentence “Pearl’s causal networks can be encoded in ZFC” until you have some familiarity with how the speaker thinks. On what basis do you claim that familiarity?