You might as well ask why all the people wondering if all mathematical objects exist, haven’t noticed the difference in style between the relation between logical axioms and logical models versus causal laws and causal processes.
If you’d read much work by modern mathematical platonists, you’d know that many of them obsess over such differences, at least in the analytical school. (Not that it’s worth your time to read such work. You don’t need to do that to infer that they are likely wrong in their conclusions. But not reading it means that you aren’t in a position to declare confidently how “all” of them think.)
Interesting. I wonder if you’ve misinterpreted me or if there’s actually someone competent out there? Quick example if possible?
Heh, false dilemma, I’m afraid :). My only point was that modern platonists aren’t making the mistake that you described. They still make plenty of other mistakes.
Mathematical platonists are “incompetent” in the sense that they draw incorrect conclusions (e.g., mathematical platonism). In fact, all philosophers of mathematics whom I’ve read, even the non-platonists, make the mistake of thinking that physical facts are contingent in some objective sense in which mathematical facts are not. Not that this is believed unanimously. For example, I gather that John Stewart Mill held that mathematical facts are no more necessary than physical ones, but I haven’t read him, so I don’t know the details of his view.
But all mathematical philosophers whom I know recognize that logical relations are different from causal relations. They realize that Euclid’s axioms “make” the angles in ideal triangles sum to 180 degrees in a manner very different from how the laws of physics make a window break when a brick hits it. For example, mathematical platonists might say (mistakenly) that every mathematically possible object exists, but not every physically possible object exists.
Another key difference for the platonist is that causal relations don’t hold among mathematical objects, or between mathematical objects and physical objects. They recognize that they have a special burden to explain how we can know about mathematical objects if we can’t have any causal interaction with them.
I’d appreciate it if you write down your positions explicitly, even if in one-sentence form, rather than implying that so-and-so position is wrong because [exercise to the reader]. These are difficult questions, so even communicating what you mean is non-trivial, not even talking about convincing arguments and rigorous formulations.
That’s fair. I wrote something about my own position here.
Here is what I called mistaken, and why:
Mathematical platonism: I believe that we can’t know about something unless we can interact with it causally.
The belief that physical facts are contingent: I believe that this is just an example of the mind projection fallacy. A fact is contingent only with respect to a theory. In particular, the fact is contingent if the theory neither predicts that it must be the case nor that it must not be the case. Things are not contingent in themselves, independently of our theorizing. They just are. To say that something is contingent, like saying that it is surprising, is to say something about our state of knowledge. Hence, to attribute contingency to things in themselves is to commit the mind projection fallacy.
My interest is piqued as well. You appear to be articulating a position that I’ve encountered on Less Wrong before, and that I would like to understand better.
So physical facts are not contingent. All of them just happen to be independently false or true? What then is the status of a theory?
I’m speculating.. perhaps you consider that there is a huge space of possible logical and consistent theories, one for every (independent) fact being true or false. (For example, if there are N statements about the physical universe, 2^N theories.) Of course, they are all completely relatively arbitrary. As we learn about the universe, we pick among theories that happen to explain all the facts that we know of (and we have preferences for theories that do so in ever simpler ways.) Then, any new fact may require updating to a new theory, or may be consistent with the current one. So theories are arbitrary but useful. Is this consistent with what you are saying?
Thank you. I apologize if I’ve misinterpreted—I suspect the inferential distance between our views is quite great.
Let me start with my slogan-version of my brand of realism: “Things are a certain way. They are not some other way.”
I’ll admit up front the limits of this slogan. It fails to address at least the following: (1) What are these “things” that are a certain way? (2) What is a “way”, of which “things are” one? In particular (3) what is the ontological status of the other ways aside from the “certain way” that “things are”? I don’t have fully satisfactory answers to these questions. But the following might make my meaning somewhat more clear.
To your questions:
So physical facts are not contingent. All of them just happen to be independently false or true?
First, let me clear up a possible confusion. I’m using “contingent” in the sense of “not necessarily true or necessarily false”. I’m not using it in the sense of “dependent on something else”. That said, I take independence, like contingency, to be a theory-relative term. Things just are as they are. In and of themselves, there are no relations of dependence or independence among them.
What then is the status of a theory?
Theories are mechanisms for generating assertions about how things are or would be under various conditions. A theory can be more or less wrong depending on the accuracy of the assertions that it generates.
Theories are not mere lists of assertions (or “facts”). All theories that I know of induce a structure of dependency among their assertions. That structure is a product of the theory, though. (And this relation between the structure and the theory is itself a product of my theory of theories, and so on.)
I should try to clarify what I mean by a “dependency”. I mean something like logical dependency. I mean the relation that holds between two statements, P and Q, when we say “The reason that P is true is because Q is true”.
Not all notions of “dependency” are theory-dependent in this sense. I believe that “the way things are” can be analyzed into pieces, and these pieces objectively stand in certain relations with one another. To give a prosaic example. The cup in front of me is really there, the table in front of me is really there, and the cup really sits in the relation of “being on” the table. If a cat knocks the cup off the table, an objective relation of causation will exist between the cat’s pushing the cup and the cup’s falling off the table. All this would be the case without my theorizing. These are facts about the way things are. We need a theory to know them, but they aren’t mere features of our theory.
Checking these references doesn’t show the distinction I was thinking of between the mathematical form of first-order or higher-order logic and model theory, versus causality a la Pearl.
So, is your complaint just that they use the same formalism to talk about logical relations and causal relations? Or even just that they don’t use the same two specific formalisms that you use?
That seems to me like a red herring. Pearl’s causal networks can be encoded in ZFC. Conversely, ZFC can be talked about using various kinds of decorated networks—that’s what category theory is. Using the same formalism for the two different kinds of relations should only be a problem if it leads one to ignore the differences between them. As I tried to show above, philosophers of mathematics aren’t making this mistake in general. They are keenly aware of differences between logical relations and causal relations. In fact, many would point to differences that don’t, in my view, actually exist.
And besides, I don’t get the impression that philosophers these days consider nth- order logic to be the formalism for physical explanations. As mentioned on the Wikipedia page for the deductive-nomological model, it doesn’t hold the dominant position that it once had.
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?
If you’d read much work by modern mathematical platonists, you’d know that many of them obsess over such differences, at least in the analytical school. (Not that it’s worth your time to read such work. You don’t need to do that to infer that they are likely wrong in their conclusions. But not reading it means that you aren’t in a position to declare confidently how “all” of them think.)
Interesting. I wonder if you’ve misinterpreted me or if there’s actually someone competent out there? Quick example if possible?
Heh, false dilemma, I’m afraid :). My only point was that modern platonists aren’t making the mistake that you described. They still make plenty of other mistakes.
Mathematical platonists are “incompetent” in the sense that they draw incorrect conclusions (e.g., mathematical platonism). In fact, all philosophers of mathematics whom I’ve read, even the non-platonists, make the mistake of thinking that physical facts are contingent in some objective sense in which mathematical facts are not. Not that this is believed unanimously. For example, I gather that John Stewart Mill held that mathematical facts are no more necessary than physical ones, but I haven’t read him, so I don’t know the details of his view.
But all mathematical philosophers whom I know recognize that logical relations are different from causal relations. They realize that Euclid’s axioms “make” the angles in ideal triangles sum to 180 degrees in a manner very different from how the laws of physics make a window break when a brick hits it. For example, mathematical platonists might say (mistakenly) that every mathematically possible object exists, but not every physically possible object exists.
Another key difference for the platonist is that causal relations don’t hold among mathematical objects, or between mathematical objects and physical objects. They recognize that they have a special burden to explain how we can know about mathematical objects if we can’t have any causal interaction with them.
http://plato.stanford.edu/entries/platonism-mathematics/#WhaMatPla http://plato.stanford.edu/entries/abstract-objects/#5
I’d appreciate it if you write down your positions explicitly, even if in one-sentence form, rather than implying that so-and-so position is wrong because [exercise to the reader]. These are difficult questions, so even communicating what you mean is non-trivial, not even talking about convincing arguments and rigorous formulations.
That’s fair. I wrote something about my own position here.
Here is what I called mistaken, and why:
Mathematical platonism: I believe that we can’t know about something unless we can interact with it causally.
The belief that physical facts are contingent: I believe that this is just an example of the mind projection fallacy. A fact is contingent only with respect to a theory. In particular, the fact is contingent if the theory neither predicts that it must be the case nor that it must not be the case. Things are not contingent in themselves, independently of our theorizing. They just are. To say that something is contingent, like saying that it is surprising, is to say something about our state of knowledge. Hence, to attribute contingency to things in themselves is to commit the mind projection fallacy.
My interest is piqued as well. You appear to be articulating a position that I’ve encountered on Less Wrong before, and that I would like to understand better.
So physical facts are not contingent. All of them just happen to be independently false or true? What then is the status of a theory?
I’m speculating.. perhaps you consider that there is a huge space of possible logical and consistent theories, one for every (independent) fact being true or false. (For example, if there are N statements about the physical universe, 2^N theories.) Of course, they are all completely relatively arbitrary. As we learn about the universe, we pick among theories that happen to explain all the facts that we know of (and we have preferences for theories that do so in ever simpler ways.) Then, any new fact may require updating to a new theory, or may be consistent with the current one. So theories are arbitrary but useful. Is this consistent with what you are saying?
Thank you. I apologize if I’ve misinterpreted—I suspect the inferential distance between our views is quite great.
Let me start with my slogan-version of my brand of realism: “Things are a certain way. They are not some other way.”
I’ll admit up front the limits of this slogan. It fails to address at least the following: (1) What are these “things” that are a certain way? (2) What is a “way”, of which “things are” one? In particular (3) what is the ontological status of the other ways aside from the “certain way” that “things are”? I don’t have fully satisfactory answers to these questions. But the following might make my meaning somewhat more clear.
To your questions:
First, let me clear up a possible confusion. I’m using “contingent” in the sense of “not necessarily true or necessarily false”. I’m not using it in the sense of “dependent on something else”. That said, I take independence, like contingency, to be a theory-relative term. Things just are as they are. In and of themselves, there are no relations of dependence or independence among them.
Theories are mechanisms for generating assertions about how things are or would be under various conditions. A theory can be more or less wrong depending on the accuracy of the assertions that it generates.
Theories are not mere lists of assertions (or “facts”). All theories that I know of induce a structure of dependency among their assertions. That structure is a product of the theory, though. (And this relation between the structure and the theory is itself a product of my theory of theories, and so on.)
I should try to clarify what I mean by a “dependency”. I mean something like logical dependency. I mean the relation that holds between two statements, P and Q, when we say “The reason that P is true is because Q is true”.
Not all notions of “dependency” are theory-dependent in this sense. I believe that “the way things are” can be analyzed into pieces, and these pieces objectively stand in certain relations with one another. To give a prosaic example. The cup in front of me is really there, the table in front of me is really there, and the cup really sits in the relation of “being on” the table. If a cat knocks the cup off the table, an objective relation of causation will exist between the cat’s pushing the cup and the cup’s falling off the table. All this would be the case without my theorizing. These are facts about the way things are. We need a theory to know them, but they aren’t mere features of our theory.
Checking these references doesn’t show the distinction I was thinking of between the mathematical form of first-order or higher-order logic and model theory, versus causality a la Pearl.
So, is your complaint just that they use the same formalism to talk about logical relations and causal relations? Or even just that they don’t use the same two specific formalisms that you use?
That seems to me like a red herring. Pearl’s causal networks can be encoded in ZFC. Conversely, ZFC can be talked about using various kinds of decorated networks—that’s what category theory is. Using the same formalism for the two different kinds of relations should only be a problem if it leads one to ignore the differences between them. As I tried to show above, philosophers of mathematics aren’t making this mistake in general. They are keenly aware of differences between logical relations and causal relations. In fact, many would point to differences that don’t, in my view, actually exist.
And besides, I don’t get the impression that philosophers these days consider nth- order logic to be the formalism for physical explanations. As mentioned on the Wikipedia page for the deductive-nomological model, it doesn’t hold the dominant position that it once had.
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?