This seems especially difficult noting that although we can claim that things are caused by certain mathematical truths, it doesn’t really make sense to include them in our Bayesian net unless we could say, for example, how anything else would be different if 2+2=3.
Well I know that when I drop something the distance it falls after time t is roughly 1⁄2 g t^2 where g~10 m/s^2. When I drop something off of a 20m high building, I can reasonably claim that the fact that it takes roughly 2s to reach the ground is a consequence of the above, and of the mathematical truth that 1⁄2 10 2^2 = 20.
That’s a physcial truth, and a local one at that. The mathematical expression of a physcial fact is not a “mathematocal truth” because, pace Tegmark, most mathematical truths don’t model physical facts. What casues objects to fall is gravity. Maths does not cause it, any more than words do.
I agree that mathematical truths do not have effects on their own. But when combined with mathematical formulations of laws of reality they do have observable consequences. The timing of a falling projectile above is a consequence of both a mathematical formulation of the law of gravity and a purely mathematical arithmetical fact. If you somehow want to describe the universe without mathematics, good luck.
An event can have more than one cause. My uncertainty about the value of some variable in an equation is related to my uncertainty about the outcome of an experiment in exactly the way that makes Pearlean methods tell me that both the value of t in the equation and the physical truth that g ≈ 10 m/s^2 are causes of the amount of time that the object takes to fall. This is just a fact about my state of uncertainty that falls directly out of the math.
Sorry, I was unclear. I meant that the causal structure where the equations of physics cause the outcome of the experiment falls out of the Pearlean causal math, not that the outcome of the experiment falls out of the physical math (though the latter is of course also true).
I’m not sure what you mean. Pearlean causality, as I understand it, is about maps. You put in a subjective probability distribution and a few assumptions and a causal structure comes out.
If it’s only about maps, its doubtful whether it deseres to be called causality.
ETA:
Indeed, EY seems to take the view that correlation is in the map, and causation in the territory:-
“More generally, for me to expect your beliefs to correlate with reality, I have to either think that reality is the cause of your beliefs, expect your beliefs to alter reality, or believe that some third factor is influencing both of them.
This is the more general argument that “To draw an accurate map of a city, you have to open the blinds and look out the window and draw lines on paper corresponding to what you see; sitting in your living-room with the blinds closed, making stuff up, isn’t going to work.”
Correlation requires causal interaction; and expecting beliefs to be true means expecting the map to correlate with the territory. ”
I’m not sure what Eliezer believes here; the way he talks about causality is why I added the “as I understand it” to the grandparent.
I think that even if it is just in the map we should still call it causality. It’s a useful concept and it is similar to the concept of causality we had before we understood these insights. It’s expected that when you come to understand something better it won’t turn out behave exactly the way you thought it did.
I don’t see any insight that causality is on the map not the territory. I think EY’s overall point is that supernatural claims can be assesed by the same episetmology as scientific ones.
Most mathematics has isomorphism to typographical or computational rules. I’m pretty sure these can be encoded into a causal diagram which connects with the real world.
Mathematics is a mental construct created to reliably manipulate abstract concepts. You can describe mathematical statements as elements of the mental models of intelligent beings. A mathematical statement can be considered “true” if, when an intelligent beings use the statement in their reasoning, their predictive power increases. Thus, ” ‘4+4=8’ is true” implies statements like “jslocum’s model of arithmetic predicts that ‘4+4=8’, which causes him to correctly predict that if he adds four carrots to his basket of four potatoes, he’ll have eight vegetables in his basket”
I’m no sure that “use the statement in their reasoning” and “their predictive power increases” are well formed concepts, though, so this might need some refining.
I find mathematics most about future physics laws who will be discovered. Math without empirical confirmation is more difficult to link, but normally is a matter of time to find a application.
You think some future experimental results will say something meaningful about whether mathematics should accept the axiom of choice? Even if the universe is inconsistent with ZFC, why does that imply studying ZFC based mathematics should stop?
Koan 4: How well do mathematical truths fit into this rule of defining what sort of things can be meaningful?
This seems especially difficult noting that although we can claim that things are caused by certain mathematical truths, it doesn’t really make sense to include them in our Bayesian net unless we could say, for example, how anything else would be different if 2+2=3.
What sort of things?
Well I know that when I drop something the distance it falls after time t is roughly 1⁄2 g t^2 where g~10 m/s^2. When I drop something off of a 20m high building, I can reasonably claim that the fact that it takes roughly 2s to reach the ground is a consequence of the above, and of the mathematical truth that 1⁄2 10 2^2 = 20.
That’s a physcial truth, and a local one at that. The mathematical expression of a physcial fact is not a “mathematocal truth” because, pace Tegmark, most mathematical truths don’t model physical facts. What casues objects to fall is gravity. Maths does not cause it, any more than words do.
I agree that mathematical truths do not have effects on their own. But when combined with mathematical formulations of laws of reality they do have observable consequences. The timing of a falling projectile above is a consequence of both a mathematical formulation of the law of gravity and a purely mathematical arithmetical fact. If you somehow want to describe the universe without mathematics, good luck.
An event can have more than one cause. My uncertainty about the value of some variable in an equation is related to my uncertainty about the outcome of an experiment in exactly the way that makes Pearlean methods tell me that both the value of t in the equation and the physical truth that g ≈ 10 m/s^2 are causes of the amount of time that the object takes to fall. This is just a fact about my state of uncertainty that falls directly out of the math.
“Falls out of the math” doens’t mean “caused by math” any more than “expressed in math” means “caused by math”.
Sorry, I was unclear. I meant that the causal structure where the equations of physics cause the outcome of the experiment falls out of the Pearlean causal math, not that the outcome of the experiment falls out of the physical math (though the latter is of course also true).
I think that still has the same problem. The (edit:) math is the map, causes are in the territtory.
I’m not sure what you mean. Pearlean causality, as I understand it, is about maps. You put in a subjective probability distribution and a few assumptions and a causal structure comes out.
If it’s only about maps, its doubtful whether it deseres to be called causality.
ETA:
Indeed, EY seems to take the view that correlation is in the map, and causation in the territory:-
“More generally, for me to expect your beliefs to correlate with reality, I have to either think that reality is the cause of your beliefs, expect your beliefs to alter reality, or believe that some third factor is influencing both of them.
This is the more general argument that “To draw an accurate map of a city, you have to open the blinds and look out the window and draw lines on paper corresponding to what you see; sitting in your living-room with the blinds closed, making stuff up, isn’t going to work.”
Correlation requires causal interaction; and expecting beliefs to be true means expecting the map to correlate with the territory. ”
I’m not sure what Eliezer believes here; the way he talks about causality is why I added the “as I understand it” to the grandparent.
I think that even if it is just in the map we should still call it causality. It’s a useful concept and it is similar to the concept of causality we had before we understood these insights. It’s expected that when you come to understand something better it won’t turn out behave exactly the way you thought it did.
I don’t see any insight that causality is on the map not the territory. I think EY’s overall point is that supernatural claims can be assesed by the same episetmology as scientific ones.
Most mathematics has isomorphism to typographical or computational rules. I’m pretty sure these can be encoded into a causal diagram which connects with the real world.
Do you mean that some of those strings are useful in modeling the “real world”? Say, by providing a way to discover further causal diagrams?
Not necessarily. I’ve realized I’m more confused than I thought I was.
Mathematics is a mental construct created to reliably manipulate abstract concepts. You can describe mathematical statements as elements of the mental models of intelligent beings. A mathematical statement can be considered “true” if, when an intelligent beings use the statement in their reasoning, their predictive power increases. Thus, ” ‘4+4=8’ is true” implies statements like “jslocum’s model of arithmetic predicts that ‘4+4=8’, which causes him to correctly predict that if he adds four carrots to his basket of four potatoes, he’ll have eight vegetables in his basket”
I’m no sure that “use the statement in their reasoning” and “their predictive power increases” are well formed concepts, though, so this might need some refining.
I find mathematics most about future physics laws who will be discovered. Math without empirical confirmation is more difficult to link, but normally is a matter of time to find a application.
Hrm?
You think some future experimental results will say something meaningful about whether mathematics should accept the axiom of choice? Even if the universe is inconsistent with ZFC, why does that imply studying ZFC based mathematics should stop?