It seems Skatche is roughly talking about what is called semantic entailment in logic and I would partially agree with your criticism… since mathematical truth is considered to be more than just that, it includes axioms that you feel good about accepting. However, I’m not sure where reality comes into the picture when considering the definition of mathematical truth.
How do we choose axioms that are good to accept as valid except for our experience with reality? This is a philosophical question so isn’t often considered within mathematics. If one reads some of Eliezer’s posts where he doubts the axiom of infinity (unless I am misunderstanding what he is doing) then it becomes clear that his argument for doubting the axiom is that it doesn’t correspond with reality.
This is generally why choice is doubted; as it is thought that non-measurable sets do not play any role in reality they are ignored in standard statistics and calculus. If non-measurable sets do play a role within the real world then some very odd things can happen that so far no one has observed happen.
I suppose if mathematical truth is changed to be whatever is provable given all possible combinations of non-contradictory axioms then reality does not play any role in math.
I think when trying to study information and its relation to mathematical truth, we must start off practical and should be talking about provability in formal systems of logic. I don’t actually know of any rigorous connections between the two notions, but I can think of an argument that “mathematical truths contain zero information” might be false based on indirect connections between existing work on proof theory and information theory. But I don’t want to give that interpretation yet because I would like to first ask Skatche if he wanted to elaborate on his statement a little better or point to some references for us to read.
The philosophical question is interesting too, and I would agree that a set theory without the axiom of infinity seems pretty adequate for describing our experiences of reality. I’m not sure if its the most harmonious, however...
It seems Skatche is roughly talking about what is called semantic entailment in logic and I would partially agree with your criticism… since mathematical truth is considered to be more than just that, it includes axioms that you feel good about accepting. However, I’m not sure where reality comes into the picture when considering the definition of mathematical truth.
How do we choose axioms that are good to accept as valid except for our experience with reality? This is a philosophical question so isn’t often considered within mathematics. If one reads some of Eliezer’s posts where he doubts the axiom of infinity (unless I am misunderstanding what he is doing) then it becomes clear that his argument for doubting the axiom is that it doesn’t correspond with reality.
This is generally why choice is doubted; as it is thought that non-measurable sets do not play any role in reality they are ignored in standard statistics and calculus. If non-measurable sets do play a role within the real world then some very odd things can happen that so far no one has observed happen.
I suppose if mathematical truth is changed to be whatever is provable given all possible combinations of non-contradictory axioms then reality does not play any role in math.
I think when trying to study information and its relation to mathematical truth, we must start off practical and should be talking about provability in formal systems of logic. I don’t actually know of any rigorous connections between the two notions, but I can think of an argument that “mathematical truths contain zero information” might be false based on indirect connections between existing work on proof theory and information theory. But I don’t want to give that interpretation yet because I would like to first ask Skatche if he wanted to elaborate on his statement a little better or point to some references for us to read.
The philosophical question is interesting too, and I would agree that a set theory without the axiom of infinity seems pretty adequate for describing our experiences of reality. I’m not sure if its the most harmonious, however...