The syllogism “All cups are green; Socrates is a cup; therefore Socrates is green” is valid within the standard system of logic, but it doesn’t correspond to anything meaningful. But the reason that we view logic as more than a curiosity is that we can use logic and true premises to reach true conclusions. Logic is useful because it produces true beliefs.
Some mathematical statements follow the rules of math; we call them valid, and they would be just as valid in any other universe. Math as a system is useful because (in our universe) we can use mathematical models to arrive at predictively accurate conclusions.
The distinction between true and valid in this sense seems useful. In practice, my first interpretation says that “5324+2326=7650” is true, while the second says it is valid.
I’d rather say it conserves true beliefs that were put into the system at the start, but these were, in turn, produced inductively.
[math statements] would be just as valid in any other universe
I’ve often heard this bit of conventional wisdom but I’m not totally convinced it’s actually true. How would we even know?
Well, what if in some other universe every process isomorphic to a statement “2 + 2” concludes that it equals “3“ instead of “4”—would this mean that the abstract fact “2 + 2 = 4” is false/invalid in that universe?
As far as I can see, this boils down to a question about where are these abstract mathematical facts stored, or perhaps, what controls these facts if not the deep physical laws of the universe that contains the calculators that try to discern these facts...
Perhaps to say “valid in any other universe” is a language mismatch, since validity doesn’t refer to universes, only to the rules of the system.
Well, what if in some other universe every process isomorphic to a statement “2 + 2” concludes that it equals “3“ instead of “4”—would this mean that the abstract fact “2 + 2 = 4” is false/invalid in that universe?
If it concluded 3 instead of 4, it would not be isomorphic to our “2+2”. Both systems of mathematics (in our and the other universes) have to be isomorphic as a whole to enable translation between them.
Since being introduced to Less Wrong and clarifying that ‘truth’ is a property of beliefs corresponding to how accurately they let you predict the world, I’ve separated ‘validity’ from ‘truth’.
The syllogism “All cups are green; Socrates is a cup; therefore Socrates is green” is valid within the standard system of logic, but it doesn’t correspond to anything meaningful. But the reason that we view logic as more than a curiosity is that we can use logic and true premises to reach true conclusions. Logic is useful because it produces true beliefs.
Some mathematical statements follow the rules of math; we call them valid, and they would be just as valid in any other universe. Math as a system is useful because (in our universe) we can use mathematical models to arrive at predictively accurate conclusions.
Bringing ‘truth’ into it is just confusing.
The distinction between true and valid in this sense seems useful. In practice, my first interpretation says that “5324+2326=7650” is true, while the second says it is valid.
I’d rather say it conserves true beliefs that were put into the system at the start, but these were, in turn, produced inductively.
I’ve often heard this bit of conventional wisdom but I’m not totally convinced it’s actually true. How would we even know?
Well, what if in some other universe every process isomorphic to a statement “2 + 2” concludes that it equals “3“ instead of “4”—would this mean that the abstract fact “2 + 2 = 4” is false/invalid in that universe?
As far as I can see, this boils down to a question about where are these abstract mathematical facts stored, or perhaps, what controls these facts if not the deep physical laws of the universe that contains the calculators that try to discern these facts...
Perhaps to say “valid in any other universe” is a language mismatch, since validity doesn’t refer to universes, only to the rules of the system.
If it concluded 3 instead of 4, it would not be isomorphic to our “2+2”. Both systems of mathematics (in our and the other universes) have to be isomorphic as a whole to enable translation between them.