“But my dear sir, if the fact of 2 + 3 = 5 exists somewhere outside your brain… then where is it?”
The truth-condition for “There are five sheep in the meadow” concerns the state of the meadow.
My guess is that the truth condition for “2 + 3 = 5” concerns the (more complex, but unproblematically material) set of facts you present: the facts that e.g.:
It’s easy to find sheep for which two sheep and three sheep make five sheep
It’s fairly easy to build calculators that model what happens with the sheep
It’s fairly easy to evolve brains that model what happens with the calculators and the sheep.
It’s fairly easy to find “formal mathematical models” that can run on these evolved brains, that model what’s going on with the sheep and the rocks and various other systems, with axioms and rules of inference that can be briefly described in English.
We have good reason to claim that “2+3 = 5” has an existence outside your mind. We have such reason because, as you point out, we see many different material systems that “correlate with each other and successfully predict one another”.
But… do we have any reason to claim that “2+3=5” has an existence outside of these correlations between material systems? My guess is “no”. My guess is that we should say that “2+3=5” is about these correlations. Once we say this, we can go ahead and investigate these correlations the way we’d investigate other aspects of material systems: we can try to spell out just what systems do “correlate with each other and successfully predict one another” in the ways we summarize with addition, and then what systems correlate with one another in the ways we summarize with Euclidean geometry, and then look for the meta-level pattern that unites the two sets of correlations.
These questions about the correlations are interesting and partially unsolved. But my guess is that they aren’t gaps in our understanding of what math is about, just gaps in our understanding of the correlated material processes that math is about. The lines of questioning needed to explain these correlations are different from the lines of questioning that tend to be invoked when someone asks “where” math is.
“But my dear sir, if the fact of 2 + 3 = 5 exists somewhere outside your brain… then where is it?”
The truth-condition for “There are five sheep in the meadow” concerns the state of the meadow.
My guess is that the truth condition for “2 + 3 = 5” concerns the (more complex, but unproblematically material) set of facts you present: the facts that e.g.: It’s easy to find sheep for which two sheep and three sheep make five sheep It’s fairly easy to build calculators that model what happens with the sheep It’s fairly easy to evolve brains that model what happens with the calculators and the sheep. It’s fairly easy to find “formal mathematical models” that can run on these evolved brains, that model what’s going on with the sheep and the rocks and various other systems, with axioms and rules of inference that can be briefly described in English.
We have good reason to claim that “2+3 = 5” has an existence outside your mind. We have such reason because, as you point out, we see many different material systems that “correlate with each other and successfully predict one another”.
But… do we have any reason to claim that “2+3=5” has an existence outside of these correlations between material systems? My guess is “no”. My guess is that we should say that “2+3=5” is about these correlations. Once we say this, we can go ahead and investigate these correlations the way we’d investigate other aspects of material systems: we can try to spell out just what systems do “correlate with each other and successfully predict one another” in the ways we summarize with addition, and then what systems correlate with one another in the ways we summarize with Euclidean geometry, and then look for the meta-level pattern that unites the two sets of correlations.
These questions about the correlations are interesting and partially unsolved. But my guess is that they aren’t gaps in our understanding of what math is about, just gaps in our understanding of the correlated material processes that math is about. The lines of questioning needed to explain these correlations are different from the lines of questioning that tend to be invoked when someone asks “where” math is.