On its own, mathematics is just the study of various systems of rules for symbol manipulation. The “unreasonable effectiveness of mathematics” in more practical areas like physics allows us to conclude something about the structure of physical law. Whatever the true laws of physics are, we conclude, they seem to be well described by certain systems of rules for symbol manipulation.
In our causal models, this is an arrow between “laws of physics” and “observations about the behavior of collections of objects” [subject to Eliezer’s caveat about a “laws of physics” node]. Once we include this arrow, we can do neat tricks like counting the numbers of rocks in piles A and B, encoding those numbers in symbols, manipulating those symbols, and finally using the result to correctly predict how many rocks we will have once we combine the two piles.
I think you’ve taken EY’s question too literally. The real question is about the status of statements and facts of formal systems (“systems of rules for symbol manipulation”) in general, not arithmetic, specifically.
If you define “mathematics” to include all formal systems, then you can say EY’s meditation is about mathematics.
On its own, mathematics is just the study of various systems of rules for symbol manipulation. The “unreasonable effectiveness of mathematics” in more practical areas like physics allows us to conclude something about the structure of physical law. Whatever the true laws of physics are, we conclude, they seem to be well described by certain systems of rules for symbol manipulation.
In our causal models, this is an arrow between “laws of physics” and “observations about the behavior of collections of objects” [subject to Eliezer’s caveat about a “laws of physics” node]. Once we include this arrow, we can do neat tricks like counting the numbers of rocks in piles A and B, encoding those numbers in symbols, manipulating those symbols, and finally using the result to correctly predict how many rocks we will have once we combine the two piles.
I think you’ve taken EY’s question too literally. The real question is about the status of statements and facts of formal systems (“systems of rules for symbol manipulation”) in general, not arithmetic, specifically. If you define “mathematics” to include all formal systems, then you can say EY’s meditation is about mathematics.