if you’re a logic freak it trains you to be comfortable with the insufficiency of symbols, memes and jokes of mathematicians and physicists disagreeing about notation reveals an underlying controversy about the question “what are symbols actually for?”
I don’t think this is an accurate description of the cultural difference between physicists and mathematicians. Tiny respective minorities of die-hard instrumentalists and formalists aside, both fields agree that the symbols are just tools for talking about the relevant objects of study more clearly and concisely than natural language permits. Plenty of published math research is formally incorrect in an extremely strong sense, but no one cares as long as all the errors can be corrected trivially. In fact, an important aspect of what’s often called “mathematical maturity” is the ability to make those corrections automatically and mostly unconsciously, instead of either falling into genuine sloppiness or getting hung up on every little “the the”.
The real core difference is the obvious one. To zeroth order: physicists study physics, and mathematicians study math. To first order: physicists characterize phenomena which definitely exist, mathematicians characterize structures which may or may not.
The universe definitely exists, it definitely has a structure, and any method which reliably makes correct predictions reflects a genuine aspect of that structure, whatever it might be. Put another way: physicists have an oracle for consistency. Mathematicians don’t have that option, because structures are the things they study. That’s what makes them mathematicians, and not physicists. They can retreat to higher and higher orders, and study classes of theories of logics for …, but the regress has to stop somewhere, and the place it stops has to stand on its own, because there’s no outside model to bear witness to its consistency.
If all known calculations of the electron mass rely on some nonsensical step like “let d = 4 - epsilon where d is the dimensionality of spacetime”, then this just means we haven’t found the right structure yet. The electron mass is what it is, and the calculation is accurate or it isn’t. But if all known “proofs” of a result rely on a nonsensical lemma, then it is a live possibility that the result is false. Physics would look very different if physicists had to worry about whether or not there was such a thing as mass. Math would look very different if mathematicians had a halting oracle.
I don’t think this is an accurate description of the cultural difference between physicists and mathematicians. Tiny respective minorities of die-hard instrumentalists and formalists aside, both fields agree that the symbols are just tools for talking about the relevant objects of study more clearly and concisely than natural language permits. Plenty of published math research is formally incorrect in an extremely strong sense, but no one cares as long as all the errors can be corrected trivially. In fact, an important aspect of what’s often called “mathematical maturity” is the ability to make those corrections automatically and mostly unconsciously, instead of either falling into genuine sloppiness or getting hung up on every little “the the”.
The real core difference is the obvious one. To zeroth order: physicists study physics, and mathematicians study math. To first order: physicists characterize phenomena which definitely exist, mathematicians characterize structures which may or may not.
The universe definitely exists, it definitely has a structure, and any method which reliably makes correct predictions reflects a genuine aspect of that structure, whatever it might be. Put another way: physicists have an oracle for consistency. Mathematicians don’t have that option, because structures are the things they study. That’s what makes them mathematicians, and not physicists. They can retreat to higher and higher orders, and study classes of theories of logics for …, but the regress has to stop somewhere, and the place it stops has to stand on its own, because there’s no outside model to bear witness to its consistency.
If all known calculations of the electron mass rely on some nonsensical step like “let d = 4 - epsilon where d is the dimensionality of spacetime”, then this just means we haven’t found the right structure yet. The electron mass is what it is, and the calculation is accurate or it isn’t. But if all known “proofs” of a result rely on a nonsensical lemma, then it is a live possibility that the result is false. Physics would look very different if physicists had to worry about whether or not there was such a thing as mass. Math would look very different if mathematicians had a halting oracle.