My experience as a math major is that most non-mathematicians, including engineering and chem/phys professors, have a working understand of math and don't think too deep about it.
> Is there some kind of axiomatic basis for how these symbols can be manipulated in a consistent way?
Yes.
> This was a guy whose research career was built on dozens of computational chemistry papers that heavily utilized DFT by the way (and by “research”, I mean plug a file with atom coordinates and atom types into the DFT software, run the software, and publish the results).
Again, large portions of academia simply stick to their small subfield, publish there, and don't think much about the broader implications, if they think about them at all. This is why so many discoveries are made by new entrants to a field with a slightly different background.
> Is there some kind of axiomatic basis for how these symbols can be manipulated in a consistent way?
Yes.
> This was a guy whose research career was built on dozens of computational chemistry papers that heavily utilized DFT by the way (and by “research”, I mean plug a file with atom coordinates and atom types into the DFT software, run the software, and publish the results).
Again, large portions of academia simply stick to their small subfield, publish there, and don't think much about the broader implications, if they think about them at all. This is why so many discoveries are made by new entrants to a field with a slightly different background.