Some areas of mathematics are applicable to physical realities, but they aren’t defined by those realities.
No problem with there being a rigorous math department. For mathematicians. But, quite literally, no one else cares. Mathematicians and everyone else are at crossed purposes, and the mathematicians are wrong to choose their purposes over their students' needs.
Uh... Not that I'm defending the previous poster. Just griping about rigor in general. Physical arguments for mathematical problems are rarely sound, and I'd assume that an undergraduate who thinks they have made a sound one is wrong until reading it to see.
For example the Italian school of algebraic geometry caused the waste of legitimately decades of work by many people because the foundations weren't right: https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge....
Part of the purpose of a maths degree is to teach the students the rigour required to be a mathematician. If you don't want to learn that rigour then thats completely fine, you can use physical or intuitive arguments, but the place to go and do that is in the physics or engineering departments.
I agree mathematicians would be wrong if they were forcing their rigour on physics students or whatever, but I think they're emphatically right to teach it to maths students.
The purpose of a published paper is to explain things to other humans, while that of a computer formalization is to explain things to computers. Just as it is generally not easy for a computer to understand a published proof, it is usually not easy for other mathematicians to understand the code you feed to Lean or whatever.
The stuff you need to focus on to get a computer to accept your proof is usually quite different to the stuff you want to focus on when explaining things to colleagues. Roughly speaking we care about why you're doing things, while the computer cares about the intricacies of what you're doing.
As someone who did an undergrad in mathematics, I have to disagree with you here. People think math is about numbers and computation, but that's like saying literature is about letters and composition. Fundamentally, mathematics is the study of things that are provably true. Without rigor it's not math.
> The idea that math is only about rigor isn't intrinsic; it's a historical accident that people seem to not realize is optional
I mean, it is by definition. To a mathematician, doing calculations is not mathematics, no more than spelling is doing poetry. Which is not to say that doing calculations is without value! I think what we have here is (ironically) an unrigorous definition.
I hope you realize that physics isn't about calculations either? You can put numbers or types in the equations, just like you can put numbers or types in the expressions or results you get in math.
Likewise, when schools focus only on teaching literature by the most artistic definition, at the cost of teaching them basic day-to-day reading and writing skills, the result is a lot of students who can do neither.
Let's treat rigorous math and lofty literature like the specialized skills that they are, and offer them to students who show particular interest in those areas. For the bulk of students, let's teach them skills that will be useful and relevant.
Why would you study such trueness if whether you know it is true or not does not have any added value in the physical world?
It's more a surprise that some rulesets map to physical world as well as they do [1].
For some there may be very important ones in the future like e.g. number theory got in cryptography. But for example just knowing whether P=?NP or if the Riemann hypothesis is true probably has no "physical reality" direct applications.
[1] https://en.m.wikipedia.org/wiki/The_Unreasonable_Effectivene...
But (integer) numbers and their behavior had huge significance in the ancient worldview, and still do even in our days if you look deep enough. And the Pythagoreans et al applied number theory in e.g. music.
The motivation of most non-applied mathematicians is aesthetic, much like the motivation of someone in the arts. It's done for its own sake ("mathematical beauty"), not because it has any practical application.
Mathematics might turn out to be "useful" more often than the average novel - but that doesn't mean the math was done with any use in mind. For example many developments in differential geometry made in the 19th century were inapplicable to anything physical for decades until Einstein applied them to create general relativity. Plenty of math becomes useful long after its development, and plenty more never finds any practical application.