I remember my first calculus exam at the University: I made a lot of explanations and justifications based of physics. Of course I did not pass. As I asked why? It was all correct, right? The prof. said “is perfect reasoning, but this is the math department, physics is one floor below”
Yeah it's not rigorous. The tragedy is that the organizations responsible for teaching math, like how to do it and how to understand things mathematically, for other purposes, would rather waste their time on rigor.
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.
I quite strongly disagree with this. Mathematicians aren't rigourous just for the hell of it. We have examples where a lack of rigour caused a bunch of people to waste a bunch of time working of stuff that ultimately didn't work at all.
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.
On the other hand, if you want full and complete rigor that means formalizing things in a computer-based proof checker. Now, that can often reveal ways to refactor the argument into a cleaner, more elegant, more intuitive proof, which is why full formalizations are often regarded as publishable work - but the other side of it is that dotting all the i's and crossing all the t's does take a whole lot of effort since some published proofs are not nearly as clear as they're supposed to be.
I agree that some published proofs are not as clear as they're supposed to be, but I disagree that formalizing things in computer-based proof checkers is the way forward there. Computer proof checking is cool, but its an augment to what we already do and not not a substitute for it.
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.
> No problem with there being a rigorous math department. For mathematicians. But, quite literally, no one else cares
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.
That's just false. What physicists (and engineers, etc) are doing is also math and it's way more useful and insightful.. and (imo) the world would be a better place if that's what everyone else was also learning in college. The idea that math is only about rigor isn't intrinsic; it's a historical accident that people seem to not realize is optional. It is also about understanding things and being able to wield concepts to accomplish goals or convey understanding to others.
Novelists and lawyers both do writing, but I wouldn't call their output the same thing. You might call legal writing more useful or novelists more insightful, but that's a matter of opinion. One is not intrinsically more valuable than the other.
> 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.
When schools focus on only teaching the most rigorous and abstract definition of math, at the cost of teaching students how to apply math to real-world problems, the result is a lot of students who can do neither.
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.
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.
Then shouldn't the professor have said that it isn't rigorous? Saying it's perfect implies that it is rigorous, just that they took issue with the association.
If some mathematics are not applicable to physical realities, to what are they applicable? Is there any un-applicable math and if so, why does it even exist?
Maybe depends on what's meant by application, but I'd say a lot of, probably most, research-level math has no (clear) practical applications and most research is not done from the PoV of applications or relationship to physical reality. Math is study of formal systems, sort of "what can and can't follow from some set of formal rules".
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.
The biggest example I can think of is Fermat's Last Theorem. Uncounted mathematicians devoted years of their lives to the theorem, and the developments they made along the way may have some applications. But I have yet to hear of any physical truths derived from the fact that a^n + b^n != c^n for n > 2. Most math is like this - beautiful, and useless.
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.
I may be wrong but this seems inaccurate. In WWII, the cryptanalysis of the Enigma (https://en.wikipedia.org/wiki/Cryptanalysis_of_the_Enigma) was related not to number theory, but to group theory. Number theory became relevant to cryptography only since the 1970’s/1980’s when Diffie and Hellman published their paper on public-key cryptography. Although number theory was related to cryptography in the past (e.g. in shift ciphers and block cyphers), it was the Diffie-Hellman paper that placed number theory in the central role.
Pretty amazing that someone discovered and explored number theory without knowing if there would be an application to it. Hopefully they lived to see its usage.
Number theory was a big deal already in ancient greece and probably got there from Babylonia, so it's probably safe to say that they didn't live to see the public key crypto usage. :)
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.
And they were right imho. Having a clear notion of a subject is good for both physics and mathematics, but what's good for physics or engineering, is not enough for math, in math, you also need rigorous reasoning, or you'll fall into subtle mistakes.
