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If when I started learning about calculus at 14/15 someone had explained why we were doing this it would have made it so much less confusing. Explaining in term

• ajkjk 3y ago
  Physics doing the work of actually teaching math is a universal embarrassment to math departments. Or, well, it should be.
  • f1shy 3y ago
    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”
    • DoughnutHole 3y ago
      He’s right - intuitive physical reasoning is useful (and often valid in the applied sciences) but it isn’t rigorous mathematical proof. Some areas of mathematics are applicable to physical realities, but they aren’t defined by those realities.
      • ajkjk 3y ago
        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.
        • eigenket 3y ago
          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. 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_geometry 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.
          • zozbot234 3y ago
            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.
            • eigenket 3y ago
              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.
      • nuancebydefault 3y ago
        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?
        • jampekka 3y ago
          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. [1] https://en.m.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences https://en.m.wikipedia.org/wiki/The_Unreasonable_Effectivene...
          • nuancebydefault 3y ago
            So if the application is not visible today, it might appear tomorrow. In fact that makes sense.
        • sampo 3y ago
          Up until about World War II and use in cryptography, number theory was pretty much an un-applicable field of mathematics.
          • nuancebydefault 3y ago
            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.
            • jampekka 3y ago
      • rendaw 3y ago
        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.
    • monista 3y ago
      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.
      • HelloNurse 3y ago
        Fundamentally, without sufficient rigor there can be no proofs; "maybe" and "someone says so" aren't good theoretical building blocks.
  • regularfry 3y ago
    It's worse than that. I remember that when I was doing physics aged 14/15/16, they weren't allowed to assume that we were taking the maths course that included calculus, so while you could see these repeating patterns crop up (half-thing-squared, you say? How peculiar) they were effectively prevented from taking the time to explain it properly. It would have made things so much simpler.
  • Workaccount2 3y ago
    Many math teahcers (and likely mathematicians) could not care less about the real world applications of math. My college calculus teacher had no interest in physics or science. She liked math in the same way someone would like doing a crossword or Sudoku. A puzzle game with a well defined rule set and the challenge of finding answers. I cannot overstate how destructive this is, because it basically presents math as this arbitrary logic game rather than as a fundamental language of the universe.
    • anon291 3y ago
      > I cannot overstate how destructive this is, because it basically presents math as this arbitrary logic game rather than as a fundamental language of the universe. You are making a clear philosophical assumption yet don't realize it. Is math really the 'fundamental language of the universe'? or is it an arbitrary logic game that, in its most common interpretation, describes the universe well? Given that we have no full formulation of the universe in terms of mathematics (no theory of everything), the claim that mathematics as it is today is the 'fundamental language of the universe' seems laughably wrong. On the other hand, there are thousands of formal mathematical systems that don't describe anything 'real' as far as we can tell. I mean, the entirety of the lambda calculus was developed before there were any computers that could realize any it in any physical system. Yet, its development has proven incredibly important in the entire foundations of mathematics (many theorem provers, etc are based on some variant of lambda calculus). I'm not denying that some people learn calculus, algebra, etc (i.e., standard high-school/early college maths) with physical systems in mind. However, mathematics is much more than that and limiting the field only to that which we can sense is actually detrimental to mathematical progress as a whole. It seems best to me to encourage purely mathematical thinking itself in the hope that the arbitrary systems humans create may one day be used to describe something. History is littered with 'useless' subfields of math later becoming fundamental to the economy (number theory and elliptic curves to cryptography, lambda calculus to computation, group theory to quantum physics, etc)
      • ajkjk 3y ago
        Personally I don't care that it's about physical things. But I care that it's about doing things: understanding concepts intuitively, solving problems, being adept at thinking mathematically. The thing that is not useful is tedious definitions and tedious proofs. A simple example is: nobody needs a proof of how differentiation or integration work. They need to know how to do it, think with it, apply it to problems, and generalize it. The proof is a tool for not being wrong, and that is a perfectly fine thing for mathematicians to do and a waste of everybody else's time (unless they really need convincing). It is an absolute travesty how many people enjoy math until they get to their first college class which is taught by a disconnected mathematician that ruins the subject for them. At my school everyone stopped at multivariable because that's the level at which mathematicians systematically ruined math -- by teaching and testing the wrong stuff. Personally, I survived by the skin of my teeth, and then later found physics and actually learned multivariable calculus and learned to love math again. Despite the efforts of the people who were paid to teach it.
• rudolph9 3y ago
  I had a professor freshman year of college who offered the question “How many times can you split the distance between me and you in half?”. Ultimately I arrived at the conclusion infinity which forever more made the general idea of calculus stick.
• fatnoah 3y ago
  > However discussing a function and talking about splitting in into small (infinitesimal) strips of delta quantities and a list of equations / proofs made it completely dry and uninteresting. This was exactly my experience as well. I spent almost an entire first semester with a C average, mostly because I couldn't grasp what I was doing or why I was doing it. I still remember my "Eureka!!!" moment of sitting in the school library and working on a series of problems about water escaping a pool through an every-widening hole in the side. At that point, it all made sense, and I a) finished with mostly A's and b) was the only one in my class to score a 5 on the AP test later that year. I eventually went on to major in EE with a focus on Signals Processing (including grad school), so I always find it ironic that I went from not getting it all, to basically 8 straight years of nothing but Calculus.
• phkahler 3y ago
  >> I don't think I had any ideas one what was going on until it was mentioned in physics several years later. This is where kids with an engineer parent have an advantage. Such a parent can offer examples of how some of the math is used in the real world. Once relevance is established it's OK to point out that this stuff is often buried in software but someone had to put that math into code form so we can all benefit from it. "You may never use it, but you use tools that use it under the hood" can be motivating when they're wondering if it's all just weird busy work.
• BeetleB 3y ago
  Although I studied calculus in a pure math perspective, I knew enough physics to tie the two concepts together and it may have benefited me some. For the most part, though, all I needed to know was "integration = area under curve" and "differentiation is computing the slope at a point on a curve" and most of the first 3 semesters in calculus made sense from that perspective. However, my concern with your approach is that those examples help only if you are interested in physics (or whatever field those examples come from). I worked at a math tutoring institute for a few years and often saw that physics examples in textbooks like Stewart's confused a lot of students who were not that interested in physics. Not only do they have to learn the math, they now have to learn physics concepts just to understand the examples! There were plenty of students who could do differentiation/integration, but struggled at the questions involving physics. Likewise, the department had a separate calculus course for people in finance, economics, etc. That textbook had applications in finance. All the tutors struggled to help those students because we had to learn basic finance concepts to understand the problems. And on the flip side, the students ended up useless at solving calculus problems on their own - but they could do the ones involving finance concepts.
• jrgoff 3y ago
  I don't know how or where you learned calculus, but when I learned it and when I taught it, the classes were filled with physical world examples to motivate and train the material. That said, my experience as a student reflected on as an instructor, was that a lot of the explanation and example stuff didn't have a lot of fertile ground in my mind yet when I was first learning the material. I'm sure it helped, but there were definitely times when as a grad student I reviewed the foundational material and thought "This makes so much sense, why didn't they teach me this when I was in high school?" only to realize that almost certainly they had taught me that in high school and I just didn't have the mathematical maturity yet to retain it. I think I was also hampered in my chances of learning the conceptual fundamentals because I was able to do most of the work through a solid ability at algebraic manipulation. While solid skills in algebraic manipulation is quite important, I do think it would be a good idea to restructure those classes so that a solid conceptual understanding is also more necessary to pass the class. Of course, easier said than done.