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Comment by thorin | Hacker News Reader

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 terms of speed/distance/acceleration seems to make complete sense now (one of several potential examples). 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. I don't think I had any ideas one what was going on until it was mentioned in physics several years later.

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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.

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.

Physics doing the work of actually teaching math is a universal embarrassment to math departments. Or, well, it should be.

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”

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.

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.

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.

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.

> doing calculations is not mathematics

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.

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.

> Study of things that are provably true.

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?

I don't even understand the question. Is there in truth no beauty? Why do anything that doesn't add value to the physical world?

It seems you did understand the question and I thank you for your answer, it is helpful.

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.

[1] https://en.m.wikipedia.org/wiki/The_Unreasonable_Effectivene...

So if the application is not visible today, it might appear tomorrow. In fact that makes sense.

Up until about World War II and use in cryptography, number theory was pretty much an un-applicable field of mathematics.

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.

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.

Fiction writing exists, should it not be enjoyed? Sometimes mathematicians invent things before scientists discover how to apply it.

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.

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.

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.

Fundamentally, without sufficient rigor there can be no proofs; "maybe" and "someone says so" aren't good theoretical building blocks.

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.

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.

> 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)

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.

> 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.

That's great. Not everyone does. Thinking simply via formal systems is not inherently inferior to 'doing' something with it. Both have their uses. In particular, as I said, quantum mechanics would not be where it's at today if it weren't for many pre-quantum mathematicians pushing around symbols on a page.

I'm very much not arguing that nobody should do pure math. I'm arguing that teaching pure math to undergrads who are trying to learn applied math is a tragedy and ruins math for a lot of people.

> nobody needs a proof of how differentiation or integration work

This "nobody" includes a lot of industry workers, e.g. those in finance and economics, not only "pure" mathematicians.

They... don't? The only reason they need proofs is to write papers proving other things. Not to do the actual work of their fields.

I think you're generalizing far too much from limited experience. Not all mathematics, even applied mathematics, corresponds neatly to intuitive physics models you build naturally by being an animal that interacts with the world. Quantum mechanics was already given as an example, but I had the opposite experience with cryptography. So much of the training as a CS student was along the lines of "accept that this works" and then practice by implementing something, and sure, it worked, but at no point did it ever become obvious why any of it should work. Until I finally went back and spent a solid year and a half learning group theory and algebraic number theory in a more rigorous way, figuring out all of the proofs we had skipped over, at least trying it myself, but giving up and reading someone else's proof if I couldn't figure it out within a day or two. And in this case, the abstraction definitely came before the application. This is math that was thought to be compeletely useless for centuries, but people worked on it anyway just because they loved math. Lo and behold, it made the Internet possible, so I'm glad someone did it and loved the tedium.

My stance in general is not that we shouldn't teach students why things work; just that rigorous proofs aren't at all a good way to do it. Indeed, as far as I could tell, most of the mathematicians who taught math classes I took had at best a rudimentary knowledge of how to use any of the theory they were teaching to do anything.

I studied physics in undergrad and the general method was, yes, it involved detailed understanding of all the theory, far better than anyone learned it in the relevant math class (differential equations, multivariable calculus, and linear algebra, in particular --- a 1-semester course in QM has to teach everyone linear algebra because the math course on the subject is garbage; a 1-semester course in E&M has to teach multivariable calculus also because the math course was garbage, etc.)

But yeah, absolutely you have to really learn e.g. group theory and algebraic number theory to do cryptography. I actually quit engineering and switched to physics precisely because I was so annoyed that an intro electrical engineering class used the Fourier transform as black magic without bothering with why it worked.

In particular many of the exact theorems that are used in formal math are totally irrelevant and historical accidents. Calculus could have been developed in a thousand different ways; we happened to pick one, the particular names of the theorems or what order the results are constructed in is, IMO, totally irrelevant to anything outside pure mathematics (particular examples: which construction of the integers you use; which construction of the real numbers you use; which definition of the derivative; which definition of the integral, stuff like that. In the case of integrals, yes, different constructions have different theoretical properties in terms of which functions can be integrated --- but, in any application, the answers intrinsically can't be affected based on your choice of definition, so the actual definition can't matter.

I would not call this "destructive." Solving problems is fun in its own right, and things would start making sense in their interconnection, regardless of whether they can be applied to sciences or not.

It is destructive to people who do not find abstract math problems inherently fun, which is (I would guess) the great majority of people. They come to the conclusion that they hate math or that they are bad at math, even though they would have grasped it better and enjoyed it had it been presented to them differently.

If your goal is teach math to the bulk of people, including those who will not go on to be mathematicians, it makes sense to tie math to something that the bulk of people can relate to. And most people DO enjoy thinking about physical objects and physical space (because we ourselves are physical beings who evolved to interact with our physical environment), so this is a really great starting point for introductory math for the average person.

If your goal is to only teach the subset of the population who prefer highly abstract puzzles, and to alienate all others, then our current methods are working fine I guess. But I don't think this is a good goal for general math education.

I think you put your finger on a challenge fundamental to teaching itself. Speaking as a (former, at least for now) teacher, the concepts that I find most intuitive were always the most difficult to teach, whereas skills I struggled to master were the ones I was best able to instruct. For problems in the second category I could diagnose the gaps in my students' understanding or application - likely because I'd faced them, or similar, myself - and strategize an approach that would help. Conversely, the subjects that I "got" without effort, left me floundering (at least at first) to find an alternate way to present to students.

(I suppose that's where the terrible canard of "those who can't, teach" comes from. There's a kernel of truth to it, of course, but the more generous and relevant point is that success through effort is better preparation for teaching than success through natural ability alone.)

My guess is not that your teacher would disagree that math is "a fundamental language of the universe", just that she spoke it so fluently that she wasn't well able to relate to people who don't.

> 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.

>> 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.

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.

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