Ask HN: Why are mathematical standards so low? (And what should they be?)

[pl922]

As a control theorist, I'm privileged to have received rigorous training in applied mathematics. In my experience, most real life systems, organizations, and even psychological patterns can be effectively and practically understood in terms of dynamical systems, feedback loops, optimization, etc., to the extent that I strongly feel all people could benefit dramatically from receiving training in applied mathematics, control, and systems theory.

Unfortunately, truly grokking these topics requires significant mathematical maturity (which is often the byproduct of graduate-level mathematics courses). My question is this: Can you imagine a hypothetical education system where students leave secondary school with an advanced-undergraduate level of mathematical maturity? Why or why not? What would such a system have to look like?

[ilayn]

Another control theorist here. I strongly disagree with your premise of rigor in control. The whole field is ripe with math snobbery and virtually plotted its own funeral to the point of absurdity.

The field's key contribution is in the ideas and the insights it can provide otherwise extremely laborious to obtain. Definitely not in senseless dry exposition of unnecessarily general theorem proof parroting.

[wenc]

As a control engineer (and former theorist), also agree. Practical control is less algebraic theory and more numerical algorithms.

We don't directly use Pontryagin's maximum principle or solve Euler-Lagrange equations (impractical for large systems) for instance. A lot of the stability proofs are nice in controller design, but rarely used in practice.

We do use Model Predictive Control, which solves a numerical optimization problem at every time step. We also use state estimators like the EKF, which is also numerical. Much of the heavy lifting here is actually in the tuning and understanding of the process, not the control theory itself.

The usefulness of control theory isn't in its actual use, but to provide a foundation for developing newer theories. There's a place for it. It's a set of building blocks.

But I submit you don't need to really know the mathematical theories to truly understand feedback loops, optimization etc. Most of us control things intuitively when we're driving a car (feedback loop with feedforward), running a business (state estimation, feedback and stochastic control), etc.

Very good businessmen -- i.e. those with acumen -- are natural control engineers despite not knowing a whit of differential geometry or state-space models.

[pl922]

Definitely agree! When reviewing papers I've come across bad (incorrect) mathematics and unnecessary posturing. I guess I'm seeing controls for what it is at it's best. Any thoughts on geometric control theory (in the vein of Jurdjevic, Sachkov)? I like the differential-geometry viewpoint. How about sum-of-squares/algebraic geometric results?

[ilayn]

I have worked heavily on SOS and LMI based methods in general, IQCs to be precise. Sum-of-squares are so stupidly explosive in the size of required conditions, (pretty much impossible to do anything more than 5-6 parameters), you can play around with it theoretically. You might feel good about it.

If you call yourself an applied mathematician and keep working on these things, I will be the last person to object. But if you say Sigma* algebra or Sobolev spaces, or infinite dimensional systems, or this or that is required to understand dynamical systems that means you are not really getting the central ideas in control and confusing the methods with the problems we are trying to solve. That is my premise.

[pl922]

Thank you for this much needed perspective. I do agree that tools shouldn't take precedence over the problems they're used to solve (like you say, unless you're a mathematician). If you're willing to spare more time, which central ideas do you think researchers should orient towards?

[ilayn]

I have rather unorthodox view on this subject, I would already concede defeat if you object to it. However, to criminally box my view into a few sentences, we should start cleaning up our undergrad curriculum from root loci, routh hurwitz, Laplace transform etc. and put industrially relevant things like PID and all kinds of heuristics. We also need to fire anyone who says yes but "that's trial and error" as if engineering is not 90% that.

This is not an easy task by any means, it would require a lot of expertise. However we have to get back to being related to the field and not hijack control engineering to do applied math. A mech/electrical engineering curriculum, with all their flaws, will leave you around the neighborhood of a working engineer. You will not be ready for the production-grade however you would be a cadet.

With control engineering, you don't learn anything about the field but learn bunch of haphazardly put together half-assed math concepts. When you get out not only you don't know anything, you also have this elite view that "oh they are just doing PID, peasants!". In other words, control design is 5% of the whole process. Understanding the system is the crucial part. Anyone can tune something that barely works. Not everyone understands the system. Strogatz' nonlinear book should be more important then Ogata.

Let me give one example; Laplace transforms or all integral transforms with different kernels are extremely hard topics at undergrad level and confuse the hell out of students. To the point that when you present them the intricacies of analytical continuation and other necessary things to treat the subject you lose them. Hence we should not pretend to teach these things. Hence "oh replace dots with powers of s and then replace s with iw" is not helping anyone.

Apologies for cutting it short but I have to get shit-faced with my guests soon :) Hence I wish all a happy new year!

[pl922]

Great points! Happy New Year to you too!

[calf]

By funeral are you saying the field is currently moribund? Is it going through some kind of crisis? I had a few control/systems professors in the past, what appealed to me about it was it was a kind of "theoretical engineering" approach to engineering.

[ilayn]

By funeral I mean, academia completely killed the applicability of this rich branch of engineering and left it as "exercise to the reader". Take MPC control, it originated from the industry and academia did not care for very long time until it has been realized that you can play around with optimization problems and now being put forth as the one of the jewels that rigorous mathematical gave birth to.

Industry, exceptions notwithstanding, pretty much got used to ignoring the the theoretical advances. Hence control theory became this testeless pages long article sillyness where you get 9 pages of integral inequalities to tune 2 parameters and call it adaptive control success.

[amelius]

I suppose they are referring to black box methods like those based on deep learning.

[max_]

I really like mathematics and I spend some time reading obscure math books for kicks.

I was a very bad math student in highschool I don't think it was because I was "ungifted" like some people would describe it.

Towards the end of high school i.e the last year.

I found out that mathematics can be studied like Biology, History (too late for it to make an impact on my final results).

And that the only way of learning mathematics was not just through practising it mindlessly until you grasp it like the teachers said.

I think the real problem with mathematics is that it has really bad comprehensive literature.

Here are the 3 main problems with maths literature. (What I think would have helped me improve my maths skills by alot)

Reason 1.

Implicitly defined notations. There is really no go to comprehensive resource that describes all math notation clearly so jumping into a new topic is difficult.

Reason 2

No clear definitions of algorithms used in respective math topics.

A great example is http://matrixmultiplication.xyz/ that describes the matrix multiplication algorithm. Step by step. Maths education would be better if all important algorithms in different math fields were described as clearly.

Reason 3

There is no comprehensive dictionary for math theorems. A great example would be something like a table of integrals in calculus, but for all fields.

If all these resources are well documented. I really don't see why someone cannot improve their maths skills even via self study up to graduate level.

[Fire-Dragon-DoL]

Oh, thank you for this. I had a similar experience as yours and when I wrote down notes when studying, I ended up with cheatsheets that described things clearly, even though it took me hours tp grok. I never understood if the problem was me or just poor books (I needed to rely solely on the teacher, which is very limiting). I see that there is indeed a problem

[pl922]

Thank you for a great response! I agree with you on communication-styles. Math education could definitely use some more clarity in exposition of procedures and notation. With regards to reason 3, this is true for me at least! Much of my learning during and after bachelors was self study.

[xigoi]

> There is really no go to comprehensive resource that describes all math notation clearly so jumping into a new topic is difficult.

Introductory literature for a new topic should always explain new notation. If not, either it’s really badly written or you accidentally chose a non-introductory book.

> There is no comprehensive dictionary for math theorems.

What’s wrong with Wikipedia?

[wageslave99]

> Can you imagine a hypothetical education system where students leave secondary school with an advanced-undergraduate level of mathematical maturity?

That would have been my dream.

> Why or why not?

Mathematics is the foundation of science, so that would have been helpful for my (then) future studies.

> What would such a system have to look like?

Not teaching algorithms to resolve problems, teaching from the source of the mathematical principles, I mean the practical issues that caused scientist to develop Math. I suppose that the Russian School of Mathematics[1] teaches Math that way, like in some Soviet books that were mainly practical.

Also, it is needed to have 5 hours per week of Math, to not have a fast pace when teaching kids. Math needs some time to "assimilate", and IMHO, 3 hours/week was not enough.

In this system, once you get out high school you could pursue the proofs of each theorem or a more inner and rigorous approach to learning Math.

[1] https://www.mathschool.com/

[pl922]

I'm a big fan of the Soviet-era books, though they often require a degree of independence to fully appreciate. Perhaps schools should focus more on developing students' ability to self-teach.

[KolenCh]

It is low probably because the meeting degree they are having already take up most of the time, and may also be because for those better equipped in math they are in research that requires more sophisticated level of math already (ie self selection).

This is extrapolated from my experience in CMB data analysis. People with better math training probably are doing theoretical work via simulation instead. Some, but not all, papers involving CMB data analysis struggles to present their research mathematically, or may perform some statistical analysis in a very naïve way.

[sgt101]

>In my experience, most real life systems, organizations, and even psychological patterns can be effectively and practically understood in terms of dynamical systems, feedback loops, optimization, etc., to the ext

So, do the understanding and then apply it and then win! Except, maybe you will find out what thousands of others have, first maths is really big, second it's not big enough to be effectively used as you describe.

[tmaly]

This question reminds me of the book A Mathematician's Lament by Paul Lockhart.

He lays out a really good case for why we don't have more mathematicians.

[CrypticShift]

I accept that enhancing fundamental math skills is important, yet I'm skeptical about the feasibility of raising the bar to more advanced levels. Considering a normal distribution, it seems improbable that a significantly larger number of individuals could be educated to this extent.

Look at the CURRENT outcomes produced by mathematical geniuses within the financial sector (and military..). My argument isn't that math's impact is solely detrimental. For me it is about balance and diversity (math is only part of it)

When it comes to systems theory itself and its buddies, back in the day, Cybernetics was more of a big tent idea. But then the math nerds took the wheel, thanks to Shannon's playbook. And that may not always represent the superior path. Again, It is about balance/diversity

[calf]

In EECS every undergrad does learn a bit of control/systems theory in their sophomore and junior requirements. I loved it but it was mathematically challenging and unfortunately the better background a student already had, the better and the more they got out of the courses. I wish there were a better way, because the result was students eventually got weeded out by an (inhumane, alienating) competitive system, rather than have each student actually learn something well on their own level.

[pl922]

Alienating students is a real problem. I think supplemental math training can help, e.g. recitations on real analysis alongside a probability course.

[mrkeen]

I read about PID controllers and got super excited to use them in a programming context. That was maybe 10 years ago and I still haven't spotted an opportunity.

[amboo7]

https://www.mg.edu.rs/en#gsc.tab=0

[glimshe]

This would be a horrendous, and perhaps inhuman system. It's already difficult to convince non-math inclined kids (the majority) to study basic algebra; forcing kids to learn a much increased level of abstraction would be borderline abusive.

[throwaway167]

What do you define as graduate-level maths courses? The scope and depth vary considerably. What I did in the first year of a maths degree may be (is, have seen) considered graduate level in some (some) engineering courses. Could you be more specific?

[pl922]

Great point! I'm thinking of analysis courses that begin with measure theory, also differential geometry and calculus on manifolds (though those definitely can be undergraduate). Some graduate level math-in-engineering topics are things like linear systems theory, convex analysis and optimization, nonlinear systems, etc.

[amelius]

In my experience, the quantitative aspects of advanced math do not bring you much extra understanding in everyday life situations. The qualitative aspects may be helpful but they can be learned without grokking advanced math.

[tlogan]

I believe that mathematics is not solely about numbers; it’s a much broader field. In fact, starting kids education with concepts like sets, discrete mathematics, and so on, instead of focusing only on numbers, might be more beneficial.

Advanced mathematics, including topics like limits, derivatives, set theory, logic, and discrete math, offers powerful tools to understand and describe everyday life.

While numbers are undoubtedly useful and interesting, the real-world applications of math extend far beyond simple numerical representation. It’s these broader applications that truly demonstrate the value and versatility of mathematics.

[pl922]

This is my perspective as well. I'm not confident I could use math to make myself a lot of money, more that I can use the style of thinking to pin down how my experiences and actions effect my habits for example.

[ganzuul]

You are also privileged not to suffer from dyslexia, ADD, ADHD, and so on. Add all the letters up and we see that so many children are affected that we should probably be looking elsewhere for their talents.

The idea that we could actually afford to customize the curriculum for each child is extremely new, but with these AI advances we might finally be able to tap into the great richness of our diversity and celebrate being different.

To have mathematical tools with which to relate to the greater cosmos is a wonderful gift but there are others.

[pl922]

Thanks for your response! I do actually suffer from ADHD, though I'm (and was) intentionally unmedicated. Really had to develop my discipline and internal motivation before I could give a good try to math, so I definitely get what you mean. You make a great point about customized education -- I definitely think it would have helped me as an easily-distractable student who struggled to build my mathematical foundation.

[ganzuul]

Please excuse my presumption. It must have been challenging!

I think the next generation will leave us in the dust if they get to tap into the potential that is there.

[jncfhnb]

How does a a cashier benefit from systems theory?

[thrill]

How does a department store bathroom janitor benefit from multivariable calculus, complex analysis, partial differential equations, linear algebra, probability, vector fields, transformations of time to frequency, ad infinita?

He goes on to become something other than a janitor - at least that worked for me.

[jncfhnb]

That’s great! But we still need janitors. and if everyone knows these things, then knowing them isn’t exactly a pathway upwards. So let’s ask again, what do they do with it.

[pl922]

I'd love to hear more about your story!

[he11ow]

There was a system like that. Roll up, roll up, I'll treat you to a story:

It all started in 1957, with the Sputnik. The US was entirely taken by surprise. The belief that the Russians were way ahead instigated an internal crisis, which, in turn, led the US to re-evaluate its national maths curriculum. Thus was born a think tank called the School Mathematics Study Group (SMSG). They developed a radical reform in mathematics education known as 'New Math'.

This was rolled out nationally, to great criticism. The teachers were ill prepared and the parents felt clueless. Look it up, there were 'Peanuts' strips from the period mocking it.

Now, New Maths focused initially on the early years, but then, in the mid '60s came a second round, and one specific initiative was the Secondary School Mathematics Curriculum Improvement Study (SSMCIS), and the guy who heads it happened to be a professor at Columbia University Teachers College. This last tiny fact probably means exactly zero to you, but it is central to why I even know all this stuff.

Anyways...The program's signature goal was to create a unified treatment of mathematics, so that instead of studying the normal curriculum you'd basically study maths the way you're taught it at university: set theory, group theory, axioms proofs and logic, all the way up to calculus. The programme was intended for grades 7 through 12, and was rolled out initially in the NY area and then later in select schools in other affluent cities in the states. It only ever targeted the top 15-20 students in the a class body. That was for sure the right call - this ties to your question, so more on that in a minute.

Eventually two things happened: one, the programme ran out of funding. Two, by the mid-seventies there was a massive backlash against New Maths and the US decided maybe it's okay to just leave it, since the Russians didn't end up winning the space race after all.

I would have known diddly squat about this whole affair were it not for a curious corollary. In 1953 one very specific individual happened to be on a mission in New York. He was a former Russian Jew who studied maths in Canada, served in WWII for the US military and eventually made it to Israel. He was an educator and had somehow caught on to what was happening in Teachers College, and upon his return to Israel, he started devising maths curriculums and translating the original SSMCIS textbooks. This was now dubbed 'The Columbia Programme'.

Fast forward almost 40 years later. In a way I've never managed to uncover, that programme survived, and was still being taught in one of Israel's gifted programmes. I entered my first maths lesson at seventh grade never realizing just how much this would end up influencing the person I'd become. Our textbooks were literally photocopies of the typewritten texts. The teachers has added to it bits of the regular curriculum plus more practice exercises, which the original textbooks lacked, but they left most of it as is.

In the first three years, no one in the class was allowed to drop out and take 'regular' maths. For many, even in a cohort that was already pre-screened for academic achievement, this was a struggle. For sure, once highschool rolled along, anyone who hated it could switch back to the regular national curriculum.

Of the people who stayed, nearly everyone went on to study Maths, Physics or Computer Science to graduate level. This tended to happen in the years above and below as well. Over the years, though, the programme got smaller and smaller. I'm not sure it still exists.

To your question:

You absolutely CAN get highschool students to leave secondary school with advanced-undergraduate level of mathematical maturity. (And, BTW, the Russians are still ahead there...) But you can only do it for a small minority. Not because of elitism, but because most people aren't a good fit for this path.

At the time I intended to write up all of this into a nonfiction essay. But other things took greater priority, and I just left it there. In a way, it's been nice telling this story here.

[pl922]

Thank you for taking the time to share this -- such a fascinating turn of events. I'll be keeping your story in mind for future reference. Happy New Year!

[lcordier]

I failed control-systems in my third year. Was too focused on other more fun subjects. Maybe it is a fun issue?

[pl922]

Good point. Was it mostly just theory? Or did you get a chance to implement anything in code or an actual machine?

[lcordier]

IIRC we did a digital control system for an electric motor drive circuit but that was a different subject: electric machines.

[lcordier]

Lots of theory, yes.