As a counterpoint to the overly exuberant article let me chime in as a graduating theoretical computer scientist. Math is hard and despite its beauty and allure academic life doesn't take place in a vacuum- ergo there's a lot of politics and pettiness from your peers. You should expect to invest years of your life into making progress on some hard problem with little encouragement in the meantime, this takes a tremendous toll on the mind and not for someone who falls easy prey to self doubt. Sometimes even after you publish your top result that took a lot out of you, it may take many years before people appreciate, let alone understand, your work- the vast majority of papers are never going to be read. To compensate for this horrible feedback mechanism you need to basically play the popularity game and try to give many talks and talk up your result when you meet people, so there's a lot of salesmanship involved here as well. So the career path of a junior scientist is pretty crushing mentally and I couldn't stomach it in the long run.
In an ideal world I might have continued in academia but the career path is so twisted you have to either be insanely good (at math and at managing time) or just hate yourself enough to sacrifice your best years working essentially in the metaphorical darkness well outside the spotlight and most likely alone and poorly paid.
I can't find the citations right now, but there was a recent study done through Elsiver's website. The group got a hold of the servers for Elsiver and took a look at page views. They wanted to know what the rates of papers being read was. It was... disheartening. They found that ~46% (again, not sure here as I can't find the source) of papers will never be looked at outside of the authors, reviewers, and the editors. The articles are not only never downloaded, but the pages are never even loaded. The stats were, to me, confusing, but hey that is what peer review is for. Still, if you take this paper (that I can't find now) to be true, ~half of all papers are effectively lecturing into the void. I admit, I drank a bit after reading that one.
In CS, most authors host "preprint" copies of their papers on their own website, and most views are of those versions. So I wouldn't be surprised if Elsiver's defunct distribution system isn't getting used much (at least in this field).
Yes, but roughly half of all papers will never even be page-loaded from the 'official' source (if you believe my recollection). I am not a CS researcher, but I would imagine that of 10 people that cite a paper, at least one will bother to download the original paper or look up the source. Hell, even 'vanity' page-views of your own work would have been counted in the paper that I mentioned, and that is with many authors on a single paper too.
I honestly don't know what to make of it really. I at least bother to look at my papers once, if only to show my family on the Holidays, and I have a lot of so-authors that may be doing the same. Are most researchers so fed-up with their own work as to not even bother looking at it again? What is going through their minds concerning their efforts? It's just ... heartbreaking. At least half of researchers don't seem to care at all, not even the people that put all that time and effort to get their research out there. Like, what are we doing with our lives?
To be fair most papers are written for readers that have the exact same viewpoint and specialty. So they are incredibly difficult to parse for people in the same field but a different sub specialty.
I think many more papers would be read if authors invested more time in learning how to write.
Isn't that plausibly the result of publish or perish? I would imagine it would lead to people publish at the cadence of their typing rate (to be cruel), not at the rate which they can generate good ideas.
Higgs said he could have not been able to operate in todays academia. Not everyone needs to have John von Neumann performance levels to do valuable work.
IIRC a lot of that has to do with spammy 'publication mill' journals. If you just look at 'decent' journals, the situation doesn't look nearly as dire.
Yes, Elsevier is far from a journal mill designed to H-index hack your way up the credential ladder. They are scum, yes, but they are at least truthful.
Maybe if Elsevier was open source & easily accessible more people would read it. I have tried reading many journal papers as a layman but in most cases got blocked by a pay wall. I dont like to rely on newspaper headline but like to go the source and because of paywall I get thwarted.
In addition, you can just email the authors too. Most, if not all, are very happy to send you a copy and help explain things too. Most scientists do want to talk about their work!
Elsevier used to demand you sign over all copyright, which would make that illegal (think about it...!!) They've also sent takedown notices to academics that posted their own articles on their own website. Oh, and by the way, Elsevier also sued Sci-hub and LibGen...
My own experience: When I have been on a University campus' wifi and needed to look at a paper, I could download and see the entire article, not just the abstracts. This has been true for all of the limited number of times I have tried this. I have also done the same at some, but not all, conferences; typically the very large ones only. I would then doubt that the numbers are being skewed towards the scihub and other 'rebellious' forms of paper acquisition. However, this is only my experience.
Maybe I skimmed the article, but my takeaway from this article was that using our abilities to think and solve mathematical problems helps us to live a life of more intellectual resilience.
In my view, this wasn't about doing mathematics as an academic career, climbing up the ivory tower, publish or perish, and all that.
Edit:
I tend to agree.
My intellectual capacity was far improved only after doing first year Engineering Maths & Physics. I did terrible gradewise, but it has helped me visualize, juxtapose mental structures, quickly iterate on different concepts etc.
I should go back to doing something again. Had to do some cartesian products in a data migration the other day, in my dayjob, it was refreshing :).
And as always, this timeless quote from Einstein motivates
"Do not worry too much about your difficulties in mathematics, I can assure you that mine are still greater."
> My intellectual capacity was far improved only after doing first year Engineering Maths & Physics. I did terrible gradewise, but it has helped me visualize, juxtapose mental structures, quickly iterate on different concepts etc.
Same here, and thanks to the internet just remembering concepts is usually enough these days.
Yesterday I had to quickly guestimate for a project how much space overhead we'd have on the server if we pre-generated an image pyramid of tiles for leaflet.js[0], instead of generating a "PNG" in RAM on the fly each time and serving that. Then I realised the number of pixels shrinks by a quarter every time we zoom out, so the worst case would be the infinite sum of 1/4^n. While I vaguely remember from my first year of physics how to calculate that, it's been over a decade. But we have Wolfram Alpha these days[0], which told me it converges on 4/3. So I knew the upper boundary would be around 4/3 of our original data, plus negligible PNG header overhead, and assuming compression is about the same.
Bachelor of Science in Mechanical Engineering with minors in Math and Physics here...
While I can probably count on one hand the times I've used what I learned in the "real" world, I can say I really value my education.
I believe the real value of a math based curriculum or even something like philosophy with an emphasis in logic is that it teaches one how to think in a reasoned, dispassionate and rational way.
Doing serious research level mathematics (or any other science for that matter) is almost impossible outside academia. Doing serious work is hard and mentally draining. Mere mortals can't motivate themselves to do it unless the incentives align just right (i.e. being in academia). Also it takes all your time so forget about it if you have another job.
I know this from personal experience. After I joined industry I thought I'll do physics research in my own time. In fact, I can't even keep myself up to date about current research. Most journal articles require a lot of effort to go through, and I don't have that kind of energy after a work day.
In math we have lots of journals devoted to recreational and more elementary problems. There is a lot of low hanging fruit out there.
For instance, compute the fibbonacci numbers modulo the square of a prime and compare to modulo a prime. Are there numbers where the first zero appears at the same point? We don't know.
I'm not saying the problem is easy. I'm saying one can make some progress without much theory.
> I'm saying one can make some progress without much theory.
Claimed without evidence.
Sure, you can make trivial progress by writing a computer program to search, and the problem is "elementary", but "elementary" doesn't mean "can be solved without studying lots of techniques and theories", it means "can be solved without calculus"
I know the feeling, but don't forget that 'human flourishing' part of your life:
... the achievement of human flourishing, a concept the ancient Greeks called eudaimonia, or a life composed of all the highest goods. Su talked of five basic human desires that are met through the pursuit of mathematics: play, beauty, truth, justice and love.
Math is different. Even complete outsiders of mortal cognitive gifts can now and again generate publishable results.( I tried to find the reference but my google-fu fails me now - I do have actual published papers in mind.)
In physics these laymen are generally called 'crackpots' for a good reason because the context one needs to understand to generate new results is so huge.
Mathematics is different - all one needs is pen, paper, a mind that's attuned to generating logically whole statements and a problem that just won't let go of ones mind. The results are likely in geometry or some other approachable field and not in some more obscure subject.
That definition is circular. Of course research-level mathematics is done in academia, but there is a much larger field of applied mathematics that is done in industry.
That stuff has very little to do with 90% of academia math. Also, if you specialized and are interested in a certain field of math, the barrier to switch to another field is enormous. On the other hand, 90% of the applied mathematics done in industry is much, much simpler, and very often a trivialized version of what you do in academia, so one could imagine switching to be possible, but what's the joy of doing that?
(I have a MS in Math, my field was algebraic geometry. I considered a career in math research, but the industry was paying too well for it to be a reasonable)
In fact, in academia, in some fields (and mathematics is certainly among them), in some sense only "the best" contribute to pushing the field forward by working on the cutting edge.
And I think the author's point was that focusing only on that is an overly narrow view, and one can engage in maths in a recreational, joyful way, without aspiring to an academic career in it.
Yeah, I'd argue a very un-Good Will Hunting thing: if you can be convinced you suck at math, you probably should not pursue a career in academia in math.
However... being a math major is really different from a career in academia. People really should be encouraged to major in math.
I would have to agree with this, but I disagree with out math is taught by most schools.
Somewhere in this thread there is a link to Francis's lectures and they are wonderful. The method/enthusiasm be has for teaching is wonderful and makes all the difference to students.
Rather than going through online courses and paying for them (many university classes are mostly online math problems, and proctored during a specific time.)
EDIT: The way math is currently taught really forces people to dislike math and think they suck at it. (in general)
"EDIT: The way math is currently taught really forces people to dislike math and think they suck at it. (in general)"
Starting in primary school with basic arithmetik.
(I help a young boy with his homework)
They don't really teach them Math, how to solve thing, they tech them to memorize different algorithms, which they can handle after a while, but don't know what they are doing at all ...
So the best mathematicians there are not the ones who can think best, but who can memorize and follow orders the best.
I think this is true in some cases and not true in others.
I met a woman from South Korea who was taught all drill and no concepts. She was solving differential equations in high school but had no intuitive concept of derivative and integral or what they were used for. In fact, she had no interest in math. She was just a diligent student focused on getting into the top university. She was obviously trained the way you describe.
I have also personally, in the United States, been in math classes where many students suffered because they couldn't string together correct calculations consistently enough to validate and reward their high-level understanding. They learned a lot of the words and pictures, could explain what an integral was for, and could listen to a lecture and feel like they got it, but if you asked them to apply what they knew to a real problem, they responded with a kind of rueful helplessness. To them, mathematics was like magic in the Harry Potter universe: anybody could explain it, but it worked for some people and not for others, for reasons that seemed to them to be innate.
Each system stressed one aspect at the expense of the other, and in each system, there were many students who picked up both, but also many students who only learned the part that was stressed by their teachers. It was certainly the case in my classes that a student who only learned the concepts, without the mechanics, was unlikely to progress much farther in the math curriculum.
A balanced method treats the two aspects as complementary, each enabling the other. Treating one as the hero and the other as the villain might make sense locally as a response to a warped system, but it can easily become a warped approach in itself.
Of course, most students do just memorize equations and follow orders. But in my experience the most successful students are always the ones who understand the material on a fundamental level and memorize very little (on some level, memorization is all but required by even the best).
The successful students do that in spite of the bad teaching. In my view, a lot more students would both enjoy and do better at maths with more appropriate teaching.
> They don't really teach them Math, how to solve thing, they tech them to memorize different algorithms, which they can handle after a while, but don't know what they are doing at all ...
Common Core in the US is supposed to solve this. I've seen some common core math homework for myself and despite the odd outrage most people have towards it, I think both the idea and execution are at least decent, and an improvement.
I can see how that can work as well, but I would argue having a deeper understanding of the problem (whatever it may be) will benefit any mathematician/student in the long run.
It works, if you don't want to think and just get results. But today there are calculators for that, so I'd like it if they teach them how to think, not to follow.
That's not a counterpoint, that is exactly the point. Maybe it becomes clearer if you read the speech. Math is hard, but it's being made much, much harder than it has to be.
What's the alternative? I mean that seriously, not sarcastically or in a defeatist way; try to explore what the alternative might actually be.
On the one hand, academia has become rather harsh and intimidating and there is room for all kinds of improvement.
On the other hand... there is no world where whoever just wants to study math can just go study whatever they want for as long as they want, regardless of how well they do it. Of course, when I spell it out, that probably seems obvious, but I suspect this may be the unexamined assumption in a lot of people's heads.
Would you say the same about exercising and sport?
"There is really no point in doing any exercise unless you're aiming for Olympics level performance. Yes, it's somewhat harsh and intimidating. But there is no world where whoever just wants to can just go do sports and exercise what they want as long as they want, regardless of how well they do it."
I think that was the point of the article - doing maths as a way of life (like exercise), not just for your career with the aspiration of being the best.
There isn't a world where you can just go do sports and exercise as long as you want, regardless of how well you do it. Eventually you're going to have to eat something.
I'm speaking in a context where we're talking about academic life and academic life being harder than it needs to be. I think that's the only way to read this context, because nobody is making self-study "harder than it needs to be", so reading this thread as being about "life in general" doesn't make any sense to me.
When you have a decent work-life balance, you can certainly engage in sport and exercise, even if it is not jour job.
I think the article is precisely not about professional mathematicians in academia. To quote: "If mathematics is a medium for human flourishing, it stands to reason that everyone should have a chance to participate in it." (my emphasis).
I solve Project Euler problems for fun. The article presents an argument that more people should do something similar, and more mathematicians and maths teachers should support it.
"there is no world" - wasn't that what University was like before the 80s and still is in many other countries? minimal fees and some casual work and you can stay for as long as you can stand the conditions of the average student flat ;-) I think you are conflating the reality of your current culture with what was/is in other cultures.
If what you describe was even remotely the case, which I don't know, then there was some other filter in place preventing "everybody" from doing it. I can tell, because not "everybody" did.
Plus I suspect you missed my "regardless of how well they do it" clause. I doubt that you can flunk every math class over and over and just keep attending. Loopholes that big get noticed, exploited, and closed very quickly.
In German universities in the 80's and 90's you had people studying for decades. It wasn't a "loophole", either, it was a feature. Universities were there to teach, basically everyone could go and study.
In recent times, the pressure to finish or drop out has increased, and studies have become more rigorously structured (or infantilised...), particularly with the introduction of the Bachelor/Master system.
There was a very strong filter. It used to be very difficult to get into university before they expanded tremendously from the 70s to now. You needed excellent grades. Now good grades are enough to get a spot somewhere.
there is no world where whoever just wants to study math can just go study whatever they want for as long as they want, regardless of how well they do it.
Certainly in Sweden, once you're in, you're basically free to hang around at University and study for as long as you want. It's not like they're going to kick you off campus.
An alternative would be a system where publishing isn't as important as obtaining results. Let small journals take care of everyday research and let only the big works end up in something that really define someone's carreer. There would be a lot less people with a "valuable" publication, but we do have other ways to judge people and assign spots as researcher. The problem is the way jobs in academia are assigned: the more you published, the more you get paid. This shifts the attention from knowledge to publishing. Just value people with the usual attitudinal tests unless thay discovered something truly remarkable.
Wow, I always shied away from academics for similar reasons you mentioned, but I just realized that my main hobby, game development, parallels what you say a lot (and creative hobbies in general, including writing).
"You should expect to invest years of your life into making progress on some meaningful game with little encouragement in the meantime...sometimes even after you release a game that took a lot out of you, it may take many years before people appreciate, let alone play, your game, and the vast majority of games aren't going to be played." "To compensate for this you basically play the popularity game and give as many interviews and talk up your game when you meet people, so there's a lot of salesmanship involved there as well."
"The career path is so twisted you have to either be insanely good or just hate yourself enough to sacrifice your best years working essentially in the metaphorical darkness well outside the spotlight and most likely alone and poorly paid." <- My experience working in the game industry, except I wasn't completely alone, I did have coworkers. However I did feel like I never had time for a relationship.
We need to do better at cultivating interest in math, as well as other "hard" fields, among people who are normally excluded from the system.
Su isn't as concerned with the amount of full-time mathematicians doing research work as he is the people who drop out far before that becomes an option. We should teach it better in high schools, and encourage more math undergraduates.
Su isn't saying math is a great field free of problems, but he's arguing we should focus more on getting people further along that path, regardless of the stage where they drop out. I also think the problem you're referring to exists in most academic fields, and that the solution (if there is one) will probably have to be more general than just relate to math.
The article is about math, not necessarily academic math. And it was far from "overly exuberant". Jesus. I have no degree, and do math all the time, precisely for the reasons listed in the article. There endless comments shitting on academic life are obnoxious and tedious, they are also becoming something of a plauge here.
Since I made this comment, I also had the chance to listen to his lecture in full. I couldn't agree more with him. What I wrote here was based on my understanding of what he meant by "doing mathematics", which as other commenters also pointed out earlier did not have anything to do with a career in math. I stand corrected.
Cool! Thanks for taking the time to circle back and add this. I was also responding to the wave of "academia sucks!!!", which, while perhaps true (for some) is getting really old to read in threads here. Appreciate you taking the time to add this comment.
Do you think it's the culture of science or the inherent uncertainty?
From my own experience I think it's the latter. Jumping off the edge of human knowledge and hoping there was a result to catch my fall was very mentally taxing. Not many other careers force you to grabble with the uncertainty of creating knowledge, and I think it takes a toll.
This has been my experience with trying to get a foothold in the technical civics communities, so I don't think it's either. I've found that the bar gets set pretty high by folks who are willing to speak loudly about their accomplishments in a ways that exaggerate the actual accomplishments. If you're not really good sales person towards your own work, you find yourself working significantly harder than what's probably needed since fewer people are willing to work with someone who hasn't had any accomplishments. It really sucks that ego and salesmanship gets in the way of trying to do civics work.
How many people can live like that in any practical sense? Everything I've ever read about motivation, learning, flow, and mastery holds quick and direct feedback as the most central necessity. Applied to software development, hence we prefer fast compilers over slow, unit tests over integration, working on a desktop over an SSH connection halfway around the world, frequent releases, etc.
It sounds like you're not being sarcastic. So let me say that when you're working on something that almost surely has zero practical impact over the course of your lifetime, and the only possible benefit to society is the advancement of the state of our knowledge, a lack of enthusiasm from your peers can be gut wrenching no matter how mentally resilient you think you are. No one does math solely for appreciation from people they admire and respect, but the lack of it, even in the short term, can be very hard on a person.
Scientists actually spending more of their time reading others' work, communicating encouragingly and giving constructive criticism (anyone in academia has read more than a few hate filled reviews of their work), better pay (a postdoc at Oxford nets you a monthly salary after taxes of about 2.4k pounds), less focus on numbers and more on quality (a big reason why people don't read papers is because they're missing out on little by not doing so i.e., much of the work is incremental, unsurprising or just badly posed or ill motivated. instead focus must be given to quality and this must come from job hiring decisions which should reflect that science is not a numbers game, but the sad sad truth is it so very is). About the encouragement part, I felt let down many times by senior scientists by their uncaring and viciously competitive nature. All this does not need to be a part of academia. I think greater science can emerge after these structural problems are addressed. But currently the only people who rise to the top are the genuinely good ones and the ones who know how to game the numbers system. Idealists who come for the beauty of math often find themselves woefully unprepared to play all these other games in addition to doing good math.
In an ideal world I might have continued in academia but the career path is so twisted you have to either be insanely good (at math and at managing time) or just hate yourself enough to sacrifice your best years working essentially in the metaphorical darkness well outside the spotlight and most likely alone and poorly paid.