Do we need more mathematicians? Rant number 1 from a maths PhD dropout.
After skimming yet another article on how important the field of mathematics is even if maths graduates don’t end up mathematicians because it develops abstract reasoning, because it’s used in a variety of fields, because it is fundamentally similar to many types of intellectual activities, e.g. software engineering and many others I must write this. I have heard this narrative before. I have heard way too much about this narrative. I’ve been listening to how maths is unfairly unpopular, how the world needs more maths graduates etc… for about 3 years from maths PhD students in my ex-PhD office. The narrative usually ends with that we should fund mathematics more, promote maths more, because maths = underrated good.
Boy, where do I even begin the rant that is about to follow…
At this point I started to suspect that there’s a secret paper somewhere that examined the causal effect of graduating in maths on improved reasoning skills, concluded that such effect indeed exists with some overwhelmingly small p-value, then it was spread amongst all vocal mathematicians all around the globe who secretly read it, memorised, and agreed to never speak about the existence of such paper ever again. However, by the looks of it, I wasn’t included.
And even if I accept the premise that there exists a causal effect of studying maths on doing overall better in life, before I start advocating that we need more maths graduates, I still need to know
1) Cost/benefit analysis of pushing someone through a 3-4 year degree designed by archaic methods of education in order to achieve the desired effect. Perhaps people without the “improved abstract reasoning” skill don’t earn as much indeed, but, considering the cost of acquiring a degree, they are still better off, financially, over the course of their careers.
2) An argument that maths degree uniquely achieves the desired effect. For example, say Joe got the impression that maths degree is useless, acquired a CS degree, still developed the “improved abstract reasoning” skill, and, in addition, found a job as a programmer in no time, compared to say, mathematicians who tried finding jobs in software for months. So Joe is still financially better off. Shouldn’t mathematicians, in the face of this scenario, back down and say we need more people in STEM except maths degrees?
3) An argument that out of the all possible ways of achieving the desired “improved abstract reasoning” effect, the most effective way is to push some people via a maths degree and that those people are currently put off from doing maths for a variety of silly reasons that we can easily address. Going back to Joe, perhaps after reading all about degrees he got put off all together, found a job as a programmer straight after school, yet still developed the “abstract reasoning” skill on the job, and he’s now considerably more financially better off than his colleagues with maths degrees.
(In all of the above examples I took earnings as some measure of the desired outcome because it is easy to reason about. However the points would still stand if I took many other measures.)
But before I get carried away but the more nuisance points 1-3, the fact that people advocating more maths graduates don’t even acknowledge that the causality of the beneficial effects of maths degree needs to be established for their points to hold, they are in “not even wrong” category.
Of course maths degrees attract clever people and those people, especially those that manage to graduate, do relatively well in life. Is that really an argument for more people having maths degrees?
The moment you translate this reasoning into a more medical setting, you realise how absurd it is. If a depressed individual starts solving sudoku puzzles and eventually gets out of the depression (like most of the depressed people do) it is infinitely erroneous for them to start pushing others depressed individuals into solving the sudoku puzzles. No, sudoku puzzles don’t treat depression. Fact.
Relevant to not considering the counterfactuals is another line of reasoning that is equally irritating to me (don’t mind me fuming…). People who look back on their lives, see mistakes they have made and say that they have no regrets: it’s those mistakes that made them who they are today, they say.
Excuse me… that’s equivalent to just saying you can never make mistakes. This devalues the entire decision process since whatever you decide to do now is valid because this will make future you whoever future you will be in the future.
So when people admit that may be doing maths degree wasn’t the best way of achieving “improved abstract reasoning” but they don’t regret picking maths as a degree anyway and other people should still pursue maths degree even if alternative paths of achieving the claimed benefits exist because they did it this way… You might as well claim it’s axiomatic at this point. “More people should do maths”. QED.
And this point I need to branch out my ran into subrants and, hopefully, I can stop branching and ranting at some point.
Subrant number one: maths degrees are pretty awful at actually teaching maths (this point stands across fields). Generally, there’s this problem that most academics that teach classes don’t have the incentives to teach classes well. Specifically in maths, however, another damaging idea amongst academics took place. The idea that students should be able to reason about maths without resorting to real-world heuristics. Professors insist, that students should be comfortable jiggling abstract definitions about without actually worrying too much about the physical reality that such definitions often represent. I believe that this thinking is inherited from the attempt to axiomise mathematics and decouple it from physics. Not only such thinking is damaging to educating about the field, it is also completely misleading about how mathematicians actually conduct research. No, they don’t operate on abstract definitions in their heads until a proof emerges. They have very good intuitions about what the symbols they operate on represent physically, and only once they see their intuition pointing towards a particular direction, they worry about the technical details of the proofs.
Just today I watched the intuitive explanation of linear algebra series on youtube by 3Blue1Brown. Not only it made me think “god why didn’t they teach it this way to me”, I also realised how terribly inefficient the teaching was. 6 months course got covered in ~1h videos and with a better outcome.
Subrant number two: why is it only mathematicians that seem to care about the issue? Is it that case that they
1) Examined various ways they could improve the humanity, figured out that maths degrees getting unpopular is a serious problem and decided to pour their efforts into advocacy of the field after doing some analysis about pouring their efforts into somewhere else.
2) They just happen to like maths, they get together with other people who happen to like maths, and they talk about the world that doesn’t seem to be as keen on liking maths as much as they do, and about how they should do something about it and they all agree with each because they are in the echo-chamber pre-selected on the principle that they all like maths.
I put my money on 2), and, as a result, that no analysis on which causes are more severe than others has been done by the advocates.
This assumes that the social advocacy is a limited resort. And I insist that it is. Not only individuals are limited in the amount of time and money that they have, they also are limited in the amount of social capital that they have.
Couldn’t this argument be applied to every cause? Well… almost every cause. There’s something suspicious when I hear about cyclists advocating for more cycling lanes (do they just like to cycle or do they know it’s a cost-effective way of lowering healthcare costs?).
However I fail to apply this criticisms to a) global warming. From my understanding many people who are advocates won’t necessarily be directly affected, but they realise that it is an important cause regardless. b) effective altruism. The whole premise of effective altruism is to care about the most cost-effective way of making the world a better place. c) (some) vegetarianism. I can’t directly see how going vegetarian benefits you unless you really really care about animals (as opposed to lowering greenhouse gas emissions).
This is not to say that every cause that seem to only benefit people that argue for the cause is somehow flawed. But I will continue to raise my eyebrow wondering whether the advocates did any kind of comparisons with other causes at all.
I will be writing more about my experience as a maths PhD and the reasons I haven’t finished, so the theme is to be continued…