Mathematicians Are Overselling the Idea That “Math Is Everywhere”
blogs.scientificamerican.com
blogs.scientificamerican.com
https://history.princeton.edu/people/michael-j-barany
I'm troubled that a Princeton history student can write about a worldwide phenomenon with so little reference to non-Western cultures. (This is a pet issue of mine, as I am an American who lived in east Asia for years after studying Chinese and sinology.) I don't think he has looked at development economics and the history of popular attitudes toward economics enough to understand how important basic understanding of mathematics is. In other words, I disagree with the conclusion of his article, summed up in the last paragraph.
"Imagining math to be everywhere makes it all too easy to ignore the very real politics of who gets to be part of the mathematical elite that really count—for technology, security, and economics, for the last war and the next one. Instead, if we see that this kind of mathematics has historically been built by and for the very few, we are called to ask who gets to be part of that few and what are the responsibilities that come with their expertise. We have to recognize that elite mathematics today, while much more inclusive than it was one or five or fifty centuries ago, remains a discipline that vests special authority in those who, by virtue of gender, race, and class, are often already among our society’s most powerful. If math were really everywhere, it would already belong to everyone equally. But when it comes to accessing and supporting math, there is much work to be done. Math isn’t everywhere."
I feel particularly deceived about this state of affairs because my parents paid a lot of money for my formal education (school, university), and I got significantly less value from it than from people simply willing to share their knowledge for free.
It might be worth thinking in this "math for and from the elites" mindset when describing public aversion to the subject. Seeing it as beyond oneself or not relevant to one's life lends little towards anyone's desire to learn anything about it.
It's easy to say "why don't they just do it," with some diffuse definition of "they" in mind, but even the most basic things can be hard to get for the poor. A proper maths education is one of them, I think.
When would they have the time and money to set aside to get deep into a subject that most of them do not have the academic background to see the value of?
Yes, everyone could learn mathematics. Just like everyone could get rich and go into space. It's just that for a huge proportion of people the pre-requisites are not there.
Ah the false analogy[1], we meet again. You don't really believe this do you? I mean you could make the same argument about gardening. Literally anyone can learn at least basic math, most people just don't have the inclination.
Imagine me or someone else that was never part of the math research establishment came up with the exact same proof Perelman came up with. Ipsis verbis.
You can rest assured that at least the slack that they cut him on sketchy details would not be extended to an outsider. (In a way the Yau debacle was a manifestation of this)
There are a few examples of this already in somewhat obscure fields.
That's assuming anyone would even give the proof the time of day.
Apart from this conflation, the article is very confused, mixes lots of unrelated topics, and is hard to follow. Maybe a rewrite is in order? Or perhaps writing logical arguments, with clear ideas that make sense is also an elitist thing we should avoid?
For example, he mentions "math" repeatedly but doesn't put boundaries around it. Is he talking about Algebra not being everywhere? Or Calculus is not necessary for everyone? Well, if you magnify the photo at the top of the article, you'll see books such as:
+ number theory
+ theory finite elements
+ topology
+ Navier-Stokes equation
Because of my tech background, my first pass at his essay made me think it was a variation of arguing against the "coding is for everyone -- everyone should learn programming". However, I don't think Barany's opinion about math is an equivalent analogy.
I too would like to know what/who Barany might be arguing against; maybe it would clarify the article.
"Math can be used to model just about everything, and so can make a meaningful contribution to almost every endeavour" is a completely different statement from "Everyone has access to learning math to do anything math-related they want".
The second argument is not one I have heard people making, and if they make it I doubt they would express it as "Math is Everywhere". I have heard people make the first argument as "Math is Everywhere".
This is a political opinion, of course, and not an assessment on the extent to which maths help people.
> Transmitted to the computer using error detecting codes (in USB)
> Being run through a CPU that essentially performs arithmetic, comparisons and branches.
> Being processed by a word processor that has been compiled in one of many of the highly mathematical languages that run on our computers.
> Written onto a disk with error detecting codes.
> Sent over the internet with error detecting codes.
> Written into a database that probably bases on relational algebra.
> I open the article and everything happens in reverse on my computer so I can view it.
And it claims maths is not everywhere. I just have to laugh.
We have to recognize that elite mathematics today, while much more inclusive than it was one or five or fifty centuries ago, remains a discipline that vests special authority in those who, by virtue of gender, race, and class, are often already among our society’s most powerful.
No, we do not. This is a factually false statement. We have to recognize that maths is open to everyone who is determined enough and has access to the internet or a library and that tribalism like this brings us nowhere. Anyone can participate in elite mathematics. If you come up with a proof for the Riemann hypothesis, go ahead and publish it! Nobody cares about your background if your work is good. Mathematics is blind to gender, race and class.
The article just conflates the field itself with these wider socio-economic conditions, constructs a weird argument out of the confusion, and comes out really bad.
Your hypothetical setup is strange. If you have all the time in the world - you can do anything.
>In other words, to even get a chance to commit time to something like mathematics at the level we're talking about, there are hard practical social and economic preconditions which have nothing to do with one's abilities, or even the field of endeavor as such.
Like what preconditions exactly?
Ok. That makes sense. Specialized knowledge is required for specialized work. What's your point?
>The author is stating that there is an advantage for some people to have the conditions to learn.
I guess this is the meat of the argument the author is making, with some ideologically-driven, and here unstated, reasons that explain this 'advantage'.
>Since there is a "society" it shouldn't be critical for everyone to have the same developed skills, which the conclusion seems a bit disconnected from.
But it isn't critical. There is a requirement to have a base level understanding of mathematics in order function in society (roughly high-school level), just as there is a requirement for literacy - but there's certainly no requirement to have in-depth knowledge of it. Nor is a deep understanding of math a prerequisite for success, financial or otherwise.
If you're asking whether an average person has a good chance of proving the Riemann hypothesis or has access to the latest ideas in the field, the answer is no. But I make my own clothes without having access to Paris or Milan or Hong Kong. We can all do rewarding non-elite things.
Math is everywhere in the same sense that I need an understanding of physics to throw a ball. That is, yes, you can find it if you look, and sometimes you may even have a trained understanding of the application of some principle, but you do not need to be able to explain it or consciously apply it step by step.
Sometimes it would help to know how to apply it, or know other ways of solving a problem, but most people need far less conscious use of maths than people with a maths background tend to think.
> No, we do not. This is a factually false statement. We have to recognize that maths is open to everyone who is determined enough and has access to the internet or a library and that tribalism like this brings us nowhere. Anyone can participate in elite mathematics. If you come up with a proof for the Riemann hypothesis, go ahead and publish it! Nobody cares about your background if your work is good. Mathematics is blind to gender, race and class.
This both manages to not address the quote you were "answering" and be a demonstration of relative privilege at the same time.
It doesn't address it because the quote is not saying that elite mathematics isn't open to everyone in theory, but that it de facto is something that is mostly accessed by a certain group.
You brush against this yourself when you write that it is open to everyone "who is determined enough". And this is where privilege comes in.
People are shaped by their surroundings. If you grow up in a working class family, it is not always that you couldn't get into a top university and aim for a top job, but that even if you can you are vastly less likely to be aware of the opportunities, and to have the support network, and to have the contacts, or to have the same expectations of being able to do something, because you constantly see people around you who has achieved it, as someone growing up more privileged.
It is not just about money. I did not grow up in a rich household. But I did grow up in a household where getting a university degree and aiming for a job requiring one was considered so self evident it was never something we discussed. It wasn't "are you planning on going to university or trade school or start working after graduation?" it was "which universities are you applying to?" Most of my close relatives are academics, with three professors amongst my aunts and uncles.
As a result it never once crossed my mind growing up to go for a career in a manual trade for example.
Subtle differences, some backed by money, some just byproducts of growing up in families with money or connections or, as in my case, an innate expectation that your future should involve studying for a degree, makes a huge difference.
I've also noted in the past on HN that one of the things that growing up in Norway did for me was that I never had a sense of worrying about unemployment.
When I did my first startup, it was never a consideration that I might be without a job. Why would it? I'd be financially secure no matter what.
In my case it was because of a strong social security net first and foremost, but access to help from family etc. does the same. Even just living in a society where making enough to save to set aside a rainy day fun is something a majority can do makes a huge difference.
Growing up in a position of relative privilege alters things in many ways:
You may be expected to study instead of being expected to start earning as early as possible. You likely grew up with enough food to avoid being damaged by malnourishment. You're less likely to have urgent, stressful financial concerns make it infeasible to set aside time to ponder more intellectual issues. You're more likely to have a job that earns you enough to allow you to spend time on other things. You're more likely to have people around you that achieve something intellectual to inspire you. And so on.
So yes, "anyone" can participate in elite mathematics. And sometimes people from unexpected backgrounds does. But in practice, more privileged people are more likely to.
This isn't specific to mathematics - it applies to pretty much any field where "payoff" is uncertain or requires lengthy investment in education first, which causes less privileged people to self-select away to a greater extent than people with more privilege.
The only people who have my sympathy in those regards are the working poor, that is, people who have to hold multiple jobs, more than full-time, just to get by and not starve. These are the people whose potential to excel in mathematics (or anything, really) is undermined.
If anything, I feel mathematics is more open than many other STEM fields. I got my degree in mathematics, and I noticed the gender gap in classes was nonexistent. In the calculus and statistics and other classes which were required for engineers, the classroom distribution would definitely skew male. But in the pure math classes, like topology, abstract algebra, real analysis, etc that would only be taken by math majors, the gender distribution was 50-50. In several of my classes the women outnumbered the men. And this was also true of the faculty: 3/6 professors I took classes with were women.
His conclusions states:
"If math were really everywhere, it would already belong to everyone equally. But when it comes to accessing and supporting math, there is much work to be done. Math isn’t everywhere."
I guess the same might be told for art or history ; they are everywhere but artists and historians are not and becoming one of them is difficult.
When it comes to math that influences public policy, we're often talking about actuarial mathematics. It’s not “elite”: it's an area of complicated applications that will determine whether our economy will sink under health-care costs or not. Math for national security? Crypto, and since good implementation of the math ideas is at least as important as the ideas, I don’t think it’s as elite as he makes out.
Gromov-Witten theory is built for the few. Elliptic cohomology is build for the few. Whittaker functions are built for the few, the elite, the more-likely-than-not Russian. This is what the NSF funds, among other math. But that doesn’t jibe with the intermittent discussion of policy and … economics?… attempted in the article.
I don’t think this author has a point. The author doesn’t distinguish between math used in and for public policy and math funded by the taxpayer. The author is trying to argue that since people aren’t good at math it maybe doesn’t exist (?) (“If math were really everywhere, it would already belong to everyone equally.” What does this mean?) The author tries to address racial/social/class inequities in math education access but only sort of randomly, at the end, without discussing causes, effects, or mechanisms, and while ignoring the international face of mathematics and the mathematicians who exist today.
Access to data, the tools to manipulate it, and the knowledge to understand it have never been more accessible. There is a growing understanding that stats is as important as calculus in the high school mathematics curriculum.
Sure - getting a job as a tenured mathematics teacher is tough. Same as an NSA researcher. But it's crazy to say they have a monopoly on mathematics.
I see math similar to many other types of knowledge. You can get by with fairly little, but the more you know, more often you will recognize opportunities to use it.
(See also: an offshoot of the Daily Mail Oncology Ontology, https://kill-or-cure.herokuapp.com/ )
1. The author's claim about "the politics of who gets to be a part of the mathematical elite" has almost nothing to do with the contention that "math is everywhere".
2. Math is everywhere. I make this claim by virtue of the facts that (1) you can, in any situation, ask questions like "how many?", "how much?", and "which one?" (which leads to encodings) and (2) Math is just our ordinary processes of reasoning, made rigorous and mechanized. Whether you're looking at someone's face, walking through the zoo, or cooking, there are underlying mathematical realities to be considered, if you're interested.
3. The author writes, "When we talk about math in public policy, especially the public’s investment in mathematical training and research, we are not talking about simple sums and measures." Except that, sums and measures and similar basic arithmetic are extremely relevant to making policy. When we talk about % growth targets, or inflation, or social security, back of the envelope arithmetic is just what you need to get an idea of what a policy actually does, or what it's results have been.
4. Examples in this article are cherry-picked and without supporting context: "Priests used astronomical calculations to mark the seasons and interpret divine will, and their special command of mathematics gave them power and privilege in their societies" But this is only true if mathematical prowess (rather than say, winning a lottery or being supposedly divinely annointed) was the key to joining the priesthood. Once accepted as a priest, what difference would it make whether you gave mathematically correct, or entirely nonsensical predictions to a crowd?
5. As another commenter writes "Mathematics is one of the most open fields of [S]cience". This is spot on. Existing free resources are really fantastic, and while you can make a completely valid point that not all children are given the relevant fundamental education to allow them to take advantage of that, the same argument applies to any field of education -- down to and including basic literacy. It makes no sense to lay the blame for this on mathematics.
tl;dr: This article is a giant non-sequitur, chock-full of poor reasoning.