Sets are probably one of the most dull objects in maths, or at least overused as the become dull. Everything is a set, a circle is the set of points with a distance r from the center c, and so on.
There are a lot more interesting things to write about than yet another article about Russels paradox. Maybe next time do the Erlanger program on why everything is a reflection.
I think they are used everywhere because you can describe everything with them easily.
I mean i still prefer a set of points over a newly introduced definition of a "circle" just to be able to express a problem in mathematical language.
It's kind of the same for functions too, they're used to describe so many things sometimes in a weird context too.
I think the best abstraction is that which creates as little mental overhead as possible. Sets and relations are pretty basic and simple in that regard.
I found it more an article about Snow, about the short 20th century (do we have a better name for it yet?), about admissible plays in Glass Bead Games, than merely about Russell, who was but the hors d'oeuvre.
(and, far from being dry —or at least, if you are having dry sets, you might consider paying more attention to your interlocutor before introducing connectives—, Comfort has as much as Sunak to do with this article)
When non-mathematicians ask me to tell them something interesting about math, I show them the lambda calculus. It's straightforward and there's a kind of narrative involved when talking about reduction. Then it's fun to blow their minds with how this simple rewriting process can be used to implement anything that a computer can do.
During my undergrad education in engineering, we were required to include a significant number of 'humanities' courses. In hindsight this was clearly beneficial but even at the time one could see the sense in that. But it struck many of my classmates and me that it was a bit odd that the converse requirement did not exist. The "arts and humanities" majors were not required to take more than a couple of "math and science" courses. Maybe one of each? While we were taking full on regular 1st, 2nd, even 3rd year courses from their world, there were watered down math and science courses on offer for them.
The asymmetry was baffling. We were told this would make us "well rounded", but it seemed that those outside the math-science-engineering world were considered Just Fine without even a 3rd semester of calculus, and were allowed to indulge a greater portion of their time at university in matters more directly aligned to their interests. This definitely put a division into the academic atmosphere, though not an acrimonious one; the two camps peaceably coexisted and as undergrads we all just did what we had to do.
That was a while ago now, maybe things have changed somewhat. But from the article, it sounds like maybe not so much.
To be fair though, at least in the USA an engineering degree requires significantly more coursework than a humanities degree. At my institution it basically added up to an extra year of courses, and was approximately equal to the added "general education" (non-engineering) requirements. And the general-ed requirements for engineers were less demanding than for the arts. So it's not like we were getting short-changed in a way that they were not.
But that said, I agree with your main point. Humanities majors should be required to do a more in-depth survey of math, science, and engineering comparable (at the very least) to the introduction engineers are required to have of their fields. If that means they need to have a 5-year degree as well, so be it.
There was once a time when mathematics was considered central to humanistic and philosophical development. It is said that at the entrance of Plato's Academy, there was the following prohibition: "Let no man ignorant of geometry enter here."
Well, most people in math and science majors can pass an arts and humanities major. The opposite is not necessarily true.
Anecdotally, I can easily engage in talks about literature, politics and art with my friends coming from a biology or physics background. I can hardly do so with my friends with a major in French or Sociology. When we mix them together, the debate is always animated when we talk about philosophy, a specific author or book, and even painting. If we speak about GPT3 or the CERN, suddenly the table is divided in two.
You can have some seriously in-depth discussions in sociology or philosophy as well, that would be hard to follow for someone with just a STEM background. The kind of sociology or philosophy one would discuss without preparation or reading with friends is not really comparable with the academic disciplines. It is just armchair philosophy/sociology.
I have a feeling many technically minded people would struggle to read and write an essay about Heidegger or Habermas. In fact, I doubt most people are even aware of that type of philosophy due to not having actually delved into it very deeply, and yet as outsiders to such fields they feel confident in making claims that those fields are "easier." (And usually there is also an implication that this being "easier" is an indication of those fields being less serious or worthwhile.)
If my non stem sample would be able to engage with armchair science in the same way that the stem sample can engage with armchair philosophy, I would agree with you.
However, it's not the case.
I also find more stem people reading entry level materials in philosophy around me (like Moral letters to Lucilius), that I will find non stem people ready entry level materials in science (like A brief history of time).
The ratio is hugely disproportionate, and while I understand this is all anecdotal, I have a large and very diverse social circle to observe, spanning on wide areas of age, social backgrounds, political preferences, sexual orientation and jobs.
Selection pressure on science courses is largely around fulfilling prerequisites for the following course or exam. With a few university programs also polling recent graduates about what they have/haven't found useful. The connection between these, and transferable understanding of the physical world, is ... problematic.
A pre-K science educator once suggested children have a human right to make sense of their world, "now" rather than several lifetimes later. Perhaps they imagined adapting existing K-12 content to pre-K.
But consider "color", widely taught around K. First-tier physical-sciences graduate students often reply to "What color is the Sun?" with answers like "it doesn't have a color" and "it's rainbow colored", reflecting foundational misunderstandings of color. Suggesting the challenge here isn't adapting working content to a younger audience, it's a lack of working content. And not just for "that falls through the gaping cracks" topics. At a first-tier university, asked what students entering intro-genetics from intro-bio most lacked, a noted professor replied "a firm grasp of central dogma". Which is foundational, and is taught, repeatedly - repeatedly unsuccessfully.
Consider someone suggesting liberal arts majors should study "science" ... as an old-school whole-class call-and-response exercise in mindless memorize-and-regurgitate. One might wonder if that would largely be a waste of their time. Not because their lives couldn't be enriched by understanding science, but because there's so little connection between this form of science education and that potential enrichment.
Science for non-majors is sort of a science education research area. But how much better are we set up to support it, than a call-and-response culture would be?
As for making sense of the physical world, for pre-K, liberal arts, or first-tier physical-sciences graduate students? What's that phrase ... Big Hairy Audacious Goal.
Who are the "knee-jerk literary math-haters" the article is aiming at? I'm not aware of any and I certainly don't think the public or the establishment, in general, "hate maths". There is a weird tribalism in this article.
I think there is context you are lacking. This is an opinion piece within a British magazine. The UK prime minister, a member of the Conservative Party, has just proposed that the standard curriculum for high/secondary school extend the requirement to take math through age 18 / grade 12. Apparently right now it is only required that math be studied through age 16 / grade 10.
Because this is a suggestion coming from the Tories, there has been in the last month or so a whole slew of opinion pieces ridiculing the suggestion that non-STEM students should be required to take maths past age 16. It's been called an elitist, out-of-touch policy change, requiring students to learn useless math they will never use, etc. etc.
By and large these were all very politically-driven opinions. I can't help but think that the same people making these arguments might have been arguing the exact opposite--in favor of more maths education--if it had been their turn in power, or if the Conservative position was different.
So yes, this opinion piece is rather tribal. But I think it is because he is responding to an intensely political issue and trying to talk some sense into people who are already forming opinions on a purely tribal basis.
“Haters” might be too strong a word, but of ugh-ers there are plenty, and humanities education does not seem to help—neither with the ugh-ing, nor with the social approval of said ugh-ing. And that is substantially harmful up to and including societal scales[1].
Not that the ugh-ers are entirely at fault! In society where nobody actually teaches anything deserving to be called “maths”[2], certainly nothing that would give any clue as to why people would actually want to do this thing[3], there’s little they could do to do better. Except maybe, as adults, wonder why there is this strange cluster of people who appreciate this apparently repulsive thing and investigate the reason for themselves.
(Not that this appreciation is universal even among experimental physicists, for example.)
This article reminds me very strongly of Lockhart’s Lament [0], which at this point I think should be required reading for every mathematician and teacher. The essential message is the same: maths is itself intrinsically fascinating as the exploration of foundational logical reasoning, so teach that instead of the boring stuff taught in maths classes now! (Lockhart writes in a specifically American context, but the general philosophy seems much more widely applicable to me.)
This has been my thinking for so long. Why teach a bunch of courses students say is boring? Can’t we instead teach basic logic and philosophy in early years and then start students on basic set theory (or some equivalent) in their second or third year and have them slowly derive mathematics themselves?
This isn’t really related to what I asked though? I don’t think drilling basic arithmetic is “teaching the students the basics of sets and having them derive mathematics themselves” and you surely don’t either
"Why won’t our artistic-literary establishment recognise that there’s mystery, beauty and humanity in maths?"
I have a maths background and I see none of mystery or beauty in it. Dunno what 'humanity' means here.
I've noticed those in humanities are terrible at debating. They can't focus, slide off the subject and tend towards handwaving arguments with little rigour. Their soft subjects are important but they neglect 'our' side of the fence.
There's quite a bit of beauty in math (see Escher's works, and Permutation City; some things are enchanting on their own like fractals and strange loops). I think it is just not explored by artists enough, potentially due to unfamiliarity.
32 comments
[ 2.8 ms ] story [ 69.4 ms ] threadThere are a lot more interesting things to write about than yet another article about Russels paradox. Maybe next time do the Erlanger program on why everything is a reflection.
(and, far from being dry —or at least, if you are having dry sets, you might consider paying more attention to your interlocutor before introducing connectives—, Comfort has as much as Sunak to do with this article)
There's a bit more going on there than just what's your favorite kind of math.
Sets really only get mentioned once in passing; the article is more a general defense of the beauty of mathematics.
https://www.stevenstrogatz.com/books/the-joy-of-x
The asymmetry was baffling. We were told this would make us "well rounded", but it seemed that those outside the math-science-engineering world were considered Just Fine without even a 3rd semester of calculus, and were allowed to indulge a greater portion of their time at university in matters more directly aligned to their interests. This definitely put a division into the academic atmosphere, though not an acrimonious one; the two camps peaceably coexisted and as undergrads we all just did what we had to do.
That was a while ago now, maybe things have changed somewhat. But from the article, it sounds like maybe not so much.
But that said, I agree with your main point. Humanities majors should be required to do a more in-depth survey of math, science, and engineering comparable (at the very least) to the introduction engineers are required to have of their fields. If that means they need to have a 5-year degree as well, so be it.
Anecdotally, I can easily engage in talks about literature, politics and art with my friends coming from a biology or physics background. I can hardly do so with my friends with a major in French or Sociology. When we mix them together, the debate is always animated when we talk about philosophy, a specific author or book, and even painting. If we speak about GPT3 or the CERN, suddenly the table is divided in two.
I have a feeling many technically minded people would struggle to read and write an essay about Heidegger or Habermas. In fact, I doubt most people are even aware of that type of philosophy due to not having actually delved into it very deeply, and yet as outsiders to such fields they feel confident in making claims that those fields are "easier." (And usually there is also an implication that this being "easier" is an indication of those fields being less serious or worthwhile.)
However, it's not the case.
I also find more stem people reading entry level materials in philosophy around me (like Moral letters to Lucilius), that I will find non stem people ready entry level materials in science (like A brief history of time).
The ratio is hugely disproportionate, and while I understand this is all anecdotal, I have a large and very diverse social circle to observe, spanning on wide areas of age, social backgrounds, political preferences, sexual orientation and jobs.
A pre-K science educator once suggested children have a human right to make sense of their world, "now" rather than several lifetimes later. Perhaps they imagined adapting existing K-12 content to pre-K.
But consider "color", widely taught around K. First-tier physical-sciences graduate students often reply to "What color is the Sun?" with answers like "it doesn't have a color" and "it's rainbow colored", reflecting foundational misunderstandings of color. Suggesting the challenge here isn't adapting working content to a younger audience, it's a lack of working content. And not just for "that falls through the gaping cracks" topics. At a first-tier university, asked what students entering intro-genetics from intro-bio most lacked, a noted professor replied "a firm grasp of central dogma". Which is foundational, and is taught, repeatedly - repeatedly unsuccessfully.
Consider someone suggesting liberal arts majors should study "science" ... as an old-school whole-class call-and-response exercise in mindless memorize-and-regurgitate. One might wonder if that would largely be a waste of their time. Not because their lives couldn't be enriched by understanding science, but because there's so little connection between this form of science education and that potential enrichment.
Science for non-majors is sort of a science education research area. But how much better are we set up to support it, than a call-and-response culture would be?
As for making sense of the physical world, for pre-K, liberal arts, or first-tier physical-sciences graduate students? What's that phrase ... Big Hairy Audacious Goal.
Terminology aside, it didn't seem that the article was trying to drive wedges, but in fact attempting to build bridges.
Because this is a suggestion coming from the Tories, there has been in the last month or so a whole slew of opinion pieces ridiculing the suggestion that non-STEM students should be required to take maths past age 16. It's been called an elitist, out-of-touch policy change, requiring students to learn useless math they will never use, etc. etc.
By and large these were all very politically-driven opinions. I can't help but think that the same people making these arguments might have been arguing the exact opposite--in favor of more maths education--if it had been their turn in power, or if the Conservative position was different.
So yes, this opinion piece is rather tribal. But I think it is because he is responding to an intensely political issue and trying to talk some sense into people who are already forming opinions on a purely tribal basis.
Not that the ugh-ers are entirely at fault! In society where nobody actually teaches anything deserving to be called “maths”[2], certainly nothing that would give any clue as to why people would actually want to do this thing[3], there’s little they could do to do better. Except maybe, as adults, wonder why there is this strange cluster of people who appreciate this apparently repulsive thing and investigate the reason for themselves.
(Not that this appreciation is universal even among experimental physicists, for example.)
[1] https://twitter.com/TeaKayB/status/1596485647115554817
[2] https://www.maa.org/external_archive/devlin/devlin_03_08.htm...
[3] https://physicstoday.scitation.org/doi/10.1063/1.2810679
[0] https://web.archive.org/web/20221201000000*/https://www.maa....
I have a maths background and I see none of mystery or beauty in it. Dunno what 'humanity' means here.
I've noticed those in humanities are terrible at debating. They can't focus, slide off the subject and tend towards handwaving arguments with little rigour. Their soft subjects are important but they neglect 'our' side of the fence.
> Who dares call maths dry?
Me. But it's useful.