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I'm a Math PhD student in pure math. Many times I have encountered people who just doesn't value pure math or does not see the beauty of pure math. Most of the time these are those people who haven't done any "serious mathematics" or any advanced level mathematics. I forgive these people because I understand that their ignorance is coming from their lack of knowledge and it's going to be hard for me explain or show them the beauty of mathematics simply because they don't know advanced mathematics.

I don't know what is the impression of pure mathematics in the industry, but my guess would be they also don't care much about pure math because they don't really need it that much.

I've also had these kind of experiences with many undergrad students who take some math courses but are not math majors. I have tried to explain it to them by saying that math is not just about using formulas and applying them in the real world, but it's also about the quantitative skills, critical thinking and problem solving skills that you develop along the way, which will be useful throughout your life and you'll need it everywhere.

But the real problem is dealing with those people who insult pure mathematics and are (have) pursuing (pursued) advanced degree in applied math or who come into mathematics from engineering background or some other more applied discipline. I find it surprising that even though these people have been exposed to quite a lot of mathematics, they still say such things. How do they not understand that just because people haven't found a real world application of some of the abstract concepts in pure math at the present that doesn't make it useless, neither does that mean real world applications can't be found? There are so many instances where people have been able to use some of these abstract concepts into the real world many years after their discovery.

(I do not intend to insult or humiliate entire group of people doing applied math or are into other more applied disciplines. I'm just surprised to hear such things from some of the people in these groups)

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  • 34
    How do you deal with ignorant people who say "literature is useless because it doesn't have any application"? Apr 5 at 3:36
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    Classic responses include: "What's the use of a newborn baby?" or "I do not know what it is good for today, but in 50 years, this will be taxed." Apr 5 at 3:45
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    Aren't expressions like "forgive these people" and "insult pure mathematics" a bit over the top? Apr 5 at 5:16
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    Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Academia Meta, or in Academia Chat. Comments continuing discussion may be removed.
    – cag51
    Apr 5 at 21:21
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    Why do you feel the need to “deal with” or change these people at all? If you yourself like what you do, that should be enough for you. Why care about the opinions of others?
    – Seb
    Apr 6 at 13:27

16 Answers 16

70

[Applied mathematician here.]

In truth, the argument these people make is not entirely wrong, and I don't think your us-vs-them approach to their them-vs-you argument is very useful.

I think an objective perspective on pure mathematics would probably have to include that the majority of what pure mathematicians (however you want to define that) work on is not driven by any current applications and is unlikely to ever find applications. Of course there are papers that 50 years later have found uses, but pointing to these one-in-perhaps-millions is not a good argument. Nor is that applied mathematicians have "frequently" drawn onto material created decades before in addressing current problems -- perhaps they have drawn onto some basic concepts, and use the language previously created, but more often than not, a lot of work still needed to be done to make this material useful to applications, and that's work applied mathematicians then did. (They were "applied" mathematicians by definition here: They wanted to solve an actual problem. Of course, they may previously have been pure mathematicians.)

A better argument, which I personally subscribe to, is to draw an analogy to other "pure" endeavors: Perhaps to people who study medieval literature, or music theory, or other forms of art; or astrophysicists; or those who research quarks and Higgs bosons. Humans are curious creatures, and we get pleasure out of figuring stuff out and "knowing". And humanity is better for it, knowledge is what we strive for.

Not everyone needs to appreciate every one of the endeavors mentioned above. My parents were scientifically educated people (my dad was a professor of biology) and appreciated music but could absolutely not find any kind of appreciation for music outside a very narrow range of classical music created between ~1700 and ~1850. I tried to explain to them some of the beauty behind modern rock and heavy metal music, but gave that up pretty quickly. But they would have appreciated the work of a literary scientist researching Dostoyevsky. The point is simply that some people like some things but not others; but most educated people can appreciate the diversity of what people contribute to the "humanity project", and one perspective you could take is to embed yourself as one small cog in this "humanity project" of all of us together creating knowledge whether it is useful or not, because that's just the kind of critter we are.

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    I agree (apart from the bit about heavy metal being beautiful). Several times I heard pure mathematics PhD students give the justification that their field sometimes finds valuable applications that nobody had predicted. But they only seemed to know a small set of examples - talking to different people almost felt like listening to a recording. And they never talked about what proportion of papers found applications or what would have happened if the pure research had not been done. It would have been better if they had talked about what motivated them and why they liked it.
    – toby544
    Apr 5 at 9:06
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    Not hard to believe that results on "semi-flasque projectively-etale $\infty$-fibered cohomology of simplicial meta-abelian sheaves on locally presentable sub-schemes" are purely self-contained and not really "for" anything else.
    – Randall
    Apr 5 at 12:49
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    (I write this as a "pure" mathematician disenchanted by the superior attitudes of many purists.)
    – Randall
    Apr 5 at 12:51
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    Arguably, humans can be conditioned to find almost anything "beautiful." I went through a noise-core phase, for example. Was enjoying that somehow really justified in some sense, or was it just conditioning? It's arguably more objective to enjoy classical and think of heavy metal as noise (though I still enjoy the latter). Your answer is excellent though. Similarly it's arguably more objective to value that which is directly useful or impactful on one's life over that which is theoretical or far away, even if they are connected. E.g. computers vs theory of computation.
    – jdods
    Apr 6 at 11:26
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    @Randall is that a real paper title?
    – justhalf
    Apr 6 at 17:04
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I'm a professor who teaches and does research in both pure math (topology) and applied math (data science + Hawkes process models). Fortunately, I've not come across the situation you're describing very often. For sure I have lots of students who are only interested in applied math and don't really see the value of "pure" math. I point out:

  1. It's essential to understand the hypotheses under which the conclusion of a theorem holds. Point out the 2008 financial crash as a warning against relying on mathematical results when the hypotheses are not met. This also justifies the need for careful and correct proofs. Resonance is also a good example, that an engineer might appreciate.

  2. Many applied math topics grew out of pure math, e.g., Fourier series, the use of Taylor polynomials, several programming languages, astrophysics uses of topology, Nash equilibria thanks to fixed point theory, Netflix problem and singular value decomposition, applications of representation theory to voting theory, higher categories and mathematical physics, etc.

  3. There are many applied math problems where no one has any idea what to do next. In some cases, e.g., topological data analysis, it's clear that more pure math knowledge can help. In others, like "what is a neural network actually doing and how can I explain why it came to the answer it did" we need a richer model, and traditionally applied math draws models from pure math.

  4. I can't be sure, but from your description it sounds plausible that the people you are talking to prefer an "us vs them" mentality that will not serve them well in life. It's better for them to befriend the pure mathematicians who might give them exactly the tool they need, when they need it, which they cannot know now. It's also plausible that they are putting down an area they find difficult and unnecessarily abstract. But, just because something is hard for you is no reason to denigrade it. I would encourage these students to approach their studies with an open mind and recognize that being challenged and struggling to make sense of the material you're studying is the norm in academia.

To summarize: instead of saying "hey, I'm doing this pure math thing because maybe it'll have an application someday" you might instead say "hey, there are tons of moments in history when applied math was stuck and then was able to draw an answer from pure math, and here's some specific applied math area that seems stuck right now, that I think my pure math might someday help."

The answers to these MO threads might help:

Unreasonable application of mathematics to the other areas

The Unreasonable Effectiveness of Physics in Mathematics

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  • The summary I got here is: "Applied math is just Pure math that has been applied to something." "Applied math is just Pure math that has found a purpose." "Pure math is a scratch ticket waiting for its future to be discovered in Applied math." Apr 5 at 17:08
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    Oh, no, I definitely don't think that! Plenty of pure math is worth doing for its own sake, regardless of applicability. Just look at my publication list! But, if you are talking to an applied math student who is trying to understand why they should learn pure math, it helps to give them a reason in terms of something they care about. Apr 5 at 17:38
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    I think the statement "Many applied math topics grew out of pure math" is historically not correct. I would suggest taking a look at the history of Fourier analysis (en.wikipedia.org/wiki/Fourier_analysis#History) for example to see that nearly every early contributor to the topic was actually solving a concrete application. These were not pure mathematical endeavors. Apr 6 at 21:19
  • The more often I read "pure math" the less sense I can make of it. I then read the sentence again and leave out the word "pure" and suddenly, the sentence makes sense.
    – Dirk
    Apr 6 at 21:42
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    I second @wolfgangbangerth s comment. Actually, the idea of something like "pure math" is pretty recent. I learned in an interview with Lanczos that it was coined in the early 1900s (and he called it a "narcissistic endeavour for the human mind"). Before that there was only math. Good times.
    – Dirk
    Apr 6 at 21:46
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The flip side of this are of course the occasional pure mathematicians who are of the mistaken belief that lower level of abstraction means that things applied people do are shallow and easy. I think that kind of arrogance in both parts of the mathematical population is damaging to both sides. My impression is that it is only a small minority of people on either side who think that way, but unfortunately they do feel like speaking up.

My view on the pure vs applied issue would be that most pure mathematics probably does not have practical applications, in the sense that the mental tools produced by pure mathematicians in most of their work are necessarily better suited to any given applications than the tools that we produce when we are trying to solve applied things. But on the other hand, it is also valid to say that any set of computational/mathematical tools that is powerful enough to do a broad amount of mathematics will be Turing-complete, which means that we can encode any applied problem we like in it, so easy dismissal of applications of something is also unjustified just because it looks like abstract nonsense.

I think one could perfectly well justify the existence of pure mathematics as an exact abstract art form, similar to games like chess or Go, which also have cultural value to an extent that probably justifies the resources poured into them.

Another point that can be made in defence of pure science in general is that science is essentially a very large system that tries to learn about the universe, and any such system has to deal with exploitation-exploration tradeoffs. Exploration always means following some of the time paths that do not lead to the desired goal. For difficult problems, the percentage of effort that an efficient agent will expend on paths leading nowhere is going to be high. In that sense, when applied research is being done well, it will also go down a lot of paths in that search tree that do not in the end prove useful. One could say therefore that applied and non-applied research are not all that different at that level.

Finally, pure research can construct versions of real problems that are much cleaner than real-world problems, and try to resolve them, and thereby either show that we have now the tools to solve these idealized problems, or that in a rigorous sense we cannot do even that. I think that kind of finding should be of direct interest for applied people. In the other direction, applied questions should also be a source of inspiration for pure research.

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TLDR (Assuming the question is about your peers, not your boss/superior). Frame challenge: You can't reliably change other people's opinions. The only thing you can directly influence is yourself and your actions. Accept that others will sometimes have (frustratingly) different opinions than yourself. It is up to you to decide whether you'd like to stay friends/friendly with such a person, or distance yourself from them.


You're asking how to "deal with people who have a negative opinion of pure mathematics" (and presumably share it loudly in front of you).

May I ask, however, what do you want to achieve by "dealing with them"? You obviously seem frustrated by your peers expressing that opinion. So, is your goal to:

  1. Convince those people that pure mathematics is useful?
  2. Convince those people that pure mathematics eventually leads to applications?
  3. Convince those people that pure mathematics is beautiful?
  4. Make those people passionate about pure mathematics?
  5. Get those people to fund you to do pure mathematics?
  6. Become friends with those people?
  7. Get those people to not insult pure maths in front of you?
  8. Feel less frustrated?

Because, if your goal is any of 1.--4., that will impossible to achieve with all of your peers. While you may be able to control some aspects of this situation (e.g. the arguments you present), fundamentally, you have no direct influence on the opinions of other people. Ultimately, if your goal is to influence something externally, you will have to accept that your influence is limited. This is also true if you're trying to achieve this as an educator -- even the best lecture in the world will leave some indifferent (which doesn't mean that you shouldn't try).

There are of course some ways you can attempt to influence the people around you with your opinions (maths is useful and beautiful, art enriches people's lives, multitasking reduces overall productivity, walkable neighbourhoods are good for mental health, Thor is the god of thunder, the Earth is flat, heavy metal is the pinnacle of musical achievement, vaccines cause autism) -- posting online, preaching, writing opinion pieces, organising rallies and events, just having passionate 1-on-1 conversations. And, it doesn't matter what the opinion in question is (or even whether it is a fact rather than an opinion) -- you will never be able to convince everybody. The fact of the matter is, people really dislike changing their opinions.

It is my opinion that 5., while related, is off-topic here. (A question on how to get funding if your research is primarily theoretical obviously contains aspects of 1.--4., but is a whole other can of worms and would be better served as a separate question. That I would also like to know the answer to. A similar consideration would be true for any case where your goal is to achieve some tangible benefit -- promotion, raise, recommendation letter. But, your question seems to be focused on peer interactions.)

If your goal is 6. (become friends), but you already tried (and failed) convincing somebody that pure mathematics is useful/beautiful/eventually applied, then it is up to you to decide whether you still want to attempt a friendship. Now, this is finally something within your control. In general, if a person holds an opinion that is different than mine, I don't mind as long as they are not insisting on changing my opinion. You think pure maths is not beautiful, I can find no deeper meaning in the visual arts, but we both like nature so let's go for a hike and be friends. On the other hand, if that person is dismissive/insulting towards you because you hold differing opinions (personal example: "I just don't understand what kind of an idiot you would have to be to spend so long in education, and then choose to spend your time teaching others."), I have to ask, why do want to be friends with that kind of person? I would handle the difference in core and casual beliefs similarly (If you like metal and I like pop, I can try to be friends with you. If your opinion about the relation between vaccines and autism drastically differs from mine... we can be acquaintances, but I don't want to be your friend). Ultimately, you can't be friends with everybody, and that's just something you have to accept.

If your goal is 7. (avoid hearing any "insults" of pure maths), first I'd say to think about how you got in a conversation where somebody is insulting pure maths in front of you. Did you insist on repeatedly discussing a topic with somebody you know has opposing views to yours? (e.g. if you ask somebody for their favourite music genre and they say "pop", but you prefer metal, nobody would consider that an insult. If then you spend the next 30 minutes trying to explain the superiority of harmonies used in metal, would you really be surprised if they asked you to "stop talking about that infernal noise?"). If you're doing so, stop initiating the same topic. Or, do they bring up and insult the topic on their own (e.g. are you joining a group of flat-Earthers in their daily discussion of all the sheeple that still believe in the floating ball in the sky theory)? If so, you could try asking them politely to stop once ("Hey, guys, just because I think it's a ball and you think it's a disk, doesn't mean we can't be nice to each other"), but if they ignore your request... move away. Don't listen. Finally, if they are approaching you on a regular basis to insult you and/or try and change your opinion ("Hi bro, have you come to your senses and accepted that Thor is the embodiment of thunder? You know he will smite you down with his hammer if you don't accept him."), and they don't stop after you've asked them to; they are acting unprofessionally and there might be a bullying case somewhere in there.

Finally, if your goal is 8., then your only option is to learn how to be less bothered by other people's opinions. Surely we've all seen some variation of the meme "Ah, time to spend hours explaining to random strangers on the internet that they are wrong". And I can't imagine that makes them feel better. Since people really dislike changing their opinions, and your external influence is limited, the only thing you can really change is how it effects you. In life, you will encounter, and often have to work with or alongside many people with differing opinions. Learning how to accept differences in opinions will ultimately make it easier for you to make friends, and also make you a more pleasant person to work with professionally.

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You say "you mean like art, music, and literature? Well, actually, pure math is much more useful than those, but that's not the point. Math is beautiful -- like art, music, and literature. If you are not capable of appreciating that beauty, that's fine. Some people don't like Shakespeare, or Mozart, or Dostoevsky. But they don't usually advertise that."

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    Mathematics is different from the arts and this is not likely to be convincing. Maybe if you gave an example of how it is beautiful, that would work.
    – toby544
    Apr 5 at 11:07
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    Pure mathematics might be termed as an art, but it is just a symbolic description, pure mathematics is the language in which science is written, it is expanded vastly with tons of applications.
    – learner
    Apr 5 at 11:31
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    "Mathematics isn't a language, it's an adventure" Paul Lockhart in A Mathematician's Lament
    – Peter Flom
    Apr 5 at 11:46
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    "pure math is much more useful than those" -- is that true? The number of people who appreciate art, music, and literature is vastly larger than the number of people who appreciate pure mathematics, to the point where many artists, musicians, and writers have become very wealthy, and support what must surely be hundreds of thousands or millions of people in these businesses. Perhaps they do not create anything that "could potentially have a use in a few decades", but I don't think that pure mathematics "is much more useful". (As I point out in my answer, this does not reduce my appreciation.) Apr 6 at 5:33
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    Is "being appreciated" the same as "useful"? I don't think so, but I can see how some people would.
    – Peter Flom
    Apr 6 at 10:12
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Well, we can always throw in the fact that a great deal of "applied math" is not actually "useful" for anything, either. :) That's a significant point, and, while sounding a little frivolous, for me it accurately hints at some inaccurate-but-traditional understandings of "application", and such.

For that matter, in various contexts, over the years, people solving PDE's (possibly numerically) declared that they were "doing applied math"... because the PDE's were abstractions of some physically meaningful process. And, then, not solving PDE's could not possibly be "applied mathematics". Plus, there was "turf". :) Some years ago, trying to hire a person who had a paper in the Annals, on number theory, and also had a few patents about cryptography, and was employed by a for-profit, ... I was told that "well, he doesn't do applied math". I asked, "Ok, it's math, and it's certainly applied, so...?" (And, of course, none of my "opponents" had any patents at all, and had no affiliation with for-profits, but that was not relevant to them...)

In fact, in a similar way, although many applied mathematicians would judge me to be "pure", the mathematics that I'm interested in seems to have a bearing on human beings' understanding of the world. (Indeed, some things are just puzzles, while other things may seem/be more serious.) In my perception, human beings (some of the time...) model the world by mathematics... making some reasonable (but sometimes volatile) assumptions about "how the math will work out".

Probably by this year there is not much concern among accountants about how compound interest works... but the mathematics of signal processing (120+ years ago), and even fancier things in mathematical physics, use mathematics that, in itself is highly non-trivial to understand. Luckily, they can do experiments, which can be very persuasive.

So, I'd say that, for some people, mathematics is a main way to understand the world in human-narrative terms. This notion of "world" does include human fantasies... and, maybe like AI's?, some hallucinations? :)

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    I've always thought that the terms "pure" and "applied mathematics" are more often used to group different mathematical sub-disciplines, rather than describe the motivations of their practitioners. I'm pushing to use the term "motivated by applications" in our next job ad, because I think that describes more accurately the kind of person I'd like my department to hire. Apr 6 at 5:46
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Maybe just agree to disagree? I pursue math because I like it, not because I care what other people think about it.

Alternatively you could explain that differential geometry is a pure math that has found application in gravity, same with topology and abstract algebras in other areas of physics.

(See Lie Algebras or Spinors in Spin Rotations)

Im just an undergraduate student in math, but I have started to feel like the distinction between pure/applied math is somewhat unhelpful, and not worth making, seeing how often "pure math" fields end up finding super cool applications. Just my 2¢ :)

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    "Agree to disagree" ... Well, OK. But what if the person disagreeing is the Dean, and what is at stake is funding for the Math department ...
    – GEdgar
    Apr 5 at 7:12
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    @GEdgar then you've already lost. If your Dean controls the budget and thinks academic departments should only exist if they're "useful", then you need to find a new academic institution.
    – Brondahl
    Apr 5 at 12:15
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    @Brondahl Or maybe not the dean, but a member of parliament.
    – gerrit
    Apr 5 at 13:02
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    Same answer. There is no value in trying to convince someone that believes "this thing should only be funded if it's useful" that maths is useful. You'll spend the rest of your life continually arguing about whether <this> particularly sub-subject is useful or not. Because, fundamentally, most of the time, most of mathematics, isn't "useful" in any way that that person is going to be able to understand. That person's ancestor from 100 years ago would never have been able to understand why the Number Theory that formed the basis of modern encryption would be useful.
    – Brondahl
    Apr 5 at 13:15
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    That person will only be able to understand that tiny slivers of the incredible universe that is mathematics, are useful. So if you engage in that argument, you'll end up losing the rest of it anyway. Just refuse to engage in the premise that you need to justify the usefulness of mathematics with them.
    – Brondahl
    Apr 5 at 13:18
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Simply tell them that pure mathematics is the language in which science is written and therefore pure mathematics is the foundation of all applications. This is the statement of Einstein, I guess.

Then take some names of famous mathematicians in the respective region or continent. For example, I live in India and I take the name of famous mathematician Ramanujan, when such stupid questions are asked to me. I know all Indian people are passionate about Ramanujan even who doesn't understand mathematics. As soon as I take his name, people say "oh he was a big mathematician". Then I ask in return to them "what did Ramanujan do in mathematics?", naturally they don't know the answer but they believe he did something great. I then tell them that there are numerous applications of pure mathematics which people don't know because we are ignorant. We just don't know, but it doesn't mean there is no applications. Immediately, I give the example of number-theory which has real world applications in cryptography, coding theory, and also calculus, differential equations are the backbone of chemistry and physics and engineering science etc. Also differential geometry has close relationship with relativity theory and Einstein said that without the development of foundational geometry of David Hilbert (then friend of Einstein), I couldn't have worked on relativity theory. At the end, people who asks those types of ackward questions calms down to sense.

Interestingly, the above method only takes not more than 5 minutes to convince those people successfully. I have tried the above method several times. You can try it.

Edit: There are numerous applications of mathematical formulas in physics and chemistry and respective engineering branches developed solely by pure mathematics. Without those formulas basic sciences couldn't advance so far. More interestingly, it is a reference showing applications of topology in molecular biology, describing shapes of DNA.

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    That absolutely works for India and Ramanujan, but that's kind of a unique circumstance? Maybe the UK and Turing or Newton would work similarly, but outside those three examples, I can't really think of any mathematicians who are iconic to the point that everyone in their home nation has heard of them. Even for incredibly influential mathematicians in Math, I'd be at least a little surprised if, for example, a random Swiss person knew Euler.
    – Idran
    Apr 5 at 15:23
  • @Idran, Newton is a global figure, undeniably, it works anywhere in the world. Carl Friedrich Gauss is a famous mathematician who has made intensive contributions in physics, e.g., the Gauss divergence theorem is known to all science students, at least. There are others as well.
    – learner
    Apr 5 at 16:14
  • Ah, I thought you meant you always made it a figure that the person knew as an icon of their own culture. That makes sense, yeah.
    – Idran
    Apr 5 at 16:29
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    "Pure mathematics is the language in which science is written." Some science, not all science.
    – toby544
    Apr 5 at 17:25
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    @learner I was commenting on your claim that "pure mathematics is the language in which science is written." Here is a recent paper from Nature: nature.com/articles/s41586-024-07130-8. It uses statistics, which uses pure mathematics, but it is not "written in" pure mathematics.
    – toby544
    Apr 5 at 20:18
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"How do they not understand that just because people haven't found a real world application of some of the abstract concepts in pure math at the present that doesn't make it useless, neither does that mean real world applications can't be found?"

This reminds me of a conversation I had with a much older close friend in Undergrad (he was a Math PhD student)

Me: "Man I wanna really have an impact! I want to change the world and find some wild theorem that alters our view on everything"

Him: "You're an idiot. The world needs toilets. Not math"

Obviously in the grand scheme he was right. People are dying from poor sanitation. In a more comfortable world people are worried about becoming middle class financially. In the ivory tower of academia (but not where the mathematicians live) people have problems involving REALITY (ex: how to make ___ physically possible, how to understand discrepancy ___ ) and the work that pure mathematicians do is distant. Maybe in centuries it will be applicable similar to how fermat's little theorem forms the bedrock of RSA. But the reality they live in is completely different than your reality.

I still think its fun to paint! It's fun to sculpt. Its important that society allocates some number of people to do these things. For similar reasons its important that we study math and society allocates people to do it. But if you get into a pissing contest of "no my thing is super useful in this immediate way ___" you're going to lose. The value that mathematical research brings to society is much more long term but no less than the value that physical art brings to civilization. I might even call mathematics as "logical art" just to drive the point home.

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You can't make someone understand something he doesn't see worth understanding (even if others claim such understanding).

1
  • I'm not sure we should see people as immutable objects - perhaps "doesn't see worth understanding" isn't always a permanent condition?
    – uhoh
    Apr 6 at 23:35
3

I've also had these kind of experiences with many undergrad students who take some math courses but are not math majors. I have tried to explain it to them by saying that math is not just about using formulas and applying them in the real world, but it's also about the quantitative skills, critical thinking and problem solving skills that you develop along the way, which will be useful throughout your life and you'll need it everywhere.

If a student chooses to take a math course and then comes up with this kind of question, you might well ask them: "Why did you choose to take this course and what do you want to get from it ?"

This question is usually answered with a silence (especially when the question is put in the presence of other students ...) since it's something that students have not clearly asked themselves - although they are always quite quick to say what's wrong with any given course.

For many students, they are advised to take course X if their major is Y. This is the official advice from professors, the blurb on the module description and the story of students in the year ahead of them. You might say that they decide to do so for fear of doing Y without X. Now, X may be essential or at least a good help to Y in certain fields. But it's not always essential and academics should be judicious in what they advise - especially where the major can be pursued independently of the minor in question and where a student has more interest in another minor. A classic example of this might be economics and math. Knowing math will help those students who eventually go down the road of economic modelling. But if a student dislikes math and has a better feel for say sociology then the latter would be a better minor choice.

As to how to deal with this category, I'd stress the importance of students being made to feel responsible for informing themselves on the requirements of their major, so they clarify own intentions and make their own conscious decisions on major-minor combination.

But the real problem is dealing with those people who insult pure mathematics and are (have) pursuing (pursued) advanced degree in applied math or who come into mathematics from engineering background or some other more applied discipline. I find it surprising that even though these people have been exposed to quite a lot of mathematics, they still say such things. How do they not understand that just because people haven't found a real world application of some of the abstract concepts in pure math at the present that doesn't make it useless, neither does that mean real world applications can't be found? There are so many instances where people have been able to use some of these abstract concepts into the real world many years after their discovery.

With these postgrad students, you might well also ask them why they chose the math course concerned and what they want to get form it.

Yet the ignorance of pure math <=> applied math exchanges is really unacceptable. Everyone knows by now that every field of pure math finds application within a generation of its development. For people with a non-pure math primary degree, this ought to be the very basis on which they are taking these courses - plus of course the recent high salaries enjoyed by math proficient staff in the financial sector.

As to how to deal with such cases, I wonder would you consider an initial no-commitment 1-week introductory set of lectures to clarify things for those who haven't yet clearly decided on their way forward ?

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A reliable way to demonstrate the value of pure math is to refer to the ongoing flow of erstwhile-pure-math results into applied areas.

As one example, you may mention to your collocutors the method of induced representations. Devised (for finite groups) by Frobenius in 1898, it was probably regarded at that time as a highly aesthetic exercise of mind. The method had to wait for 41 years to be extended to the Poincaré group in a celebrated work by Wigner (1939) who demonstrated that the method is needed to classify species of elementary particles. And that was not the end of story. In 2018, exactly 120 years after its discovery, the method found an application in the theory of steerable group-convolutional neural networks.

Speaking of Group Theory at large:
When Galois (1829) published his prophetic work, who could imagine that his super-abstract ideas would eventually develop into tools critical for investigation of crystals, atoms, and elementary particles? (Aware of the existence of crystals, his contemporaries were still unsure about the existence of atoms, not to mention elementary particles.)

Among other examples of the kind, I would also mention Number Theory, which used to be a paragon of abstract mathematics, but eventually became the basis of cryptology.

A historian of mathematics would surely be able to offer dozens of such examples.

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@Michael_1812 mentioned group theory, I want to add that finite group theory as developed in the 20th century mainly by pure mathematicians, namely the CFSG, was used a few years ago by Lazlo Babai to anwer a longstanding problem in computer science on the computational complexity of the The Graph Isomorphism Problem. The proof gave an algorithm and has practical consequences. Also in other instances, combinatorial results, like the Sunflower Lemma, are often used in algorithms, in particular in the field of Parameterized Complexity. And mentioning computer science, I am pretty sure, should the NP vs P problem ever be solved, it will involve some interesting and new "pure" mathematics.

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    When complex numbers were invented/discovered, the guy who did them said something about "And these will never have any practical application for anything." Unfortunately, an EE prof sat in on his talk, and said "This is EXACTLY what I need for AC circuit analysis." Apr 6 at 0:20
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    @WolfgangBangerth No, I looked it up. The next day the professor expanded on it and talked about having three imaginary units and quaternions. He talked how it unifies mathematical physics, but also admitted "it will never have any practical applications". (Un-)Fortunately, a computer animator sat in the audience and said "This is EXACTLY what I need to smooth out my animations."
    – StefanH
    Apr 6 at 13:17
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    @WolfgangBangerth But seriously, the story of the quaternions is another nice example how something that was even firstly despised as too theoretical found applications 100 years later in computer graphics.
    – StefanH
    Apr 6 at 13:19
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    @WolfgangBangerth Of course, I know the history. I thought it was pretty obvious that my comment was meant to be a joke.
    – StefanH
    Apr 7 at 13:43
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    @StefanH I'm sorry I didn't get the joke. I'm pretty sure the first comment above was not a joke. Apr 7 at 18:16
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Your critic reminds me of the old lady who asked Faraday what was the use of his work on electricity: "Madam, what is the use of a newborn baby?". My advice to give people like that the attention they deserve: ignore them unless you can think of a witty put-down.

During my career (applied mathematician turned software engineer), I frequently turned to mathematics to find tools to solve problems. If someone works with new problems, they may find they need mathematics that hasn't been applied before: if a pure mathematician has already done the work because he finds it interesting, that saves the engineer doing it.

I suggest that you look at Pevzner et al(2001). We can safely assume that Euler did not think: "I really must work on this topological problem because someone will need it in 250 years to sequence DNA". Should he have worked on a problem that didn't have a practical application at the time? Does your critic have an infallible method of predicting what maths will and won't be useful at some time in the future?

Another example of maths that proves useful after the fact is Riemannian geometry, which found an application when Einstein invented General Relativity. And what is the use of General Relativity, Dr Einstein? Well, one day it will make the GPS in your smartphone work.

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  • +3 for Faraday (yes, I mentioned that one above, but only in a comment), GPS and topology. Apr 6 at 12:25
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How do you deal with ignorant people who say X?

I believe that either wittingly or unwittingly, sometimes these things are said to get under your skin.

Other times they are said for the benefit of the sayers - to feel better about not understanding that huge (and exciting) body of knowledge and research, or not taking the time to try.

In other words, it's about them, not you. So instead of focusing on how to deal with them focus on not dealing with them, at the moment.

You could engage with "when did these feelings begin?" or "what part of pure math in particular do you feel is most useless, and how were you initially exposed to it?" or just "Well obviously there's a whole lot of people who don't see it that way." Depending on if you'd like to help them explore their thinking or just end the converstaion.

When life gives you lemons, make lemonade!

There may be a blogpost, and article in Quanta Magazine (or similar) a lecture, a class, or even a book in your future!

Start recording these, everything you can remember from past interactions and encourage new ones. Try to find recurring themes. Is it certain aspects of mathematics about which people feel this way most? Is it folks who are struggling with something academically? Is it social conversation or within an academic context?

That's your chapter 1, the motivation of your writing.

Then choose several wonderful examples of where what was once thought to be pure mathematics collided with the real world. Just for example:

Of course complex numbers are a good example, we use discrete mathematics in some areas of industry, and the discovery of quasicrystals actually drove advancements in basic theory. You get the idea.

Use this frustration to write up anything between a short essay or handout, to a special class curriculum, to an article in popular press, to a book.

That's how to deal with them - teach them!

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This is how I explain to my relatives and friends when they ask the same question to me:

What distinguishes human beings from other animals (at least most of them, cf. chimpanzees) is that we have the intelligence and energy to work on purely useless things to expand the boundary of our knowledge --- a creature should be proud of, rather than ashamed of, doing useless things, with pure math being one of them.

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  • Handicap principle at work here, haha.
    – StefanH
    Apr 7 at 14:52

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