Better beware of notions like genius and inspiration; they are a sort of magic wand and should be used sparingly by anybody who wants to see things clearly.(José Ortega y Gasset, “Notes on the novel”)

Does one have to be a genius to do mathematics?

The answer is an emphatic **NO**. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the “big picture”. And yes, a reasonable amount of intelligence, patience, and maturity is also required. But one does **not** need some sort of magic “genius gene” that spontaneously generates *ex nihilo* deep insights, unexpected solutions to problems, or other supernatural abilities.

The popular image of the lone (and possibly slightly mad) genius – who ignores the literature and other conventional wisdom and manages by some inexplicable inspiration (enhanced, perhaps, with a liberal dash of suffering) to come up with a breathtakingly original solution to a problem that confounded all the experts – is a charming and romantic image, but also a wildly inaccurate one, at least in the world of modern mathematics. We do have spectacular, deep and remarkable results and insights in this subject, of course, but they are the hard-won and cumulative achievement of years, decades, or even centuries of steady work and progress of many good and great mathematicians; the advance from one stage of understanding to the next can be highly non-trivial, and sometimes rather unexpected, but still builds upon the foundation of earlier work rather than starting totally anew. (This is for instance the case with Wiles‘ work on Fermat’s last theorem, or Perelman‘s work on the Poincaré conjecture.)

Actually, I find the reality of mathematical research today – in which progress is obtained naturally and cumulatively as a consequence of hard work, directed by intuition, literature, and a bit of luck – to be far more satisfying than the romantic image that I had as a student of mathematics being advanced primarily by the mystic inspirations of some rare breed of “geniuses”. This “cult of genius” in fact causes a number of problems, since **nobody** is able to produce these (very rare) inspirations on anything approaching a regular basis, and with reliably consistent correctness. (If someone affects to do so, I advise you to be *very* sceptical of their claims.) The pressure to try to behave in this impossible manner can cause some to become overly obsessed with “big problems” or “big theories”, others to lose any healthy scepticism in their own work or in their tools, and yet others still to become too discouraged to continue working in mathematics. Also, attributing success to innate talent (which is beyond one’s control) rather than effort, planning, and education (which are within one’s control) can lead to some other problems as well.

Of course, even if one dismisses the notion of genius, it is still the case that at any given point in time, some mathematicians are faster, more experienced, more knowledgeable, more efficient, more careful, or more creative than others. This does not imply, though, that only the “best” mathematicians should do mathematics; this is the common error of mistaking absolute advantage for comparative advantage. The number of interesting mathematical research areas and problems to work on is vast – far more than can be covered in detail just by the “best” mathematicians, and sometimes the set of tools or ideas that you have will find something that other good mathematicians have overlooked, especially given that even the greatest mathematicians still have weaknesses in some aspects of mathematical research. As long as you have education, interest, and a reasonable amount of talent, there will be some part of mathematics where you can make a solid and useful contribution. It might not be the most glamorous part of mathematics, but actually this tends to be a healthy thing; in many cases the mundane nuts-and-bolts of a subject turn out to actually be more important than any fancy applications. Also, it is necessary to “cut one’s teeth” on the non-glamorous parts of a field before one really has any chance at all to tackle the famous problems in the area; take a look at the early publications of any of today’s great mathematicians to see what I mean by this.

In some cases, an abundance of raw talent may end up (somewhat perversely) to actually be *harmful* for one’s long-term mathematical development; if solutions to problems come too easily, for instance, one may not put as much energy into working hard, asking dumb questions, or increasing one’s range, and thus may eventually cause one’s skills to stagnate. Also, if one is accustomed to easy success, one may not develop the patience necessary to deal with truly difficult problems (see also this talk by Peter Norvig for an analogous phenomenon in software engineering). Talent is important, of course; but how one develops and nurtures it is even more so.

It’s also good to remember that **professional mathematics is not a sport** (in sharp contrast to mathematics competitions). The objective in mathematics is not to obtain the highest ranking, the highest “score”, or the highest number of prizes and awards; instead, it is to increase understanding of mathematics (both for yourself, and for your colleagues and students), and to contribute to its development and applications. For these tasks, mathematics needs all the good people it can get.

Further reading:

- “How to be a genius“, David Dobbs, New Scientist, 15 September 2006. [Thanks to Samir Chomsky for this link.]
- “The mundanity of excellence“, Daniel Chambliss, Sociological Theory, Vol. 7, No. 1, (Spring, 1989), 70-86. [Thanks to John Baez for this link.]

## 476 comments

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19 July, 2018 at 2:29 am

My CAREER PLANS after university. – María Alegría[…] [1]“Does one have to be a genius to do maths”, by Terrence Tao […]

9 August, 2018 at 1:59 pm

João CortesI would say that suffering doesn’t enhance any kind of creative thinking, in particular it doesn’t enhance one’s mathematical abilities. But I think that there is a fraction of mathematicians/scientists which are drawn to there fields of work as a response to some kind of suffering, wich diffuses into the way they work resulting in the mentioned dash.

So, even if suffering doesn’t make one a best mathematician, maybe it can make a mathematician in the first place.

23 August, 2018 at 12:34 am

Mike IQAre you Terence Tao? I have just seen him on a news about top highest IQ

Nice to meet you.

6 September, 2018 at 2:13 pm

Meet the partial differential equations tamer - BGSMath[…] famous mathematician, Terry Tao, says that ‘the answer is an emphatic no’. I agree with him. It’s neither necessary nor sufficient. A great deal of math research is just […]

15 September, 2018 at 10:31 am

11th class result 2018Thanks for sharing this :)

15 October, 2018 at 5:03 am

A Hypothesis of Understanding Human Intelligence through Quantum Cognition | Cortical Chauvinism[…] “Does one have to be a genius to do mathematics? The answer is an emphatic no.” – Terry Tao, Fields Medalist (https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/) […]

15 October, 2018 at 5:21 pm

yuvalleventalHello again,

You probably see the backlink to your blog, but I have been thinking of this topic for a while now. It is now my belief that the correct way to understand human abilities is not in terms of genius, it is in terms of the ability to “let go” of measuring people by their intelligence. The more somebody has this ability, the more capable they are. It sounds paradoxical but true, and the reasoning stems from quantum cognition.

I will email it to you if you would like to read it. Hopefully you will enjoy it.

Yuval

https://corticalchauvinism.com/2018/10/15/a-hypothesis-of-understanding-human-intelligence-through-quantum-cognition/

23 October, 2018 at 7:06 am

yuvalleventalAdditionally, you say innate talent is beyond one’s control, that is true. But having less innate talent could prove to be an advantage in the long term. Because autism interfered with my life, I discovered I was vitamin D deficient, and learned a lot about myself in the process https://corticalchauvinism.com/2018/06/11/yuval-levental-vitamin-d-and-autism/

3 December, 2018 at 10:10 am

JozefThank you.

3 December, 2018 at 10:38 pm

Geometry Solving TricksGeometry is fun and even more fun when you score the desired marks. With only a set of rules and theorems to memorize, you are on your way of earning the grades you have always wished for.

18 February, 2019 at 3:51 pm

AnonymousVery nice essay. But…Ramanujan. ;-)

[See my previous comment at https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/#comment-3277 – T.]3 June, 2019 at 5:19 am

yuvalleventalI haven’t commented in a while, but looking back, I think the problem here is that a lot of the advice you give is good, but most people already have heard this advice.

Obviously, this is not a complete list. For instance, as most people know already, diet is very important in success. If one were to eat deep-fried McDonald’s food three times a day, that person would be far worse at math than someone that had whole grains for breakfast and salads for lunch and dinner with light dressing.

In this case, the trick is to look at thousands of years of historical practices, and see if we can find something new that we didn’t think about before.

In the past, there were no refrigerators. So people had to store food by fermenting it. The fermentation process created probiotics, good bacteria that provide nourishment and cognitive benefits.

Only in the past 50 years did the western world abandon probiotics, which suddenly caused a dramatic cognitive decline in America and Europe. In fact, on page 31 of this report, it says that top 5% of American students are only as good as the top 50% of Japanese students (As you definitely know, pickled vegetables are very common in Asian cuisine). http://www.cogsci.ucsd.edu/~deak/classes/EDS115/Stevenson_Stigler_1992.pdf

People can be much better mathematicians and thinkers, but there needs to be an international campaign to bring probiotics back as a daily food. It is a basic human right. My favorites are kombucha, Greek yogurt, and saltwater pickles.

More information: https://corticalchauvinism.com/2019/04/15/yuval-levental-probiotics-prebiotics-and-autism/

27 December, 2019 at 6:39 pm

yuvalleventalI have re-thought your statement “innate talent is beyond one’s control”…

That statement is somewhat true, but far from 100% true. Obviously, everyone has their own set of limitations. But the brain can become more or less powerful to an extent depending on certain actions that we perform: https://en.wikipedia.org/wiki/Neuroplasticity

For instance, in the 1950s, American engineering was the best in the world. Since then, in the following decades, Asian engineering became the best in the world, and American engineering is not very strong today in most areas. The reason is that most American foods became far more processed and less nutritious for various reasons, whereas Asian countries kept eating healthily. This change in lifestyle caused Americans to be far less cognitively capable. It’s not biologically possible that Americans suddenly became far less genetically capable in half a century.

24 March, 2020 at 12:27 am

riderrijuSir what is formality conjecture .

3 May, 2020 at 6:04 am

Hollis WilliamsI think at least in the UK the idea of the ‘lone genius’ is due in part to Isaac Newton but the role of mathematics is really to contribute to mathematics present and future and not ‘to be a genius’. In that sense, Newton’s genius actually had quite a negative impact, as his influence on mathematics in the UK swamped contributions from other good mathematicians from outside for centuries to the point that we still feel the damage perhaps even today several centuries later.

4 July, 2020 at 3:23 am

Who can learn math? – Not So Short Notes[…] Terence Tao on “Does one have to be a genius to do maths?“ […]

21 July, 2020 at 10:39 pm

Rupsa Sahui respect you sir. thank you

23 July, 2020 at 1:26 am

ثرثرة عفوية #5 : الرياضيات، وثقافة العبقرية. - فارس.[…] للمزيد: Does one have to be a genius to do maths? […]

24 July, 2020 at 6:37 am

SanjuVery helpful blog…

We feel more easier if we find something interesting,so the basics should be strong enough in mathematics which will make one to feel comfortable. UCMAS offers Abacus mental math program which helps remove math-related anxiousness through active learning process designed carefully by child development program experts.

3 September, 2020 at 9:44 am

Server Bug Fix: How should I proceed when a (famous) professor says I'm not good enough for research? - TECHPRPR[…] perceive as being ‘geniuses’ who actually did harm to their subject. Please read the blog post by Terry Tao on this […]

11 November, 2020 at 2:54 am

BrunoOut of 10 000 PhD a year, 1 will get the field medal.

The chances for the 50 people who get the golden medal at IMO to get this field medal, including all those who wont study math at research level, is 1 in 100 or 100 times more than the PhD persons

But it appears then out of 2/3 people who got a perfect score, 8% will get a Medal Fields including the only woman who got it.

IMO being a math IQ test (very restricted knowledge and very astute thinking required), it shows an extreme predictive power of math IQ to math research potential

11 November, 2020 at 9:37 am

Anonymous (@Dinostraurio)nice observation; let us see it the other way around (which I think is the point of Terence): there are 10K new mathematicians every year which will participate with at least one theorem published in a journal.

9.999K of those are not considered genius by the IMU.

9.95K will not even get a gold medal in the IMO…

Furthermore, if you find one of them in the street, you may not even give a penny for him :)

Besides Terence —the only genius in this discussion— we, the most majority of mathematicians alive, are normal street people who decided to have a challenging profession, not genius at all.

23 November, 2020 at 3:56 am

AnonymousDo “the most majority of mathematicians” contribute anything new (and quite important as well) to math?

23 November, 2020 at 9:07 am

AnonymousCurrently “unimportant” mathematical contribution may become “important” in the future.

24 November, 2020 at 10:30 pm

AnonymousRight, this happened in the history too. But what I was mainly asking was do they have the ability to contribute new things while not being genius, mostly with the help of hardwork?

26 November, 2020 at 5:31 am

Anonymous (@Dinostraurio)Exactly, that is the hole point of the discussion here; most of the theorems proved and published out there are the result of hard work, not of a genius idea…

30 December, 2020 at 8:34 am

Mike SmithI am in a PhD science program and have been told that my GPA isn’t high enough to continue. I can graduate with a master’s, but I need to pay the rest of the funds. How can I make up for this pitfall, while being able to follow my passions?

30 December, 2020 at 10:34 am

AnonymousGet a job, and follow your passion in your spare time.

30 December, 2020 at 10:40 am

Mike SmithCitizen scientist then, it is

5 March, 2022 at 2:09 am

AnonymousWho is selling you that crap? Find a place where you will continue.

28 February, 2021 at 1:20 pm

fatherofdragons23Grothendieck told deligne that, math is not a sport.

2 March, 2021 at 1:22 am

DanielHello, Professor Tao. I’m an Engineering student in my second year, but with a very avid interest in higher mathematics. Is there any more efficient method of self-study and also is it a good idea to restrict my study to certain fields only at this stage?

2 March, 2021 at 12:33 pm

NikDo you think it is possible for anyone (with normal mental capacity) to become a mathematician given dedication, hard work, patience, etc.?

19 March, 2021 at 7:44 am

YuvalHello, I have done a lot of research on diet and intelligence, and the results that I discovered are astounding. Essentially, anti-nutrients in food and artificial sugars/fibers dramatically lower intelligence. These foods are all too common in America. I would say that these discoveries could even change the world as a whole for the better.

https://corticalchauvinism.com/2020/07/27/the-best-diet-plan-for-autistic-people-and-for-everyone-else/

https://corticalchauvinism.com/2021/03/15/opposition-to-artificial-sugars-impact-on-autism-and-general-health/

13 July, 2021 at 12:07 pm

CalcuquackI am also a member of the cult of geniuses and the cult of prime numbers(tm).

28 July, 2021 at 10:57 am

Chris Smith“The popular image of the lone (and possibly slightly mad) genius – who ignores the literature and other conventional wisdom and manages by some inexplicable inspiration (enhanced, perhaps, with a liberal dash of suffering) to come up with a breathtakingly original solution to a problem that confounded all the experts…”

That is true in some cases. John Nash developed Nash Equilibrium using this method as a graduate student. Some of the professors thought that he wouldn’t be able to initially succeed.

28 July, 2021 at 11:43 am

AnonymousOther famous examples are Galois and Ramanujan.

3 November, 2021 at 9:47 pm

AnonymousRamanujan was a genius so he can’t be an example of non-genius who contributed to mathematics.

28 July, 2021 at 5:25 pm

AnonymousNash wasn’t lone at the time though—he was in contact with the other experts in the field like von Neumann. He also wasn’t mad or suffering quite yet.

29 July, 2021 at 8:08 am

AnonymousHe wasn’t schizophrenic yet, but many of his fellow graduate students thought he was very strange (see A Beautiful Mind by Sylvia Nasar). He got along with some faculty members, but was still an outsider compared to the faculty’s expectations as a whole.

22 September, 2021 at 1:18 pm

jercaiguitarThanks for article. I am going to start university soon to do a 4 year maths degree. Throughout my life especially in high school I was surrounded with math “geniuses”,people who aced at olympiads, and competitions without trying much. While I had to study very hard just to get top marks in the normal exams. I’ve always loved maths but sometimes I just think that there are so many people out there greater than me and maybe I’m not really needed.

But thanks for the post anyways, I will continue having fun with math in the next 4 years.

3 November, 2021 at 5:19 am

RiemannHypothesisProofYou are certainly wrong! To contribute to mathematics, one has to be either a GENIUS (like you – Terry Tao) or one has to be in mathematics profession in ivy league western university.

There is are no exceptions.

Because, if ome is not a genius or one does not go to do pure maths in top university, then you are simply of no use to mathematics.

Reasons:

1. Even if one solves some big long standing problem, like Riemann Hypothesis, no one from genius-maths-community (like you) ever wants to even read the paper.

2. You and maths geniuses like you, publicly declared that you will not respond to amateur’s emails even, and do not want non-genius-mathematician to send you emails.

3. Websites like arXiv and blah blah want to upload maths preprints only from geniuses or from pure mathematicians from top universities.

4. Asking non-geniuses and non-mathematicians to go and publish their findings in Peer Reviewed Maths journals is a JOKE. No editor of top journal, that includes you when you were editor for maths journal, even properly reads the paper and even send it to a referee to review, if the paper is about significant maths result and it’s not from a GENIUS or from TOP UNIVERSITY MATHEMATICIANS.

5. There can’t be any contribution to mathematics if no reputed mathematicians like yourself is ever interested in reading a proposed paper from non genius etc.

6. You saying you would consider looking into to someone’s finding after that someone’s work has been recognized by some other mathematicians, either through publication in reputed journal or by some recognized maths genius personally endorsing some non-genius’s work.

But the the other mathematicians like you have more or less same attitude and same policy. So non-maths-genius’s work will never reach the mainstream maths community.

7. Even if some mathematician reads non-maths-genius’s work on problem like Riemann Hypothesis, and finds no error, they don’t want to accept the fact that some non-genius non-pure-maths professional can ever be right. After all if genius-mathematicians like you couldn’t prove Riemann Hypothesis, then how can a non-genius be right! So let’s just non bother reading or listening to what non-genius found.

You see, there can’t be any meaningful significant contribution by non-genius in mathematical field.

There is no room for non geniuses in mathematics.

You may be genius in mathematics, but your opinion, that one doesn’t have to be genius to contribute to maths, is certainly flawed and is not reflective of what happens in the real (maths) world.

3 November, 2021 at 9:51 am

AnonymousReasonable arguments.

3 November, 2021 at 10:04 am

AnonymousYou assume that the problem is malevolence or ego, rather than a lack of time to read thousands of dubious proofs of famous problems that are extremely unlikely to be correct.

3 November, 2021 at 1:58 pm

RiemannHypothesisProofMalevolence, Ego, Lack of time,

or Assumption that a proof from non-genius-mathematician is unlikely to be correct… … Whatever the reasons , good or bad,… …. the fact is that a non-genius-mathematician CAN NOT make significant contribution to the field of mathematics!… Because even if we go by your argument, if the genius-mathematicians do not have time to read non-genius’s work assuming that it would be extremely unlikely to be correct, then the original problem still remains.

Also, there are not really thousands of dubious proofs of famous problems.

For example, for Riemann Hypothesis, there are about 15 proposed proofs printed on arXiv and some useless journals which are of no value.

These proofs are know to be wrong because people have read found flaws.

However, there are about 180-190 proposed proofs on places like researchgate, vixra, and what not.

Yes, the majority of these would be wrong too!

But there are 2000 mathematician-geniuses out there in USA and the likes.

Even if 1 maths-genius genuinely investigates one proposed proof from one of those 180 non geniuses, just once in a lifetime, that would be enough. And that would actually save time too, because same 180 people will not then keep writing to 2000 geniuses one after another and then repeating the process of years, giving feeling that there are 1000’s of dubious proofs.

Whatever the reasons, whatever the numbers, IF YOU ARE NON-GENIUS-MATHEMATICIAN, whatever work you do, even if worth the Nobel prize, will go waste, and you would not have contributed anything to the mathematical world, BECAUSE YOUR WORK WILL BE LABELLED AS DUBIOUS WORK EXTREMELY UNLIKELY TO BE CORRECT, WITHOUT ANY SERIOUS CONSIDERATION OR REVIEW.

3 November, 2021 at 7:47 pm

AnonymousMaybe we should randomly generate strings of math characters and have Terence Tao check each one to see if it is a correct proof of the Riemann hypothesis. That way you can complain that random generators are not being taken seriously as mathematicians.

3 November, 2021 at 8:02 pm

RiemannHypothesisProofWhat you are saying is crap, and looks like you don’t understand the main point. No one is asking Terry Tao to check random strings, or to check anything.

The comment is simple, that it’s not worth for any non-maths-genius to waste his/her own time trying to contribute anything meaningful to the field of mathematics, because no one would read it!.

And people like you, if you are maths-genius yourself, have time to argue uselessly on forum but don’t have time to read a proposed mathematical solution. You need to first learn to understand before arguing, and especially without hiding under “anonymous” skin.

3 November, 2021 at 8:06 pm

AnonymousDon’t be stupid!

3 November, 2021 at 8:43 pm

VickyDid Terry Tao posted an article saying that “Random Generators can contribute to the field of mathematics” ?

If he does use his position and make such comments, then, of course one can complain that random generators are not taken seriously!…(despite Terry Tao proclaiming that random generators can contribute to maths).

Here it seems that Terry Tao is telling the world that even if you are not a mathematical genius you can still contribute to the Nobel field of mathematics. But then such people who take Terry Tao’s word for it, are never taken seriously,…unless of course they are mathematical genius too.

3 November, 2021 at 9:44 pm

AnonymousSeriously!

You are comparing ‘random generators’ with ‘amateur mathematicians’, who are human beings, even if most of their work may be flawed, they may be hardworking, serious and nice human beings. Just because some people who may not be professional mathematicians but may still be interested in mathematics, you start comparing them with “random generators”. That’s derogatory.

Didn’t your mother teach you to respect other human beings?

I bet you are not even a mathematician, let alone a decent human being.

4 November, 2021 at 10:26 pm

AnonymousI do think amateurs can contribute to math. At the same time it’s unreasonable to get angry at mathematicians for not reading their work given they are right a vanishingly small fraction of the time.

By the way, this article is not about amateur mathematicians. It’s about professional mathematicians who aren’t geniuses. He says the phrase “professional mathematics” in the article. I don’t know what his opinion is about whether amateurs can contribute.

By the way I don’t have a problem getting my papers in serious journals. Maybe learn the conventions of mathematical writing before submitting. (And yes I looked at your proof on vixra.)

Finally, I did not start the hostility here. This gentleman should be respectful to Terence Tao. No one is entitled to Tao’s attention.

5 November, 2021 at 11:19 pm

RiemannHypothesisProofWith so many people posting comments as ‘Anonymous’, it’s hard to track who made reasonable civil arguments and who was the one who started hostility. However to the person who claimed that he made effort to read a paper, I want to say Thank You!

4 November, 2021 at 10:37 pm

AnonymousOne more remark: it is absolutely a waste of one’s time to attempt famously difficult problems like RH. This is true for amateurs but even for most professional mathematicians. Tao is not saying “non-geniuses can contribute to math by proving RH out of nowhere with no track record.” He’s saying mathematicians who are not geniuses can contribute to the discipline as a whole. And probably this extends to highly motivated amateurs. If you are an amateur mathematician who wants to do math research, pick something that isn’t one of the most famous unsolved problems. This is much more likely to be taken seriously and you are much more likely to succeed.

3 November, 2021 at 8:54 pm

AnonymousIt seems that the field of mathematics is indeed reserved for deserving genius.

Like someone above highlighted, ordinary people (i.e. those who are neither genius not mathematician) should find interest in some other fields in the world, they are not welcome in the mathematicians world.

10 November, 2021 at 1:20 am

No nameTo be honest, most smart and intelligent individuals are not interested in career in mathematics. There are plenty of options for brilliant people, including medicine, engineering, infotech, mining, and what not. And the importance of mathematics nowadays is felt important only by the community of mathematicians. Even the famous open maths problems are overrated and really doesn’t matter much to economy or to the development of the world as a whole…the importance is only overrated by mathematicians themselves, again.

Mathematics has become so unimportant that Clay institute had to announce million dollar prize per a maths problem, because except for some nerds and crankpots, no one is interested, and that includes most mathematicians. Glorifying open maths problems and attaching a price tag to it was a marketing stunt to revive people’s interested in maths. Doesn’t work in long term.

And , no, you definitely do not have to be genius to be a mathematician. Not at all. You do have to be a genius to be Bill Gates, Kerry Packer, or Warren Buffett. And they don’t care about mathematics.

So I partly agree with Terry Tao, that you don’t have to be a genius to be a mathematician.

I understand that most people reading comments on this blog are mathematician or wannabe mathematician, and so may not like my comment, and that ok!

12 March, 2022 at 5:08 am

Understand advanced mathematics? | Since 1989[…] process, a sudden insight comes, but it would not be possible without the painstaking groundwork [https://terrytao.wordpress.com/ca… ].Indeed, most of the bullet points here summarize feelings familiar to many serious students of […]

23 March, 2022 at 11:51 pm

Rinika MukhopadhyayI think that there is a fraction of mathematicians/scientists which are drawn to there fields of work as a response to some kind of suffering, which diffuses into the way they work resulting in the mentioned dash.