8 ms·

How to become a pure mathematician or statistician (2008)

111 points by ipnon 1y ago

nphardon 1y ago
"Just follow this simple road map". I think about it differently. If you want to become a pure mathematician, you have to publish research in pure mathematics. There are many different routes one can take to accomplish this, and the route that you can stick with and enjoy is the best one.
- kbrkbr 1y ago
  I could not agree more. I am on my second try to master mathematics (30 years after the first), and I can see, understand and appreciate mathematics mainly from the constructive standpoint. Nothing wrong with classical mathematics, as also used in this roadmap. Having axioms and drawing logical conclusions or searching proof does just not click for me. Give me 0: N and suc: N -> N and I see how to construct stuff. Induction makes sense right away as a case distinction on those two constructors.
- kronicum2025 1y ago
  As a young person growing up in India, enrolled in engineering in a small desolate city, when I wanted to be a mathematician, with no real path in sight, this blog post is actually what I followed, and learnt everything that is mandatory on this blog post during my undergraduate on my own, and then applied for entrance examinations for grad institutes in India, and now have successfully finished my PhD nicely in US. So, in that sense, this blog post is for people like me, who had no exposure to research scientists in around 2012 in the places we are growing up. (I didn't yet have stable internet then, only once an hour every evening). They still have some years before they could actually come in contact with researchers who can give them good problems to work on. and the local universities are horribly out of whack in their skill/desire to teach the students properly. And so, in the meantime, till you get access to good research problems, you prepare yourself to be ready for the problem when the day finally comes. And this blog post along with another one: https://webspace.science.uu.nl/~hooft101/theorist.html https://webspace.science.uu.nl/~hooft101/theorist.html was what kept motivating me when I was studying. The list of books is as important as anything else. To be fair, in 2012, math.stackexchange.com had grown up quite a bit, but I was not aware of it for various reasons.
  - yu3zhou4 1y ago
    Very inspiring story, congrats on your journey!
- guyman16 1y ago
  What different routes are there to publish research besides academia? I would love to work on publications but it is not practical for me to return to an institution right now.
  - Ivan92 1y ago
    I don't know what your current industry is, but Google publishes their research and collaborates with academic institutions is one that immediately comes to mind. The government (NASA, NSA for one), tech companies (IBM, Microsoft, etc.), medical companies, aerospace/defense (JPL), just to name a few. I am sure there are way more than I could think of and I am sure others will care to fill in as well. Research is not exclusive to academia.
    - unixbeard1337 1y ago
      Yeah, but this is specifically about pure mathematics. Do Google and medical companies publish pure math research?
      - nativeit 1y ago
        Sure, this was the first result: https://www.nature.com/articles/s41586-021-04086-x https://www.nature.com/articles/s41586-021-04086-x
  - throwaway652368 1y ago
    I've never seen a math journal that requires academic credentials or affiliation to submit a paper, and I've published several math papers without an academic affiliation. You can put your employer as your affiliation or even "Independent Researcher". The hard part is writing the papers themselves. Getting a paper in a decent math journal as an outsider is rare, not because the journals ban or are biased against outsiders, but just because it's rare for an outsider to write a decent math paper.
  - gus_massa 1y ago
    The sibling comments are good. Some additional info: It's hard, there are a lot of i to dot and t to cross (and remember the j and f too). The paper must have the correct length, a relevant introduction, conclusions and a lot of other form stuff, detailed but not too much detailed, the correct references, ... And you must solve an interesting problem. The famous problems are too hard, a lot of people already tied to solve them. And "interesting" means what the community consider to be interesting, that sometimes it's what is interesting for the application and sometimes is something completely different. And must be correct, in my university we got a few independent people (5?) claiming to have solved the Goldbach conjeture a few years ago. Unsurprisingly all had big mistakes. My suggestion is to partner with someone already publishing math papers. Do you have a friend that is mathematician? Do you have a friend that has a friend that is mathematician? If your friends work in an uninteresting topic, you can ask for help to contact someone else that is working in something you like. Be aware that there are abusive people, but most horror cases I heard are about experimental physics where a few extra hands are more helpful, mathematician usually don't lock people in the basement until they finish the experiments. Once you understated how to publish, which journals are good, which journals are run by idiots, you can try to publish independently.
hiAndrewQuinn 1y ago
Before anything one should probably check or at least ballpark their IQ score. The median IQ for mathematics PhD students probably hovers somewhere around 145, about the top 0.2% of the population, correlated with about a 1510/1600 on the SATs, a 34 on the ACTs, etc. Those aren't perfect correlates but you're much more likely to have an SAT or ACT score than a professional IQ score handy. Math is infamously g-loaded, pure math even more so. An unfortunate fact of life. On the bright side, math is very much a "shoot for the moon and you'll land among the stars" subject to pursue if you even loosely keep industrial or business applications in mind.
- crinkly 1y ago
  I did consider doing a mathematics postgrad qualification but my IQ is high enough to realise I liked actually getting paid decent money.
  - hiAndrewQuinn 1y ago
    Yes, exactly! One reason it's valuable to look at these numbers is because it makes you take a step back and say "Wait - I'm in what percentile?" That naturally leads many people to ask whether making only $200,000 a year as a professor somewhere is really a price you're willing to pay, as opposed to making multiples of that as the smartest guy in the room in any number of private industries. Opportunity cost matters!
    - crinkly 1y ago
      Bit more complicated here in the UK with salaries in academia. But in private sector I was earning 1.5x average mathematics professor salary before I even had a mathematics degree. And equity on top of that. Not the smartest guy in the room by far but the most useful.
    - ndriscoll 1y ago
      If you're making multiples of $200k for being a smart guy, there's a decent chance you're doing something that a lot of people would be ethically uncomfortable with (HFT, ads/surveillance tech, etc.). Being happy with what you do in the world also matters, and $200k/yr is easily enough to support a family on a single income pretty much anywhere already.
barrenko 1y ago
Also a great way to lose your mind.
- deleted 1y ago
  [deleted]
jcbe 1y ago
Wow the way this renders on mobile is dizzying
- wellthisisgreat 1y ago
  I thought / hoped it’s intentional
cobbechampo 1y ago
[dead]
Caelan22 1y ago
Unfortunately, most links are dead. At least the pages related to real analysis.
atulvi 1y ago
What timing. I just got Schuam's Trigonometry to practice over the weekend and start a pure mathematics journey today and I see this article.
FranzFerdiNaN 1y ago
These kinds of lists are just completely worthless. Like ok, let’s look at “Stage 1 Elementary Stuff”. It’s a list of 18 books. So what are you supposed to do with that? Figure out which ones are good and useful? Take the next five years working through them all? Either write a good guide, explaining why you pick each book, what it will teach you and why it’s needed, or just post a link to a university degree and say “just finish all these courses, good luck”.
- racl101 1y ago
  You should checkout a YouTube channel called the Math Sorcerer. It's exactly what you are requesting. Some guy, who's passionate about math, who reviews textbooks and breaks down why you should be reading them.
up2isomorphism 1y ago
You can mix mathematician with statistician, especially if you put pure in front of it.
- tvb12 1y ago
  Pure mathemistician
lsc4719 1y ago
This sounds like "Just read the TAOCP, then you could be a programmer".
opensandwich 1y ago
Wow, this is a wild ride. I remember coming across this page because the author was from my alma mater and we were pursing the same (undergrad) degree. At the time, we could do a double major in Pure Mathematics and Statistics so long as we completed the coursework requirements, which is probably why that page even exists. The page is ~15 years old now, and I think it should be read as though its written by a 22 yr old, more reflecting on their recent university education than a guide to how to become a working mathematician. --- With that note, I would say if someone is eager to engage in mathematics and statistics _at an undergrad level_ (at the time at my university, it was _unusual_ for people to pursue machine learning as a major, and it was in computer science school). I would recommend really focussing on Real Analysis, and the higher statistics courses, try to find the links and the commonality between the proofs and the key ideas. I would also tell myself to not to shy away from martingale theory and link it to measure theory. Pure mathematics is a weird world. In the moment I hated myself for choosing it in undergrad, it absolutely tanked my grades because of the weird mental state I was in. At the same time when I got to my PhD/research everything starting really started to click. It's immensely difficult to digest and consume all the content in the 12-14 odd weeks that the coursework typically demands.
pgustafs 1y ago
Nah, just study linear algebra (Shilov or Hoffman & Kunze) and baby Rudin. Then read the most famous books in geometry, analysis, and algebra (do proofs + get a mentor). All these roadmap things are meaningless. It’s like “how to join the NBA.” Lift weight, condition, and practice fundamentals. Nothing else matters.
- 2snakes 1y ago
  What do you think of this list? https://web.archive.org/web/20240222200132/https://sheafification.com/the-fast-track/ https://web.archive.org/web/20240222200132/https://sheafific...
  - pgustafs 1y ago
    The books are good, but way too many and wildly varying in difficulty. No one can read all that in 2 years starting without knowledge of linear algebra. just worry about the fundamentals first and then pick a couple good books in areas you’re interested in. The main thing is deep understanding, not superficial breadth.
    - cap11235 1y ago
      Lol, Landau and Lifshitz as an intro
- kronicum2025 1y ago
  Getting a good mentor is the most difficult part for most people following this list.

111 points by ipnon 1y ago

"Just follow this simple road map". I think about it differently. If you want to become a pure mathematician, you have to publish research in pure mathematics. There are many different routes one can take to accomplish this, and the route that you can stick with and enjoy is the best one.

kbrkbr 1y ago
I could not agree more. I am on my second try to master mathematics (30 years after the first), and I can see, understand and appreciate mathematics mainly from the constructive standpoint. Nothing wrong with classical mathematics, as also used in this roadmap. Having axioms and drawing logical conclusions or searching proof does just not click for me. Give me 0: N and suc: N -> N and I see how to construct stuff. Induction makes sense right away as a case distinction on those two constructors.
kronicum2025 1y ago
As a young person growing up in India, enrolled in engineering in a small desolate city, when I wanted to be a mathematician, with no real path in sight, this blog post is actually what I followed, and learnt everything that is mandatory on this blog post during my undergraduate on my own, and then applied for entrance examinations for grad institutes in India, and now have successfully finished my PhD nicely in US. So, in that sense, this blog post is for people like me, who had no exposure to research scientists in around 2012 in the places we are growing up. (I didn't yet have stable internet then, only once an hour every evening). They still have some years before they could actually come in contact with researchers who can give them good problems to work on. and the local universities are horribly out of whack in their skill/desire to teach the students properly. And so, in the meantime, till you get access to good research problems, you prepare yourself to be ready for the problem when the day finally comes. And this blog post along with another one: https://webspace.science.uu.nl/~hooft101/theorist.html https://webspace.science.uu.nl/~hooft101/theorist.html was what kept motivating me when I was studying. The list of books is as important as anything else. To be fair, in 2012, math.stackexchange.com had grown up quite a bit, but I was not aware of it for various reasons.
- yu3zhou4 1y ago
  Very inspiring story, congrats on your journey!
guyman16 1y ago
What different routes are there to publish research besides academia? I would love to work on publications but it is not practical for me to return to an institution right now.
- Ivan92 1y ago
  I don't know what your current industry is, but Google publishes their research and collaborates with academic institutions is one that immediately comes to mind. The government (NASA, NSA for one), tech companies (IBM, Microsoft, etc.), medical companies, aerospace/defense (JPL), just to name a few. I am sure there are way more than I could think of and I am sure others will care to fill in as well. Research is not exclusive to academia.
  - unixbeard1337 1y ago
    Yeah, but this is specifically about pure mathematics. Do Google and medical companies publish pure math research?
    - nativeit 1y ago
      Sure, this was the first result: https://www.nature.com/articles/s41586-021-04086-x https://www.nature.com/articles/s41586-021-04086-x
- throwaway652368 1y ago
  I've never seen a math journal that requires academic credentials or affiliation to submit a paper, and I've published several math papers without an academic affiliation. You can put your employer as your affiliation or even "Independent Researcher". The hard part is writing the papers themselves. Getting a paper in a decent math journal as an outsider is rare, not because the journals ban or are biased against outsiders, but just because it's rare for an outsider to write a decent math paper.
- gus_massa 1y ago
  The sibling comments are good. Some additional info: It's hard, there are a lot of i to dot and t to cross (and remember the j and f too). The paper must have the correct length, a relevant introduction, conclusions and a lot of other form stuff, detailed but not too much detailed, the correct references, ... And you must solve an interesting problem. The famous problems are too hard, a lot of people already tied to solve them. And "interesting" means what the community consider to be interesting, that sometimes it's what is interesting for the application and sometimes is something completely different. And must be correct, in my university we got a few independent people (5?) claiming to have solved the Goldbach conjeture a few years ago. Unsurprisingly all had big mistakes. My suggestion is to partner with someone already publishing math papers. Do you have a friend that is mathematician? Do you have a friend that has a friend that is mathematician? If your friends work in an uninteresting topic, you can ask for help to contact someone else that is working in something you like. Be aware that there are abusive people, but most horror cases I heard are about experimental physics where a few extra hands are more helpful, mathematician usually don't lock people in the basement until they finish the experiments. Once you understated how to publish, which journals are good, which journals are run by idiots, you can try to publish independently.

hiAndrewQuinn 1y ago

Before anything one should probably check or at least ballpark their IQ score. The median IQ for mathematics PhD students probably hovers somewhere around 145, about the top 0.2% of the population, correlated with about a 1510/1600 on the SATs, a 34 on the ACTs, etc. Those aren't perfect correlates but you're much more likely to have an SAT or ACT score than a professional IQ score handy. Math is infamously g-loaded, pure math even more so. An unfortunate fact of life. On the bright side, math is very much a "shoot for the moon and you'll land among the stars" subject to pursue if you even loosely keep industrial or business applications in mind.

crinkly 1y ago
I did consider doing a mathematics postgrad qualification but my IQ is high enough to realise I liked actually getting paid decent money.
- hiAndrewQuinn 1y ago
  Yes, exactly! One reason it's valuable to look at these numbers is because it makes you take a step back and say "Wait - I'm in what percentile?" That naturally leads many people to ask whether making only $200,000 a year as a professor somewhere is really a price you're willing to pay, as opposed to making multiples of that as the smartest guy in the room in any number of private industries. Opportunity cost matters!
  - crinkly 1y ago
    Bit more complicated here in the UK with salaries in academia. But in private sector I was earning 1.5x average mathematics professor salary before I even had a mathematics degree. And equity on top of that. Not the smartest guy in the room by far but the most useful.
  - ndriscoll 1y ago
    If you're making multiples of $200k for being a smart guy, there's a decent chance you're doing something that a lot of people would be ethically uncomfortable with (HFT, ads/surveillance tech, etc.). Being happy with what you do in the world also matters, and $200k/yr is easily enough to support a family on a single income pretty much anywhere already.

barrenko 1y ago

Also a great way to lose your mind.

deleted 1y ago
[deleted]

jcbe 1y ago

Wow the way this renders on mobile is dizzying

wellthisisgreat 1y ago
I thought / hoped it’s intentional

cobbechampo 1y ago

[dead]

Caelan22 1y ago

Unfortunately, most links are dead. At least the pages related to real analysis.

atulvi 1y ago

What timing. I just got Schuam's Trigonometry to practice over the weekend and start a pure mathematics journey today and I see this article.

FranzFerdiNaN 1y ago

These kinds of lists are just completely worthless. Like ok, let’s look at “Stage 1 Elementary Stuff”. It’s a list of 18 books. So what are you supposed to do with that? Figure out which ones are good and useful? Take the next five years working through them all? Either write a good guide, explaining why you pick each book, what it will teach you and why it’s needed, or just post a link to a university degree and say “just finish all these courses, good luck”.

racl101 1y ago
You should checkout a YouTube channel called the Math Sorcerer. It's exactly what you are requesting. Some guy, who's passionate about math, who reviews textbooks and breaks down why you should be reading them.

up2isomorphism 1y ago

You can mix mathematician with statistician, especially if you put pure in front of it.

tvb12 1y ago
Pure mathemistician

lsc4719 1y ago

This sounds like "Just read the TAOCP, then you could be a programmer".

opensandwich 1y ago

Wow, this is a wild ride. I remember coming across this page because the author was from my alma mater and we were pursing the same (undergrad) degree. At the time, we could do a double major in Pure Mathematics and Statistics so long as we completed the coursework requirements, which is probably why that page even exists. The page is ~15 years old now, and I think it should be read as though its written by a 22 yr old, more reflecting on their recent university education than a guide to how to become a working mathematician. --- With that note, I would say if someone is eager to engage in mathematics and statistics _at an undergrad level_ (at the time at my university, it was _unusual_ for people to pursue machine learning as a major, and it was in computer science school). I would recommend really focussing on Real Analysis, and the higher statistics courses, try to find the links and the commonality between the proofs and the key ideas. I would also tell myself to not to shy away from martingale theory and link it to measure theory. Pure mathematics is a weird world. In the moment I hated myself for choosing it in undergrad, it absolutely tanked my grades because of the weird mental state I was in. At the same time when I got to my PhD/research everything starting really started to click. It's immensely difficult to digest and consume all the content in the 12-14 odd weeks that the coursework typically demands.

pgustafs 1y ago

Nah, just study linear algebra (Shilov or Hoffman & Kunze) and baby Rudin. Then read the most famous books in geometry, analysis, and algebra (do proofs + get a mentor). All these roadmap things are meaningless. It’s like “how to join the NBA.” Lift weight, condition, and practice fundamentals. Nothing else matters.

2snakes 1y ago
What do you think of this list? https://web.archive.org/web/20240222200132/https://sheafification.com/the-fast-track/ https://web.archive.org/web/20240222200132/https://sheafific...
- pgustafs 1y ago
  The books are good, but way too many and wildly varying in difficulty. No one can read all that in 2 years starting without knowledge of linear algebra. just worry about the fundamentals first and then pick a couple good books in areas you’re interested in. The main thing is deep understanding, not superficial breadth.
  - cap11235 1y ago
    Lol, Landau and Lifshitz as an intro
kronicum2025 1y ago
Getting a good mentor is the most difficult part for most people following this list.