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.
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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.
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.
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
It's not comfortable, but this seems to be what the priors point to. I suspect that pure mathematics is one of the most intelligence-dependent fields; one where hard work, practical solving problems and a large knowledge base is less of a substitute.
> one where hard work, practical solving problems and a large knowledge base is less of a substitute
Collaboration remains an important skill – I had an REU mentor who said that, given the explosion of mathematics that one had to learn to do cutting-edge work in a field, she had to end up "pooling experience..."
> one where hard work, practical solving problems and a large knowledge base is less of a substitute.
I have seen this first hand. I remember when I was in university doing my math major. This one older adult lady (she seemed 40yrs old, and very attractive too), she had decided for some reason or other she wanted to do a major in mathematics. Not for a job or anything but just to do it.
Whereas the rest of us, let’s face it, we just wanted a good job in STEM.
Bless this lady, she was so determined and hard working. She would show up to every lecture, first in, last out, and she would show up to every study session and give it her all.
But unfortunately, she was not good at grasping the concepts nor solving the problems. It was shocking how little she grokked the introductory concepts for the amount of effort she put in. She worked harder than anyone in our group.
I don’t think any of us had the heart to tell her that maybe a math major was not in the cards.
I never saw her on campus in my 3rd year and on so imagine she dropped off.
I think what happens is, IQ is a sort of speed. They teach the class assuming they can get most people through it at whatever pace they are used to teaching at. If you're quick, you can keep up with a below average effort, and vice versa.
As you get higher and higher up in the stratosphere, the balance between "we need enough students" and "we need to go faster" ends up favouring the few super intelligent people, along with the people who can arrange their lives to put in the hours.
That's not to say you can't learn something if you are slow. You just can't learn it at the pace they are teaching, and you might not have the wherewithal to learn it at your own pace.
So to you it looks like this lady would never learn it, but I would guess if she had a personal tutor they would be able to pace it.
Same lol. By the OP's logic, every student pursuing this field in a university as an undergrad/graduate student should be taking an IQ test before proceeding to the upper level math courses covering these topics. Anything less than the threshold will mean they have to focus on something different.
> The median IQ for mathematics PhD students probably hovers somewhere around 145
Does that mean the 145 figure is only a guess on your end ?
Second, as far as I know, an individual's IQ is not something set in stone, and can absolutely be improved with training. I remember reading (that's an anecdote so correct me if I'm wrong) that rewarding a good score with money was able to improve the outcome by up to 20 points. It doesn't sound absurd to me that someone with a slightly above average IQ could get close to 140 after 6, 7 years of high level math training.
>Does that mean the 145 figure is only a guess on your end ?
EDIT: Mea maxima culpa, confusion crept in, my 145 number was supposed to be a much looser guess for actual working full time mathematicians. I miswrote this in the original post as applying to math PhD students, which are much lower. Closer to a 130 median.
ORIGINAL: It's not quite a guess, but I don't have precise data on this exact thing either. Previous studies in this field have consistently found a range of between 140 and 150, and you can probably find those with some Googling if you want to corroborate it yourself. I have a long cached memory of seeing a study where theoretical physics PhD students had an average IQ of 150, which also loosely supports this, since theoretical physics is almost its own form of pure mathematics.
>an individual's IQ is not something set in stone, and can absolutely be improved with training
Most psychological research I've seen says no such thing, unfortunately. Believe me, I would love for that to be the case - one extra point of IQ correlates to roughly $1000 extra income per year in the US, and so if your 20 point claim were really true we could potentially cause a double digit spike in GDP over the next few months just by implementing it in smart ways. But my baseline belief is that study is almost certainly an outlier in a sea of similar studies which support the null hypothesis.
Note that it's an IQ of 128 vs 125 for humanities. With the small sample size, it's basically noise. And given that this is Oxford, I would expect the average PhD student to have less than these numbers.
Oh, hold on, I may have gotten my numbers mixed up slightly. 145 might actually be for working, full time pure mathematicians - sorry about that, I'm double checking now.
EDIT: My massive bad, it looks like I accidentally bumped everything up a standard deviation in my head somewhere. Jesus. I should update the numbers in the original post. Maybe I should also consider getting a math PhD after all, apparently I'd be ahead of the pack in that case.
Kudos to you for acknowledging your error and correcting it. We certainly do need more high IQ (>128) math PhDs with integrity, like yourself. Please do consider pursuing higher math in a full time capacity.
Maybe, but let say you're right. I still don't understand your suggestion of doing an IQ test before you decide to study math. If you can't go further than a masters, like you said there are still a lot of industries you can go too and have an interesting, lucrative job. And if you do succeed to finish your PhD then that's great news. There are no benefits that I can see in doing an IQ test like you suggested before you make your decision. If you love math and are good at it, chances are you're moderately smart at least. Might as well go as far as you can if that's what you want.
My experience with pure math is that this is not necessary to get job as one, even at good institutions, but you will be terrorized by the arrogance of the ones you mention. Learning to deal with the "brilliant jerk" is a problem in many fields, but the ones I've met in pure math are some of the weirdest (and most vicious)
At least from my point of view (in the industry, not academia) this is actually the opposite. Math graduates tend to be smart and humble and I respect them a lot. Sometimes it almost feels like math and physics are the last "real" degrees left.
Who's to say that you can't go into industry and not be a researcher? You don't have to stay in academia to do research. Many companies and industries tend to publish papers and some even work with universities for research.
The definition of professional mathematics is research. That’s what they are trained in and that’s what they are competent at. I don’t understand your comment.
This is what I've observed as well. By my own metrics and grades, I was a somewhat bright math minor (near-perfect score in abstract algebra, etc), would have been middle of the pack as a PhD student, may have been below par if I managed to complete the PhD, and almost certainly would have been deadweight as a pure mathematician myself. That's just how the scaling and competitive dynamics have worked out; it's not really something to feel personally bad about, any more than you might feel personally bad about not having the potential to be a competitive figure skater.
EDIT: Uh, actually, it looks like I may have underestimated myself at basically every point here and would have become a basically okay mathematician based on updated priors.
The silly thing about this is that context is everything. I bet it's extremely easy to be a top-tier figure-skater in, say, a small tropical island nation? In a similar way, I very much doubt that you'd really need to be in the top 0.2% of the population to complete a phd. Do you need to be in the top 0.2% of people to compete as a contributor with absolutely everyone else in the whole world at the same time? Well yeah, but at that point the statement is so obviously true that it doesn't mean much.
You're right in a sense, but I took the context we're working in as somewhat of a given based on the title of the post. Our goal is to work, full time, as a professional pure mathematician; that naturally puts us in the labor market for pure mathematicians. We can't know that market exactly, of course, but it's far from arbitrary. We are in competition with other market participants, and we can study their properties and use that knowledge to guide our actions productively - including making the decision to exit the market if that's what makes sense to us.
> Our goal is to work, full time, as a professional pure mathematician; that naturally puts us in the labor market for pure mathematicians. We can't know that market exactly, of course[...]
For what it's worth, my classmates from college who have completed PhDs, based their postgrad career decisions on completely different factors – mostly their families and partners, and whether they're willing to move around (especially to rural areas) to target an extremely shrinking academic job pool.
EDIT: example that came to mind – I had a classmate who postdoc'd at Chicago, who decided to stay in town and work in finance rather than pursue some tenure-track offers at R1s, because his young one went to a prestigious UChicago Lab School and didn't want to uproot her.
The dumbest people I’ve ever encountered in university were the math and physics majors who thought they could score some easy points by taking humanities classes, because just like you they considered that below their level. I’m sure they were smart on an IQ test but they couldn’t reason their way out of a paper bag, and their writing skills were just laughable.
The smartest ones were usually the philosophy majors. Also some of the weirdest (in a good way) folks.
Apart from the fact that IQ tests are racist bunk, there's no need to do some fancy self-discovery journey or anything to determine whether you're cut out for pure math or not: if you have to ask, then it's not for you.
Or, alternatively, you could just skip over "general intelligence" and get to your second measure that's directly validated against college outcomes – SAT score.
(My understanding is that the "general" GRE – not the math subject test – is less of a predictor of completing a PhD, but I think we can come up with a hundred reasons why.)
The interest in pure mathematics probably sets inquiring individuals in a higher intelligence bracket already. If somebody got through high school enjoying and succeeding at geometry and calculus, then they probably could stomach most undergraduate work in the same manner.
It seems a bit like gatekeeping to make people question whether they are smart enough when they will figure out pretty quickly if they have the aptitude or will to do it just by being exposed.
>they will figure out pretty quickly if they have the aptitude or will to do it just by being exposed
We disagree here, most people are not very good at figuring this out for themselves at all ime. It's always wise to compare yourself to known or semi-known metrics before you take the plunge into any given career, to make sure it really does seem like a good fit for you, or to make sure you can justify why you want to do it anyway even if the metrics paint an unflattering story.
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!
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.
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.
Imagine if Richard Feynman used his IQ as a metric for deciding whether he should become a physicist. Physics would not be the same.
I am certain that there are mathematicians below, near, and above an IQ of 145 that all have great research productivity. IQ tests do not approximate the creativity, effort, and collaboration required in a mathematician. Not to mention the dubious nature of the 145 claim.
Of course, there are some people that will have a greater aptitude for mathematics than others. But you do not need to be a genius, and this is echoed by Terence Tao [0].
Just to complement your post, Richard Feynman's quote on the topic:
“I was an ordinary person who studied hard. There are no miracle people. It happens they get interested in this thing and they learn all this stuff, but they’re just people.”
I dunno man but I always believed Feynman was expressing a very “aw shucks” everyman type of sensibility to motivate his students but really he’s a genius who just never saw himself on par with the other genius demigod scientists of his time but still far removed from common people like me for example. Or he knew he was exceptional but he just liked to distinguish himself from the more square academic types by appealing to the regular people.
Either way, I never bought his claim that he was not exceptional.
Also possible that Feynman had superb verbal-mathematical ability and bad visual-spatial ability and took a visual-spatial test. It's unusual but not incredibly so. I am the same way.
I misspoke, 145 should have been my loose estimate for the median of actual working mathematicians and taken with many more grains of salt. Mathematics PhD students cluster around a much more attainable average of 128-130. Per [1] this would map to a much easier SAT score of only around 1280.
My general point still stands that you probably want to look at this and consider your potential career in math against this, but the skill curve is less punishing than I initially thought.
But why would you do an IQ test, when you could just do a math test? Surely a math test is a better indicator of math ability than a test that is merely correlated?
That approach would work too, but generally (not always) math tests require you to already know a fair bit of math to do them. IQ tests can and have been designed with don't even require the use of language, let alone any conceptual machinery that strong.
Ha, yeah, it's a weird saying. I think it makes more sense if you imagine the sky as like a static painting you shoot yourself into with a cannon or something.
I was about to ask if there is any way we can bridge the mathematics the same way we did with programming ; I love going-on with an LLM to learn math and apply them directly to problems I'm facing ; and to be honest I often feel like a musician who never actually learned his tuning lately ; especially in topics correlated to balancing, monitoring, and even simply in projection of cost ; It's like I have been over-focused on complexity of algorithms without actually realizing that it's only one part of the problem - so I have a huge potential usage of mathematics and one can really easily leverage them thanks to AI - especially considering tests ; but that's where my road ends without a more sophisticated approach - and even then it's a very dark place to wonder by (eg: lot of time spent for seemingly unknown appliance) it does however start to feel even more attracting for these exact reasons - I feel like having problems to solve with just makes this a lot easier - but I'd love to be able to ground-up things even more - and especially be able to take shortcuts.
I mean a lot of people just run a database but don't know wtf it does - but it still useful to them - maths however need to be understood to be really useful -
Is there not a way to make this lot more navigable ? Are there bridge concepts that are important enough that we can spend some time to learn them ? (there are ofc) - and how deep shall we go ?
Unfortunately, I have heard logic like this throughout my life, leading me to decide that because I struggled with (some subset of) math, I am not an intelligent person, which led to forcing myself to do pure math in college to prove my intelligence. This led to many significant and awful mental health issues. While this is a bit of a fallacious logical leap, it's not impossible that other people have went through this because of this sort of information being hammered into their head.
I choose to believe succeeding at anything is mostly about persistence and interest, barring other immense structural factors. I have zero interest after doing difficult pure math classes, so I stopped. I now think I am good at what I do, but everyone's intelligence and interests are different.
I think this sort of quantification of intelligence is really harmful to people. I don't want to exclude people from pursuing their interest because their SAT score wasn't high enough. I have met math PhD students with bad GPAs and poor math class grades in their undergrad.
On a tangent, CS undergraduate programs are insanely competitive and filter in crazy ways, and most of my friends who were passionate about CS (especially systems CS and SWE) did ECE just to avoid the competition and dispassionate culture. Your GPA and SAT scores had to be insane to get into almost any undergrad school for CS.
In a shocking twist of fate, people who do abstract reasoning and patterns oriented exercises all day long score high on a test measuring abstract reasoning and patterns manipulation. /s
You have your cause and effect reversed. You don’t need high IQ score to do maths, doing a ton of maths will naturally increase your IQ score. IQ is not general intelligence.
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”.
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.
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.
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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.
"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.
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 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.
Very inspiring story, congrats on your journey!
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.
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.
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.
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.
Yeah, but this is specifically about pure mathematics. Do Google and medical companies publish pure math research?
Sure, this was the first result: https://www.nature.com/articles/s41586-021-04086-x
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.
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.
Getting a good mentor is the most difficult part for most people following this list.
What do you think of this list? https://web.archive.org/web/20240222200132/https://sheafific...
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.
Lol, Landau and Lifshitz as an intro
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.
I got rage baited by this so hard, cant comprehend thinking this way.
Hung out with PhD's, economists, bankers, trust find kids, scientists, and artists - who maybe weren't top tier enough, but none thought this way.
Literally the weirdest take on a forum filled with dreamers, but every take is valid.
It's not comfortable, but this seems to be what the priors point to. I suspect that pure mathematics is one of the most intelligence-dependent fields; one where hard work, practical solving problems and a large knowledge base is less of a substitute.
> one where hard work, practical solving problems and a large knowledge base is less of a substitute
Collaboration remains an important skill – I had an REU mentor who said that, given the explosion of mathematics that one had to learn to do cutting-edge work in a field, she had to end up "pooling experience..."
> one where hard work, practical solving problems and a large knowledge base is less of a substitute.
I have seen this first hand. I remember when I was in university doing my math major. This one older adult lady (she seemed 40yrs old, and very attractive too), she had decided for some reason or other she wanted to do a major in mathematics. Not for a job or anything but just to do it.
Whereas the rest of us, let’s face it, we just wanted a good job in STEM.
Bless this lady, she was so determined and hard working. She would show up to every lecture, first in, last out, and she would show up to every study session and give it her all.
But unfortunately, she was not good at grasping the concepts nor solving the problems. It was shocking how little she grokked the introductory concepts for the amount of effort she put in. She worked harder than anyone in our group.
I don’t think any of us had the heart to tell her that maybe a math major was not in the cards.
I never saw her on campus in my 3rd year and on so imagine she dropped off.
But I was rooting for her.
I think what happens is, IQ is a sort of speed. They teach the class assuming they can get most people through it at whatever pace they are used to teaching at. If you're quick, you can keep up with a below average effort, and vice versa.
As you get higher and higher up in the stratosphere, the balance between "we need enough students" and "we need to go faster" ends up favouring the few super intelligent people, along with the people who can arrange their lives to put in the hours.
That's not to say you can't learn something if you are slow. You just can't learn it at the pace they are teaching, and you might not have the wherewithal to learn it at your own pace.
So to you it looks like this lady would never learn it, but I would guess if she had a personal tutor they would be able to pace it.
Same lol. By the OP's logic, every student pursuing this field in a university as an undergrad/graduate student should be taking an IQ test before proceeding to the upper level math courses covering these topics. Anything less than the threshold will mean they have to focus on something different.
Two things :
> The median IQ for mathematics PhD students probably hovers somewhere around 145
Does that mean the 145 figure is only a guess on your end ?
Second, as far as I know, an individual's IQ is not something set in stone, and can absolutely be improved with training. I remember reading (that's an anecdote so correct me if I'm wrong) that rewarding a good score with money was able to improve the outcome by up to 20 points. It doesn't sound absurd to me that someone with a slightly above average IQ could get close to 140 after 6, 7 years of high level math training.
>Does that mean the 145 figure is only a guess on your end ?
EDIT: Mea maxima culpa, confusion crept in, my 145 number was supposed to be a much looser guess for actual working full time mathematicians. I miswrote this in the original post as applying to math PhD students, which are much lower. Closer to a 130 median.
ORIGINAL: It's not quite a guess, but I don't have precise data on this exact thing either. Previous studies in this field have consistently found a range of between 140 and 150, and you can probably find those with some Googling if you want to corroborate it yourself. I have a long cached memory of seeing a study where theoretical physics PhD students had an average IQ of 150, which also loosely supports this, since theoretical physics is almost its own form of pure mathematics.
>an individual's IQ is not something set in stone, and can absolutely be improved with training
Most psychological research I've seen says no such thing, unfortunately. Believe me, I would love for that to be the case - one extra point of IQ correlates to roughly $1000 extra income per year in the US, and so if your 20 point claim were really true we could potentially cause a double digit spike in GDP over the next few months just by implementing it in smart ways. But my baseline belief is that study is almost certainly an outlier in a sea of similar studies which support the null hypothesis.
Did a Google search, and the only actual, definitive thing I found was this:
https://pmc.ncbi.nlm.nih.gov/articles/PMC5008436/#tbl1
Note that it's an IQ of 128 vs 125 for humanities. With the small sample size, it's basically noise. And given that this is Oxford, I would expect the average PhD student to have less than these numbers.
Oh, hold on, I may have gotten my numbers mixed up slightly. 145 might actually be for working, full time pure mathematicians - sorry about that, I'm double checking now.
EDIT: My massive bad, it looks like I accidentally bumped everything up a standard deviation in my head somewhere. Jesus. I should update the numbers in the original post. Maybe I should also consider getting a math PhD after all, apparently I'd be ahead of the pack in that case.
Kudos to you for acknowledging your error and correcting it. We certainly do need more high IQ (>128) math PhDs with integrity, like yourself. Please do consider pursuing higher math in a full time capacity.
Maybe, but let say you're right. I still don't understand your suggestion of doing an IQ test before you decide to study math. If you can't go further than a masters, like you said there are still a lot of industries you can go too and have an interesting, lucrative job. And if you do succeed to finish your PhD then that's great news. There are no benefits that I can see in doing an IQ test like you suggested before you make your decision. If you love math and are good at it, chances are you're moderately smart at least. Might as well go as far as you can if that's what you want.
My experience with pure math is that this is not necessary to get job as one, even at good institutions, but you will be terrorized by the arrogance of the ones you mention. Learning to deal with the "brilliant jerk" is a problem in many fields, but the ones I've met in pure math are some of the weirdest (and most vicious)
At least from my point of view (in the industry, not academia) this is actually the opposite. Math graduates tend to be smart and humble and I respect them a lot. Sometimes it almost feels like math and physics are the last "real" degrees left.
Similar - I found math majors to be fairly humble. Yes, there's always the exception, but I found them to be fairly fun folks.
Physics majors, in my experience, had a significantly higher arrogance level.
I'm talking about professors at R1 schools and Ivies to be clear.
Yes, and those are the ones who didn’t make it to researcher.
Who's to say that you can't go into industry and not be a researcher? You don't have to stay in academia to do research. Many companies and industries tend to publish papers and some even work with universities for research.
No I’m saying the top tier mathematicians tend to work in academia because that’s where the most math is done (general trend with exceptions).
I think that is a fair statement to make. Thanks for clarifying :)
I don't think that's what GP was saying, but I could be wrong.
A corollary of this is that many professional mathematicians are not actually competitive in research.
It’s just different leagues of intelligence: social studies undergrad vs math undergrad vs math grad vs competitive researcher.
The definition of professional mathematics is research. That’s what they are trained in and that’s what they are competent at. I don’t understand your comment.
Lots of professors aren’t leading their field in research - they aren’t competitive with those that do.
So yes, they are teachers or administrators or make minor research contributions.
This is what I've observed as well. By my own metrics and grades, I was a somewhat bright math minor (near-perfect score in abstract algebra, etc), would have been middle of the pack as a PhD student, may have been below par if I managed to complete the PhD, and almost certainly would have been deadweight as a pure mathematician myself. That's just how the scaling and competitive dynamics have worked out; it's not really something to feel personally bad about, any more than you might feel personally bad about not having the potential to be a competitive figure skater.
EDIT: Uh, actually, it looks like I may have underestimated myself at basically every point here and would have become a basically okay mathematician based on updated priors.
The silly thing about this is that context is everything. I bet it's extremely easy to be a top-tier figure-skater in, say, a small tropical island nation? In a similar way, I very much doubt that you'd really need to be in the top 0.2% of the population to complete a phd. Do you need to be in the top 0.2% of people to compete as a contributor with absolutely everyone else in the whole world at the same time? Well yeah, but at that point the statement is so obviously true that it doesn't mean much.
You're right in a sense, but I took the context we're working in as somewhat of a given based on the title of the post. Our goal is to work, full time, as a professional pure mathematician; that naturally puts us in the labor market for pure mathematicians. We can't know that market exactly, of course, but it's far from arbitrary. We are in competition with other market participants, and we can study their properties and use that knowledge to guide our actions productively - including making the decision to exit the market if that's what makes sense to us.
> Our goal is to work, full time, as a professional pure mathematician; that naturally puts us in the labor market for pure mathematicians. We can't know that market exactly, of course[...]
For what it's worth, my classmates from college who have completed PhDs, based their postgrad career decisions on completely different factors – mostly their families and partners, and whether they're willing to move around (especially to rural areas) to target an extremely shrinking academic job pool.
EDIT: example that came to mind – I had a classmate who postdoc'd at Chicago, who decided to stay in town and work in finance rather than pursue some tenure-track offers at R1s, because his young one went to a prestigious UChicago Lab School and didn't want to uproot her.
The dumbest people I’ve ever encountered in university were the math and physics majors who thought they could score some easy points by taking humanities classes, because just like you they considered that below their level. I’m sure they were smart on an IQ test but they couldn’t reason their way out of a paper bag, and their writing skills were just laughable.
The smartest ones were usually the philosophy majors. Also some of the weirdest (in a good way) folks.
I didn’t say that and I also have a philosophy degree.
What's much more sensible than taking an IQ test is looking at your experience with math to date.
Yes.
Apart from the fact that IQ tests are racist bunk, there's no need to do some fancy self-discovery journey or anything to determine whether you're cut out for pure math or not: if you have to ask, then it's not for you.
Or, alternatively, you could just skip over "general intelligence" and get to your second measure that's directly validated against college outcomes – SAT score.
(My understanding is that the "general" GRE – not the math subject test – is less of a predictor of completing a PhD, but I think we can come up with a hundred reasons why.)
[1] https://journals.plos.org/plosone/article?id=10.1371/journal...
The interest in pure mathematics probably sets inquiring individuals in a higher intelligence bracket already. If somebody got through high school enjoying and succeeding at geometry and calculus, then they probably could stomach most undergraduate work in the same manner.
It seems a bit like gatekeeping to make people question whether they are smart enough when they will figure out pretty quickly if they have the aptitude or will to do it just by being exposed.
>they will figure out pretty quickly if they have the aptitude or will to do it just by being exposed
We disagree here, most people are not very good at figuring this out for themselves at all ime. It's always wise to compare yourself to known or semi-known metrics before you take the plunge into any given career, to make sure it really does seem like a good fit for you, or to make sure you can justify why you want to do it anyway even if the metrics paint an unflattering story.
I did consider doing a mathematics postgrad qualification but my IQ is high enough to realise I liked actually getting paid decent money.
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!
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.
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.
Imagine if Richard Feynman used his IQ as a metric for deciding whether he should become a physicist. Physics would not be the same.
I am certain that there are mathematicians below, near, and above an IQ of 145 that all have great research productivity. IQ tests do not approximate the creativity, effort, and collaboration required in a mathematician. Not to mention the dubious nature of the 145 claim.
Of course, there are some people that will have a greater aptitude for mathematics than others. But you do not need to be a genius, and this is echoed by Terence Tao [0].
[0] https://terrytao.wordpress.com/career-advice/does-one-have-t...
Just to complement your post, Richard Feynman's quote on the topic:
“I was an ordinary person who studied hard. There are no miracle people. It happens they get interested in this thing and they learn all this stuff, but they’re just people.”
― Richard Feynman
I dunno man but I always believed Feynman was expressing a very “aw shucks” everyman type of sensibility to motivate his students but really he’s a genius who just never saw himself on par with the other genius demigod scientists of his time but still far removed from common people like me for example. Or he knew he was exceptional but he just liked to distinguish himself from the more square academic types by appealing to the regular people.
Either way, I never bought his claim that he was not exceptional.
I think Feynman was bullshitting you, sorry to say. This is just a manifestly crazy claim from a guy who scored literally #1 on the Putnam.
Also possible that Feynman had superb verbal-mathematical ability and bad visual-spatial ability and took a visual-spatial test. It's unusual but not incredibly so. I am the same way.
!!!! IMPORTANT COMMENT !!!!,
because the edit button is now gone:
I misspoke, 145 should have been my loose estimate for the median of actual working mathematicians and taken with many more grains of salt. Mathematics PhD students cluster around a much more attainable average of 128-130. Per [1] this would map to a much easier SAT score of only around 1280.
My general point still stands that you probably want to look at this and consider your potential career in math against this, but the skill curve is less punishing than I initially thought.
[1]: https://www.iqcomparisonsite.com/satiq.aspx
But why would you do an IQ test, when you could just do a math test? Surely a math test is a better indicator of math ability than a test that is merely correlated?
That approach would work too, but generally (not always) math tests require you to already know a fair bit of math to do them. IQ tests can and have been designed with don't even require the use of language, let alone any conceptual machinery that strong.
> "shoot for the moon and you'll land among the stars"
I´ve never been able to wrap my mind around this saying.
Ha, yeah, it's a weird saying. I think it makes more sense if you imagine the sky as like a static painting you shoot yourself into with a cannon or something.
That's not a bad outcome, especially if the world ends up being one large Truman show and you escape! ;)
I was about to ask if there is any way we can bridge the mathematics the same way we did with programming ; I love going-on with an LLM to learn math and apply them directly to problems I'm facing ; and to be honest I often feel like a musician who never actually learned his tuning lately ; especially in topics correlated to balancing, monitoring, and even simply in projection of cost ; It's like I have been over-focused on complexity of algorithms without actually realizing that it's only one part of the problem - so I have a huge potential usage of mathematics and one can really easily leverage them thanks to AI - especially considering tests ; but that's where my road ends without a more sophisticated approach - and even then it's a very dark place to wonder by (eg: lot of time spent for seemingly unknown appliance) it does however start to feel even more attracting for these exact reasons - I feel like having problems to solve with just makes this a lot easier - but I'd love to be able to ground-up things even more - and especially be able to take shortcuts.
I mean a lot of people just run a database but don't know wtf it does - but it still useful to them - maths however need to be understood to be really useful -
Is there not a way to make this lot more navigable ? Are there bridge concepts that are important enough that we can spend some time to learn them ? (there are ofc) - and how deep shall we go ?
Here is a project in that direction: https://www.math.inc/gauss
I think that yes, math will become much more accessible, and pure brain power will become much less important to use and understand math successfully.
I don't like attitudes like that of hiAndrewQuinn. If you like math, just do it, there is no need for an IQ test.
Not true at all, we need mathematicians and scientists from all backgrounds, and creativity comes in many shapes
Unfortunately, I have heard logic like this throughout my life, leading me to decide that because I struggled with (some subset of) math, I am not an intelligent person, which led to forcing myself to do pure math in college to prove my intelligence. This led to many significant and awful mental health issues. While this is a bit of a fallacious logical leap, it's not impossible that other people have went through this because of this sort of information being hammered into their head.
I choose to believe succeeding at anything is mostly about persistence and interest, barring other immense structural factors. I have zero interest after doing difficult pure math classes, so I stopped. I now think I am good at what I do, but everyone's intelligence and interests are different.
I think this sort of quantification of intelligence is really harmful to people. I don't want to exclude people from pursuing their interest because their SAT score wasn't high enough. I have met math PhD students with bad GPAs and poor math class grades in their undergrad.
On a tangent, CS undergraduate programs are insanely competitive and filter in crazy ways, and most of my friends who were passionate about CS (especially systems CS and SWE) did ECE just to avoid the competition and dispassionate culture. Your GPA and SAT scores had to be insane to get into almost any undergrad school for CS.
In a shocking twist of fate, people who do abstract reasoning and patterns oriented exercises all day long score high on a test measuring abstract reasoning and patterns manipulation. /s
You have your cause and effect reversed. You don’t need high IQ score to do maths, doing a ton of maths will naturally increase your IQ score. IQ is not general intelligence.
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Your personal experience does not line up with the empirical evidence at all.
Some people are (much) smarter than others. It sucks, but that's life.
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”.
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.
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.
This sounds like "Just read the TAOCP, then you could be a programmer".
Wow the way this renders on mobile is dizzying
I thought / hoped it’s intentional
Also a great way to lose your mind.
You can mix mathematician with statistician, especially if you put pure in front of it.
Pure mathemistician
Unfortunately, most links are dead. At least the pages related to real analysis.
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