How to become a pure mathematician or statistician (2008)
hbpms.blogspot.com
hbpms.blogspot.com
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.
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.
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.
Research is not exclusive to academia.
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.
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.
Physics majors, in my experience, had a significantly higher arrogance level.
It’s just different leagues of intelligence: social studies undergrad vs math undergrad vs math grad vs competitive researcher.
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.
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.
So yes, they are teachers or administrators or make minor research contributions.
The smartest ones were usually the philosophy majors. Also some of the weirdest (in a good way) folks.
Some people are (much) smarter than others. It sucks, but that's life.
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.
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 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 ?
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.
> 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.
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.
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.
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.
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.
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.
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.
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..."
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!
I´ve never been able to wrap my mind around this saying.
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.
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...
“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
Either way, I never bought his claim that he was not exceptional.
(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...
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.
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.
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.
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”.