I have a PhD in math, did some successful research, but left math for a variety of personal reasons. My advice:
1. Network, network, network. Get famous or established mathematicians interested in your work.
2. Be a minority. Due to the prevailing ideology of academia, it's much easier to be successful if you are a minority. (I am not saying it doesn't take hard work, but the competition is not as extreme due to intense equity measures.) I will laugh if anyone gets mad at this point. Equity rules make this point an incontrovertible fact.
3. Solve problems related to the work of established mathematicians. Interesting or novel problems that are not related to the work of others will be ignored.
4. Forget about being curious. Okay, this one has some dark humour about it...but seriously, if you just explore your curiosity, you might solve cool problems but you won't be "career successful". It's much better just to make useless investigations into ever-more esoteric problems that validate the "successful" work of others which is becoming increasingly technical.
A note: this might sound a bit cynical, but I thought I'd give another viewpoint to show a bit more of the balanced reality of the math world.
I'm curious: what field did you work in? One of my reasons for not pursuing math was going to a summer undergrad research mentoring thing by a well-known-enough American algebraic topologist (the field that Eugenia works in), and finding out that even his students were having problem getting positions in higher ed in the US that weren't temporary – that was simply the state in general academia, with an ageing professoriate.
Maybe I should add a 5th point, but I need to phrase it very cautiously.
5. If you're interested in doing mathematics in an institution of higher ed in the US, focus your work on teaching, and be prepared to live in the middle of nowhere.
> Equity rules make this point an incontrovertible fact
Seems like flawed reasoning to me. If a group as a whole is underrepresented in a highly competitive environment like academia, then presumably they have a harder time getting in or staying in.
Despite equity measures, whites in the us are not underrepresented. It follows that it is not harder for whites, but still more difficult specifically for black and Latino people to make it in academia, no?
> If a group as a whole is underrepresented in a highly competitive environment like academia, then presumably they have a harder time getting in or staying in.
No, that is fallacious. It could also mean that fewer members of that group try to get in in the first place. That is because there are multiple stages to getting into academia. But the question I was referring to was about the final stage: getting into research math, and then minorities are already under-represented in the pool of people thinking about becoming a research mathematician.
To put it simply, IF a person is a minority and is already in grad school, they will have an easier time going further simply because of equity hiring. That's all I meant.
I appreciated Eugenia Cheng's youtube lectures on category theory and I know interviews aren't the best format for sustained exposition, but the remark about mathematical rigor being a European colonialist thing seemed pretty forced. If anything, it underestimates the strangeness of the history of math in an effort to cram it into a simple frame: "Europeans bad".
There was something distinctive about work in foundations of math in the late 19th / early 20th centuries, work which came out of European cultures (or more accurately Latin Christendom). Plausibly, this work was centuries in the making, the culmination of a trend towards increasingly analytic and systematizing modes of thought that begins with the recovery of Greek philosophy in the middle ages.
But, ok, even if you don't buy that, how do you get from the Congo Free State to Frege, Hilbert and Goedel? It's a radical claim that would be utterly fascinating if it were true. But Chang just kind of tosses it off at the end and asks us to accept it as an unremarkable truism.
I've talked about this in a replay to another comment, but I don't think she's making any value judgment in the article, or (at least in the article) actually articulating any claim about colonialism and mathematics. In fact, I don't even think she's saying "Europeans bad":
We might marvel at, say, Amish people raising a barn or Inuit people building a kayak without any European formal maths training. We might declare that this is maths — and I think it is. But it’s problematic if we think we’re doing them some great honour by bestowing the label of maths on what they do.
I think that if I were being interviewed and my interviewer said "you also discuss how colonialist ideology has influenced modern mathematics?", but I didn't think that colonialist ideology has influenced modern mathematics, I would say: "What? I don't think colonialist ideology has influenced modern mathematics at all."
Rather than insinuating that Europe must have stolen math from other parts of the world and pawned it off as its own, or been really mean about rigor, or whatever.
There were several totalizing ideologies in the 20th century that attempted to make all of human affairs about themselves: everything from literature to science to birdwatching had some sort of party line: Did this literature say the right things? Was this science done by the right kind of scientist? Is this bird the right kind of bird to be watching? Judging by their results, I wouldn't look forward to a do-over of that line of thinking.
That's not very actionable advice. I've found this article [0] from Terence Tao very insightful:
[A]ctual solutions to a major problem tend to be arrived at by a process more like the following (often involving several mathematicians over a period of years or decades, with many of the intermediate steps described here being significant publishable papers in their own right):
1. Isolate a toy model case x of major problem X.
2. Solve model case x using method A.
3. Try using method A to solve the full problem X.
4. This does not succeed, but method A can be extended to handle a few more model cases of X, such as x’ and x”.
5. Eventually, it is realised that method A relies crucially on a property P being true; this property is known for x, x’, and x”, thus explaining the current progress so far.
6. Conjecture that P is true for all instances of problem X.
7. Discover a family of counterexamples y, y’, y”, … to this conjecture. This shows that either method A has to be adapted to avoid reliance on P, or that a new method is needed.
8. Take the simplest counterexample y in this family, and try to prove X for this special case. Meanwhile, try to see whether method A can work in the absence of P.
(... 15 more steps)
I loved this book and respect the author but to be honest, for anyone that doesn't already know, it is geared towards highschool level mathematics. Still a great book though.
My real analysis instructor drily pointed out “You really like proof by contradiction huh” and my defense was that my main life skill is spotting why plausible-seeming things don’t work correctly
Watercooler guy has never heard of Mathematical Engineering or Edsger Dijkstra. If he had then he'd know that proof by construction is the method par excellence for showing an algorithm is correct.
From the number of math channels I follow, the common advice I see for handling a complex problem is to start with a simpler version of it and solve it. Try simplifying it in a few different ways, find a few different ways to solve each simplification, and then see which ways of solving the problem might work to solve the more complex problem. With experience one gets better at finding the right simplification with less effort/time invested.
And then erase all intermediate steps and just publish the end result - which no one will understand (including yourself a couple of years later), but that's OK /s
I've lamented this effect of the paper writing process for some time. I think a paper should present the steps taken by the authors to arrive at the new results, not just the new results in isolation. Those steps often provide so much useful insight.
Increasingly this evolution is spread across multiple papers, and is not enclosed within a single paper. So you can partially follow the chain of thought.
Tao’s advice is characteristically excellent but I think Eugenia Cheng’s advice (not just the article title but as a whole) is actionable, just not so much for someone who is ready for specific advice like Tao’s. Her message seems to be targeted towards those who are just getting started and who may not otherwise have been interested in studying math.
This is a great way to go about implementing features that deal with complex situations in NP-Complete problems, where you know a subset is trivially solvable, you build patterns / approaches for said problem, and for the remaining cases, just use approximate solutions.
That's about how to solve problems, but to do research you need to find good problems, and that turns out to be more important, imo. How do you pick "major problem X" anyway? This strategy from Tao just leads to incremental results...
1. If Tao (a Fields medalist) is telling you a process to follow, a dismissal of "just leads to incremental results" requires more evidence than just a bald claim. IMO, he's giving away his working process for free.
2. If "all" you want is tenure at a research 1 institution in the us, there's a lot to be said for this process. Another ingredient is "pick an area where other people will be interested in the results."
PS I don't know what to make of your username, but perhaps you have more to say about picking a problem... in which case I'm sure many of us would love to hear it.
The whole tirade on rigorous proofs and colonial mindset puzzles me.
From my understanding (I am European ...), rigorous proofs were increasingly seen as a need because as things like calculus progressed, any loose definition had a tendency to become a minefield. Think of the definition of derivatives, continuity. Also, at a time where things were getting mechanize because we could and it was a gain, it was natural to check if we could somehow mechanize math. After all, Pascal made a machine that could do sums, why not a machine for proofs ?
I don't think 'colonial' is the right word; I think it's more to do with theology and Europe's very twisted (and ongoing) history with it.
A very clear example is S Ramanujan's way of doing mathematics, where proofs were secondary to 'calculate, calculate, calculate, then hypothesize, then maybe prove' approach. This was not a one off thing, nor was his strange approach unique.
Indian traditional mathematics also follows SR's approach of compute & then generalize & then prove (which most working mathematicians actually follow to some degree anyway). India was ahead of much of the world in Math until the cusp of renaissance(finally culminating in the dev. of calculus a good hundred years before Newton, and possibly having traveled to and 'inspired' him with the Jesuit missions), and its outcome's are simply extraordinary.
Of course, the tradition was not given any credit whatsoever due to the prevalent racial-thinking in old-Europe; this now continues because Indologists continue to hold onto their inherited toxic racial thinking and prevent anyone else from touching this subject because 'Hindus are bad/dirty/nazis/<add adjective> people' (you see this with Yoga, NSDR, Mindfullness,... to this day).
Regardless of all the 'credit allocation' business, and somewhat more importantly, it's actually really sad that this tradition is no longer alive, and is instead replaced with a cheap mimicry of Europe. C K Raju has written extensively on this.
It's also not like this way of doing math. is inherently worse. A lot of actual formal math, starting from the construction of real-numbers is actually very 'unreal' when you get down to it. The kinds of hiccups thrown up by real-numbers in the theory of computation needs to be seen to be believed. Chaitin even has a paper by the title 'How real are real numbers'... We actually get around many of these weird issues because the construction of formal mathematics is actually based on ensuring that discretization is stable (this doesn't work in TOC). Still one also can't deny the strides that modern mathematics has made, in large part built on formalization.
All said, to destroy any existing tradition (even Western) is extremely detestable. OTOH given that it is a jealous-monoculture (all of academia has become one big church-like monoculture at this point), the creation of a new tradition needs active support. I wish people would spend time on the latter instead of pointless race-wars (or DEI non-sense).
I've just finished reading Feyerabend's Tyranny of Science which goes down a very similar path to OP's interview:
>In this wide-ranging and accessible book Feyerabend challenges some modern myths about science, including the myth that ‘science is successful’. He argues that some very basic assumptions about science are simply false and that substantial parts of scientific ideology were created on the basis of superficial generalizations that led to absurd
misconceptions about the nature of human life. Far from solving the pressing problems of our age, such as war and poverty, scientific theorizing glorifies ephemeral generalities, at the cost of confronting
the real particulars that make life meaningful. Objectivity and generality are based on abstraction, and as such, they come at a high price. For abstraction drives a wedge between our thoughts and our
experience, resulting in the degeneration of both. Theoreticians, as opposed to practitioners, tend to impose a tyranny on the concepts they use, abstracting away from the subjective experience that makes
life meaningful. Feyerabend concludes by arguing that practical experience is a better guide to reality than any theory, by itself, ever could be, and he stresses that there is no tyranny that cannot be resisted, even if it is exerted with the best possible intentions.
In other words: there's a 'tacit knowledge' that we ignore by just writing down what we can measure in numbers. Feyerabend died before it became en vogue to talk about colonialism's impact, but his points are similar. (This ties in with her gut feeling: the scientific literature pretends the gut feeling does not exist, we invent a logical story of how we've deduced an insight after we've stumbled over the insight)
The scientific literature pretending that gut feeling does not exist, is, in my own opinion, a legend. Most discoveries/invention were stumbled upon by collective efforts, a bit of intuition, lots of work, and a share of luck. Once something is found/made, then we clean up so as it can be shared and taught.
Scientists always had gut feelings, routinely shared and share their feelings, it's what makes most of discussion between scientists. It was always ok to have feelings. However, to build something somewhat reliable, feelings are not enough, so you do the work to turn a feeling into a falsifiable theory that ypu put to the test.
But do "we" really ignore the tacit knowledge? I think you are wrong that "scientific literature" pretends those things.
The issue of "context of discovery" vs "context of justification" is a hundred years old. And modern accounts of measurement and modeling make it quite clear that there is much more going on than simply writing down numbers. Theories/hypotheses can come from anywhere, I don't think that is even disputed.
However, when doing science, maths, engineering, the question is often what people are even saying, and what counts as evidence that something works. You cannnot do this without abstractions, and at some point we'll have to agree what those abstractions are. Should we leave it to individual experience if LK-99 is a superconductor?
You don't need formal maths training to raise a barn or build a kayak because there is brutal feedback if you are wrong. You'll need maths training to learn how to be right about things that will not give you any feedback.
I'm a little confused by your description – it doesn't seem like a tirade at all (and indeed "colonial" is the interviewer's phrase), or even a judgment on anything: more like a sober conversation about the history of mathematics, the introduction of proof, the role of the Western tradition, and what "mathematical thought" has looked like elsewhere.
I might suggest we're seeing the word "colonialism" and jumping to assumptions.
I think there's a quote that illustrates her argument more accurately:
We might marvel at, say, Amish people raising a barn or Inuit people building a kayak without any European formal maths training. We might declare that this is maths — and I think it is. But it’s problematic if we think we’re doing them some great honour by bestowing the label of maths on what they do.
Being a successful research mathematician, although I am not one, sounds nowadays like more about figuring out the market for your skills among multitudes of opaque research projects so I imagine following your gut you might just get into a good post if you're lucky.
We are truly in a time where there are plenty of (bad, unsubstantial) ideas out there, i reckon it takes a good instinct to be a good mathematician (or any other thing that does not rely on public opinion).
This is a philosophical viewpoint but in some sense all of them are lucky.
Louis de Branges claimed for years to have a proof of Riemann Hypothesis. No one really agreed with his conclusion. He is certainly a first rate mathematician and had he came up with a proof that others accepted he’d be lauded as a great mathematician.
Abhyankar claimed that no one has ever really understood Hironaka’s resolution of singularities paper. Hironaka is a Field’s Medalist and Abhyankar a great mathematician in his own right. He was never able to find a simpler proof for resolution of singularities. If he had gotten lucky then he would have.
The point is, that luck plays a role in terms of whether or not the right idea pops into your head. How many brilliant people labored over problems that simply have no solution and thus aren’t considered one of the greats? Newton wrote more about alchemy than math or physics. He’s not considered a great chemist. One thing the greats have in common is spending a great deal of time thinking about problems. That increases the probability of coming up with a brilliant insight.
Of course, it is not all luck. You do have to have good intuition. Here’s a quote by Chaitin:
Gödel's incompleteness theorem tells us that within mathematics there are statements that are unknowable, or undecidable. Omega tells us that there are in fact infinitely many such statements: whether any one of the infinitely many bits of Omega is a 0 or a 1 is something we cannot deduce from any mathematical theory. More precisely, any maths theory enables us to determine at most finitely many bits of Omega.
In some sense we get a survivorship bias when talking about the greats. They happened to work on a problem that was solvable. I suggest there are many more equally brilliant people who didn’t get lucky and thus are unknown.
The author teaches maths to art students. The paragraph on maths research:
But now I realize that, in maths research, you don’t just follow logical steps. If you do, you’ll never get anywhere interesting. You have to use your gut instinct and feel your way through something first, and then back it up with logic afterwards.
Opposite to TFA, in school, I gave up on maths because explanations were not provided; you had to take things on faith (e.g. algebra of limits, epsilon-delta proofs were not given), and the questions were carefully manufactured to be solvable by the given methods. i.e. puzzle-solving, not general powerful methods. Also, no proof of the uniqueness of prime factorization.
TBF, it's hard to load proofs on to school students, and since Euclid was dropped, the focus is on technicians and results - not truth. But it can be done: spend the time on building foundations; or omit subjects where you can't. I'd like an axiomatic approach.
Alternatively, just do whatever obscure research you want outside of the university system. There won't be anyone to judge your 'success' or lack of it.
When was the last time these mathematicians received feedback from the outside world instead of just deciding within their group who to declare as #1? They literally get paid by tax payers to do what is often the equivalent of glorified crosswords full time then congratulate themselves that they're good at it.
Readit News
1. Network, network, network. Get famous or established mathematicians interested in your work.
2. Be a minority. Due to the prevailing ideology of academia, it's much easier to be successful if you are a minority. (I am not saying it doesn't take hard work, but the competition is not as extreme due to intense equity measures.) I will laugh if anyone gets mad at this point. Equity rules make this point an incontrovertible fact.
3. Solve problems related to the work of established mathematicians. Interesting or novel problems that are not related to the work of others will be ignored.
4. Forget about being curious. Okay, this one has some dark humour about it...but seriously, if you just explore your curiosity, you might solve cool problems but you won't be "career successful". It's much better just to make useless investigations into ever-more esoteric problems that validate the "successful" work of others which is becoming increasingly technical.
A note: this might sound a bit cynical, but I thought I'd give another viewpoint to show a bit more of the balanced reality of the math world.
Maybe I should add a 5th point, but I need to phrase it very cautiously.
5. If you're interested in doing mathematics in an institution of higher ed in the US, focus your work on teaching, and be prepared to live in the middle of nowhere.
Seems like flawed reasoning to me. If a group as a whole is underrepresented in a highly competitive environment like academia, then presumably they have a harder time getting in or staying in.
Despite equity measures, whites in the us are not underrepresented. It follows that it is not harder for whites, but still more difficult specifically for black and Latino people to make it in academia, no?
No, that is fallacious. It could also mean that fewer members of that group try to get in in the first place. That is because there are multiple stages to getting into academia. But the question I was referring to was about the final stage: getting into research math, and then minorities are already under-represented in the pool of people thinking about becoming a research mathematician.
To put it simply, IF a person is a minority and is already in grad school, they will have an easier time going further simply because of equity hiring. That's all I meant.
Won't help if you're Asian IMO.
There was something distinctive about work in foundations of math in the late 19th / early 20th centuries, work which came out of European cultures (or more accurately Latin Christendom). Plausibly, this work was centuries in the making, the culmination of a trend towards increasingly analytic and systematizing modes of thought that begins with the recovery of Greek philosophy in the middle ages.
But, ok, even if you don't buy that, how do you get from the Congo Free State to Frege, Hilbert and Goedel? It's a radical claim that would be utterly fascinating if it were true. But Chang just kind of tosses it off at the end and asks us to accept it as an unremarkable truism.
Rather than insinuating that Europe must have stolen math from other parts of the world and pawned it off as its own, or been really mean about rigor, or whatever.
There were several totalizing ideologies in the 20th century that attempted to make all of human affairs about themselves: everything from literature to science to birdwatching had some sort of party line: Did this literature say the right things? Was this science done by the right kind of scientist? Is this bird the right kind of bird to be watching? Judging by their results, I wouldn't look forward to a do-over of that line of thinking.
Summary: https://www.math.utah.edu/~alfeld/math/polya.html
2. If "all" you want is tenure at a research 1 institution in the us, there's a lot to be said for this process. Another ingredient is "pick an area where other people will be interested in the results."
PS I don't know what to make of your username, but perhaps you have more to say about picking a problem... in which case I'm sure many of us would love to hear it.
Deleted Comment
From my understanding (I am European ...), rigorous proofs were increasingly seen as a need because as things like calculus progressed, any loose definition had a tendency to become a minefield. Think of the definition of derivatives, continuity. Also, at a time where things were getting mechanize because we could and it was a gain, it was natural to check if we could somehow mechanize math. After all, Pascal made a machine that could do sums, why not a machine for proofs ?
A very clear example is S Ramanujan's way of doing mathematics, where proofs were secondary to 'calculate, calculate, calculate, then hypothesize, then maybe prove' approach. This was not a one off thing, nor was his strange approach unique.
Indian traditional mathematics also follows SR's approach of compute & then generalize & then prove (which most working mathematicians actually follow to some degree anyway). India was ahead of much of the world in Math until the cusp of renaissance(finally culminating in the dev. of calculus a good hundred years before Newton, and possibly having traveled to and 'inspired' him with the Jesuit missions), and its outcome's are simply extraordinary.
Of course, the tradition was not given any credit whatsoever due to the prevalent racial-thinking in old-Europe; this now continues because Indologists continue to hold onto their inherited toxic racial thinking and prevent anyone else from touching this subject because 'Hindus are bad/dirty/nazis/<add adjective> people' (you see this with Yoga, NSDR, Mindfullness,... to this day).
Regardless of all the 'credit allocation' business, and somewhat more importantly, it's actually really sad that this tradition is no longer alive, and is instead replaced with a cheap mimicry of Europe. C K Raju has written extensively on this.
It's also not like this way of doing math. is inherently worse. A lot of actual formal math, starting from the construction of real-numbers is actually very 'unreal' when you get down to it. The kinds of hiccups thrown up by real-numbers in the theory of computation needs to be seen to be believed. Chaitin even has a paper by the title 'How real are real numbers'... We actually get around many of these weird issues because the construction of formal mathematics is actually based on ensuring that discretization is stable (this doesn't work in TOC). Still one also can't deny the strides that modern mathematics has made, in large part built on formalization.
All said, to destroy any existing tradition (even Western) is extremely detestable. OTOH given that it is a jealous-monoculture (all of academia has become one big church-like monoculture at this point), the creation of a new tradition needs active support. I wish people would spend time on the latter instead of pointless race-wars (or DEI non-sense).
(edit: expand)
>In this wide-ranging and accessible book Feyerabend challenges some modern myths about science, including the myth that ‘science is successful’. He argues that some very basic assumptions about science are simply false and that substantial parts of scientific ideology were created on the basis of superficial generalizations that led to absurd misconceptions about the nature of human life. Far from solving the pressing problems of our age, such as war and poverty, scientific theorizing glorifies ephemeral generalities, at the cost of confronting the real particulars that make life meaningful. Objectivity and generality are based on abstraction, and as such, they come at a high price. For abstraction drives a wedge between our thoughts and our experience, resulting in the degeneration of both. Theoreticians, as opposed to practitioners, tend to impose a tyranny on the concepts they use, abstracting away from the subjective experience that makes life meaningful. Feyerabend concludes by arguing that practical experience is a better guide to reality than any theory, by itself, ever could be, and he stresses that there is no tyranny that cannot be resisted, even if it is exerted with the best possible intentions.
In other words: there's a 'tacit knowledge' that we ignore by just writing down what we can measure in numbers. Feyerabend died before it became en vogue to talk about colonialism's impact, but his points are similar. (This ties in with her gut feeling: the scientific literature pretends the gut feeling does not exist, we invent a logical story of how we've deduced an insight after we've stumbled over the insight)
Scientists always had gut feelings, routinely shared and share their feelings, it's what makes most of discussion between scientists. It was always ok to have feelings. However, to build something somewhat reliable, feelings are not enough, so you do the work to turn a feeling into a falsifiable theory that ypu put to the test.
The issue of "context of discovery" vs "context of justification" is a hundred years old. And modern accounts of measurement and modeling make it quite clear that there is much more going on than simply writing down numbers. Theories/hypotheses can come from anywhere, I don't think that is even disputed.
However, when doing science, maths, engineering, the question is often what people are even saying, and what counts as evidence that something works. You cannnot do this without abstractions, and at some point we'll have to agree what those abstractions are. Should we leave it to individual experience if LK-99 is a superconductor?
You don't need formal maths training to raise a barn or build a kayak because there is brutal feedback if you are wrong. You'll need maths training to learn how to be right about things that will not give you any feedback.
I might suggest we're seeing the word "colonialism" and jumping to assumptions.
I think there's a quote that illustrates her argument more accurately:
Louis de Branges claimed for years to have a proof of Riemann Hypothesis. No one really agreed with his conclusion. He is certainly a first rate mathematician and had he came up with a proof that others accepted he’d be lauded as a great mathematician.
Abhyankar claimed that no one has ever really understood Hironaka’s resolution of singularities paper. Hironaka is a Field’s Medalist and Abhyankar a great mathematician in his own right. He was never able to find a simpler proof for resolution of singularities. If he had gotten lucky then he would have.
The point is, that luck plays a role in terms of whether or not the right idea pops into your head. How many brilliant people labored over problems that simply have no solution and thus aren’t considered one of the greats? Newton wrote more about alchemy than math or physics. He’s not considered a great chemist. One thing the greats have in common is spending a great deal of time thinking about problems. That increases the probability of coming up with a brilliant insight.
Of course, it is not all luck. You do have to have good intuition. Here’s a quote by Chaitin:
Gödel's incompleteness theorem tells us that within mathematics there are statements that are unknowable, or undecidable. Omega tells us that there are in fact infinitely many such statements: whether any one of the infinitely many bits of Omega is a 0 or a 1 is something we cannot deduce from any mathematical theory. More precisely, any maths theory enables us to determine at most finitely many bits of Omega.
In some sense we get a survivorship bias when talking about the greats. They happened to work on a problem that was solvable. I suggest there are many more equally brilliant people who didn’t get lucky and thus are unknown.
TBF, it's hard to load proofs on to school students, and since Euclid was dropped, the focus is on technicians and results - not truth. But it can be done: spend the time on building foundations; or omit subjects where you can't. I'd like an axiomatic approach.
When was the last time these mathematicians received feedback from the outside world instead of just deciding within their group who to declare as #1? They literally get paid by tax payers to do what is often the equivalent of glorified crosswords full time then congratulate themselves that they're good at it.
Deleted Comment