How to be successful as a research mathematician? Follow your gut
nature.com
nature.com
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.
Won't help if you're Asian IMO.
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.
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.
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.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.
I'm sorry if I come off as a little obtuse, so I might as well say what I'm really thinking. My thinking leans towards Eugenia's, and it's a lot more conservative: the question itself is worth thinking about, but it's a lot to think through, and the link may as well be tenuous. But, in my thinking, if NATO once funded pure mathematics (as Grothendieck once protested against), if DARPA also currently funds pure mathematics, then it is worth asking what historical narrative explains it, and whether it applies to the slightly more distant past, too.
Above all I don't believe that the academic enterprise has ever been a neutral and disinterested endeavor, either – it's a lot of public money, and wherever there's public money politics comes before or after. I don't think that's a totalizing way of understanding academia.
> “Now, at the same time as all this, it is true that mathematics as a field has grown carefully and rigorously according to a carefully constructed framework of logic and rigour. It is, arguably, the framework that is keeping some maths in and keeping some maths out. But... there are questions about the values built into the framework of mathematics. That framework is geared towards the principles of ‘progress’ and ‘development’. Deep down I have an uncomfortable suspicion that those principles are indelibly linked to colonialism, imperialism and the urge to conquer others.” (p. 320-1)
That's the argument in its entirety. What humanity was up to in the ten millennia of recorded history before formal mathematics is left unexplained.
The most parsimonious explanation for the military's funding of mathematics research is that mathematics often has applications to military technology. But even here, we have to be careful: it's typically applied math that's the most, well... applicable, which is precisely the sort of thing that pure mathematicians are supposed to look down upon as "not really math". (The pure math work at DARPA is AFAICT more a consequence of DARPA's idiosyncratic governance structure that gives program managers wide latitude to swing for the fences, accepting that there will be some strikeouts.)
But the fact that modern Western mathematics is indispensable for advanced engineering projects at all, in a way that Sanskrit prosody and Inuit kayak architecture are not, is pretty crucial to understanding the funding situation: it's funded because it's useful, not useful because it's funded. If we don't grant that math is actually unreasonably effective in explaining / predicting / manipulating the world then sure, what math gets funded looks like another courtly intrigue. But it seems that the unreasonable effectiveness is precisely what needs to be explained away on Cheng's view.
I'm all ears for the contrary view, but we've already written more on the topic than Cheng did in the entire book, which is light on argument and regrettably heavy on insinuation.
That being said, you're right –– this has gotten into a bit of a side track, and I'm not about to re-scour through the book to see precisely what argument she's making, and the level of generality she stated it, either! (My summer reading list is full.)
I wonder if Eugenia's comments should actually be a historical argument – that way, multiple explanations can co-exist on different levels, rather than trying to find the most parsimonous theory and excluding the rest. But I'm not about to find out, and, even then, I figure a good historical explanation does need at least a dissertation's worth of work and evidence :)
[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)
Then again proving code works isn't everything either. There's a reason Knuth once stated "Beware of bugs in the above code; I have only proved it correct, not tried it."
Type checking is a nice intermediate, though not all languages allow all properties you care about to be encoded in types.
you can apply the techniques anywhere though - I think the book is much more philosophy than math
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.
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)
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.>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.
BTW TFA uses "feelings" in two different senses: hunch, inkling, intuition, suspicion, gut instinct vs. emotion (as in music).
In mathematics, there is even the formal term "conjecture" for an unproven theorem.
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.
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.
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.
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.
1. Write down the problem. 2. Think very hard. 3. Write down the solution.
If your maths problem is actually a specific instance of a known broader problem then just stating that will point the way forward. In fact a proof that your problem is an instance of a broader problem can generally be a worthy paper in its own right.
I'm in CS research (with most of my work being applied to biological problems), so I'm a few steps removed from pure math, but I can completely related to this. It's why I find writing grants so difficult -- because they generally require you to identify and describe the problem and propose some potential solutions. But that's like 90% of the work for the whole thing!
One effect is like how Latin used to be the language of professionals, so you'd better learn Latin if you want to access the knowledge base.
* Forget about glory and peer recognition. One day you'll die and as time passes you'll be forgotten, including anything you made. Don't waste energy fighting it, just embrace it and carry on. Do it for that 1 mn moment of bliss when you found something nice.
* Accept repeated failure. It's ok.
* Spend time to learn why attempt N failed before going to attempt N+1,
* Before attempt N+1, go out for a walk, cook something nice, do something with your kids, have a beer, watch a movie