And this blog is the best guidance for me to learn the wholehearted of mathematics. Not only technical or motivations, now my priority moves to the struggling of math. Thanks for your aspirational posts.

]]>– I read the aforementioned link and it definitely applies to both Andrew Wiles and Grigori Perelman. Both had produced important results prior to going rark to work in their respective problems. However, that is not the case for Yitang Zhang. Zhang didn’t have any meaningful career in mathematics (other than getting a PhD and teaching undergraduate math after having held a series of menial jobs) prior to publishing his result on primes.

– My most important objection though is that mathematics is very different from other scientific endeavors in that all you need to produce good mathematical results is access to the relevant research -which has never been easier thanks to the internet-, a pen and a piece of paper. In terms of resources, it has a relatively low barrier of entry although ironically, in terms of intellectual rigor, it has a pretty high barrier of entry. Given the high stakes and that, as you mention, mathematicians have had their reputations damaged for publishing erroneous results, why would anyone want to work publicly on an important problem? I remember watching NOVA’s documentary “The Proof” on the work of Andrew Wiles and I remember him saying -I am paraphrasing- that the work he did in secret was very fulfilling and enjoyable. No matter how tough things seemed, he enjoyed every minute of it. On the other hand, when he had to work in the public eye for around 1 year fixing his original proof he experienced a lot of pain. Peter Higgs also mentioned that under the current “publish or perish” environment in academia he would probably have not discovered the particle that bears his name and for which he got a Nobel prize. To be clear, I am not saying that publish or perish should be abolished or that a person’s publication record shouldn’t be the main criteria when making hiring and promotion decisions in academia, what I am saying is that given that in academia the “publish or perish” mindset creates a particular environment that is not for everyone, perhaps that environment is not even the optimal when it comes to enabling work like that of Wiles, Perelman or Zhang. Of the three only Wiles had tenure or a tenured like job when they did the work for which they are celebrated. And none of them did said work as part of a traditional “tenure track” process. Further, given mathematics low barrier of entry -in terms of resources- perhaps an alternative path is more conductive for those who love math but who lack the stamina -or simply don’t want- to put up with the rigors of the tenure track process.

]]>It’s also a good idea to take survivorship bias into account when trying to extrapolate from only the most successful examples of any given endeavour.

]]>– Andrew Wiles: Fermat’s Last Theorem

– Grigori Perelman: Poincare’s Conjecture

– Yitang Zhang: establishing the existence of infinitely many pairs of prime numbers separated by 70 million or less

Also, while still under scrutiny, Shinichi Mochizuki’s Inter-universal Teichmüller theory increasingly seems to be one of these.

Given these examples, shouldn’t it be clear by now that the best advice anybody interested in working on a very difficult and challenging problem should have is this: “work in secrecy, don’t tell any one about what you are doing and only when you are 300% sure your work is correct, put your work in arxiv.org and let others fight for the credit of understanding your work”.

The evidence seems pretty overwhelming that the advice contained in these pages is about careerism in mathematics, or any other field for that matter, not about producing great results. To put an analogy with the business world, this advice is about how one can climb the corporate ladder, not about how one can become a successful entrepreneur. I am not claiming that the intersection between careerism and being a great entrepreneur is an empty set -there are examples of successful entrepreneurs who started their careers as careerists- but careerism and entrepreneurship seem to be very different beasts.

*[The cases of Wiles, Perelman, and Zhang are discussed at https://terrytao.wordpress.com/career-advice/dont-prematurely-obsess-on-a-single-big-problem-or-big-theory/ . In particular, it is instructive to look at the earlier work of these three mathematicians, prior to their most well known breakthroughs, as they all followed the more traditional path of working openly within the mathematical community, while also laying the groundwork for their later work. -T]*

INDIAN ]]>