On proof and progress in mathematics

behavior of that program, precise and rigorous enough that a computer can convert the understanding embodied in that code into that actual behavior. But, crucially, it's human readable: the reason we don't program in machine code is to maximize our and other people's understanding of what exactly the program (or system) does. From this perspective, when we write code, articles, etc., we should be highly focused on whether our intended audience would even understand what we are writing (at least, in the way that we, the writer, seem to). Thinking about cognitive load seems to be good, because it recognizes this ultimate objective. On the other hand, principles like DRY -- at least when divorced from their original context -- don't seem to implicitly recognize this goal, which is why they can seem unsatisfactory (to me at least). Why shouldn't I repeat myself? Sometimes it is better to repeat myself!? When should I repeat myself?? If you want to see an example of a fabulous mathematician expressing the same ideas in his field (with much better understanding and clarity than I could ever hope to achieve), I highly recommend Bill Thurston's article "On proof and progress in mathematics" <https://arxiv.org/abs/math/9404236>.

mind on how to think about the whole field. I'm sure we've come across abstruse books where every theorem has a rabbit being pulled out of a hat, whereas other readers find it intuitive. It's because the latter has developed a mental model for the discipline, and you haven't. So he set about slowing down, and focusing on holding lots of seminars where he worked with other mathematicians to explain the thought process. Eventually others started publishing proofs of key theorems. When people publish in a journal, they are not merely doing it to show the result. They are having a conversation with other mathematicians. If they cannot explain their own proof, they're not having a conversation. This is why even decades after the Four Color Theorem was proved, plenty of mathematicians don't consider it "mathematics".

(which ones they elide depends on the target audience), for the simple reason that it a) massively shortens the book b) maintains the "flow" well, instead of stopping every few sentences to distract the reader with ideas not directly relevant to the argument c) acts as a natural method to help people choose materials: if you're hitting things you can't understand or even parse every two lines, maybe you'd be better served by coming back to it a while later? In other words, the skipped details and "why would that be the case?" are going to be there even in books that are considered masterpieces of mathematical writing, and everyone encounters them and learns to deal with them. So what do you do when you hit the inevitable "yeah, right, that's trivial, suuuuuure" next time? To paraphrase someone I can't place right now, you either go under it, over it, around it, and -- if all those fail -- then ($deity willing or no) you push right through it. (Exercise for the cough careful cough reader: what do I mean?) Less faux-metaphorically, try * skipping forward a little -- new ideas do not have to be learned and absorbed linearly; case in point: the next paragraph goes into detail on this :) * checking out another book on the same topic * asking a question on "MSE" / math.stackexchange.com (the /r/math subreddit is good too) * checking out the extremely friendly MSE chatroom, which I credit with fixing a ton of my "I am just stupid" moments * looking for lecture notes (shameless plug: [0]) online * (failing everything above) spending a day walking around your house with your hands squeezing your head from the sides, trying to mimic the romantic picture of the tortured-genius lonewolf-mathematician-in-training and somehow trying to become more intelligent by increasing the density of your brain matter until you've triumphed over whatever puny margin-note was keeping you occupied Also note that mathematical understanding and intuition always, always grow organically. You often end up understanding something "properly" years after you stop struggling with it, and it only gets better as you learn things and see how disparate ideas fit together or even become special cases of a general concept, or variations upon a basic theme. This is touched upon in Terry Tao's excellent career advice essays[3], which are aimed at students learning to make sense of the (mathematical) world: in particular, I'd recommend * https://terrytao.wordpress.com/career-advice/there%E2%80%99s... * https://terrytao.wordpress.com/career-advice/does-one-have-t... Grothendieck talks about his journey (see [4]) himself, and his writing is beautiful (if also prone to creating a ton of hope in young people encountering this anecdote for the first time). It's profound, but not an authoritative account of what it means to do mathematics, by any means (just like Hardy's A Mathematician's Apology). > Since then I’ve had the chance in the world of mathematics that bid me welcome, to meet quite a number of people, both among my “elders” and among young people in my general age group who were more brilliant, much more ‘gifted’ than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle–while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured) things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates almost by sleight of hand, the most forbidding subjects. In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still from the perspective or thirty or thirty five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve done all things, often beautiful things in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have to rediscover in themselves that capability which was their birthright, as it was mine: The capacity to be alone. Some of the stuff on this page[1] is very helpful, whatever your current level of knowledge. Ctrl-F "tendrils". You might find Thurston's essay "On Proof and Progress in Mathematics" (arXiv link: [2]) interesting too. Good luck! [-1] There are some well-known books that students are almost always advised to come back to later (and there are probably more that they're advised to not bother with, ha). [0]: http://refinator.herokuapp.com/ [1]: http://math.stanford.edu/~vakil/potentialstudents.html [2]: https://arxiv.org/abs/math/9404236 [3]: https://terrytao.wordpress.com/career-advice/ [4]: https://quomodocumque.wordpress.com/2010/07/05/the-capacity-...

in the production function during knowledge work. 2. That production function and the associated sociological process spans years or even decades, and must outlast many of the artifacts that are produced during the early years of the function. 3. You cannot get the right lines of code or the right theorems proved without running that sociological process alongside the artifact production process. 4. It is impossible to completely separate the sociological process from the artifact construction process. If you just iterate on artifacts then too much of the required hidden state is lost to make progress in the right direction. This is true even if you include distilled artifacts capturing pieces of the sociological process (eg meeting notes, documentation, commit logs, prompts). 5. So you need that sociological process, or something like it, to still happen. 6. For a lot of knowledge work that process plays out in extremely high-fidelity social interactions [3] that we have not yet captured in the datasets that would be required to reproduce those dynamics. 7. And even if we do collect that data, our current architectures and training algorithms and hardware would be useless given the size of the datasets. So: the technology today gives us the ability to iterate on the production of artifacts. But it does not sufficiently simulate the social process which gives rise to the Right artifacts. This isn't exactly what I actually think, but it's a version of the thing that I intuit when I watch heavy use of AI in both software projects and formalization projects. And simulating that process feels way harder than people are currently assuming. [1] https://arxiv.org/pdf/math/9404236 Section 5. [2] https://www.nationalacademies.org/read/11587/chapter/11 pp 166-168. [3] there is a reason we still gather in-person around white boards, and why doing so is more crucial for some types of work than others.

so conceptually deep as mathematics of the article (while still having a few great ideas). But understanding the source would require familiarization with 'tedious details' which incidentally, what this article is also about. So the issue, stated this way, is not so much raw talent as time and effort. This leads to the topic of motivation - finding something great in an idea can lead to investment in the subject. For those more talented, the journey might be easier. Alan Kay's maxim is crucial - a change of perspective can simplify things relative to 80 extra IQ points. A long sequence of technical steps can be compressed into a single good intuitive definition/idea like the complexity of navigating a path becomes clear from a higher viewpoint. Grothendieck, considered by many to be the best mathematician of the past century, made the point that there were several colleagues who were more proficient at symbolic manipulation. But, he was able to look at things with a more foundational perspective and could make great contributions. Here's a good essay by Thurston "Proof and Progress in mathematics". https://arxiv.org/pdf/math/9404236 He discusses this problem of being intimidated by jargon.

and philosophers and containing both formal and informal reasoning: http://www.thatmarcusfamily.org/philosophy/Course_Websites/R... I don't know how one could "remove the philosophy" from this work without making it far less useful to mathematicians. The entire reason the T-schema is used in model theory is because of Tarski's philosophical argument that it provides a meaningful definition of truth.

more everyday level, it is common for people first starting to grapple with computers to make large-scale computations of things they might have done on a smaller scale by hand. They might print out a table of the first 10,000 primes, only to find that their printout isn’t something they really wanted after all. They discover by this kind of experience that what they really want is usually not some collection of “answers”—what they want is understanding."

of how the bulk of research gets done. Of course, even if it doesn't require genius, making serious research contributions in mathematics does typically require many thousands of hours of training and hard work (usually in the form of a PhD) which most people are not going to dedicate their lives to. Given this, I really like Bill Thurston's perspective, from the famous essay "On proof and progress in mathematics" (http://arxiv.org/abs/math/9404236), which is that proving new theorems is only a small part of the project of mathematics. The real goal is to increase humanity's mathematical understanding, which does not necessarily grow monotonically: there are plenty of results proved 100 years ago that are not known or understood by any currently living mathematician, thus have been "lost" to the mathematical community. And of course there are new people born every day who know no math as all, so understanding of mathematics will tend to decrease unless people actively work to sustain it. In this view, understanding and communicating mathematical insights is just as important and valid a part of the intellectual process as proving new theorems, and requires a different set of talents. Plenty of great researchers are not great communicators, and vice versa. (Terry Tao actually is an excellent writer, but again he's only one person and there's a ton of math out there to be shared). You may not know as much math as Terry Tao, but anyone on HN probably knows more than the vast majority of adults (the bar here is really low), and the world at this moment is certainly not suffering from a surplus of mathematical thinking. So whatever your background and abilities, there are almost certainly lots of useful ways to contribute.