I think most people commenting in this thread are reading the headline and/or reading the article as outside the context of mathematics. This is Terry Tao's blog and not a VC's microblog. The very first block of text is a quote from "I want to be a mathematician". He is purely talking about how to approach mathematics at a level sufficiently close to the frontier. What he is saying is that when you are taught something like optimization theory, the definition of convexity is presented to you in the first lecture. A "stupid" question to ask is what makes convex functions special. After thinking a lot about this question, I've come to the conclusion that convex functions are functions that are trivial to optimize but the converse is not true. Not all function that are trivial to optimize are convex functions. There are then plenty of questions like which other classes are easy to optimize. In Terry's words, "the answers to these questions will occasionally lead to a surprising conclusion, but more often will simply tell you why the conventional wisdom is there in the first place, which is well worth knowing.". Another example I ran into was the notion of metric spaces and completeness. It is "stupid" to ask what happens if you get rid of completeness. Stupid because you were just taught completeness for a reason. Trying to understand gives you a whole new appreciation than being presented the formal definition refined over many years.
> Trying to understand gives you a whole new appreciation than being presented the formal definition refined over many years.
This is so true, and which is why makes learning from certain professors way more enjoyable than others. What you have explained is in some sense an explanation of 'thinking from the first principles' or 'intuition behind the concept', at least imo.
And it would be great if a lot of this (why these first principles) was at least mentioned in passing in the teaching. This is not just a question of being close to the frontier but about understanding why and where. Why some precept and where we are on the map. Necessarily in teaching this would be "just in passing" but still useful. Useful in all subjects but all the more so in highly abstract ones.
I asked gpt4 about this first and it didn’t really know. What is it about “First Principles“ that’s cringeworthy or pretentious? I think I’m out of the loop on this one.
> I've come to the conclusion that convex functions are functions that are trivial to optimize
I think a lot of maths is created like this. I'm trying to solve problem P in domain D. I couldn't do that, but I've noticed that I could do it in subset D1 of the domain. If problem P was important enough and D1 large enough, I become very famous within mathematics and people start studying D1 by itself, without connection to P. After a generation or more a lot of experts in D1 might not know the original reason why someone decided to study it. Eventually some historians of science re-discover the original motivation and publish it as a curiosity.
Very true, combined with the fact that deep understanding of D1 turns out, centuries later, to be of practical use in building some sort of physical machinery.
One of the things that separates ordinary people from smarter people is the topic of this article, the ability to imagine new concepts, questions, ideas. Colloquially we call this creativity, and it stems from a large degree of playfulness and enjoyment of the subject at hand.
What separates geniuses from the crazies is the ability to test those ideas and the humility to dismiss the ones that do not work. It’s that combination that makes a great thinker.
One bit of advice for problem solving that I've taken to heart and find very effective is to not shy away from trying solutions you'd be embarrassed to tell other people because they'd probably think you were crazy/an idiot. Sometimes that direction is where the truly effective stuff is. Of course you should also try the obvious, simple, "normal" solutions - but don't be afraid to experiment.
A bit like "write drunk, edit sober" but instead of drunk, you're a raving lunatic.
Man, I think feeling crazy for trying stuff even when completely alone, is something not enough people talk about. It's clear when you do it that most artists and creatively successful people have done it and gotten over it, but it's like something that you actually don't hear about or maybe don't register until you start doing it, and do it enough to get used to the feeling and carry on with whatever you were thinking of.
General research advice: I would also go in with the mindset "I am trying to disprove this approach as quickly as possible". This allows you to iterate quickly across hypotheses and not get dogmatically fixated on one approach. Cast the net wide in the initial search, including both normal and crazy solutions, then measure what approach is working, and then narrow the search based on that, learning lessons along the way. Don't start with a narrow search. Also, be sure that the different approaches you're trying are uncorrelated and ideally thematically/qualitatively different. Many researchers get tripped up trying "different" things that are really just superficial iterations of the same core concept.
Is this sort of that Warren Buffet negative problem solving thing or are is it just like stochastic trial and error. It sounds similar in spirit if not in exact form
I haven't read Buffet on this but I wouldn't call it stochastic because each hypothesis should be driven roughly by theory, sort of multiple educated guesses that are each uncorrelated and unique
Even being around other people can conform your thoughts to normal terms: get them back "on track", where your only choices are which track, which direction. It's not just the thoughts, but the terms, the dimensions.
I think a most fragile stage is when you've part-imagined a new track, and unclear if it's a good idea, if it works, even what it is. This can collapse like a house of card in the lightest breeze, a dream upon waking.
Human beings are automatically conformist - crucial for learning, language, custom, law, cooperation and trade, from the beginning - so it's good we need special circumstances for "creativity".
The way I frame this for myself is - just get the stupid solutions out of the way first. Do them so you can see how dumb they are, and move on to the better ones.
And that comes with the proviso that sometimes, the stupid solution is actually just fine, and you didn’t even need to make it so complicated.
I try to go one step further and only talk publicly about solutions I've tried that make me look like an idiot. But that's because my main motivation in speaking to others is to entertain, not to impress :)
This is actually something I discovered when I had a mountain of dishes and it had me down for months.
One day, I thought, I don't even want all this crap, I would rather just have one of each essential "form" (large spoon, large fork, cup, plate, etc) and I certainly can't get them washed. So I grabbed a garbage bag and gently placed everything in (clearing the sink) and disposed of the weight.
Haven't had a problem since and I live much more peacefully and unburdened by all that.
> One of the things that separates ordinary people from smarter people is the topic of this article, the ability to imagine new concepts, questions, ideas. Colloquially we call this creativity, and it stems from a large degree of playfulness and enjoyment of the subject at hand.
Pretty sure we call that imagination. Creativity, shockingly enough, involves creation, not just imagination.
You can creatively destroy things as well. It’s kind of pedantic to say that creating in the mind does not count as “creating”, just imagining. What about solving abstract problems?
Crazies say they validate and test their idea, but they actually suffer from severe confirmation bias where they summarily dismiss any counterexample or problem.
> Would you be able to spot a Ramanujan from the dross?
Yes. We would do this by analyzing their arguments for logical errors, then testing their new theory, then hopefully proving their new theory. The difference between a crazy and genius is the results their new ideas produce.
I don't think you appreciate how crazy Ramanujan was in his time. While he was somewhat appreciated in India (not at any high level though), when he reached out to professors in the UK, the first several thought he had no ability to become a mathematician. They were used to seeing mathematical proofs, and Ramanujan not providing any made him appear fraudulent. The only one who understood his brilliance was G.H. Hardy, also considered an eccentric by his peers.
In Ramanujan's words, "an equation for me has no meaning unless it expresses a thought of God". In mathematician circles of the early 20th century, this was an obscene statement. Even today, can you imagine a researcher in any field saying something like that, and being taken seriously?
Yet Ramanujan is probably one of the greatest mathematicians of all time. In Hardy's word, he had "never met his equal, and can compare [Ramanujan] only with Euler or Jacobi".
To think that "we" (whoever you mean by that) can recognize sheer talent of Ramanujan's type is, frankly, arrogant. If anything, the ivory towers are even more enclosed - research is more and more about the quantity of publications, not about the quality of ideas. A recent Nobel Laureate stated that if he had not received the Nobel Prize, his university would have probably fired him, for lack of a consistent publication record.
So no, I don't agree with your statement. While the scientific method you describe for approaching nature has worked tremendously well for understanding the world, it still completely fails to capture the core of what it is that makes a person a true genius.
Which only proves the point further - Hardy could have easily selected one of the other 999 cranks, yet by some intuition he knew that Ramanujan was the real deal.
After seeing Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardy said the theorems "defeated me completely; I had never seen anything in the least like them before", and that they "must be true, because, if they were not true, no one would have the imagination to invent them".
Ideas are cheap and proof is expensive. It’s impossible to give all ideas equal consideration. You have to apply a filter, it’s inevitable. The question is just what filter you apply.
Ramanujan was missed by at least one maths professor. Also, there are mathematical proofs such as that of Fermat's last theorem, or the claimed proof of the abc conjecture by Shinichi Mochizuki for which verification is very difficult.
I'll agree that the costs don't come close to replicating work by LIGO or CERN.
There's a difference between a flawed proof (e.g. Wiles's original proof of FLT which was found to contain a gap) and outright crankery, though. There are more mathematical cranks than most people would think, and their arguments usually fall apart rather easily.
yes, but it depends on the field. ramanujan had the great advantage of working in a field where ideas can be tested with pencil and paper. if you show a purported ramanujan a lacuna in his proof, or if you can't find one, that's pretty good evidence
it's much harder to skim the dross off a leonardo, to correct your mixed metaphor, because siege engines and flying machines require empirical testing, and that's expensive. within the hypothetico-deductive method you can tell whether someone's logic is incorrect (just as with ramanujan) but if they're proposing a different hypothetical basis, you often have to test it empirically
often you can proceed from their proposed hypotheses to obviously empirically false conclusions. but historically that hasn't been an especially good guide; in the 19th century kelvin proved that the sun was younger than the earth, demonstrating that something was wrong with his hypothetico-deductive framework, but it took quite a while to figure out what. and today relativity and quantum mechanics are widely considered irreconcilable
Would you, though? Ramanujan famously left out his proofs quite often, relying on his intuition which was especially phenomenal. Many accomplished mathematicians did not see his genius for what it is and it took the support of Hardy to really further Ramanujan's career and accomplishments.
Maybe not me personally, but more accomplished, smarter mathematicians would. His results were checked as being correct and he did publish papers in which he showed some of the steps and this was before Hardy. T
Ramanujan was so ahead of his peers that they didn’t even know how to assess him or his work.
After seeing Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardy said the theorems "defeated me completely; I had never seen anything in the least like them before",[76] and that they "must be true, because, if they were not true, no one would have the imagination to invent them".[76] Hardy asked a colleague, J. E. Littlewood, to take a look at the papers. Littlewood was amazed by Ramanujan's genius. After discussing the papers with Littlewood, Hardy concluded that the letters were "certainly the most remarkable I have received" and that Ramanujan was "a mathematician of the highest quality, a man of altogether exceptional originality and power".[74]: 494–495 One colleague, E. H. Neville, later remarked that "not one [theorem] could have been set in the most advanced mathematical examination in the world".
The current norms of pedagogy permit incremental improvements, but they fail to handle talent that is far and away better than the current top of the field. In fact they find ways to prohibit people from shaming them in such a way, which is understandable. But unfortunately, a Ramanujan today would probably be relegated to “remedial education” or institutionalized.
As for his results being checked, they were - after nearly a century in some cases:
As late as 2012, researchers continued to discover that mere comments in his writings about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle number theory results that remained unsuspected until nearly a century after his death.
What I learned from reading about Ramanujan is that true geniuses are not just incrementally ahead of their field - they are so far ahead, it is literally unfathomable to laypeople and experts alike. The best approach is to step aside and try to nurture the talent without interfering in their methods and ways.
Ooh, please define "ordinary" as opposed to "smarter".
The article is about not artificially constraining your thoughts when considering a problem. Mathematics and by extension - mathematicians - are famous for looking kindly on a line of enquiry that subsequently turned out to be wrong.
To be fair to mathematicians: Anyone can trot out any sort of bollocks and call it maths, rather cheaply, by jotting down some vaguely plausible looking ramblings on paper. If you try to fiddle up, say: cold fusion, you have to at least persuade your readers on archive that you have found the open end of a test tube and work on from there!
There is a damn good reason why scientists, mathematicians and everyone else who works via publishing papers (for want of a better inclusive term) are rather dismissive of "dumb questions". Rather often the question ... is dumb. I hate that term and being British, I prefer: "stupid".
This is all about "thinking outside of the box", which is nice until you note how people get treated for doing exactly that. I suggest you keep your thoughts inside your head when you are thinking outside of it! Once you have your thoughts all lined up and you are not wearing your underwear on your head and have stopped dribbling, you might be in a good place to deliver your pontifications.
Your mother may well have given you the advice: "think before you speak" - she was a wise woman. She probably doesn't blog about it.
I think a lot of the (over)reaction to “dumb” questions is that people of average capability can’t explain the conventional wisdom, and therefore feel an ego threat when it’s questioned.
To quote the point:
> So one should be unafraid to ask “stupid” questions, challenging conventional wisdom on a subject; the answers to these questions will occasionally lead to a surprising conclusion, but more often will simply tell you why the conventional wisdom is there in the first place, which is well worth knowing.
I’m always amazed at how many people insist the world is round, but can’t explain how we know that. Or explain the experiments you can see yourself, eg pendulums or round pools.
Everyone can explain how we know that. The base of ships disappear before the mast from far away. You can see the tops of skyscrapers, but not the bottom, from the right distance across a body of water. We have photos and video of the Earth. We've flown in planes. We can call up people in different latitudes and compare lengths of shadows or positions of the sun and stars.
I'm guessing when you say they "can't explain how we know that", you're discounting photo and video evidence that they've seen; you're discounting the flight trajectory maps they see on planes; you're discounting the nature shows and documentaries they've watched; you're discounting the fact that any other explanation makes no fucking sense; you're discounting the history they learned about Magellan. There's no reason to say that none of that is evidence. They're not blindly following like sheep, they have an enormous body of evidence in front of them and there is no counter-evidence. Stop acting like they're stupid.
I haven’t tried asking many people that, have you? I wouldn’t be surprised if there was a noticeable population for whom that sort of example just didn’t come to mind. I don’t think they are stupid, we’re just naturally lazy about essentially meaningless questions that everybody knows the answer to already, in general.
I think that quite reasonably, many just trust what someone told them — eg, of your answers:
- trust someone’s photo/video
- trust someone’s map
- trust someone’s narrative in a video
- “everyone else is just wrong!”
- trust someone’s narrative about long ago
I think it’s fascinating that your examples to drive home how ridiculous you find the question were all about how we just need to trust authorities. (Including you!)
There’s nothing wrong with trusting people; I’m just pointing out that most people believe the Earth is round because they trust an authority, and not because they did any of the physical examples in your first paragraph personally. Or understand the logic of why that’s an example.
> we just need to trust authorities. (Including you!)
That's not actually what I'm saying at all. They're not putting 100% trust in one authority. They're putting a very small amount of trust in millions of others. It's nonsense to say that someone who has never independently replicated Eratosthenes's test doesn't really know the world is round. That's absolute nonsense. Of course they do. The individual tiny pieces of evidence -- yes, including maps and videos and pictures -- form a complete whole that can only be explained by a round Earth. The probability that millions of people are conspiring to deceive them is much much lower than the probability that we're all simulated brains in a twisted alien experiment, or other such crazy hypotheticals. Humans are capable of piecing together reality from millions of data points without needing a White Man In A Lab Coat to tell them. It is not irrational for someone to believe the Earth is round without being able to devise an experiment to prove it. These people are being logical in believing that, and walking around with your upturned nose telling them they're stupid because they can't "justify" their beliefs is not helping scientific literacy in the world.
Humans routinely believe wrong things, collectively, without it being a collective conspiracy to trick you.
The false dichotomy you present (“they’re right or they’re lying!”) isn’t reflected in the real world — and it’s worth knowing that you’re depending on trust in authorities rather than direct evidence.
> It is not irrational for someone to believe the Earth is round without being able to devise an experiment to prove it.
And I never said it was: just that it’s something you believe based on trust in authorities, not because you can (personally) justify the belief.
> These people are being logical in believing that, and walking around with your upturned nose telling them they're stupid because they can't "justify" their beliefs is not helping scientific literacy in the world.
Pretending that belief in authorities without being able to explain the proof is the same as scientific literacy is what’s actually harming scientific literacy — and it’s a confusion many people of average capability engage in.
> I’m always amazed at how many people insist the world is round, but can’t explain how we know that.
I've played a few times this game with people on a topic that generates strong feelings (climate change) enough to keep play for awhile, and I have yet to find someone going further than "scientists say". Each person wasn't able to cite a single name of a scientist or paper.
In general there is a lot of assumed "knowledge" about everything everywhere and it's better not asking questions because it can make people really angry.
Unlike philosophy or religion, science does not rest on the laurels of one accomplished prophet. The idea that something is only true if it comes out of the words of a famous scientist is antithetical to science. Compare the temperature of the inside of your car to the outside on a hot sunny day if you want an example of the greenhouse effect.
scientific conclusions rest on specific arguments in specific papers by specific people, including specific evidence. the people don't have to be famous or prophets; they just have to provide evidence
but it is the antithesis of science to repeat that 'scientists say' something without knowing which scientists say them and why. it's just as easily used to support false statements as true ones; for example, climate change deniers often say that scientists say we are in more danger of global cooling, or that scientists say that volcanic emissions of carbon dioxide vastly exceed anthropogenic emissions
> but it is the antithesis of science to repeat that 'scientists say' something without knowing which scientists say them and why
There are a few issues mixed up in this.
First, we all hold core beliefs about the world that we can’t justify from first hand experience. Many of those beliefs boil down to “scientists say”, but they’ve been in the public consciousness for long enough that the beliefs no longer evoke deference to scientific figures, e.g. very few people believe the earth is flat, but would not be able to tell you why, or who was involved in proving this.
While climate change deniers may pull the “scientists say” card, and while I think it’s worth being extra wary of claims that falsely invoke science, I think that this kind of science denialism is orthogonal to the ongoing practice/reality of “trusting science”, which is something that most people must do for practical reasons.
Put another way, most of us have to “trust the science”, mostly blindly, and there’s very little we can do about that. Helping people discern who they should trust seems like the most important area of focus.
> Many of those beliefs boil down to “scientists say”
The beliefs don't "boil down" at all: they form an interconnected web of theory and explanations that support and reinforce each other. You're acting like anything that one hasn't independently verified is something one is taking on faith, but that's not true. These aren't independent propositions that can be falsified on their own without dragging down an entire body of scientific understanding -- including things you have verified yourself.
I think my use of "boil down" didn't come across as intended. My point was that many/most people do not have the ability to independently verify the claims of experts in various scientific fields, nor do they care to. They instead trust the institutions that have been built throughout history and entrusted with the task of getting things right. Their beliefs really are roughly "this is what scientists say so I believe it". I'm not saying they don't have a good reason to believe it, just that this is the reality of the dynamic.
I think this is important because it underscores the criticality of institutions, and the importance of maintaining the health of the ones we have.
> You're acting like anything that one hasn't independently verified is something one is taking on faith
To be clear, this is definitely not what I'm saying.
i don't think it's orthogonal at all; i think this kind of science denialism is the inevitable result of the ongoing practice of "trusting science" instead of actually doing science, which is how you can discern who you should trust
I'd say their reaction depends mostly on how you approach them and the discussion. People generally have more pressing issues in their life and prefer delegating them to recognised authorities.
It's indeed the opposite of science, but the average person doesn't have the time, patience or energy to be doing science. Thus the need to put the trust on people that do because they're often right. If anything, when you see them appealing to the wrong or vague authority, you could provide better options or become one yourself.
there's a world of difference between 'the crc handbook says pentamethonium frangonide has a specific heat of 2951 kilojoules per mole per kelvin, but its molar mass is only 502 grams per mole, so that would give a surprisingly high specific heat of 5.9 kilojoules per gram per kelvin, so i'd better check the nist webbook and google scholar, but probably this grammatical-error-filled paper i found in google scholar from three researchers i've never heard of published in a journal i've never heard of isn't a very reliable source and they were probably just copying the crc handbook' and 'scientists say there's a missing link'
being able to, in principle, rederive things from scratch gives you a lot of information about who to trust
A difference in degree maybe, but not in kind. Trust is a central part of science, without it no advance would be possible.
> being able to, in principle, rederive things from scratch
Basically no scientist nowadays has any hope of being able to rederive everything from scratch. Not even in mathematics, where experiments are cheap, can you avoid relying on theorems from outside of your area of expertise.
I find it unsurprising that many people have no direct awareness of the underlying science behind basic things. That lack of awareness is not an indictment of the science or even the unaware person. Most of us know almost nothing about the world around us relatively speaking. We behave as if we know, but for the most part we’re just repeating/perpetuating beliefs embedded by the current iteration of culture. Direct knowledge is increasingly rare.
Yes I completely agree with you. I just don't think that knowing scientists names is a good proxy for having direct knowledge, as the post I originally replied to seemed to imply.
i think it is. in any area that i know well, i can name some important ideas, who came up with them, what their arguments were, who was opposed, what arguments they presented against those positions, and so on
if you can't name any papers or authors you probably can't explain any of the arguments in the papers either, and if you happen to have chosen the correct authorities to put your trust in, that's most likely purely a question of luck, not epistemic virtue
I once thought like this. Still do. But I had known one guy who didn't seem to exercise any limit in thinking stupid questions. He was like the antichrist to me, everything i hate. But he said some things that i still remember today cause it's just pure genius, that perfectly encapsulates the reality in so little words. And there's some sick natural barriers that prevent 'normal' people from knowing them, cause the language, while precise, also sounds dumb. But it's really not.
i'm kinda hesitant to say most of them. Like I said, they sound silly and reductive at first. And none of them is exactly what people here will find interesting. One thing he said that a game becomes engaging when 'things that were once hard, is now easy'. I like the formulation cause it works on just right abstraction level. It lends to many interpretation and it works as long as the 'rule' is satisfied. And I've obsessed over it and just how true it is. Another thing he said that fantasy is about 'sky upon sky'. Which is one of the more abstract things he said. I swear I was fuming when he said it with such elation as if he's discovering electricity. The refusal to elaborate was also infuriating, as if I was supposed to understand what he meant. And I did, eventually. The abstraction works perfectly, at least for me...
Related tangentially perhaps, but I find this to be a particularly helpful application of things like ChatGPT. Instead of asking it to give me the answer, I tell it what I’m trying to do and ask it to ask me questions about it. It’s great for gathering thoughts for writing, and I’ve been using it a little to try to help me think through software problems more quickly.
It’s very good at asking “dumb” (ew) or at least simple, building-block questions, and it almost always ends up asking me about something I hadn’t considered that ends up contributing to what I’d have done without the machine. It’s also kind of fun.
Just something like, “I need to write an abstract for a presentation about XYZ. It need to this that and the other thing. Ask me questions about it.”
Or
“I have this data in this format and I need to write a program to transmit it using this system to an another system that accepts this format. I’m using Python. Ask me questions to help me figure out an approach.”
> It’s also acceptable, when listening to a seminar, to ask “dumb” but constructive questions to help clarify some basic issue in the talk.
I think this is one of the reasons I hit a wall in my study of mathematics forty-five years ago. In the master's program I was enrolled in, the students never asked questions during class. The teachers would probably have welcomed questions, but the culture among the graduate students was never to show ignorance by asking. The result was that I often got lost in the lectures, was unable to catch up by studying the material on my own outside of class, and never became a mathematician.
Somewhere between college and university I also lost my courage to ask "dump questions". Maybe it was for the same reason you mentioned. I don't want to look ignorant. I'm trying to get that ability back in my work life. I have the attitude asking dump questions and getting the answer helps yourself and most likely everyone else in the room. So don't feel ashamed of asking in the first place.
Even more productive: ask yourself, is this really the best way to accomplish this? Why are we doing it this way? Is the standard solution bound by some constraint that was valid in its time but can now be discharged?
My canonical example for this is standard mathematical notation, which was invented before computers and optimized for quill pens and chalk boards. How much leverage could you get by discarding it and starting over with something a computer can more easily process, like, say, s-expressions?
Or... most programming languages use ascii for their standard syntax but nowadays we have unicode. How much leverage can you get by using more non-ascii characters, like, say, «balanced quotation marks»?
I beg to differ. A lot of mathematics is written in LaTeX which is written for computers to render into traditional notation. No one uses LaTeX directly because it has a horrific syntax, but it doesn't have to be that way. There is no reason why one could not invent a notation that didn't require complex rendering. Programming languages are an existence proof.
> No one uses LaTeX directly because it has a horrific syntax
I’ve interacted with quite a few mathematicians who readily mix raw LaTex into their emails! I have to squint but it seems to be a somewhat normal practice.
Well, duh, nothing is consumed by a computer for its own sake. The only reason to feed anything into a computer is so that what you feed in can be processed to produce something that is useful for humans. That doesn't change the fact that a notation developed when paper and ink were state-of-the-art technology may no longer be optimal in an age of keyboards and screens.
I read mathematics on screens but when it comes to working out exercises etc I still prefer to do it with pen and paper, so I prefer succinct notation.
I'm not saying that traditional notation should be abandoned altogether. Pen and paper are still useful for some things even in the age of computers. Nonetheless, computers exist, and you might get more leverage out of them if you don't restrict yourself to using a notation designed for pen and paper.
> Well, duh, nothing is consumed by a computer for its own sake.
Except, you know, proofs that are automatically checked by a proof system.
It's really hard to take a suggestion like "mathematical notation should be replaced by s-expressions" seriously. There's approximately 0% of working mathematicians who would think that this is a good idea (hell, not even most programmers think it's a good idea to write everything in s-expressions).
Prior to any major advance in any field of intellectual endeavor there are approximately zero percent of workers in that field who thought that the thing that led to the advance was a good idea. This is exactly why major advances are rare.
BTW, you might want to look up the work of Steven Wolfram and Gregory Chaitin. They both thought using s-expressions was a good idea, and they got quite a bit of mileage out of it.
> Prior to any major advance in any field of intellectual endeavor there are approximately zero percent of workers in that field who thought that the thing that led to the advance was a good idea. This is exactly why major advances are rare.
This argument would make sense if nobody had ever come up with an alternative system of maths notation, but MathML, Mathematica, Coq, Isabelle, Lean, and many others exist, and yes, probably even some notation based on s-expressions (you could do mathematics in Pie[0], although it's probably not super pleasant since it's a toy language). They get their use for specific situations, as I mentioned before, but they haven't replaced mathematical notation wholesale.
I think you're missing the point. The s-expression thing was just intended to be an illustrative example. I'm not actually advocating for it (at least not here). All I'm saying is that asking specifically why things are the way they are is more likely to lead to good ideas than just "asking dumb questions".
(And note too that even with a heuristic that improves your odds, coming up with good novel ideas is really hard and happens only rarely.)
I thought it had a noble goal but it fell far short of the mark. I see at least two big problems with it.
First, their notation is a definite improvement on traditional notation in that it can in principle be translated into s-expressions and thus at least potentially avoids the ambiguity of traditional notation. However, it still looks like traditional notation and so it still inherits many of the problems of that notation, not least of which is that parsing it is really hard, even for humans. The fact that all of the formulas in the on-line version of the book are rendered as gifs is symptomatic of this. If you want to actually follow the program of the book and run the code that corresponds to the formulas, you have to render the formulas into Scheme by hand. Well, OK, you could just use the on-line code, but that's cheating. The whole point (or at least a big part of the point) of the notation was supposed to be that mapping it onto s-expressions was supposed to be simple, and it's not.
Second, and much more seriously, they don't actually follow their own rules. Look at equation 1.1. I can't reproduce it here because it's a gif, but I can reproduce a reasonable facsimile of some of the following explanatory text, which goes:
... where F[gamma] is a function of time ...
The problem is that F[anything] is not the notation for a function. Functions are denoted simply by letters, and the values of functions are denoted by juxtaposing the name of the function with a list of arguments delimited by parens (which overloads the meaning of parens because they also denote up-tuples, but we'll let that slide). So F is a function, but F[gamma] is not, it is ... what? A function juxtaposed with a 1-element down-tuple?
If you look carefully, you will notice a footnote that kinda sorta explains this:
"Traditionally, square brackets are put around functional arguments. In this case, the square brackets remind us that the value of may depend on the function in complicated ways, such as through its derivatives."
So they very carefully describe a precise notation, and then in the very first equation in the book they abandon that notation and use a completely different notation that indicates that something may depend on something else in unspecified but potentially complicated ways. Worse, it now adds even more ambiguity to the notation, because now both parens and square brackets are ambiguous with respect to whether they are denoting function arguments or up/down tuples. So we've already run off the rails before we've even begun.
This is awesome. Glad we can improve on the “thought leadership” of theoretical people like Terrance Tao with some real titans’ advice like yours (I see you were a software engineer at Google!!!). Thanks for sharing.
Yeah, true. I was being subtle. This narcissist is framing things as "even better" in comparison to someone who is likely, literally, the most intelligent person currently alive.
There is at least one person commenting that mathematical notation is stupid in every maths submission on HN, and the author usually never gives any indication as to having made any contributions towards mathematical research. It's an inevitability at this point.
> likely, literally, the most intelligent person currently alive.
Why do people always seem to say this about Tao? What makes him different from, say, other recent Fields medalists?
I feel like it has something to do with having been a child prodigy more than anything else (and because he’s somehow known to laypeople worldwide when other names aren’t). I’m not even sure what the most intelligent person is supposed to mean, especially when the domain of expertise is so narrow.
I have no particular opinion on the man; it’s a genuine question.
Once we had a very smart young guy on work. But he always asked things, even if he knew the answer. It was like a sickness, people refused to work with him, because of his many also stupid questions. So I told him he can ask me anything, but he allways has to present at least two ideas to solve the problem. He always came with a possible solution and if I asked him witch solution then the better would be, he always picked the right one. After a very short time he asked much less and it was much more calm to work with him.
I worked with a similar-type person. I gave them 15 minutes at the end of every day to ask whatever they wanted.
Usually by the end of the day they had worked out the answers and had very few, if any, questions.
I like your solution just as much, but we weren't in a 'many people could benefit' environment.
I have found that many people don't respond well to one asking dumb questions. It can quickly marginalize you in a room of people trying to be the smartest in the room.
(one of the distinguishing features of a good hack is that it leaves the Sherpas behind and takes a different —usually more direct— route to the solution than the siege approach that Conway's-Law-compliant enterprise team(s) would take)
Great read. As a past adjunct, you can learn this through your students. No question is a dumb question. I eventually got to a point where I couldn't wait for the "dumbest" question of the session so that we could explore it as a class.
Could someone recommend a good maths research book that covers the whole messy discovery process, rather than just the end result?
I don't mind in which field, and it's fine if it's quite formal (not looking for pop science). But it would be good if it is relatively self-contained (doesn't require too much prior knowledge, it establishes some foundations and builds from them).
I'm a fan of On Numbers and Games by Conway for instance, and the more literary Surreal Numbers by Knuth (written earlier but inspired by Conway's research before publication).
I don't know if you would classify this as a "maths research book", but it's a wonderful personal and emotional description of the process of math discovery:
Birth of a Theorem - A mathematical adventure by Cedric Villani
I'd go further to say that if you don't ask to yourself a single question while you're reading a substantial part of a book (a section, a chapter), then you don't really internalize what you think you're learning.
The questions don't need to be answered perfectly, nor they need to be "good" questions. The questions can be general and simple, like:
- Does it have to be this way? Why?
- How is it related/connected to what I've learned previously (last chapter, last year, last decade)? Do I need to unlearn something or adjust my previous mental model?
Answering popular questions like the "contrarian question" Thiel poses is a good exercise. Flesh out the answers to a degree you would feel good presenting it in an interview.
