[HN Gopher] Ask yourself dumb questions and answer them (2020)
___________________________________________________________________
Ask yourself dumb questions and answer them (2020)
Author : vector_spaces
Score : 201 points
Date : 2023-12-23 23:50 UTC (23 hours ago)
(HTM) web link (terrytao.wordpress.com)
(TXT) w3m dump (terrytao.wordpress.com)
| vegetablepotpie wrote:
| 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.
| gerdesj wrote:
| 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.
| justanotherjoe wrote:
| 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.
| cookie_monsta wrote:
| Hmm, I would like to hear an example of one of the
| Antichrist's thoughts
| whatshisface wrote:
| The Eurasian region must (bleh)* be engaged in protracted
| war to ensure that the populace is distracted from the
| Omicron program.
|
| * Frog emerges from mouth
| justanotherjoe wrote:
| 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...
| zmgsabst wrote:
| 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.
| gwervc wrote:
| > 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.
| feoren wrote:
| > a single name of a scientist or paper
|
| 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.
| kragen wrote:
| 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
| haswell wrote:
| > _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.
| maksimur wrote:
| 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.
| kragen wrote:
| naturally, but obviously _argumentum ad verecundiam_ is
| the opposite of science
| maksimur wrote:
| 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.
| blackbear_ wrote:
| > someone going further than "scientists say"
|
| What does "going further" mean here, what kind of answer
| would make you happy?
| kragen wrote:
| literally the next sentence in the comment you're
| replying to, after the one you quoted, is an implicit
| answer to that question:
|
| > _Each person wasn 't able to cite a single name of a
| scientist or paper._
| blackbear_ wrote:
| I really hoped that there was something more substantial
| than name-dropping before dismissing people's opinions.
| haswell wrote:
| Are people's opinions being dismissed here?
|
| 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.
| blackbear_ wrote:
| 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.
| feoren wrote:
| 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.
| bee_rider wrote:
| 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.
| BHSPitMonkey wrote:
| You can't prove to us that "everyone" understands a thing
| simply by demonstrating that you understand that thing.
| soundarana wrote:
| Crazies say they validate and test their idea, but they
| actually suffer from severe confirmation bias where they
| summarily dismiss any counterexample or problem.
| gerdesj wrote:
| "Crazies"
|
| Would you be able to spot a Ramanujan from the dross?
|
| You say that "they" ... something ... but ... they ...
| something ... disagree with the world. Care to flesh this out
| with an anonymised example?
| swagmoney1606 wrote:
| > 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.
| mulmen wrote:
| 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.
| l33t7332273 wrote:
| In math, proof is not generally so expensive, so perhaps
| Ramanujan was not the best example.
| penteract wrote:
| 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.
| Tainnor wrote:
| 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.
| syndicatedjelly wrote:
| 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.
| soundarana wrote:
| For one Ramanujan you have 1000 cranks.
| keeganpoppen wrote:
| 1,000,000 cranks
| syndicatedjelly wrote:
| 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".
|
| https://en.wikipedia.org/wiki/Srinivasa_Ramanujan#Contact
| ing...
| paulpauper wrote:
| _Would you be able to spot a Ramanujan from the dross?_
|
| yes, because math has a logical flow and rules, and most
| errors would be noticed
| kevindamm wrote:
| 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.
| kragen wrote:
| 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
| FPGAhacker wrote:
| > 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.
| mym1990 wrote:
| 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?
| phito wrote:
| What's the difference? OP could have said "create new
| concepts, questions and ideas" and it would be exactly the
| same
| p1necone wrote:
| 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.
| broscillator wrote:
| 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.
| mock-possum wrote:
| 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.
| hackerlight wrote:
| 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.
| Obscurity4340 wrote:
| 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
| hyperthesis wrote:
| 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".
| hiAndrewQuinn wrote:
| 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 :)
| Obscurity4340 wrote:
| You are my people XD
| Obscurity4340 wrote:
| 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.
| jmbwell wrote:
| 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.
| gerdesj wrote:
| Good skills! Never thought of that, nor seen anyone else
| espousing it.
| sebg wrote:
| Do you have a simple example prompt set? I am having trouble
| figuring out how to try it on my own? Thanks!
| jmbwell wrote:
| 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."
| lisper wrote:
| 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>>?
| constantly wrote:
| 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.
| electrondood wrote:
| Dropped the /s
| constantly wrote:
| 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.
| Tainnor wrote:
| 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.
| lisper wrote:
| Dude, it was just a suggestion. Lighten up.
| kragen wrote:
| i'm interested to hear what you think of _structure and
| interpretation of classical mechanics_
| lisper wrote:
| 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.
|
| I give it an A for effort, a D- for execution.
| Tainnor wrote:
| Most mathematics isn't written for computers, but for humans,
| so that's why mathematical notation is optimised for that use
| case.
|
| If you want to write mathematics for computers, you can use
| Coq, Lean, etc. - or, in a different direction, Mathematica.
| lisper wrote:
| > Most mathematics isn't written for computers
|
| 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.
| Tainnor wrote:
| That is not what I mean by "for humans". Also, LaTeX !=
| mathematical notation.
| lisper wrote:
| What did you mean then? (And it's true that LaTeX is not
| equal to mathematical notation, but it does contain a
| mathematical notation as a subset.)
| Tainnor wrote:
| Mathematical papers, textbooks, lecture notes, in-person
| and video lectures, etc. are produced and consumed by
| humans and not computers.
| tkgally wrote:
| > 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.
| hasoleju wrote:
| 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.
| ipsin wrote:
| Is this different from Rubber Duck Debugging?
| 082349872349872 wrote:
| > _if a simple result is usually proven by method X, you can ask
| whether it can be proven by method Y instead_
|
| in particular, see
| https://en.wikipedia.org/wiki/Mathematics_Made_Difficult
|
| (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)
| j7ake wrote:
| Another useful technique in understanding a topic is to find is
| to simple (but nontrivial) examples and make sure one can solve
| them fully and easily.
|
| Then slowly add complexity from that example to gain insight to
| more challenging cases.
| balaji1 wrote:
| 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.
| ironmagma wrote:
| To me the existence of this post is particularly relevant given
| the contemporary context:
| https://twitter.com/StefanFSchubert/status/17383270562391696...
| hervature wrote:
| 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.
| alberth wrote:
| Sounds like "First Principles" thinking for mathematics.
| kragen wrote:
| all of mathematics is first principles thinking, and that's
| not a brand name
|
| now get your damn car off my lawn
| kdwikzncba wrote:
| All maths is first principles, without the pretentiousness of
| using the phrase "first principles".
| creer wrote:
| 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.
| kdwikzncba wrote:
| > 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.
| beltsazar wrote:
| 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?
| zubairq wrote:
| Very good thought process here... I always try to steel man
| oppositional arguments of things that I may think are true
| bmitc wrote:
| 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.
| _trampeltier wrote:
| 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.
| oersted wrote:
| 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).
| ghufran_syed wrote:
| 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
|
| https://us.macmillan.com/books/9780374536671/birthofatheorem
| spacecadet wrote:
| 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.
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