Ask yourself dumb questions and answer them (2020)
terrytao.wordpress.com
terrytao.wordpress.com
now get your damn car off my lawn
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.
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.
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.
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.
* Frog emerges from mouth
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.
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.
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
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.
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 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.
being able to, in principle, rederive things from scratch gives you a lot of information about who to trust
> 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.
What does "going further" mean here, what kind of answer would make you happy?
> Each person wasn't able to cite a single name of a scientist or paper.
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.
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'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.
- 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.
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.
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.
Which was my original point.
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?
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'll agree that the costs don't come close to replicating work by LIGO or CERN.
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.
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".
yes, because math has a logical flow and rules, and most errors would be noticed
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".
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.
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
Pretty sure we call that imagination. Creativity, shockingly enough, involves creation, not just imagination.
A bit like "write drunk, edit sober" but instead of drunk, you're a raving lunatic.
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 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".
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.
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.
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.”
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.
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»?
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.
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.
If you want to write mathematics for computers, you can use Coq, Lean, etc. - or, in a different direction, Mathematica.
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.
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).
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.
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.
(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’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.
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)
Then slowly add complexity from that example to gain insight to more challenging cases.
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).
Birth of a Theorem - A mathematical adventure by Cedric Villani
https://us.macmillan.com/books/9780374536671/birthofatheorem
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?