6 ms·

There's more to mathematics than rigour and proofs (2007)

195 points by haraball 2y ago

ffhhj 2y ago
Could modern AI help amateur mathematicians to build proofs?
- ykonstant 2y ago
  To an extent; they can give hints and suggest directions, but you need to treat them as an unreliable narrator: think of them as entities that can help or deceive you at random. That being said, we are researching tailored LLMs and other architectures to assist mathematical research that are more geared towards accuracy at the expense of freedom ("imagination"). The Lean FRO has some related information and links.
- drpossum 2y ago
  I was trying to coerce gpt-4o to talk about the gcd and lcm in terms of sets of the prime factors where the product is the union of the sets, gcd is the intersection, and the lcm is the union less the intersection and it kept telling me I was incorrect and being "non standard". It has a long, long way to go.
  - wizzwizz4 2y ago
    If you meant multiset, then you were correct. (Not that I expect GPT-4o to make the distinction.)
- BeetleB 2y ago
  Not sure why you're downvoted. From a few days ago: https://news.ycombinator.com/item?id=40646909 https://news.ycombinator.com/item?id=40646909
hyperman1 2y ago
The first time I really felt I understood math in depth was my uni linear algebra course. Distance and orthogonality were replaced with a more abstract but better inner product. It behaved like an IT interface: As long as some basic properties were fulfilled, aal of linear algebra came along. Half o the examples were the usual numeric vectors and matrices, the others were integrals, etc...
- zeroonetwothree 2y ago
  I didn’t get much out of linear algebra. It felt too computational. I only really got it once I “relearned” it as part of abstract algebra
  - topaz0 2y ago
    There is a big range of approaches out there for teaching linear algebra. I really enjoyed my quite abstract linear algebra course, but there were a lot of other kids in it that really struggled and did not get much out of it. Takes all kinds.
csmeyer 2y ago
The kind of follows the standard midwit meme progression one sees in programming as well 1. Making stuff is fun and goofy and hacky 2. Coding is formal and IMPORTANT and SERIOUS 3. What cool products and tools can I make? I feel like this pattern probably happens in many fields? Would be fun to kind of do a survey/outline of how this works across disciplines
jbandela1 2y ago
> One can roughly divide mathematical education into three stages: Similarly with programming. 1. Write programs that you think are cool 2. Learn about data structures and algorithms and complexity and software organization. 3. Write programs that you think are cool. But since you know more, you can write more cool programs. If things are working as they should, the end stage of mathematics and programming should be fun, not tedious. The tedious stuff is just a step along the way for you to be able to do more fun stuff.
- arketyp 2y ago
  This is true but I think it's iterative, cyclic. It applies to any art and craft, really. You alternate between perceiving and projecting, receiving and creating.
  - zeroonetwothree 2y ago
    Any skill really. You alternate between theory and practice. For example in sports you play for fun, then do some coaching to get better, then play for fun using your new skills and so on.
- nadam 2y ago
  Or alternatively: 1. Programming in very concrete/practical terms because you do not know how to think in precise and abstract terms (do not know math) 2. Thinking more precisely and abstractly (more mathematical way) 3. Only do some key important abstractions, and being a bit hand-wawy again in terms of precision. The reason: important real-world problems are usually very complex, and complex problems resist most abstractions, and also being totally precise in all cases is impossible due to the complexity. All-in-all it is due to increased complexity in my opinion. Example: 1. Writing some fun geometry related programs 2. learn about geometry more seriously 3. write software based on a multiple hundred thousand line CAD kernel. Other example: 1. Write fun games on C64 2. Learn about computer graphics in University 3. Contribute to the source code of Unreal Engine with multiple million lines of code with multiple thousand line class declaration header files.
- richrichie 2y ago
mgaunard 2y ago
Is there?
- jfengel 2y ago
  Absolutely. We have machines that can crank out true theorems, rigorously proven, all day. It takes a mathematician to know what is worth working on. And that is fundamentally an intuitive decision. Computers don't care whether a proof is interesting or not.
  - zeroonetwothree 2y ago
    It’s a bit tautological since we are defining interesting as what human mathematicians work on. Perhaps if computers ran the show they wouldn’t agree with our definition.
    - rishabhjain1198 2y ago
      Regardless of what computers find interesting, humans want to progress math in stuff humans find interesting. We can't fully rely on computers to do that yet since they can't seem to judge that very well rn as good as human mathematicians. Don't necessarily see a tautology here.
    - abdullahkhalids 2y ago
      Think of pets. We, humans, run the show. There are some things the pets think are interesting because we humans are doing it, but by and large pets like what is dictated by their genes + individual preferences.
    - bee_rider 2y ago
      Maybe it is a feedback loop, rather than a tautology? The things many mathematicians find interesting are the things that the general mathematician community is working on. And the way you become a mathematician is by publishing things that the community finds interesting enough to let through the peer review process. Ultimately though this is all funded by, in the end, the belief that they’ll be able to dumb down the good stuff for us scientists, engineers, and other folks who build actual physical things when we hit the point that we need it. (Of course it is an exploration process so not everything needs to be directly applicable). If computers ran the show, we would probably stop plugging them in if they used more power than their theories saved us, or whatever.
sherburt3 2y ago
I want to be Terrance Tao when I grow up
- ramraj07 2y ago
  I think the ship sailed when you were 4..
- paulpauper 2y ago
  Looking at his success it's hard to not believe that some people are objectively better than others.
  - margorczynski 2y ago
    Nobody who has a grasp on basic biology and a honest mind wouldn't believe that. The thing is this completely goes against the liberal "tabula rasa" worldview.
  - 082349872349872 2y ago
    Whether some people are objectively better at maths and communication than others, and whether they all get equal treatment under the law are two different things, right? Right? (I don't know where you grew up; where I grew up we were always obliged to chant "with liberty and justice for all")
    - yipbub 2y ago
      Right.
- theshaper 2y ago
  At this point, Terence Tao is already worthy of a list of facts, much like those about Chuck Norris and Bruce Schneier. For example: When Terence Tao solves a problem, the problem appreciates the solution.
taeric 2y ago
I wish people had more exposure to building mathematical models of things. I am fairly convinced that the only real exposure I was given was to models that we knew worked. So much so, that we didn't even execute many. Specifically, parabolic motion is something you can obviously do by throwing something. You can, similarly, plot over a time variable where things are observed. You can then see that we can write an equation, or model, for this. For most of us, we jump straight to the model with some discussion of how it translates. But nothing stops you from observing. With modern programming environments, you can easily jump people into simulating movement very rapidly and let people try different models there. We had turtle geometry years ago, but for most of us that was more mental execution than it was mechanical. Which is probably a great end goal, but no reason you can't also start with the easy computer simulations.
- nextaccountic 2y ago
  Something I really like is that the curve that a rope or thread makes when fixed in two points but not under tension, it's not a parabola. It really looks like one though, but it isn't. It's a catenary. That's something you can verify by writing some simulation code, then drawing the curve, and then drawing the best matching parabola on top. It doesn't fit. To model the issue mathematically you need some not-too-advanced calculus. On both the computer simulation and the mathematical model, you model the rope as being made of very small elements that are linked together (like a chain). In the simulation those elements are small, but finite. In the math you take the limit as the volume of the element tends to zero. It's the same way of thinking but math gives some different tools, enabling you to solve the curve analytically
  - taeric 2y ago
    Exactly! I think this scenario alone would be an amazing set of lessons for many grade schools. Move this into modeling and then guessing stuff like bridge tensions, and you can easily show what many of the maths are good for. We used to have this with the attempts at building tooth pick bridges and such. Which I still think is very illuminating. But, I think there are a lot of questions you can expose with models that were often only seen by the more advanced students. And again, I agree that getting people to mentally model these things is a good goal. Right now, people rarely ponder things on paper, it seems.
ji_zai 2y ago
"The intuitive mind is a sacred gift and the rational mind is a faithful servant.” - Einstein
- InDubioProRubio 2y ago
  The problem is though, that with half the data, your mind considers glueing another base to the seasaw, to balance things out and restore symmetry and intuitive beauty.
grape_surgeon 2y ago
I love how well-spoken Tao is. I've enjoyed lots of his lectures before; even if you're not an expert in whatever he's discussing he knows how to explain it just right to get you up to speed as best as he can. His communication and math skills are phenomenal.
- deleted 2y ago
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- xanderlewis 2y ago
  Yes! He’s a great counterexample to the popular view that mathematical/pure logical reasoning ability is negatively correlated (even zero-sum) with communication ability. Yes, there are people that are crap at one and quite good at the other… but you can’t make much of an inference when given one without the other.
  - tadala 2y ago
    Is this a popular view? I think mathematicians can be odd, but usually they communicate quite well. I think as far as popularization of their fields go, mathematics is probably doing the best out of the lot: numberphile, 3blue1brown etc.
    - xanderlewis 2y ago
      The examples you list are not known as mathematicians; they’re popularisers who (sometimes) happen to have qualifications and a history of studying the subject. 3B1B is absolutely brilliant but Grant Sanderson is not a ‘mathematician’ in the sense of someone who does research in mathematics. Ironically, the fact that mathematics popularisation is as visible as it is is itself a sign of how much it is needed and therefore how unpopular and misunderstood the subject is. Branches of science like, say, astrophysics don’t need popularisation; people already think they’re cool. The view of ‘people who are good at mathematics’ being bad at English is a relatively common one, in my experience. At least at the level of university students. People think there’s some sort of conservation of ability or equilibrium in the universe that means that if you have a ‘maths brain’ then you’re no good at much else, and vice versa. If anything, I think there’s a positive correlation between mathematical and communication ability — after all, mathematics is basically just the science of clever notation and clear-headed thinking.
rokob 2y ago
> The distinction between the three types of errors can lead to the phenomenon ... of a mathematical argument by a post-rigorous mathematician which locally contains a number of typos and other formal errors, but is globally quite sound, with the local errors propagating for a while before being cancelled out by other local errors I was initially amazed at this when I was in graduate school, but with enough experience I started to do it myself. Handwaving can be a signal that someone doesn't know what they are doing or that they really know what they are doing and until you are far enough along it is hard to tell the difference.
- richrichie 2y ago
  Did anyone whisper in your ears, “Welcome to the dark world!”?
- sigmoid10 2y ago
  >Handwaving can be a signal that someone doesn't know what they are doing or that they really know what they are doing and until you are far enough along it is hard to tell the difference. I found it's very easy to distinguish these two when you have another expert ask questions. But if you don't have someone like that in the audience it might take forever. Or at least until you become an expert yourself.
  - bogeholm 2y ago
    “Learn the rules like a pro, so you can break them like an artist.” - Pablo Picasso [0] [0]: https://www.goodreads.com/quotes/558213-learn-the-rules-like-a-pro-so-you-can-break https://www.goodreads.com/quotes/558213-learn-the-rules-like...
- solikeppl 2y ago
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- deleted 2y ago
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- topologie 2y ago
munchler 2y ago
Good article, but should mention that it's not new. First copy in the Wayback Machine is from 2018, but there are comments all the way back to 2009. https://web.archive.org/web/20180301000000*/https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ https://web.archive.org/web/20180301000000*/https://terrytao...
- paulpauper 2y ago
  his old articles are shared a lot on here
tsaixingwei 2y ago
“Before I learned the art, a punch was just a punch, and a kick, just a kick. After I learned the art, a punch was no longer a punch, a kick, no longer a kick. Now that I understand the art, a punch is just a punch and a kick is just a kick.” - Bruce Lee
- mr_mitm 2y ago
  People have pointed out similarities to a few things in this thread now. It's pretty much the bell curve meme: https://knowyourmeme.com/memes/iq-bell-curve-midwit https://knowyourmeme.com/memes/iq-bell-curve-midwit
  - FrustratedMonky 2y ago
    man, sometimes things are obvious and right in front of us. I've known the Dogen saying for years. Have even meditated on it. I've been enjoying Bell Curve meme for sometime. Very funny. Never put together that these were the same thing. Today I am awakened.
    - karmakurtisaani 2y ago
      At first, the Dogen saying and the bell curve meme seem like different things, then..
      - ocodo 2y ago
        first there is the bell curve, then there is no bell curve, then there is the bell curve... If you learn anything significantly deeply, this is a repeating pattern.
  - paulpauper 2y ago
    This is the best meme ever / So accurate in many ways
    - 082349872349872 2y ago
johnwatson11218 2y ago
The idea of the 3 levels really resonated with the ideas in "Bernoulli's Fallacy" as well. Right now we are seeing a resurgence of Bayesian reasoning across all fields that deal with data and statistical reasoning. I think many errors of modern civilization were caused by people at a level 2 understanding attempting to operationalize their knowledge for others at level 1. We need it to become much more common to operate at level 3, especially in fields like enterprise software development.
JoeyBananas 2y ago
The worst thing is when someone who thinks that "math is 100% infallible and all about rigor, you gotta show your work and include all the steps" yet they think that set theory is good enough and it doesnt have problems they say things like "Everything in math is a set," but then you ask them "OK, what's a theorem and what's a proof?" they'll either be confused by this question or say something like "It's a different object that exists in some unexplainable sidecar of set theory" They don't know anything about type theory, implications of the law of excluded middle, univalent foundations, any of that stuff
- johnwatson11218 2y ago
  My favorite was when a manager tried to get me to agree with the statement that "math was just for the numbers right?". Meaning not character strings nor dates. I was dumbstruck by the question.
  - burnished 2y ago
    Math is for.. numbers? Thats engineer talk right there
- qbit42 2y ago
  Modern set theory is sufficient for most mathematicians. That other stuff is interesting, but you can do great mathematics without it.
- Tainnor 2y ago
  It's 100% possible to base logic and proof theory off of set theory. For example, you can treat proofs as natural numbers via Gödel encoding (or any other reasonable encoding) and we know that natural numbers can be represented by sets in multiple different ways. You may prefer type theory or other foundations, but set theory is definitely rigorous enough and about as "infallible" (or not) as other approaches.
- red_trumpet 2y ago
  Yeah, they should have heard about ZFC and have a notion what a formal proof is. On the other hand, I'm not sure your last sentence is really that relevant. > They don't know anything about type theory, implications of the law of excluded middle, univalent foundations, any of that stuff I'm doing a PhD in algebraic geometry, and that stuff isn't relevant at all. To me "everything is a set" pretty much applies. Hell, even the stacks-project[1] contains that phrase! [1] https://stacks.math.columbia.edu/tag/0009 https://stacks.math.columbia.edu/tag/0009
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staticshock 2y ago
> The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. This is a key insight; it's something I've struggled to communicate in a software engineering setting, or in entrepreneurial settings. It's easy to get stuck in the "data driven" mindset, as if data was the be-all and end-all, and not just a stepping stone towards an ever more refined mental model. I think of "data" akin to the second phase in TFA (the "rigor" phase). It is necessary to think in a grounded, empirical way, but it is also a shame to be straight-jacketed by unsafe extrapolations from the data.
- hun3 2y ago
  > It's easy to get stuck in the "data driven" mindset, as if data was the be-all and end-all, and not just a stepping stone towards an ever more refined mental model. Yes. "Data driven" either includes sound statistical modelling and inference, or is just a thiny veiled information bias.
- paulpauper 2y ago
  rigour is is not about destroying bad intuition, but rather formalizing good intuition, imho. The ability to know good from bad is somewhere in-between total newb and expert.
red_admiral 2y ago
This argument is very close to one by Whitehead in an essay called "The Rhythm of Education". The stages back there are called Romance, Precision, and Generalisation - but I'd argue there is an isomorphism (in a suitable category) between that and Tao's three stages.
dang 2y ago
Related: There’s more to mathematics than rigour and proofs (2007) - https://news.ycombinator.com/item?id=31086970 https://news.ycombinator.com/item?id=31086970 - April 2022 (90 comments) There’s more to mathematics than rigour and proofs - https://news.ycombinator.com/item?id=13092913 https://news.ycombinator.com/item?id=13092913 - Dec 2016 (2 comments) There’s more to mathematics than rigour and proofs - https://news.ycombinator.com/item?id=9517619 https://news.ycombinator.com/item?id=9517619 - May 2015 (32 comments) There’s more to mathematics than rigour and proofs - https://news.ycombinator.com/item?id=4769216 https://news.ycombinator.com/item?id=4769216 - Nov 2012 (36 comments)
highfrequency 2y ago
“The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture. Without one or the other, you will spend a lot of time blundering around in the dark.” Well put! In empirical research, there is an analogy where intuition and systematic data collection from experiment are both important. Without good intuition, you won’t recognize when your experimental results are likely wrong or failing to pick up on a real effect (eg from bad design, insufficient statistical power, wrong context, wrong target outcome, dumb mistakes). And without experimental confirmation, your intuition is just untested hunches, and lacks the refinement and finessing that comes from contact with the detailed structure of the real world. As Terry says, the feeling of stumbling around in the dark suggests you are missing one of the two.
EVa5I7bHFq9mnYK 2y ago
What? Americans learn proper calculus in later undergraduate years? Really?
- nyssos 2y ago
  Tao is Australian. That said, yes, real analysis is often a third-year class.
qnleigh 2y ago
I think entire research subfields can go through a similar process. Plenty of mathematics was done before mathematical rigor really existed. Then axiomatization became more and more important. The intuition never went away, but I have heard of 'Nicholas Bourbaki' (https://en.m.wikipedia.org/wiki/Nicolas_Bourbaki https://en.m.wikipedia.org/wiki/Nicolas_Bourbaki), the movement to right mathematics in purely formal language while eschewing intuitive language. And then more recently I read a prominent mathematician describing this phases having been a bit of a mistake. But maybe it was just a necessary part of the fields transition. I've definitely gone through a parallel transition in physics, but replacing 'rigor' with 'calculation' and 'intuition' for 'physical intuition/simple pictures.' In physics there is the additional aspect that problems directly relate to the physical world, and one can lose and then regain touch with this. I wonder what other fields have an analogous progression.
- philipov 2y ago
  > I wonder what other fields have an analogous progression. "Before one studies Zen, mountains are mountains and waters are waters; after a first glimpse into the truth of Zen, mountains are no longer mountains and waters are no longer waters; after enlightenment, mountains are once again mountains and waters once again waters."
hilux 2y ago
I think one of our biggest problems in society (business, politics, etc.) is that Stage 1 superficially resembles Stage 3. This means that many people in Stage 1 (or Stage 0, if that's a thing) believe that they're as good as Stage 3 thinkers. AKA Dunning-Kruger. In other words, complete bullshit, confidently delivered, has come to dominate informality-born-of-rigor. And the audience can't tell the difference.
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topologie 2y ago
Tao... What can I say... Always great work. Haven't read a single bad contribution from him. And I've read quite a bit...

195 points by haraball 2y ago

ffhhj 2y ago

Could modern AI help amateur mathematicians to build proofs?

ykonstant 2y ago
To an extent; they can give hints and suggest directions, but you need to treat them as an unreliable narrator: think of them as entities that can help or deceive you at random. That being said, we are researching tailored LLMs and other architectures to assist mathematical research that are more geared towards accuracy at the expense of freedom ("imagination"). The Lean FRO has some related information and links.
drpossum 2y ago
I was trying to coerce gpt-4o to talk about the gcd and lcm in terms of sets of the prime factors where the product is the union of the sets, gcd is the intersection, and the lcm is the union less the intersection and it kept telling me I was incorrect and being "non standard". It has a long, long way to go.
- wizzwizz4 2y ago
  If you meant multiset, then you were correct. (Not that I expect GPT-4o to make the distinction.)
BeetleB 2y ago
Not sure why you're downvoted. From a few days ago: https://news.ycombinator.com/item?id=40646909 https://news.ycombinator.com/item?id=40646909

hyperman1 2y ago

The first time I really felt I understood math in depth was my uni linear algebra course. Distance and orthogonality were replaced with a more abstract but better inner product. It behaved like an IT interface: As long as some basic properties were fulfilled, aal of linear algebra came along. Half o the examples were the usual numeric vectors and matrices, the others were integrals, etc...

zeroonetwothree 2y ago
I didn’t get much out of linear algebra. It felt too computational. I only really got it once I “relearned” it as part of abstract algebra
- topaz0 2y ago
  There is a big range of approaches out there for teaching linear algebra. I really enjoyed my quite abstract linear algebra course, but there were a lot of other kids in it that really struggled and did not get much out of it. Takes all kinds.

csmeyer 2y ago

The kind of follows the standard midwit meme progression one sees in programming as well 1. Making stuff is fun and goofy and hacky 2. Coding is formal and IMPORTANT and SERIOUS 3. What cool products and tools can I make? I feel like this pattern probably happens in many fields? Would be fun to kind of do a survey/outline of how this works across disciplines

jbandela1 2y ago

> One can roughly divide mathematical education into three stages: Similarly with programming. 1. Write programs that you think are cool 2. Learn about data structures and algorithms and complexity and software organization. 3. Write programs that you think are cool. But since you know more, you can write more cool programs. If things are working as they should, the end stage of mathematics and programming should be fun, not tedious. The tedious stuff is just a step along the way for you to be able to do more fun stuff.

arketyp 2y ago
This is true but I think it's iterative, cyclic. It applies to any art and craft, really. You alternate between perceiving and projecting, receiving and creating.
- zeroonetwothree 2y ago
  Any skill really. You alternate between theory and practice. For example in sports you play for fun, then do some coaching to get better, then play for fun using your new skills and so on.
nadam 2y ago
Or alternatively: 1. Programming in very concrete/practical terms because you do not know how to think in precise and abstract terms (do not know math) 2. Thinking more precisely and abstractly (more mathematical way) 3. Only do some key important abstractions, and being a bit hand-wawy again in terms of precision. The reason: important real-world problems are usually very complex, and complex problems resist most abstractions, and also being totally precise in all cases is impossible due to the complexity. All-in-all it is due to increased complexity in my opinion. Example: 1. Writing some fun geometry related programs 2. learn about geometry more seriously 3. write software based on a multiple hundred thousand line CAD kernel. Other example: 1. Write fun games on C64 2. Learn about computer graphics in University 3. Contribute to the source code of Unreal Engine with multiple million lines of code with multiple thousand line class declaration header files.
richrichie 2y ago

mgaunard 2y ago

Is there?

jfengel 2y ago
Absolutely. We have machines that can crank out true theorems, rigorously proven, all day. It takes a mathematician to know what is worth working on. And that is fundamentally an intuitive decision. Computers don't care whether a proof is interesting or not.
- zeroonetwothree 2y ago
  It’s a bit tautological since we are defining interesting as what human mathematicians work on. Perhaps if computers ran the show they wouldn’t agree with our definition.
  - rishabhjain1198 2y ago
    Regardless of what computers find interesting, humans want to progress math in stuff humans find interesting. We can't fully rely on computers to do that yet since they can't seem to judge that very well rn as good as human mathematicians. Don't necessarily see a tautology here.
  - abdullahkhalids 2y ago
    Think of pets. We, humans, run the show. There are some things the pets think are interesting because we humans are doing it, but by and large pets like what is dictated by their genes + individual preferences.
  - bee_rider 2y ago
    Maybe it is a feedback loop, rather than a tautology? The things many mathematicians find interesting are the things that the general mathematician community is working on. And the way you become a mathematician is by publishing things that the community finds interesting enough to let through the peer review process. Ultimately though this is all funded by, in the end, the belief that they’ll be able to dumb down the good stuff for us scientists, engineers, and other folks who build actual physical things when we hit the point that we need it. (Of course it is an exploration process so not everything needs to be directly applicable). If computers ran the show, we would probably stop plugging them in if they used more power than their theories saved us, or whatever.

sherburt3 2y ago

I want to be Terrance Tao when I grow up

ramraj07 2y ago
I think the ship sailed when you were 4..
paulpauper 2y ago
Looking at his success it's hard to not believe that some people are objectively better than others.
- margorczynski 2y ago
  Nobody who has a grasp on basic biology and a honest mind wouldn't believe that. The thing is this completely goes against the liberal "tabula rasa" worldview.
- 082349872349872 2y ago
  Whether some people are objectively better at maths and communication than others, and whether they all get equal treatment under the law are two different things, right? Right? (I don't know where you grew up; where I grew up we were always obliged to chant "with liberty and justice for all")
  - yipbub 2y ago
    Right.
theshaper 2y ago
At this point, Terence Tao is already worthy of a list of facts, much like those about Chuck Norris and Bruce Schneier. For example: When Terence Tao solves a problem, the problem appreciates the solution.

taeric 2y ago

I wish people had more exposure to building mathematical models of things. I am fairly convinced that the only real exposure I was given was to models that we knew worked. So much so, that we didn't even execute many. Specifically, parabolic motion is something you can obviously do by throwing something. You can, similarly, plot over a time variable where things are observed. You can then see that we can write an equation, or model, for this. For most of us, we jump straight to the model with some discussion of how it translates. But nothing stops you from observing. With modern programming environments, you can easily jump people into simulating movement very rapidly and let people try different models there. We had turtle geometry years ago, but for most of us that was more mental execution than it was mechanical. Which is probably a great end goal, but no reason you can't also start with the easy computer simulations.

nextaccountic 2y ago
Something I really like is that the curve that a rope or thread makes when fixed in two points but not under tension, it's not a parabola. It really looks like one though, but it isn't. It's a catenary. That's something you can verify by writing some simulation code, then drawing the curve, and then drawing the best matching parabola on top. It doesn't fit. To model the issue mathematically you need some not-too-advanced calculus. On both the computer simulation and the mathematical model, you model the rope as being made of very small elements that are linked together (like a chain). In the simulation those elements are small, but finite. In the math you take the limit as the volume of the element tends to zero. It's the same way of thinking but math gives some different tools, enabling you to solve the curve analytically
- taeric 2y ago
  Exactly! I think this scenario alone would be an amazing set of lessons for many grade schools. Move this into modeling and then guessing stuff like bridge tensions, and you can easily show what many of the maths are good for. We used to have this with the attempts at building tooth pick bridges and such. Which I still think is very illuminating. But, I think there are a lot of questions you can expose with models that were often only seen by the more advanced students. And again, I agree that getting people to mentally model these things is a good goal. Right now, people rarely ponder things on paper, it seems.

ji_zai 2y ago

"The intuitive mind is a sacred gift and the rational mind is a faithful servant.” - Einstein

InDubioProRubio 2y ago
The problem is though, that with half the data, your mind considers glueing another base to the seasaw, to balance things out and restore symmetry and intuitive beauty.

grape_surgeon 2y ago

I love how well-spoken Tao is. I've enjoyed lots of his lectures before; even if you're not an expert in whatever he's discussing he knows how to explain it just right to get you up to speed as best as he can. His communication and math skills are phenomenal.

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xanderlewis 2y ago
Yes! He’s a great counterexample to the popular view that mathematical/pure logical reasoning ability is negatively correlated (even zero-sum) with communication ability. Yes, there are people that are crap at one and quite good at the other… but you can’t make much of an inference when given one without the other.
- tadala 2y ago
  Is this a popular view? I think mathematicians can be odd, but usually they communicate quite well. I think as far as popularization of their fields go, mathematics is probably doing the best out of the lot: numberphile, 3blue1brown etc.
  - xanderlewis 2y ago
    The examples you list are not known as mathematicians; they’re popularisers who (sometimes) happen to have qualifications and a history of studying the subject. 3B1B is absolutely brilliant but Grant Sanderson is not a ‘mathematician’ in the sense of someone who does research in mathematics. Ironically, the fact that mathematics popularisation is as visible as it is is itself a sign of how much it is needed and therefore how unpopular and misunderstood the subject is. Branches of science like, say, astrophysics don’t need popularisation; people already think they’re cool. The view of ‘people who are good at mathematics’ being bad at English is a relatively common one, in my experience. At least at the level of university students. People think there’s some sort of conservation of ability or equilibrium in the universe that means that if you have a ‘maths brain’ then you’re no good at much else, and vice versa. If anything, I think there’s a positive correlation between mathematical and communication ability — after all, mathematics is basically just the science of clever notation and clear-headed thinking.

rokob 2y ago

> The distinction between the three types of errors can lead to the phenomenon ... of a mathematical argument by a post-rigorous mathematician which locally contains a number of typos and other formal errors, but is globally quite sound, with the local errors propagating for a while before being cancelled out by other local errors I was initially amazed at this when I was in graduate school, but with enough experience I started to do it myself. Handwaving can be a signal that someone doesn't know what they are doing or that they really know what they are doing and until you are far enough along it is hard to tell the difference.

richrichie 2y ago
Did anyone whisper in your ears, “Welcome to the dark world!”?
sigmoid10 2y ago
>Handwaving can be a signal that someone doesn't know what they are doing or that they really know what they are doing and until you are far enough along it is hard to tell the difference. I found it's very easy to distinguish these two when you have another expert ask questions. But if you don't have someone like that in the audience it might take forever. Or at least until you become an expert yourself.
- bogeholm 2y ago
  “Learn the rules like a pro, so you can break them like an artist.” - Pablo Picasso [0] [0]: https://www.goodreads.com/quotes/558213-learn-the-rules-like-a-pro-so-you-can-break https://www.goodreads.com/quotes/558213-learn-the-rules-like...
solikeppl 2y ago
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topologie 2y ago

munchler 2y ago

Good article, but should mention that it's not new. First copy in the Wayback Machine is from 2018, but there are comments all the way back to 2009. https://web.archive.org/web/20180301000000*/https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ https://web.archive.org/web/20180301000000*/https://terrytao...

paulpauper 2y ago
his old articles are shared a lot on here

tsaixingwei 2y ago

“Before I learned the art, a punch was just a punch, and a kick, just a kick. After I learned the art, a punch was no longer a punch, a kick, no longer a kick. Now that I understand the art, a punch is just a punch and a kick is just a kick.” - Bruce Lee

mr_mitm 2y ago
People have pointed out similarities to a few things in this thread now. It's pretty much the bell curve meme: https://knowyourmeme.com/memes/iq-bell-curve-midwit https://knowyourmeme.com/memes/iq-bell-curve-midwit
- FrustratedMonky 2y ago
  man, sometimes things are obvious and right in front of us. I've known the Dogen saying for years. Have even meditated on it. I've been enjoying Bell Curve meme for sometime. Very funny. Never put together that these were the same thing. Today I am awakened.
  - karmakurtisaani 2y ago
    At first, the Dogen saying and the bell curve meme seem like different things, then..
    - ocodo 2y ago
      first there is the bell curve, then there is no bell curve, then there is the bell curve... If you learn anything significantly deeply, this is a repeating pattern.
- paulpauper 2y ago
  This is the best meme ever / So accurate in many ways
  - 082349872349872 2y ago

johnwatson11218 2y ago

The idea of the 3 levels really resonated with the ideas in "Bernoulli's Fallacy" as well. Right now we are seeing a resurgence of Bayesian reasoning across all fields that deal with data and statistical reasoning. I think many errors of modern civilization were caused by people at a level 2 understanding attempting to operationalize their knowledge for others at level 1. We need it to become much more common to operate at level 3, especially in fields like enterprise software development.

JoeyBananas 2y ago

The worst thing is when someone who thinks that "math is 100% infallible and all about rigor, you gotta show your work and include all the steps" yet they think that set theory is good enough and it doesnt have problems they say things like "Everything in math is a set," but then you ask them "OK, what's a theorem and what's a proof?" they'll either be confused by this question or say something like "It's a different object that exists in some unexplainable sidecar of set theory" They don't know anything about type theory, implications of the law of excluded middle, univalent foundations, any of that stuff

johnwatson11218 2y ago
My favorite was when a manager tried to get me to agree with the statement that "math was just for the numbers right?". Meaning not character strings nor dates. I was dumbstruck by the question.
- burnished 2y ago
  Math is for.. numbers? Thats engineer talk right there
qbit42 2y ago
Modern set theory is sufficient for most mathematicians. That other stuff is interesting, but you can do great mathematics without it.
Tainnor 2y ago
It's 100% possible to base logic and proof theory off of set theory. For example, you can treat proofs as natural numbers via Gödel encoding (or any other reasonable encoding) and we know that natural numbers can be represented by sets in multiple different ways. You may prefer type theory or other foundations, but set theory is definitely rigorous enough and about as "infallible" (or not) as other approaches.
red_trumpet 2y ago
Yeah, they should have heard about ZFC and have a notion what a formal proof is. On the other hand, I'm not sure your last sentence is really that relevant. > They don't know anything about type theory, implications of the law of excluded middle, univalent foundations, any of that stuff I'm doing a PhD in algebraic geometry, and that stuff isn't relevant at all. To me "everything is a set" pretty much applies. Hell, even the stacks-project[1] contains that phrase! [1] https://stacks.math.columbia.edu/tag/0009 https://stacks.math.columbia.edu/tag/0009

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staticshock 2y ago

> The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. This is a key insight; it's something I've struggled to communicate in a software engineering setting, or in entrepreneurial settings. It's easy to get stuck in the "data driven" mindset, as if data was the be-all and end-all, and not just a stepping stone towards an ever more refined mental model. I think of "data" akin to the second phase in TFA (the "rigor" phase). It is necessary to think in a grounded, empirical way, but it is also a shame to be straight-jacketed by unsafe extrapolations from the data.

hun3 2y ago
> It's easy to get stuck in the "data driven" mindset, as if data was the be-all and end-all, and not just a stepping stone towards an ever more refined mental model. Yes. "Data driven" either includes sound statistical modelling and inference, or is just a thiny veiled information bias.
paulpauper 2y ago
rigour is is not about destroying bad intuition, but rather formalizing good intuition, imho. The ability to know good from bad is somewhere in-between total newb and expert.

red_admiral 2y ago

This argument is very close to one by Whitehead in an essay called "The Rhythm of Education". The stages back there are called Romance, Precision, and Generalisation - but I'd argue there is an isomorphism (in a suitable category) between that and Tao's three stages.

dang 2y ago

Related: There’s more to mathematics than rigour and proofs (2007) - https://news.ycombinator.com/item?id=31086970 https://news.ycombinator.com/item?id=31086970 - April 2022 (90 comments) There’s more to mathematics than rigour and proofs - https://news.ycombinator.com/item?id=13092913 https://news.ycombinator.com/item?id=13092913 - Dec 2016 (2 comments) There’s more to mathematics than rigour and proofs - https://news.ycombinator.com/item?id=9517619 https://news.ycombinator.com/item?id=9517619 - May 2015 (32 comments) There’s more to mathematics than rigour and proofs - https://news.ycombinator.com/item?id=4769216 https://news.ycombinator.com/item?id=4769216 - Nov 2012 (36 comments)

highfrequency 2y ago

“The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture. Without one or the other, you will spend a lot of time blundering around in the dark.” Well put! In empirical research, there is an analogy where intuition and systematic data collection from experiment are both important. Without good intuition, you won’t recognize when your experimental results are likely wrong or failing to pick up on a real effect (eg from bad design, insufficient statistical power, wrong context, wrong target outcome, dumb mistakes). And without experimental confirmation, your intuition is just untested hunches, and lacks the refinement and finessing that comes from contact with the detailed structure of the real world. As Terry says, the feeling of stumbling around in the dark suggests you are missing one of the two.

EVa5I7bHFq9mnYK 2y ago

What? Americans learn proper calculus in later undergraduate years? Really?

nyssos 2y ago
Tao is Australian. That said, yes, real analysis is often a third-year class.

qnleigh 2y ago

I think entire research subfields can go through a similar process. Plenty of mathematics was done before mathematical rigor really existed. Then axiomatization became more and more important. The intuition never went away, but I have heard of 'Nicholas Bourbaki' (https://en.m.wikipedia.org/wiki/Nicolas_Bourbaki https://en.m.wikipedia.org/wiki/Nicolas_Bourbaki), the movement to right mathematics in purely formal language while eschewing intuitive language. And then more recently I read a prominent mathematician describing this phases having been a bit of a mistake. But maybe it was just a necessary part of the fields transition. I've definitely gone through a parallel transition in physics, but replacing 'rigor' with 'calculation' and 'intuition' for 'physical intuition/simple pictures.' In physics there is the additional aspect that problems directly relate to the physical world, and one can lose and then regain touch with this. I wonder what other fields have an analogous progression.

philipov 2y ago
> I wonder what other fields have an analogous progression. "Before one studies Zen, mountains are mountains and waters are waters; after a first glimpse into the truth of Zen, mountains are no longer mountains and waters are no longer waters; after enlightenment, mountains are once again mountains and waters once again waters."

hilux 2y ago

I think one of our biggest problems in society (business, politics, etc.) is that Stage 1 superficially resembles Stage 3. This means that many people in Stage 1 (or Stage 0, if that's a thing) believe that they're as good as Stage 3 thinkers. AKA Dunning-Kruger. In other words, complete bullshit, confidently delivered, has come to dominate informality-born-of-rigor. And the audience can't tell the difference.

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topologie 2y ago

Tao... What can I say... Always great work. Haven't read a single bad contribution from him. And I've read quite a bit...