[HN Gopher] There's more to mathematics than rigour and proofs (...
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There's more to mathematics than rigour and proofs (2007)
Author : haraball
Score : 162 points
Date : 2024-06-19 09:51 UTC (13 hours ago)
(HTM) web link (terrytao.wordpress.com)
(TXT) w3m dump (terrytao.wordpress.com)
| ffhhj wrote:
| Could modern AI help amateur mathematicians to build proofs?
| ykonstant wrote:
| 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 wrote:
| 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 wrote:
| If you meant multiset, then you were correct. (Not that I
| expect GPT-4o to make the distinction.)
| BeetleB wrote:
| Not sure why you're downvoted.
|
| From a few days ago:
|
| https://news.ycombinator.com/item?id=40646909
| hyperman1 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| > 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| Also (in C++ lingo):
|
| 1. Start by writing programs with vectors and maps.
|
| 2. Learn all about data structures, algorithms, cache misses,
| memory efficiency etc
|
| 3. And then write programs with vectors and maps.
| jiiam wrote:
| Also in Haskell:
|
| 1. Start by doing everything in ReaderT Env IO
|
| 2. Learn all about mtl (or monad transformers, free monads,
| freer monads, algebraic effects, whatever)
|
| 3. Do everything in ReaderT Env IO
| jbandela1 wrote:
| > 3. And then write programs with vectors and maps.
|
| But the maps this time are absl::flat_hash_map (or another
| C++ alternative hash map such as Folly F14, etc) instead of
| std::map (or even std::unordered_map).
| prmph wrote:
| It's kind of like how people who are really, _really_ good at
| something approach it with a certain simplicity and
| straightforwardness. Superficially, it looks like how a novice
| would approach things. But look under the covers they are doing
| similar things but with a much deeper understanding why they
| are doing things that way.
|
| Example, (1) You start programming with the simplest
| abstractions and in a concrete way. (2) You learn about all the
| theory and mathy stuff: data structures, algorithms, advanced
| types, graphs, architecture, etc. Eventually you become very
| skilled with these, but at a certain point you start to bump up
| against their limitations. Technical disillusionment and
| burnout may set in if you are not careful (3) You return to
| using abstractions and architecture that are as simple as
| possible (but no simpler), but with a much deeper understanding
| of what is going on. You can still do very complex stuff, but
| everything is just part of a toolbox. Also, you find yourself
| able to create very original work that is elegant in its
| seeming simplicity.
|
| I've noticed the same thing in other fields: the best approach
| their work with a certain novice-like (but effective)
| simplicity that belies what it took for them to get to that
| point.
| JumpCrisscross wrote:
| > _3. Write programs that you think are cool. But since you
| know more, you can write more cool programs_
|
| The integration phase goes much deeper. The first stage is
| about learning how to write programs. The second is about
| writing programs well. The third is to intuitively reason about
| how to solve problems well using well-written programs; you can
| still code, but it's no longer where the lifting is.
| GiovanniP wrote:
| > 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.
|
| Hegel :-)
| mgaunard wrote:
| Is there?
| jfengel wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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.
| edflsafoiewq wrote:
| Math existed for thousands of years before modern rigor.
| xanderlewis wrote:
| True. It's also interesting to note that a lot of Newton and
| Leibniz's original reasoning about infinitesimals itself went
| through the same sort of three step process: first it was
| accepted because it was all there was, then rigour became
| fashionable and it was thrown out. Finally, only last
| century, it was shown that such ideas can be made perfectly
| rigorous in the right setting [see: nonstandard analysis].
| sherburt3 wrote:
| I want to be Terrance Tao when I grow up
| ramraj07 wrote:
| I think the ship sailed when you were 4..
| paulpauper wrote:
| Looking at his success it's hard to not believe that some
| people are objectively better than others.
| margorczynski wrote:
| 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 wrote:
| 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_
| ")
| theshaper wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| "The intuitive mind is a sacred gift and the rational mind is a
| faithful servant." - Einstein
| InDubioProRubio wrote:
| 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 wrote:
| 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.
| xanderlewis wrote:
| 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.
| rokob wrote:
| > 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 wrote:
| Did anyone whisper in your ears, "Welcome to the dark world!"?
| sigmoid10 wrote:
| >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 wrote:
| "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...
| munchler wrote:
| 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...
| paulpauper wrote:
| his old articles are shared a lot on here
| tsaixingwei wrote:
| "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 wrote:
| 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
| FrustratedMonky wrote:
| 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 wrote:
| At first, the Dogen saying and the bell curve meme seem
| like different things, then..
| paulpauper wrote:
| This is the best meme ever / So accurate in many ways
| 082349872349872 wrote:
| The bell curve meme classifies, and it classifies the
| classifier.
| dkarl wrote:
| The bell curve meme is a static taxonomy of people, and if
| the link you posted is correct, its origins were explicitly
| political. The version described by Bruce Lee, and anybody
| who has achieved a high level of skill, is about the process
| of learning and mastery.
|
| In the bell curve version, the wisdom of grug brain and the
| wisdom of the monk are presented as equal, and the struggle
| of the midwit is framed as a pretentious, unnecessary
| aberration. The lesson it teaches is, don't seek out
| knowledge and new ideas. Education confuses the "midwit"
| people who are smart enough to partially grasp it but not
| smart enough to see through it like the monk.
|
| In contrast, the "a punch is just a punch" version frames the
| conceptual struggle as a necessary phase in a process that
| leads to mastery. The beginner cannot engage directly with
| the simplicity of the master, so the beginner must engage
| through concepts and through practice. The more they do so,
| the more simple things begin to feel.
|
| Since this started with a Bruce Lee quote, we can use him to
| see that it is not just a linear process that passes through
| conceptual education and ends in mastery. That leads to a
| dead end, because mastery can only be complete in a limited
| context. Bruce Lee kept searching outside his zone of mastery
| to find ways to get better. He studied techniques from other
| martial arts and fighting sports, even though in doing so he
| had to engage at a conceptual level since he had not mastered
| those arts.
|
| For example, his art included trips and throws, and he was a
| master of his art. Yet when he got the chance later, he
| practiced judo with expert judoka. To do so, he had to back
| off from "a trip is just a trip, a throw is just a throw" and
| learn the techniques of judo. I don't think anybody has ever
| suggested he was a master at judo, which suggests he had to
| be engaging with it on a conceptual level. Yet he believed
| that his practice with judo improved the skills he had
| already achieved mastery at.
|
| Not only did Bruce Lee preach constant assimilation of new
| ideas, he also preached simplification by discarding what is
| not useful. If a punch is just a punch, what do you discard?
| The whole punch? In order to find something to discard, you
| must look past the apparent simplicity of the internalized
| skill and dissect it conceptually.
|
| In this view of things, conceptual thinking is not just a
| phase you go through on the way to mastery, but rather a
| complementary way of engaging with a skill. It is a tool for
| refining and elevating your intuitive mastery. Simplification
| and desimplification are the tick-tock of learning. A
| "mastered" skill is not like a video game sword that, once
| forged, always has the exact same stats, but is more like a
| Formula 1 car that is continually disassembled, analyzed, and
| rebuilt.
|
| The bell curve meme does occasionally get used to express a
| linear ignorance-struggle-mastery story of learning, but its
| origin and most common use is to caricature the pursuit of
| knowledge as pretentious foolishness.
| FrustratedMonky wrote:
| Is this a subtle joke on the discussion?
|
| Is this a joke by mimicking someone on the mid-point of the
| 'bell curve meme' explaining the 'bell curve meme'?
| dkarl wrote:
| If you want a shorter version, "grug brain, monk brain"
| pretends to profound, like "Zen mind, beginner's mind,"
| but it's just a lazy take. If "grug brain, monk brain"
| was poetry, it would say: We shall not
| cease from exploration And the end of all our
| exploring Will be to arrive where we started
| And it'll look exactly the same because travel is
| pointless.
| srik wrote:
| For the interested, the original Dogen zen koan goes something
| like this -- Before I began to practice, mountains were
| mountains and rivers were rivers. After I began to practice,
| mountains were no longer mountains and rivers were no longer
| rivers. Now, I have practiced for some time, and mountains are
| again mountains, and rivers are again rivers.
| spicymaki wrote:
| And Dogen is quoting a Zen (Chan) Buddhist Qingyuan Weixin
| [Seigen Ishin].
| penguin_booze wrote:
| There's a quote I've heard attributed to Einstein: "Now that
| the mathematicians have invaded relativity, I myself don't
| understand it".
| auggierose wrote:
| You can see people go through that process right here on HN,
| slowly realizing that 1 the integer is the same as 1 the real
| number.
| cjk2 wrote:
| I hope they aren't all realising zero isn't zero somewhere as
| well. I could really do without that headache today.
| johnwatson11218 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| Math is for.. numbers? Thats engineer talk right there
| qbit42 wrote:
| Modern set theory is sufficient for most mathematicians. That
| other stuff is interesting, but you can do great mathematics
| without it.
| Tainnor wrote:
| It's 100% possible to base logic and proof theory off of set
| theory. For example, you can treat proofs as natural numbers
| via Godel 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 wrote:
| 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
| bubblyworld wrote:
| Maths, when done correctly, _is_ 100% infallible by its own
| design. It's just that reality isn't obliged to play by your
| rules =P
| staticshock wrote:
| > 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 wrote:
| > 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 wrote:
| 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.
| dang wrote:
| Related:
|
| _There's more to mathematics than rigour and proofs (2007)_ -
| 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 - Dec 2016 (2
| comments)
|
| _There's more to mathematics than rigour and proofs_ -
| 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 - Nov 2012 (36
| comments)
| red_admiral wrote:
| 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.
| highfrequency wrote:
| "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 wrote:
| What? Americans learn proper calculus in later undergraduate
| years? Really?
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