There's more to mathematics than rigour and proofs (2007)
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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.
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.
[0]: https://www.goodreads.com/quotes/558213-learn-the-rules-like...
I think handwaving comes in different flavors:
- Handwaving and not knowing what they are doing, when they know they don't know:
This is arrogance and/or fear of people thinking you are a fool. Bad practice. Professionals who do this are status chasers and not fun to be around. Students who do this are mostly insecure, and they might just need some help with their self-esteem. Help them by letting them feel comfortable with being wrong. Foster a good environment so that the arrogance and fear fade away.
- Handwaving and not knowing what they are doing, when they don't know they don't know:
I believe this is a good thing, in particular for Students, if they are within a nurturing environment. It can lead to interesting ideas and to discussions of innovative ways to move forward. I believe this to be a way of actually "training your intuition muscle" both for Students and Professionals. It lets them know not to fear moving on, tackling the thing that captures their attention the most at first, and later on filling some of the gaps, which I feel is common practice for people who have been working on the field for a while. However, if the gaps are left unattended it can lead to bad things... Environment matters.
- Handwaving and knowing what they are doing, when they know they don't know:
For trained Professionals only... :) This modality kinda kicks in when deep in mathematical work. It's the path that leads to the Eureka moments... Pure trained intuition acting almost as a separate entity to oneself. We are facing the unknown and something tells us that certain aspect can be handwaived, we don't fully know why but we feel it is. Later on it becomes clear why we could do the handwave. It works itself out.
- Handwaving and knowing what they are doing, when they don't know they don't know:
For trained Professionals only... Kind of a stretch, but might be where our intuition either fails us completely, or completely takes us by the hand to turn the unknown unknowns into known unknowns, then it goes back to the previous category.
This isn't set in stone by the way, just some thoughts I had while reading the article...
Any ideas or suggestions for modifications more than welcomed.
As San Juan climbs Monte Carmelo, he finds nothing at the base of the monte, then he finds nothing at the middle, but then, at the cusp, he finds Nothing (capital N Nothing).
See the following image for reference: https://commons.wikimedia.org/wiki/File:Monte_Carmelo_Juan_d...
There's quite a bit of parallels between proper "Catholic Mystics" and Zen teachers...
I highly recommend both the San Juan de la Cruz works, in particular Ascent of Mount Carmel and The Dark Night, along with The Cloud of Unknowing, which was an inspiration for him.
For those curious about learning more about Koans, I cannot recommend this other book highly enough: https://www.amazon.com/Two-Zen-Classics-Gateless-Records/dp/...
The book contains a collection of Koans along with Mumon's (et al) Commentary, Mumon's Verse, as well as modern day notes that help us understand some of the concepts hidden behind what looks like "poetical nonsense" at first, as well as giving the context for historical and mythological figures that are mostly unknown for those "outside the loop." What I love about the notes is that they still leave you with the opportunity to explore the koan further, properly, so they don't really take away all the fun.
Example Koan from the book:
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Case 4 The Western Barbarian with No Beard
Wakuan said, "Why has the Western Barbarian no beard?"
Mumon's Comment:
Study should be real study, enlightenment should be real enlightenment. You should meet this barbarian directly to be really intimate with him. But saying you are really intimate with him already divides you into two.
Mumon's Verse:
Don't discuss your dream before a fool. Barbarian with no beard Obscures clarity.
NOTES (abridged)
- The Western Barbarian: The Western Barbarian stands for Bodhidharma, who brought Zen to China from India. He is always depicted with a beard. The case therefore means, "Why doesn't Bodhidharma, who has a beard, have no beard?"
- Meet this barbarian directly: This is not really a meeting but a becoming. You should yourself become Bodhidharma. Then if you have a beard, Bodhidharma has a beard; if you have no beard, then neither had Bodhidharma. But can you say that you are in truth Bodhidharma?
[...]
- Obscures clarity: Words, concepts, and other inventions of the mind only obscure the truth. Do not cling to shadows, but catch hold of truth itself.
##########################################################################
You get the idea.
Enjoy!
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.
then there is no bell curve,
then there is the bell curve...
If you learn anything significantly deeply, this is a repeating pattern.
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.
Is this a joke by mimicking someone on the mid-point of the 'bell curve meme' explaining the 'bell curve meme'?
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.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.
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.
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.
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.
1. Hack programmatic-functionality in a first language
2. Master the intricacies of a first language, understanding all programmatic concepts through the lens of that languages specific implementation-details. Pedantically argue with those familiar with different language implementations, due to a kind of implementation-plurality/ essential-form blindness
3. Learn additional languages, and 'see past' specific implementation details and pitfalls of each; develop a less biased understanding of the essence of any task at hand
> 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 :-)
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.
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).
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
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.
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.
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.
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
You may prefer type theory or other foundations, but set theory is definitely rigorous enough and about as "infallible" (or not) as other approaches.
> 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!
Yes, exactly. These are topics that 99% of legit mathematicians don't know or care about.
It's like saying "I'm a computer expert" when you only know Python, and then a computer engineer that designs CPUs starts laughing at you
It's not as rigorous as type theory (yes, this is an umbrella term) because type theory can be verified by a computer. Homotopy type theory is an example of the type of math that set theory can't handle
There are so many layers of ignorance to unpack here and I don't care to be your unpaid tutor
proofs in FOL can be checked by a computer without any need for type theory - just look at metamath.
For example:
When Terence Tao solves a problem, the problem appreciates the solution.
(I don't know where you grew up; where I grew up we were always obliged to chant "with liberty and justice for all")
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.
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
https://en.wikipedia.org/wiki/Catenary#Catenary_bridges
> Comparison of a catenary arch (black dotted curve) and a parabolic arch (red solid curve) with the same span and sag. The catenary represents the profile of a simple suspension bridge, or the cable of a suspended-deck suspension bridge on which its deck and hangers have negligible weight compared to its cable. The parabola represents the profile of the cable of a suspended-deck suspension bridge on which its cable and hangers have negligible weight compared to its deck. The profile of the cable of a real suspension bridge with the same span and sag lies between the two curves. The catenary and parabola equations are respectively, y = cosh x and y = x²( (cosh 1) − 1) + 1
https://www.quora.com/How-do-you-tell-the-difference-between...
> If the chain is carrying nothing other than its own weight, the resulting shape is a "catenary". If the chain is like a suspended cable carrying a deck below it, and its own weight is nothing compared to that of the deck, the resulting shape is a "parabola".
Which shows that sometimes your model (either using pure math or a simulation) is too simple to capture whatever is going on in the real world. (it gets further complicated when one considers elasticity etc)
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.
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.
Yes. "Data driven" either includes sound statistical modelling and inference, or is just a thiny veiled information bias.
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.
"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."
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
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)
We need it to become much more common to operate at level 3, especially in fields like enterprise software development.
https://web.archive.org/web/20180301000000*/https://terrytao...
Haven't read a single bad contribution from him. And I've read quite a bit...
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.
That said, yes, real analysis is often a third-year class.
It has a long, long way to go.
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.
From a few days ago:
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.
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.
Don't necessarily see a tautology here.