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

> 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.

“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

> 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.

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.

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...

“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.

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

I want to be Terrance Tao when I grow up

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.

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.

> 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.

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), 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.

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

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)

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

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.

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...

Tao... What can I say... Always great work.

Haven't read a single bad contribution from him. And I've read quite a bit...

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.

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

Could modern AI help amateur mathematicians to build proofs?

Is there?