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

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

#81

Post by qnleigh »

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.

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

#82

Post by hilux »

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.

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

#83

Post by philipov »

qnleigh wrote: post #81

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 th…

> 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."

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

#84

Post by tadala »

xanderlewis wrote: post #67
grape_surgeon wrote: post #12

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.

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.

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.

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

#85

Post by xanderlewis »

tadala wrote: post #84
xanderlewis wrote: post #67

Earlier quoted context omitted.

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.

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.

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.

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

#86

Post by JoeyBananas »

red_trumpet wrote: post #45
JoeyBananas wrote: post #27

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…

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,…

> I'm doing a PhD in algebraic geometry, and that stuff isn't relevant at all.

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

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

#88

Post by nextaccountic »

nextaccountic wrote: post #62
taeric wrote: post #10

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 equ…

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-adva…

That's totally a tangent but I was reading a bit and it happens that, in practice, real cables bend in an inbetween curve between catenaries and parabolas

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)

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

#89

Post by yipbub »

082349872349872 wrote: post #70
paulpauper wrote: post #55

Earlier quoted context omitted.

Looking at his success it's hard to not believe that some people are objectively better than others.

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 ")

Right.

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

#90

Post by ocodo »

srik wrote: post #34
tsaixingwei wrote: post #22

“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

For the interested, the original Dōgen 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.

I came here looking for this Zen Koan
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