The paper that keeps showing up • Hacker News Board

Re: The paper that keeps showing up

#41

Post by JadeNB » Mon, Oct 31, 2022, 4:59 PM UTC

asciimov wrote: post #30

club_tropical wrote: post #2

I have a question for people who read this and whose eyes aren't immediately glazing over: I am incredibly put off by math that feels like "symbol manipulation". Equations that I can't easily put in terms of geometry or statistics. I think something in me "broke" with the constant "i" (square root of -1) during my education - a purely synthetic concept that I used to manipulate equations and get good grades without a…

> did I just kinda hit the limit of my IQ / abstract thinking skills? No, the biggest problem is you haven't been exposed to enough of theory side of maths to have a intuition about how these things combine into the bigger picture this article is covering. Imagine for a moment, that you are a good basic JavaScript programmer. You don't use typescript, so you don't understand types, just variables (vars), but you get…

As a math professor and sub-dilettante programmer, I found this a fantastic and instructive analogy. Thank you!

Re: The paper that keeps showing up

#42

Post by whatshisface » Mon, Oct 31, 2022, 4:59 PM UTC

club_tropical wrote: post #2

I have a question for people who read this and whose eyes aren't immediately glazing over: I am incredibly put off by math that feels like "symbol manipulation". Equations that I can't easily put in terms of geometry or statistics. I think something in me "broke" with the constant "i" (square root of -1) during my education - a purely synthetic concept that I used to manipulate equations and get good grades without a…

There are a lot of nice answers here talking about math generally, but I can offer a specific answer. This is a paper that generalizes a lot of things, and you have to understand the specific things for it to come across as an insight instead of a pointless abstract thing. Not knowing any specific thing and trying to learn the generalization is like trying to know Netwon's laws without relating them to any physical situations.

Re: The paper that keeps showing up

#43

Post by whatshisface » Mon, Oct 31, 2022, 5:14 PM UTC

ivanhoe wrote: post #19

club_tropical wrote: post #2

I have a question for people who read this and whose eyes aren't immediately glazing over: I am incredibly put off by math that feels like "symbol manipulation". Equations that I can't easily put in terms of geometry or statistics. I think something in me "broke" with the constant "i" (square root of -1) during my education - a purely synthetic concept that I used to manipulate equations and get good grades without a…

Complex numbers are extensively used in electronics so I never had a feeling of it as something purely synthetic, but most of n-dimensional math was for me beyond my capabilities to understand it any level higher then treating at as a purely mathematical abstraction of our 3d world - and thus all I'd do was just crunching the equitations. Probably if I was exposed to real world applications of that math it would be e…

Well, suppose you have a circuit with four mesh currents. Now you have dynamics going on in four dimensions.

Re: The paper that keeps showing up

#44

Post by derefr » Mon, Oct 31, 2022, 6:19 PM UTC

club_tropical wrote: post #2

I have a question for people who read this and whose eyes aren't immediately glazing over: I am incredibly put off by math that feels like "symbol manipulation". Equations that I can't easily put in terms of geometry or statistics. I think something in me "broke" with the constant "i" (square root of -1) during my education - a purely synthetic concept that I used to manipulate equations and get good grades without a…

The interesting thing about doing mathematics — in a research sense — is that often there is no intuition. Instead, there's a projection from one domain into another, where that projection might have an intuitive proof yet is itself completely unintuitive; and then there are people using that domain-mapping equivalence to prove things in one domain using completely unintuitive statements about them that can only be made in the other domain.

I might liken it to what LLVM's IR optimizer passes do. You put in source code (intuitive), get out object code (also intuitive, for those who know the processor) — but in between, the code passes through various projective and derived forms that are not clean representations of either the source or target form, but instead are just models amenable to certain static analyses — i.e. proofs! — that then allow/deny certain rewrites to happen.

A mathematician is someone who has to do what those IR optimization passes do, in their head. They know that they can derive a projective model, that they can then poke at symbolically, to derive more lemmas that will be able to be mapped back into lemmas about the model outside the projection. That doesn't mean that they intuitively understand what those within-projection steps mean in terms of the outer model. (If they did, they wouldn't need the projected model; they'd just make their deductions in terms of the outer model!) Instead, they just know how the inner model works as a model — how to poke it to make it do things, etc — and they know general rules of math and logic that allow them to prove things within that model, and to transform and map proofs between models. But, in "feeling grounded" terms, they're mostly working blind.

Re: The paper that keeps showing up

#45

Post by Nzen » Mon, Oct 31, 2022, 6:44 PM UTC

vrnvu wrote: post #8

Beautiful explanation. After reading this I wonder if anyone knows about homomorphic encryption and ZKP uses for role/policy management? Or something similar? They main uses I know for ZKPs are in open distributed scenarios but I wonder if it could be used to simplify other spaces where public/private keys are involved. For example, instead of having a user creating roles and policies and storing all of this info in…

You've described the problem that macaroons [0] intend to solve by amending an hmac access token with constraints on its use. The described scenario of appending a hme circuit to evaluate policy requests sounds much more expensive (both in terms of effort to answer 'can this request send an email?' and size of the token) than a key:value pair.

[0] https://fly.io/blog/api-tokens-a-tedious-survey/#macaroons

Re: The paper that keeps showing up

#46

Post by fjeifisjf » Mon, Oct 31, 2022, 6:56 PM UTC

at_compile_time wrote: post #22

lordnacho wrote: post #12

Earlier quoted context omitted.

I guess whoever taught you complex numbers didn't do it with a focus on diagrams? I agree math should be taught as visually as possible, with algebra-only explanations being minimised. Even things like the expansion of (a+b)^2 = a^2 + 2ab + b^2 or a^2 - b^2 = (a + b)(a - b) should be motivated by diagrams. Anyway it turns that multiplication of complex numbers is adding the polar angles. This explains why multiplying…

It doesn't help that i is given names like "imaginary" and "complex". We may as well call them "too-hard-for-you" numbers, and laugh condescendingly when students ask about them. Same with quaternions and dual quaternions. They perform rotation and scaling in 3D space. Calling them "hypercomplex numbers" makes it sound like an advanced concept only to be understood after years of dedicated study. I get that naming th…

No post body was provided.

Re: The paper that keeps showing up

#47

Post by fjeifisjf » Mon, Oct 31, 2022, 6:58 PM UTC

club_tropical wrote: post #2

I have a question for people who read this and whose eyes aren't immediately glazing over: I am incredibly put off by math that feels like "symbol manipulation". Equations that I can't easily put in terms of geometry or statistics. I think something in me "broke" with the constant "i" (square root of -1) during my education - a purely synthetic concept that I used to manipulate equations and get good grades without a…

No post body was provided.

Re: The paper that keeps showing up

#48

Post by cestith » Fri, Nov 04, 2022, 7:14 PM UTC

cestith wrote: post #31

cyberbanjo wrote: post #27

Earlier quoted context omitted.

"If we call +1, -1, and √-1 had been called direct, inverse and lateral units, instead of positive, negative, and imaginary (or impossible) units, such an obscurity would have been out of the question." --Gauss

One of the most helpful things for me when handling imaginary and complex numbers came not from a math class, but a physics class. My HS physics instructor said rather than "imaginary" we could think of -1 as a "hidden" number, because after you do the math fairly often you see where the number would be, but you just can't see it among the "real" numbers. This was literally an aside to something else he was talking a…

Somehow I missed the √ symbol in √-1 there but I think it was understood I meant i, not -1.