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The paper that keeps showing up

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Re: The paper that keeps showing up

#21

Post by jacobolus »

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…

If you want to understand the geometric basis for the square root of –1, start with http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

A complex number is most intuitively thought of as a quotient of Euclidean planar vectors, i.e. a quantity z = v / u with which a Euclidean vector can be multiplied to scale and rotate it into another vector: zu = (v / u)u = v(u \ u) = v.

A unit bivector i then represents the quotient of two perpendicular vectors of the same magnitude. It naturally has the orientation of the plane spanned by the two vectors. Multiplying i by any vector in that plane serves to rotate it by a quarter turn. Multiplying a vector in the plane by i twice rotates by a half-turn.

Re: The paper that keeps showing up

#22

Post by at_compile_time »

lordnacho wrote: post #12
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…

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 things is hard, but they could have gone with something that didn't sound like it was intended to stroke the egos of the learned few who understand.

I prefer the term rotor, because they perform rotation. The actual math involved isn't that difficult if you've learned the basics of geometric/Clifford algebra.

Re: The paper that keeps showing up

#23

Post by lordnacho »

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…

Someone on HN proposed "Lateral Number" which I thought was a good one. It's a number, just to the side of the numbers you know.

Re: The paper that keeps showing up

#24

Post by JW_00000 »

lordnacho wrote: post #12
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…

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…

To add to this, a website I've found useful is betterexplained.com. Specifically for complex numbers, take a look at [1] and [2]. But they also have good articles with visualizations to help you understand the Fourier transform, the number e, integrals and derivatives, linear algebra, etc. A starting point is [3]. One useful idea is that they color the different parts of math equations to explain each part separately, see [4].

Another great source of math visualizations is the YouTube channel 3Blue1Brown, in particular the series on linear algebra [5]. (I didn't intuitively understand what a determinant represented or what the determinant of a transformation being zero really meant, until watching [6].)

[1] https://betterexplained.com/articles/a-visual-intuitive-guid... [2] https://betterexplained.com/articles/intuitive-arithmetic-wi... [3] https://betterexplained.com/articles/developing-your-intuiti... [4] https://betterexplained.com/articles/colorized-math-equation... [5] https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x... [6] https://www.youtube.com/watch?v=Ip3X9LOh2dk&list=PLZHQObOWTQ...

Re: The paper that keeps showing up

#25

Post by omginternets »

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…

Friend, I submit that you and I were both taught poorly. I think you will find this video [0] extremely helpful. It provided me with the missing piece for understanding the need for complex numbers, and what we're actually doing when we manipulate them.

[0] https://youtu.be/Bhf6W-j5O7s

Re: The paper that keeps showing up

#26

Post by d_tr »

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…

I think it is mostly about motivation. Intuition will come with practice and will remain even if you forget some details.

Why limit yourself just to models that map well to everyday life? Everyday life sucks in a lot of ways after all, and imaginary worlds are intriguing. People like fiction.

In the case of abstract math, you have a world with varied and intricate structure but lots of real life applications as well.

Re: The paper that keeps showing up

#27

Post by cyberbanjo »

lordnacho wrote: post #23
at_compile_time wrote: post #22

Earlier quoted context omitted.

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…

Someone on HN proposed "Lateral Number" which I thought was a good one. It's a number, just to the side of the numbers you know.

"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

Re: The paper that keeps showing up

#28

Post by red_admiral »

It's a beautiful piece of theory, but when the IACR tried to build a voting scheme based on that to elect their board annually (https://vote.heliosvoting.org/), they got caught by a non-obvious bear trap when you make the whole thing non-interactive (which is the way this is mostly used in practice).

Details here: https://eprint.iacr.org/2016/771

Basically, if you do this with a hash function, you need to HASH ALL THE THINGS, not just some of them.

As to why this paper keeps showing up without everyone knowing that they're using it: the result in this paper is not, to my knowledge, a new invention of Maurer, rather it's something that everyone working with Sigma protocols more or less knew at the time, but no-one had written it down in its generality (or at least, no-one else got a paper on that accepted; some reviewers might have rejected such a paper as not novel enough). You'll note that Maurer itself got the paper into AFRICACRYPT 2009, which is not quite in the same league as CRYPTO and friends - for example, https://sites.google.com/site/conferenceranking/ calls it "unranked", and the CORE ranking page doesn't seem to list it either.

It's a neat little result and it's very useful to be able to cite, but it's not ground-breaking.

Re: The paper that keeps showing up

#29

Post by throwawaywork0 »

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…

I've done abstract reasoning tests and score top 0.1% and none of this makes sense to me without pouring in effort. Its interest related.

Re: The paper that keeps showing up

#30

Post by asciimov »

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 the big concepts (algorithms, classes, strings).

Now say you go look at a large technical C program, like a compiler. On the surface you can see a lot of things you do understand, but there is a lot you don't. What are pointers and doubles, long doubles, how do unsigned ints play into this, what is malloc, why am I mallocing? What is this namespace doing here, how about this struct?

Sure given enough time and a dictionary you could probably make sense of the program and you will have to learn a bit to get there.

Higher level maths are the same way. You have to have a ton of foundational knowledge to start having intuition on the bigger things. Unfortunately for most of us computer people, we just don't even get to scratch the surface of maths theory. Most of us stop just short of the intro to maths theory classes. The equivalent for us would be learning some logic, a few programming languages (algebra, geometry, trig, calculus), maybe intro to algorithms, but no theory of automata, programming language theory, etc (number theory for maths).

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