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The paper that keeps showing up

cronokirby.com

11–20 of 48 posts

Re: The paper that keeps showing up

#11

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…

Adding to the chorus - you'll probably figure it out with enough time.

With the rise of neural nets, there is some solid circumstantial evidence that human intelligence is a re-purposing of parts of the brain evolved for vision processing (dunno what the biologists have figured out but hopefully they agree). Most of the people who I've listened to talk about how there is usually a sharp change-over moment when they go from being unable to do understand something to grasping it.

Get used to maths being impossible to understand for a long period of time, then suddenly easy to understand after that. IQ is relevant if you have to try and figure it out from first principles, but after that it probably only governs speed of learning rather than what can be done. This math in particular is not hard in the grand scheme, it is basically multiplication but a bit spiced up with multiple perspectives. And outrageously easy to implement wrong.

Although if you try to approach math expecting to be able to 'see' the answer with concepts you already understand it'll be painful. Real Math (TM, taught by Real Scotsmen) is learning to see things that you couldn't see before. The expectation is to encounter concepts that initially you can only access by symbol manipulation and in time, with training, become intuitive suddenly after a long period of practice.

Re: The paper that keeps showing up

#12

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 two negative real numbers gives you a positive despite multiply two positive also giving a positive. If you then go from that and say "what number when multiplied by itself would give me minus 1" you get i.

Also I don't think you've hit an IQ limit, it's not like you're lifting some weight that's too big for you. You just haven't seen an explanation that links to your existing corpus of knowledge.

Re: The paper that keeps showing up

#14

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’m glad you commented this (because the replies are encouraging me). I feel similarly to you in that I never had someone explain i to me in a way that I could intuitively grasp. I’m realizing now that this likely had more to do with not having a teacher who could explain it! Also, shoutout to non-Euclidean math for confusing me to no end and making me think I’m dumb (but also probably suffering from the same problem of not having a great teacher).

I’ve finally gotten to a point where I’m the one writing the confusing symbols for others to read after publication. Here’s something I never really realized before. Confusing equations, maybe more so in informally mathematic disciplines like computer science, are mostly an alternative way of trying to explain a complex paradigm. In a paper that I’m working on right now, we basically state something in plain English and then create an equation to communicate it in an alternative way. It’s a formal representation, but more than that, it’s just an attempt to communicate something that is inherently confusing.

It’s an oversimplification to say this, but: I think a lot of the confusion stems from humanity having discovered most of the simple stuff, what’s left to be communicated is inherently confusing.

Re: The paper that keeps showing up

#15

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…

Yes. Manipulating abstract symbols should be easy and natural - after all, that's what arithmetic is.

Re: The paper that keeps showing up

#16

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…

[deleted]

Re: The paper that keeps showing up

#17

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 don't think it's related to IQ or abstract thinking skills.

Maybe you have astonishing skills in other domains but are not suited to mathematical thinking for whatever reasons. Also, the current mathematical language is just a compromise that suits a majority, maybe you are not in it.

I find the formulas on the page quite easy to read and to "sense" (I also have a background in cryptography, which helps a lot). I like these expressions, because they are very simple to manipulate, which helps understanding them better, and also to infer more results. However, because I "sense" them, I can always dig down every symbols and know what's behind (eg. the scalar multiplication, k.G, I know it is actually like doing the sum of G+G k times, which in turns results in more operations in the particular field or group that G belongs to, etc).

If you study electricity where complex numbers are used a lot for instance, the i symbol becomes very tangible. It is more than a symbol, but a real object with well defined properties, and that you can describe in multiple ways. If you start to learn all these properties and multiple ways, you will understand it better, until it becomes very familiar. Also, you have to understand that i is just not a fancy object, it is actually an object mathematicians created because some operations didn't have results in the set of real (in particular not all polynomial equations had roots). So we extended the set of real with a bigger set (the set of complex, which is actually a two-dimensional set), where these problem disappeared. Later, mathematicians and physicists start to use complex numbers a lot because they could describe many useful properties in a very compact way.

Re: The paper that keeps showing up

#18

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…

Yes you have hit your limit in this particular area, but do not worry about it.

There is a saying that “A change of perspective is worth 80 IQ points.”, frequently attributed to Alan Kay.

In this particular case, you just need to look at 2D complex numbers, and what their multiplication looks like on a plane, and you will then have an intuitive understanding about i, as others have already pointed out.

Re: The paper that keeps showing up

#19

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 easier to reason about it. I found that to be the biggest problem with my academic curriculum - there was a lot of abstract math concepts that they teach you in first 2 years, and you discover their actual usefulness only years later.

Re: The paper that keeps showing up

#20

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…

> "Young man, in mathematics you don’t understand things. You just get used to them."

- John von Neumann

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