This is SO TRUE.

The same thing happens to me regularly, and not just with "computer science" but with other technical fields, hard sciences, and mathematics. The purpose of most academic papers is not to explain (let alone teach!) ideas in an intuitive manner, but rather to express them in formal, correct, unambiguous terms -- that is, to make them as accurate and critique-proof as possible for publication in some journal.

Worse, they also like to use the same symbol to denote different things in different fields. I'm sure it's extremely convenient to have a shorthand when working and sharing "code" with some peers in the know but for stuff like wikipedia articles it makes things appear more complex and obscure than they should.

I don't mind if people call a local function variable "n" in some code, it's usually non ambiguous. But if you export a variable "n" in an external API you will be screamed at. Why is it ok for maths?

To give a quick example, the letter R in maths can mean the set of real numbers. It can also (with a different typeface!) mean Ramanujan summation. Oh wait, you're doing physics? Then R is the gas constant, silly.

Also, i is the imaginary unit. Except in physics, then it's j, because i is used for currents. Makes sense.

It's an interesting discussion because from the mathematician's perspective, they don't see why they should cater to anyone who doesn't bother absorbing the lingua franca and the method of delivery. Countering that is the philosophical argument that information should be as available as possible. Countering that is how practical and useful that is, and whether the cost / benefit would be worth it.

I still think we can have our cake and eat it too, but I'm not sure. I think if the purpose is merely to transmit proofs and axioms unambiguously, I think we can have a language that performs just that and nothing else. I think stuff like this exists, but I don't know why it isn't the standard to publish with it.

Then the explanation can reside alongside this unambiguous description, and can take whatever liberties it pleases.

It's like this all the time in math papers. It often seems in the end like the ideas themselves are fairly straightforward, and it shouldn't have taken that long to understand. I think, though, that if you actually sat down and tried to explain it in more intuitive terms, you'd find that you might not be able to. True, you could find a way to convey the general idea, but without the technical details, (1) while non-mathematicians may get a surface-level understanding faster, even mathematicians will not grasp the technical aspects, and (2) it will be very hard for anyone to extend your work or for anyone to apply it to another situation, so it will only be applied in the specific contexts that you explained it in.

Language isn't built to communicate math, so doing so effectively will be either difficult to understand or imprecise. Many people claim that it would be easy to explain deep math concepts with "a couple of block diagrams and a few short paragraphs"s, but I'd challenge them to write a textbook on abstract algebra, or topology, or something like that before they make that claim.

Firstly, it's broadly considered to be the case that mathematical ideas are not understood until you've gotten them "from a few different angles". Math builds upon itself so much that an idea may be almost useless on its own and produce true value in lying at the nexus between many convergent ideas. For instance, statistics as a field enjoys a very nice convergent point between logic, measure theory, and information theory (among others). Approaching it from all of these positions can lead to important mental breakthroughs and a paper or book author desires to appeal to these various "roads" to their topic. Without providing that context it might be said that the presentation is very sparse.

Secondly, almost conversely, each reader is likely to be more familiar with a subset of the possible roads to the author's topic of interest. By covering as many roads as possible as they approach their goal topic they provide more chances for the reader to pick an approach they find most comfortable and follow it (lightly ignoring the rest) to the goal. A simple block diagram might be the best way to present it to you, but only a private tutor could specialize their presentation so much.

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There's an art to reading a math paper when you're an outsider to the primary topic. You want to breeze through the paper at a high level first, slowly collecting the exposition points which are most applicable to your own method of understanding. After that, iteratively deepen your reading while looking up topics which you feel you almost-but-perhaps-not-quite-enough understand. You can very easily read a paper and get enormous value while failing to connect to 60-70% of what's written.

I systematically ignore maths in CS papers. If the concept isn't described with code or pseudo-code at least, I will look at who cites the paper, and look for someone who has replicated enough in code. 90%+ of the time I can avoid dealing with the maths entirely, or get enough details that I can glean the rest without bothering trying to actually understand the maths properly.

Usually I find the maths tends to obscure vast amounts of missing details.

Of course, how well that works depends on the specific field.

You have the benefit of hindsight. Everything is obvious in hindsight. All the research I've done is obvious and straight forward, if only I had known what I know now and would be able to draw a few simple diagrams.

That is, until you realise it's not. In those days and weeks spent deciphering mathematics you are actually learning a lot. I cannot count how many times I've been reading mathematics and struggled for weeks on a concept. Then one day it clicks and it all makes sense. Then I re-read the description again and the answer is clear as day. The answer was always there, I just hadn't learnt enough to appreciate it.

>So many ideas seem closed to mortals because of the nature of mathematics.

I disagree with this statement 100%. No ideas are closed because of mathematics. The ideas are only closed if you are not willing to put in the time.

Anyone with a ugrad pure math major is supposed to know what a group is and the early, standard theorems.

Group theory was used by E. Wigner for the quantum mechanics of molecular spectroscopy so that at times some chemistry students want to know some of group theory and group representations.

My ugrad honors paper was on group theory.

I published a paper showing how a group of measure preserving transformations could lead to a statistical hypothesis test useful for 'zero-day' monitoring for anomalies in server farms and networks.

Group theory pops up occasionally. Get a good, standard text in abstract algebra or two or three and spend a few evenings. If you get to Sylow's theorem, you are likely deep enough for starters. Group theory is very clean, polished stuff and can be fun. Go for it.

(My personal mission is to find/share the aha! moments that actually make the details click. Why do we force everyone to laboriously discover them for themselves? Can't want talk about the underlying insights directly?)

Maybe not the corruption angle so much; but special interests. Now every book is designed to not offend the politically correct Californians, or the religious right in Texas, and to convince people that they are keeping up with the latest fads - http://www.edutopia.org/muddle-machine

Which is why everyone, but especially programmers, should learn a lot more math.

I've been a secondary math teacher in the UK and I want to defend this point a little.

The job of a secondary math teacher at this level is to teach everyone math, particularly including a majority who don't have a strong interest and won't go on to study more mathematics. You probably underestimate how difficult a job this is.

With that in mind, this is already quite a long list of diverse topics. You neglected to mention an introduction to number, up to the real numbers, perhaps because you now think it is obvious. You were taught that.

I personally try to teach 'looking forward': explaining how these concepts link towards what direction you might take in the future.

However, it's very difficult to cover the whole scope of mathematics and mathematical subjects. For example, I personally knew only a little about what's relevant to EE (although I've learned over time). But, it's not that you could skip anything from the school curriculum anyway - my general advice is that potential EE students need to show interest enough to study independently outside of school.

Something like algorithmic complexity, you should be learning from Knuth. Well done for that: there are not many educational experiences better than learning independently from someone who has devoted his life to making his subject accessible.

For the benefit of those not familiar with the UK school system:

- Compulsory education goes up to the age of 16 (roughly 11 'grades')

- Many people continue at school until around 18 (an additional 2 years, called the 'sixth form')

- These final 2 years of school can be taken at a secondary school (~= high school in the US) or at a college (which may specialise in these 2 years only, or may go beyond). This college is not the same as a university, although there are some areas where they may compete for the same students

- Most people who enter university as undergraduates do so after completing these additional 2 years, and the exams which partially determine university acceptance are taken when people are about 17 or 18

The thing is, I use very little abstract math in programming. Doing analytic philosophy actually did more than mathematics. The math stuff I do in programming is super basic stuff like trigonometry (sometimes trig with differentiation if I want to be fancy and use inflexion on curves) for GPS triangulation etc. I've never had to do anything more complex than that

I'm a big fan of A Levels (my experience in late 90s) because of the depth they can bring. Between the ages of 16 and 18 I was exposed to group theory, n dimensional linear algebra, (basic) proofs, curvature, complex numbers, various series and convergence. It was very eye opening. And the same is true of virtually all disciplines. The insight into a discipline A Levels gives, at a quite young age, cannot be achieved with a broad syllabus where depth is diluted.

Im also trying to understand crypto better, from a mathematical perspective. Apart from group theory, what else should I have an understanding of before going further? Do you have any resources that you would recommend?

"If I had more time, I would have written a shorter letter." -- attributed to Pascal.