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Ask HN: How did you learn math notation?

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Re: Ask HN: How did you learn math notation?

#81

Post by throwaway31338 »

pjc50 wrote: post #74
throwaway31338 wrote: post #56

Earlier quoted context omitted.

One can look at the source code to a program, the libraries it uses, the compiler for the language, and the ISA spec for the machine language the compiler generates. You can know that there are no hidden unspecified quantities because programs can't work without being specified. When you get down to the microcode of the CPU that implements the ISA you might have an issue if it's ill-specified. You might be talking ab…

> because programs can't work without being specified. Someone hasn't read the C spec, with all its specified as undefined behavior. Programs working on real systems is very different from those systems being formally specified. I suspect that if you only had access to the pile of documentation and no real computer system - if you were an alien trying to reconstruct it, for example - you'd hit serious problems.

Undefined behavior isn't a feature. A spec isn't an implementation, either.

All behavior in an implementation can be teased-out if given sufficient time.

> if you were an alien trying to reconstruct it, for example - you'd hit serious problems.

I can't speak to alien minds. Considering the feats of reverse-engineering I've seen in the IT world (software security, semiconductor reverse-engineering) or cryptography (the breaking the Japanese Purple cipher in WWII, for example) I think it's safe to say humans are really, really good at reverse-engineering other human-created systems from close-to-nothing. Starting with documentation would be a step-up.

Re: Ask HN: How did you learn math notation?

#82

Post by OneTimePetes »

Through a really nice and helpful math prof who took time out of her day to explain it to those in the "im in trouble" additional course. Forever grateful for that, would have failed otherwise.

Math notation becomes very readable, as soon as the teacher writes a example out on the black board, and that is why i will never forgive wikipedia / wolfram / latex for not having a interactive "notation to example expansion". They had such a chance to reform the medium - to make it more accessible to beginners and basically forgot about them.

Re: Ask HN: How did you learn math notation?

#83

Post by kergonath »

tomerv wrote: post #17
ColinWright wrote: post #14

Earlier quoted context omitted.

You say "There's a formula with a triangle ..." without telling me where. That's not real helpful, and you're making me do the work to find out what you're talking about. If you want assistance to get started, you need to be more explicit. However, I have done that work, so I've looked, and in the second column of page 210 there's a "formula with a triangle": t_c = 5 \middot 10^{-5} \sqrt( V / Dt ) ... where the "D"…

I think that this is indeed the formula in GP's question. And indeed sometimes math notation is obtuse like that. It looks like 2 terms, but the triangle goes together with the t as a single term. At other times it might be called "dt" and despite looking like a multiplication of 2 variables (d and t, or triangle and t in this case) it's just a single variable with a named made of 2 characters. The important thing he…

That’s because “Δ” means “a change of” or “an interval of”. So, Δt is “an interval of time”. It is like a compound word, really. It conveys more information than giving it an arbitrary, single-letter name.

This convention is used in a whole bunch of scientific fields, like quantum mechanics, chemistry, biology, mechanics, thermodynamics, etc.

It’s also very useful in how it relates to derivatives, which is a crucial concept in just about any kind of science you could care to mention.

So yes, there is a learning curve, but we write things this way for good reasons, most of the time.

Multiplication should be represented by a (thin) space in good typography, to avoid this sort of things. Not doing it is sloppy and invites misreading. Same with omitting parenthesis around a function’s argument most of the time (e.g. sin 2πθ instead of sin(2 π θ) ).

Re: Ask HN: How did you learn math notation?

#84

Post by ColinWright »

throwaway31338 wrote: post #77
ColinWright wrote: post #69

Earlier quoted context omitted.

You seem to be complaining that math isn't programming, that it's something different, and you've discovered that you don't like how mathematicians do math. Math notation is the way it is because it's what mathematicians have found useful for the purpose of doing and communicating math. If you are upset and disappointed that that's how it is then there's not a lot we can do about it. If there was a better way of doin…

I really, really appreciate your reply and its tone. Thank you for that. You've given me some things to think about. I often wish people were more like computers. It probably wouldn't make the world better but it would make it more comprehensible.

Thanks for the pingback ... I appreciate that. And thanks for acknowledging that I'm trying to help.

It might also help to think of "scope" in the computing sense. Often you have a paragraph in a math paper using symbols one way, then somewhere else the same symbols crop up with a different meaning. But the scope has changed, and when you practise, you can recognise the change of scope.

We reuse variable names in different scopes, and when something is introduced exactly here, only here, and only persists for a short time, sometimes it's not worth giving it a long, descriptive name. That's also similar to what happens in math. If I have a loop counting from 1 to 10, sometimes it's not worth doing more than:

    for x in [1..10] {
      /* five lines of code */
    }
If you want to know what "x" means then it's right there, and giving it a long descriptive name might very well hamper reading the code rather than making it clearer. That's a judgement call, but it brings the same issues to mind.

I hope that helps. You may still not like math, or the notation, but maybe if gives you a handle on what's going on.

PS: There are plenty of mathematicians who complain about some traditional notations too, but not generally the big stuff.

Re: Ask HN: How did you learn math notation?

#85

Post by Hermitian909 »

ColinWright wrote: post #58
cabalamat wrote: post #50

Earlier quoted context omitted.

> I think a real problem in this area is the belief that there is "one true notation" and that everything is unambiguous and clearly defined. One main cause for this belief is that in a programming there is one true noation (or rather, a separate one for each language) that is unambiguous and clearly defined. I dislike maths notation as I find it lacks rigour.

> I dislike maths notation as I find it lacks rigour. I see this a lot from programmers, but in essence, you seem to be complaining that maths notation isn't what you want it to be, but is instead something else that mathematicians (and physicists and engineers) find useful.

As someone who's studied math and CS extensively, it's not that mathematicians don't need that rigor it's only certain sub-fields have a culture of this kind of notational rigor. You absolutely see little bubbles of research, 2-4 professors, get sealed off from the rest of the research community because their notational practices are so sloppy that no one wants to bother whereas others make it easy to understand their work.

CS as a field just seems to have a higher base standard for explaining their notation and ideas. It helps in cross-collaboration by making it significantly easier to self study.

Related to this, I'd say math books have a significantly worse pedagogical culture in regards to both notation and defining pre-requisites. It's very common for a math book to say "we expect readers to have taken a discrete math course" and not defining notation despite knowing the topics covered in discrete math vary greatly from school to school and may not overlap. Math professors frequently have to paper over these problems at Uni as they realize the class does not understand some notation. CS are just better about this, and I can only explain it as a part of the culture and tradition.

Re: Ask HN: How did you learn math notation?

#86

Post by nicoburns »

bajsejohannes wrote: post #75
ColinWright wrote: post #2

I think a real problem in this area is the belief that there is "one true notation" and that everything is unambiguous and clearly defined. Yes, conventions have emerged, people tend to use the same sort of notation in a given context, but in the main, the notation should be regarded as an aide memoire , something to guide you. You say that you're struggling because of "the math notations and zero explanation of it i…

> I think a real problem in this area is the belief that there is "one true notation" and that everything is unambiguous and clearly defined. Just to back up this point: In probably every university-level math book I've read, they introduce and explain all the notation used. In the preface and/or as concepts are introduced. There are lists at wikipedia [1] and other places, but I'm not sure how valuable it is out of…

Ha, you've clearly used better books than me. I've read plenty where they glossed over the notation and expected you to guess.

Re: Ask HN: How did you learn math notation?

#87

Post by kergonath »

kens wrote: post #66
rullopat wrote: post #11

Earlier quoted context omitted.

I was trying to grasp some of the papers linked in the Valhalla DSP block, for example this one: http://www2.ece.rochester.edu/courses/ECE472/resources/Paper... There is a formula with a triangle and I don't get what's that about, for example.

Everyone is talking about the Δ symbol, but the real problem that you'll encounter will be later in the paper where they start talking about H(ω), which is the Fourier transform of the impulse function (equation 4 and following). You'll need to know a fair bit about Fourier transforms and impulse responses and filter design to get through this section. The notation is the least of the problems. One place to start is…

Just like with programming, knowing the meaning of keywords is not enough to understand something.

Re: Ask HN: How did you learn math notation?

#88

Post by randomNumber7 »

colechristensen wrote: post #76
randomNumber7 wrote: post #60

Earlier quoted context omitted.

Formulas would also be easier to read if they would not name all their variables and functions with one character. If programmers would write code like that (even fortran programmers use 3 characters), noone would be able to understand the code...

Scientists and engineers write code with single variables all the time and don’t seem to have any large amount of trouble with it. Long variable names seriously limit the ability to put complexity in one place and make it understandable. Take a look at this and tell me it would be easier to understand if all the symbols were words instead of single characters: https://www.grc.nasa.gov/www/k-12/airplane/nseqs.html

This is pretty understandable because:

1. They explain all abbreviations at the top.

2. There is a lenghty text explaining the formula.

3. It's mathematically pretty easy if you know partial differentation.

Also scientists and engineers write pretty horrible code....

Re: Ask HN: How did you learn math notation?

#89

Post by ColinWright »

nicoburns wrote: post #86
bajsejohannes wrote: post #75

Earlier quoted context omitted.

> I think a real problem in this area is the belief that there is "one true notation" and that everything is unambiguous and clearly defined. Just to back up this point: In probably every university-level math book I've read, they introduce and explain all the notation used. In the preface and/or as concepts are introduced. There are lists at wikipedia [1] and other places, but I'm not sure how valuable it is out of…

Ha, you've clearly used better books than me. I've read plenty where they glossed over the notation and expected you to guess.

Often books assume some prerequisites, the question here is the level of those prerequisites. Some books try to include all the necessary background, others assume a pre-existing base level of knowledge.

Different authors, different books, different audiences, and different contexts.

Re: Ask HN: How did you learn math notation?

#90

Post by anthomtb »

This block post, which has been referenced several times on HN, was a god send for me: https://www.neilwithdata.com/mathematics-self-learner

I also used get hung up on “mathematical notation”. But it turns out the problem wasn’t the notation. I was just bad at math. Well, out-of-practice is more like it.

Once you have the fundamentals clearly explained and you’re doing some math on a regular basis the notation, even obscure non-standard notation becomes relatively intuitive.

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