[HN Gopher] Ask HN: How did you learn math notation?
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Ask HN: How did you learn math notation?
Hi all! I'd really like to learn "higher level than highschool"
math as a (long time ago) college drop out, but I find it really
hard to read anything because of the math notations and zero
explanation of it in the context. I didn't find on the web any good
resource on the topic, do you any advice / link? Thanks!
Author : rullopat
Score : 105 points
Date : 2021-11-25 13:24 UTC (9 hours ago)
| phtrivier wrote:
| Do you mean all the introductory mathematics books you tried fail
| to properly explain the notation ?
|
| Or that the notation differs from books to books ?
|
| (In my case, I learned the notation via French math textbooks,
| and in the first day of college/uni we litteraly went back to
| "There is a set of things called natural numbers, and we call
| this set N, and there is this one thing called 0, and there is a
| notion of successor, and if you keep taking the successor it's
| called '+', and..." etc..
|
| But then, the French, Bourbaki-style of teaching math is
| veeeeeeeery strict on notations.
| Syzygies wrote:
| I'm a math professor, and my students find it revelatory to
| understand math as I talk and draw.
|
| Math notation is not math, any more than music notation is music.
| Notably, the Beatles couldn't read sheet music, and it didn't
| hold them back.
|
| The best comparison would be is reading someone else's computer
| code. At its best computer code is poetry, and the most gifted
| programmers learn quickly by reading code. Still, let's be
| honest: Reading other people's code is generally a wretched
| "Please! Just kill me now!" experience.
|
| Once you realize math is the same, it's not about you, you can
| pick your way forward with realistic expectations.
| j7ake wrote:
| Great insight! I've definitely encountered mathematically
| inclined people but who cannot read or write math. Now it makes
| sense to me.
|
| Also I've found the converse true. There are people who can
| manipulate mathematical symbols very well but actually don't
| understand the big picture or general direction. The analogy
| would be that there are people who can write and read music
| notes (even transpose to different keys) without hearing it in
| their head (I was one of them).
| eointierney wrote:
| Super answer! I wish you were one of my professors, and I had
| excellent professors.
|
| If I may humbly add, try making your own notation and playing
| around with it. Very rapidly one realizes just how hard a
| problem good notation is.
| wenc wrote:
| I learned it by asking peers in grad school what stuff meant. And
| working through the math myself (it was a slog at first) and then
| writing stuff out it in LaTeX. When one is forced to learn
| something because one needs to take courses and to graduate, the
| human brain someone figures out a way.
|
| A lot of it is convention, so you do need a social approach - ie
| asking others in your field. For me it was my peers, but these
| days there's Math stack exchange, google, and math forums. Also,
| first few chapters of an intro Real Analysis text is usually a
| good primer to most common math notation.
|
| When I started grad school I didn't know many math social norms,
| like the unstated one that vectors (say x) were usually in column
| form by convention unless otherwise stated (in undergrad calc and
| physics, vectors we're usually in row form). I spent a lot of
| time being stymied by why matrix and vector sizes were wrong and
| why x' A x worked. Or that the dot product was x'x (in undergrad
| it was x.x). It sounds like I lacked preparation but the reality
| was no one told me these things in undergrad. (I should also note
| that I was not a math major; the engineering curriculum didn't
| expose me much to advanced math notation. Math majors will
| probably have a different experience.)
| ivan_ah wrote:
| As a starting point you can check out the notation appendices
| from my books:
| https://minireference.com/static/excerpts/noBSmathphys_v5_pr...
| https://minireference.com/static/excerpts/noBSLA_v2_preview....
| You can also see this excerpt here on set notation
| https://minireference.com/static/excerpts/set_notation.pdf
|
| That covers most of the basics, but I think your real question is
| how to learn all those concepts, not just the notation for them,
| which will require learning/reviewing relevant math topics. If
| you're interested in post-high-school topics, I would highly
| recommend linear algebra, since it is a very versatile subject
| with lots of applications (more so than calculus).
|
| As ColinWright pointed out, there is no one true notation and
| sometimes authors of textbooks will use slightly different
| notation for the same concepts, especially for more advanced
| topics. For basic stuff though, there is kind of a "most common"
| notation, that most books use and in fact there is a related ISO
| standard you can check out:
| https://people.engr.ncsu.edu/jwilson/files/mathsigns.pdf#pag...
|
| Good luck on your math studies. There's a lot of stuff to pick
| up, but most of it has "nice APIs" and will be fun to learn.
| sumnole wrote:
| The rhino book is a good dead tree reference.
|
| https://www.amazon.com/Mathematical-Notation-Guide-Engineers...
| fsloth wrote:
| I think good first resource would be the book and lecture notes
| in an introductory university course treating the specific domain
| you are interested in because often lots of things in notation
| are domain specific. Lots of good open university lectures out
| there, if not sure from where to start the MIT open courseware
| used to be a good first guess for accessing materials.
|
| As a sidenote I have MSc in Physics with a good dollop of maths
| involved and I am quite clueless when looking at a new domain so
| it's not as if university degree in non-related subject would be
| of any help...
| CornCobs wrote:
| Mathematics is a lingo and notations are mostly convention.
| Luckily people generally follow the same conventions, so my best
| advice if you want to learn about a specific topic is to work
| through the introductory texts! If you want to learn calculus
| find an introductory college text. Statistics? There are
| traditional textbooks like Introduction to Statistical Learning.
| The introductory texts generally do explain notation which may
| become assumed knowledge for more advanced texts, or as you seem
| to be wanting to read, academic papers. If those texts are still
| too difficult, then maybe move down to highschool text first.
|
| Think about it this way. A scientist, wanting to communicate his
| ideas with fellow academics, is not going to spend more than half
| the paper on pedantics and explaining notations which everyone in
| their field would understand. Else what is the purpose of
| creating the notations? They might as well write their formulas
| and algorithms COBOL style!
|
| Ultimately mathematics, like most human-invented languages, is
| highly tribal and has no fixed rules. And I believe we are much
| richer for it! Mathematicians constantly invent new syntax to
| express new ideas. If there was some formal reference they had to
| keep on hand every time they need to write an equation that would
| hamper their speed of thought and creativity. How would one even
| invent something new if you need to get the syntax approved
| first!
|
| TL;DR: Treat math notation as any other human language. Find some
| introductory texts on the subject matter you are interested in to
| be "inducted" into the tribe
| gammalost wrote:
| I learned most of my university math through "Calculus a Complete
| Course". But it's a bit expensive so I would recommend you buy an
| older version of the book where you can find a free solution pdf.
|
| But you'll have to be a bit realistic when going through the
| book, it's going to take a good while.
| wizardforhire wrote:
| 1] learn the greek alphabet if you haven't already.
|
| 2] dive deep into the history of math.
|
| 3] youtube... 3 blue 1 brown, stand up maths, numberphile, kahn
| academy. These channels are your friends.
|
| 4] don't give up and make it fun. Once you're bit by the bug of
| curiosity and are rewarded with understanding you'll most
| probably be unstoppable but still, its a long road. Better to
| focus on the journey.
|
| Lastly, the notation is what it is because of the nature of math
| itself coupled with the history of who was doing the solving
| exacerbated by the cultural uptake. There have been and will
| continue to be new notation. Its unfortunate that often to learn
| a new concept the barrier is with parsing the syntax. Stick with
| it and stay curious and those squiggles will take on new magical
| and profound meanings.
| teawrecks wrote:
| By doing math. Khanacademy has a lot of higher than highschool
| math courses you could check out.
| ColinWright wrote:
| 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 in the context." Can you give us some
| examples? Maybe getting a start on it with a careful discussion
| of a few examples will unblock the difficulty you're having.
| Grustaf wrote:
| There pretty much is one true notation. There could be some
| slight variations, like bolding vectors, putting an arrow over
| them or not distinguishing them at all from scalars. But 95% of
| the time everyone uses the same notation.
| ColinWright wrote:
| I don't know your background, but I wonder how broad it is in
| terms of mathematical topics. The notations used in Algebraic
| Topology vs Category Theory vs Algebraic Number Theory vs
| Analytical Combinatorics vs Complex Analysis.
|
| This isn't a criticism, it's just that notations vary wildly
| in those areas, and there's lots of cross-over of notations,
| not all of which agree with each other.
|
| I'm not an expert, but I've had some exposure to the
| problem(s).
| Grustaf wrote:
| I studied diff geo at phd level and met stat at undergrad
| level, plus a sprinkling of category theory, some discrete
| mathematics and some physics, so I've been exposed to most
| of these.
|
| I presumed we were talking about basic mathematics here
| since new notation is the least of your worries when your
| thinking about fibre bundles and cohomologies, but I can't
| really think of any significant overlap in notation that
| would be different between the fields I've come across.
| Could you give some examples?
| ColinWright wrote:
| I'm trying to be more general than specific questions at
| the mid-undergrad level, because looking in from the
| outside, people seem to thing that if only the notation
| weren't so mysterious then they could understand
| everything. But this comment --
| https://news.ycombinator.com/item?id=29344238 -- gives a
| flavour, talking about coming across "p" in different
| contexts and having to give different interpretations.
|
| But I remember sketching an algorithm to someone and just
| inventing notations on the fly as I did so, knowing that
| they would simply be ways to remember the underlying
| ideas.
|
| Even so, at 1st year undergrad the notations used in
| Mathematical Physics vary from those used in Introductory
| Graph Theory, and again from Real Analysis. But once the
| reader knows the underlying semantics, the actual
| notation is mostly a non-issue (as you know).
| Grustaf wrote:
| Alright, but are there really any overlapping concepts
| between graph theory and analysis? There can't be many!
|
| The comment you linked to is pretty strange, given the
| limited number of symbols in the Greek and Latin
| alphabets, there's obviously going to be a lot of reuse,
| but I can't see how that could really cause any confusion
| though, unless you're just grabbing books from the shelf
| and opening them at random. And even then, it should
| almost always be clear from context if pi is a number or
| a plane, and if it's a function that will be visually
| distinguished.
|
| I've seen non-mathematicians use words as names of
| variables and functions, it always makes me shudder. I
| unsuccessfully tried to introduce Hebrew letters as an
| alternative,when I discovered how to use the in Latex,
| but it never caught on...
|
| I actually find math notation incredibly intuitive and
| effective, I think it's close to optimal. In fact it's
| only after getting into programming that it even occurred
| to me how elegant and magical it is. I understand what
| things mean and can write things myself, without being
| able to exactly explain how, or to translate it into a
| fully specified system that a computer would understand.
| bajsejohannes wrote:
| > 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 context.
|
| [1]
| https://en.wikipedia.org/wiki/Glossary_of_mathematical_symbo...
| nicoburns wrote:
| Ha, you've clearly used better books than me. I've read
| plenty where they glossed over the notation and expected you
| to guess.
| bajsejohannes wrote:
| It's not entirely unlikely that I am remembering just the
| good stuff :) But I was surprised how many books would
| define even the most common notation, like [?], [?], and
| [?].
|
| I guess if you call your book "Introduction to..." you
| ought to do that. And it seems that all books were called
| that, regardless of how narrow and advanced the rest of the
| title was :)
| ColinWright wrote:
| 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.
| cabalamat wrote:
| > 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.
| throwaway31338 wrote:
| Came here to say the same thing harshly and laced with
| profanity. I guess I can back off a bit from that now.
|
| I was filled with crushing disappointment when I learned
| mathematical notation is "shorthand" and there isn't a formal
| grammar. Same goes for learning writers take "shortcuts" with
| the expectation the reader will "fill in the gaps".
| Ostensibly this is so the writer can do "less writing" and
| the reader can do "less reading".
|
| There's so much "pure" and "universal" about math, but the
| humans who write about it are too lazy to write about it in a
| rigorous manner.
|
| I can't write software w/ the expectation the computer "just
| knows" or that it will "fill in the gaps". Sure-- I can call
| libraries, write in a higher-level language to let the
| compiler make machine language for me, etc. I can inspect and
| understand the underlying implementations if I want to,
| though. Nothing relies on the machine "just knowing".
|
| It's feels like the same goddamn laziness that plagues every
| other human endeavor outside of programming. People can't be
| bothered to be exact about things because being exact is hard
| and people avoid hard work.
|
| "We'll have a face-to-face to discuss this there's too much
| here to put in an email."
| irchans wrote:
| There are formal grammars. The formal grammars are really
| hard to understand in my humble opinion. The best examples
| I think are COQ (see e.g.
| https://en.wikipedia.org/wiki/Coq) and Lean (see e.g
| https://en.wikipedia.org/wiki/Lean_(proof_assistant) ).
|
| Yes, we are too lazy to be 100% formal and many times we
| are too lazy to be mostly formal. This is mostly because we
| target our writing to other mathematicians who have no need
| to see every small step and including every step makes the
| proofs long. On the other hand, I do feel that generally
| speaking mathematicians should show more of their work and
| skip fewer steps.
|
| I find your statement "People can't be bothered to be exact
| about things because being exact is hard and people avoid
| hard work." to be very true. Being precise is difficult.
| kmill wrote:
| Here's a take from a mathematician-in-training, and it's
| biased toward research-level math, or at least math from
| the last hundred years:
|
| Math is difficult, and a lot of what we have is the result
| of the sharpest minds doing their best to eke out whatever
| better understanding of something they can manage. Getting
| any sort of explanation for something is hard enough, but
| to get a clear theory with good notation takes an order of
| magnitude more effort and insight. This can take decades
| more of collective work.
|
| Imagine complaining about cartographers from a thousand
| years ago having sketchy maps in "unexplored" regions. Maps
| are supposed to be precise, you say, there's actual earth
| there that the map represents! But it takes an
| extraordinary amount of effort to actually send people to
| these places to map it out -- it's hardly laziness.
| Mathematics can be the same way, where areas that are
| seemingly unrigorous are the sketches of what some
| explorers have seen (and they check that their accounts
| line up), then others hopefully come along and map it all
| in detail.
|
| When reading papers, there's a fine balance of how much
| detail I want to see. For unfamiliar arguments and
| notation, it's great to have it explained right there, but
| I've found having too much detail frustrating sometimes,
| since after slogging through a page of it you realize "oh,
| this is the standard argument for such-and-such, I wish
| they had just said so." You tend to figure that something
| is being explained because there is some difference that's
| being pointed out.
|
| I've been doing some formalization in Lean/mathlib, and it
| is truly an enormous amount of work to make things fully
| rigorous, even making it so that all notation has a formal
| grammar. It relies on Lean to fill in unstated details, and
| figuring out ways to get it to do that properly and
| efficiently, since otherwise the notation gets completely
| unworkable.
| CogitoCogito wrote:
| > There's so much "pure" and "universal" about math, but
| the humans who write about it are too lazy to write about
| it in a rigorous manner.
|
| Are you sure it's laziness? Maybe it's a result of there
| not actually being any universal notation (not even within
| subfields) or the exactness you refer to really isn't
| necessary. This doesn't mean that unclear exposition is a
| good thing. Mathematical writing (as with all writing)
| should strive towards clarity. But clarity doesn't require
| some sort of minutely perfectly consistently notation which
| would be required by a computer because humans are better
| than computers at handling exactly those kinds of
| situations.
|
| > People can't be bothered to be exact about things because
| being exact is hard and people avoid hard work.
|
| I think you have it wrong. People can't be bothered to be
| as exact because they don't need to. People can understand
| things even if they are inexact. So can mathematicians.
| Honestly this is a feature. If computers would just
| intuitively understand what I tell them to do like a human
| assistant would, that would be a step up not a step down in
| human computer interfaces.
| Tronno wrote:
| People can also understand each other through
| combinations of obscure slang, garbled audio, thick
| accents, and drunken slurring. It's still an unpleasant
| way to communicate.
|
| Shall we be satisfied with the same low standards in a
| technical field, because _it is how it is_?
|
| Hands-on users of math notation are complaining that it
| sucks. I'm not sure why a dismissive "works for me" is so
| often the default response.
| CogitoCogito wrote:
| > Hands-on users of math notation are complaining that it
| sucks. I'm not sure why a dismissive "works for me" is so
| often the default response.
|
| Are you sure this is because the notation is
| unclear/imprecise or because you just don't like it? I
| like certain programming languages and certain
| programming styles and really don't like others. But in
| none of the cases (those I like nor those I don't) are
| they not 100% "clear". The code compiles and executes
| after all so there really isn't much of an argument that
| somehow it's underspecified.
|
| The same thing exists in mathematics. There are certain
| fields of math whose traditional
| notation/style/approach/etc. are totally incomprehensible
| to me. There are also many mathematicians who would say
| the same about my preferences as well.
|
| So my point is that all people are _different_. Some
| people like certain things and some people like others.
| How can you hope to please everyone simultaneously? In my
| experience, there is no field at all that is as precise
| as mathematics. Sure "code" is precise, but (imo)
| professional programmers are nowhere near as precise in
| any general design or conversation than mathematicians.
| So I find the attack on supposedly bad mathematical
| notation a bit odd.
|
| Mathematicians constantly try to come up with better
| methods of explaining things. They put more effort into
| it than basically any field in my experience. The
| problems are really that we as humans don't all think the
| same and that mathematics is just plain hard. We've
| improved mathematical communication immensely throughout
| history and we will continue to do so. But we'll never
| reach some sort of perfect communication style because no
| single such style could ever exist.
| Jensson wrote:
| > Hands-on users of math notation are complaining that it
| sucks. I'm not sure why a dismissive "works for me" is so
| often the default response.
|
| It is really easy to complain. People also complain about
| every popular programming language, but it is really hard
| to make something that is actually better. It is easy to
| make something that you yourself think is better, but it
| is hard to make something that is better in practice.
| webmaven wrote:
| _> But clarity doesn 't require some sort of minutely
| perfectly consistently notation which would be required
| by a computer_
|
| I made this point in another comment, but I think it
| bears repeating and elaboration: Consistency isn't
| required (at least outside any single paper), but
| explicitness would be a tremendous boon.
|
| Software incorporates outside context all the time, but
| it pretty much always does it explicitly (though the
| explicitness may be transitive, ie. dependencies of
| dependencies). Math papers often assume context that is
| _not_ explicitly noted in the citations, nor those papers
| ' citations, etc.
|
| Instead, some of the context might only be found in other
| papers that cite the same papers you are tracking down.
| You sometimes need to follow citations both backward and
| forward from every link in the chain. And unlike
| following citations backward (ie. the ones each author
| considered most relevant), the forward links aren't
| curated and many (perhaps most) will be blind alleys
| (there also may be cycles in the citation graph, but
| these are relatively rare). But somehow you have to
| collect knowledge (or at least passing familiarity) with
| an encyclopedic corpus in order to at least recognize and
| place the context left implicit in any one paper in order
| to understand it.
|
| It's maddening.
| CogitoCogito wrote:
| I totally agree. I think that many mathematical papers
| aren't explained as well as they can be. My advisor was
| pretty adamant that papers should not be written in some
| proof-chasing style like you describe and that the author
| should clearly include the arguments they need (citing
| those authors they might have learned it from) unless
| those arguments are truly standard. No "using a method
| similar to [author] in Lemma 5 of [some paper]" and
| instead just including it in your paper and making sure
| if fits in well.
|
| That is just an example of bad exposition in my opinion.
| It's also not technically "unclear" in any notational
| sense so it's a bit of an aside from this argument. But I
| agree with you 100% that it is bad bad bad. This is a
| perfect example of why arguments like "does this proof
| make coq happy" totally misses the point.
| ColinWright wrote:
| 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 doing it, people
| would be jumping on it. If a different way of doing it
| would let you achieve more, people would be doing it.
|
| It's not laziness, and I think you very much have got the
| wrong idea of how it works, why it works, and why it is as
| it is. Your anger comes across very clearly, and I'm
| saddened that your experience has left you feeling that
| way.
|
| Maths is very much about communicating what the results are
| and why they are true, then giving enough guidance to let
| someone else work through the details should they choose.
| Simply giving someone absolutely all the details is not
| really communicating why something is true.
|
| I'm not good at this, but let me try an analogy. A computer
| doesn't have to understand _why_ a program gives the result
| it does, it just has to have the exact algorithm to
| execute. On the other hand, if I want you to understand
| _why_ when n is an integer greater than 1, { n divides
| (n-1)!+1 } if and only if { n is prime } then I can sketch
| the idea and let you work through it. Giving you all and
| every step of a proof using Peano axioms isn 't going to
| help you understand.
|
| Similarly, I can express in one of the computer proof
| assistants the proof that when p is an odd prime, { x^2=-1
| has a solution mod p } if and only if { p = 4k+1 for some k
| }, but that doesn't give a sense of _why_ it 's true. But I
| can sketch a reason why it works, and you can then work out
| the details, and in that way I'm letting you develop a
| sense of _why_ it works that way.
|
| Math isn't computing, and complaining that the notation
| isn't like a computer program is expressing your
| disappointment (which I'm not trying to minimise, and is
| probably very real) but is missing the point.
|
| Math isn't computing, and "Doing Math" is not "Writing
| Programs".
| throwaway31338 wrote:
| 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.
| ColinWright wrote:
| 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.
| JohnHaugeland wrote:
| > We reuse variable names in different scopes
|
| This example works against you. Scope shadowing is nearly
| universally considered bad practice, to the point that
| essentially every linter is pre-configured to warn about
| it, as are many languages themselves (eg prolog, erlang,
| c#, etc)
|
| To a programmer, you're saying "see, we do it just like
| the things you're taught to never ever do"
|
| .
|
| > You may still not like math, or the notation,
|
| The notation is probably fine
|
| What I personally don't like is mathematicians' refusal
| to provide easy reference material
|
| Programmers want mathematicians to make one of these:
| https://matela.com.br/pub/cheat-sheets/haskell-cs-1.1.pdf
|
| It doesn't have to be perfect. We don't need every
| possibility of what y-hat or vertical double bars means.
| An 85% job would be _huge_.
| Jensson wrote:
| > This example works against you. Scope shadowing is
| nearly universally considered bad practice
|
| So you never used the same variable name in two different
| scopes ever? Like, if a function takes argument "name",
| no other function you ever write again in any program can
| have a variable named "name" unless it is the same exact
| usage?
|
| Or, as is commonly complained about in math, every
| programmer in the world then use the variable "name" only
| for that usecase and otherwise comes up with a new name
| for it?
|
| Having different scopes doesn't imply shadowing, it just
| means that you define it and then use it and then scope
| goes out and it no longer exists. No mathematician knows
| even close to every domain, so different domains of math
| uses notation differently. It is like how different
| programmers programs in different programming languages.
| It is such a waste to have so many programming languages,
| but people still do it for legacy reasons.
| shlurpy wrote:
| > Math notation is the way it is because it's what
| mathematicians have found useful for the purpose of doing
| and communicating math.
|
| That's only really a good description for the most well
| trod areas, where people habe bothered to iterate. I
| think a more realistic statement would be:
|
| "Math notation is the way it is because some
| mathematician found it sufficient to do and communicate
| math, and others found it tolerable enough to not bother
| to change."
|
| Personally, though, my problem has always been where
| publications use letters and symbols to mean things that
| are just "known" in some subfield that isn't directly
| referenced. It's not a problem for direct back and forth
| communication during development, true, but it
| dramatically increases the burden on someone who wants to
| jump in.
| JohnHaugeland wrote:
| I mostly agree with you.
|
| That all said, it would still be quite nice if it was
| somehow more accessable. A lot of papers containing
| material that's probably actually quite standardizable
| remain opaque to me, and the notation invariably falls by
| the wayside if there's a code or language description
| available.
|
| Many times, math notatons have been thought to be
| minimal, or most clear possibly, only to fall by the
| wayside
|
| Whereas this notation serves domain specialists well, it
| still leaves people like me somewhat confused
|
| A cheat sheet - even to the practical norms - would go a
| long way
| randomdata wrote:
| _> One main cause for this belief is that in a programming
| there is one true noation_
|
| And then there is SQL.
| bigbillheck wrote:
| > in a programming there is one true noation (or rather, a
| separate one for each language) that is unambiguous and
| clearly defined
|
| Yes this is why we all use Hungarian notation and GNU
| indentation.
| ColinWright wrote:
| > _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.
| Hermitian909 wrote:
| 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.
| Jensson wrote:
| > CS are just better about this, and I can only explain
| it as a part of the culture and tradition.
|
| CS professors writes just as incomprehensible math as
| everyone else, as you can see many here brings up
| examples of CS professors writing incomprehensible math
| in their papers.
| kazinator wrote:
| Moreover, you might think that Lisp notation would
| improve it, but CS papers using S-expressions are just as
| incomprehensible, even to a seasoned Lisp programmer.
|
| Math notations are two-dimensional and don't suffer very
| badly from structural ambiguities, so that actually fixes
| almost nothing.
|
| The problem in unfamiliar math notations is rarely the
| chunking of which clump is a child of which clump.
|
| E.g say that some paper uses, say, angle brackets, with
| some deep meaning that you can learn about if you recurse
| three levels down in the list of references.
|
| I'm not confused that in <Ap>, the Ap thing is a child of
| the angle brackets; and calling it (frob (A p)) doesn't
| help much in this regard.
|
| However, at least you can search literature for the word
| _frob_ more easily than for angle brackets.
| mistrial9 wrote:
| I read some graph theory papers from mid-20th Century and
| yes, this is true
| JuliusBranson wrote:
| I think "useful" is doing a lot of work here. A lot of math
| notation exists clearly to gate keep. It's often
| nonsensical. It's a shame because it really makes
| mathematicians look bad (re:annoying) to those who can see
| through it. It's not hard to see through it or anything,
| but it is obnoxious. All you need is an english explanation
| of the notation, and then you're good, but often all of the
| sources on the topic are written in the same obnoxious
| babble language.
|
| Take sequential Monte Carlo / sequential importance
| sampling for instance. This powerpoint on it is clownishly
| bad: http://people.eecs.berkeley.edu/~jordan/courses/260-sp
| ring10...
|
| This is supposed to be an algorithm implemented in code.
| It's essentially illegible without code examples, which it
| doesn't feature. Code examples tell you what the cipher
| signifies; at no point does the cipher provide any value to
| the learner. Fanciful bayes-theoretical statements and so
| on basically reduce to "iteratively build enlarging valid
| states." Given the fact that this simple statement is
| missing, I question if the professor has some sort of
| communication disorder or if they're just a troll. Similar
| to pomo philosophers, it's probably a mix.
| Jensson wrote:
| Lecture powerpoints are bad everywhere since you are
| meant to listen to the lecturer speaking about them, they
| aren't meant to be read independently like this.
|
| Try to understand programming based on a programming
| lecture powerpoint, it is usually impossible.
|
| Edit: Also you can't write code for what he is talking
| about in that lecture. Code cannot deal with infinities
| or continuous values. You'd get approximations which
| isn't the same thing, then you'd need to prove that those
| approximations are good enough which would have to be
| done without code anyway.
| klibertp wrote:
| Yes. What's wrong with changing math notation? Why wouldn't
| you do it if _you know_ that it would make it easier for
| others to approach? What 's the rationale behind doing
| exactly nothing to make the notation more approachable for
| the masses?
| ColinWright wrote:
| Math notation has evolved to be what it is because it is
| useful for the actual doing of math, and the
| communication of math to those who have sufficient
| background. It's not deliberately designed to keep people
| out, and there are literally hundreds of thousands of
| books that introduce people to the notations used, to
| help on-board them.
|
| Haskell is unreadable to one who has not trained in it or
| similar languages ... why don't they make the syntax more
| readable? Or C++ with its modern templating ... why don't
| they change the syntax to make it more readable?
|
| You might be tired of wandering into someone else's area
| of expertise and telling them:
|
| _You must change! You must make it more accessible!_
|
| Believe me, mathematicians are tired of non-
| mathematicians wandering up and saying:
|
| _Look! Computer programs are easy and intuitive and
| everyone can understand them, even without training! Make
| math like that!_
|
| Do you _really_ believe that math notation is
| deliberately designed to make it hard for people
| untrained in math to learn how to use it? Do you _really_
| believe that no one has tried to make it more accessible?
|
| Do you _really_ believe you know more about why math
| notation is what it is than mathematicians and trained
| mathematics educators do?
| klibertp wrote:
| > It's not deliberately designed to keep people out,
|
| It looks that way, to many people, even in this thread.
|
| > why don't they change the syntax to make it more
| readable?
|
| They do, actually. Quite often at that. It's called
| releasing new version.
|
| > Look! Computer programs are easy and intuitive and
| everyone can understand them, even without training! Make
| math like that!
|
| No. Computer code is as far from intuitive as it can be.
| Nobody says otherwise. So you don't need to do anything
| to get there, the notation's good on that front (meaning:
| completely non-intuitive).
|
| That's where the IDEs come in. And debuggers. And other
| tools. Lots of tools. They really help. You could _use_
| them, because the IDEs-for-math already exist. In college
| I had exactly one semester to familiarize myself with one
| of them, and it was _never mentioned again_ until
| graduation.
|
| > Do you really believe that math notation is
| deliberately designed to make it hard for people
| untrained in math to learn how to use it?
|
| Why, do you believe it's not possible for it to be that
| way? See: https://en.wikipedia.org/wiki/Pythagoras#Prohib
| itions_and_re...
|
| > Do you really believe that no one has tried to make it
| more accessible?
|
| Why did they fail? (If they didn't - where's the
| exponential growth of first years' mathematicians in
| training)
|
| > Do you really believe you know more about why math
| notation is what it is than mathematicians and trained
| mathematics educators do?
|
| I'm 100% _not_ interested in why it is like this, it 's
| not my problem, so I really wouldn't know. Would you be
| interested in how at some point you had to write `class
| X(object):` and that it later changed to simply `class
| X:`? Would you go hunt on the mailing list to see who
| exactly came up with the idea? Or why they thought it
| would be better that way? Would you be interested in that
| if you just had to write a 10-lines of Python, to scrape
| some web site?
| Jensson wrote:
| > Why, do you believe it's not possible for it to be that
| way? See: https://en.wikipedia.org/wiki/Pythagoras#Prohib
| itions_and_re...
|
| Did you just use an example from 2600 years ago to make a
| point? You don't think much changed since then?
| klibertp wrote:
| > Did you just use an example from 2600 years ago to make
| a point?
|
| Yes? What's wrong with that?
|
| I'm pointing out the most widely known example, to make a
| point, which is: "it is possible to design notation
| specifically for keeping outsiders out". I'm not saying
| that modern math notation is like that. I think, as a
| layman, that it probably evolved over a long time and so
| is full of idiosyncrasies that made perfect sense back
| when they were introduced (my GP seems to describe it in
| similar terms, so I hope I'm not _that_ far removed from
| reality).
| JohnHaugeland wrote:
| > It's not deliberately designed to keep people out
|
| Surely you must realize that you're protesting this
| because it has this reputation, though?
|
| And surely you must realize that it has this reputation
| for a reason?
|
| When I was a teenager and took my first calculus course,
| I struggled with summation for three days. When I finally
| went to my dad he looked at me funny and said "your
| teacher is an idiot, isn't he? It's a for loop."
|
| I had been writing for loops for seven years at that age.
| I almost cried. It was like a lightswitch.
|
| The problem was always that nobody had ever actually
| explained what the symbol meant in any practical way.
| Every piece of terminology was explained with other
| terminology, when there was absolutely no reason to do
| so.
|
| Mathematics has the reputation for impermeability and
| unwelcomingness for a reason.
|
| It's because you guys are ignoring us saying "we want to
| learn, please write out a cheat sheet" and saying "yes,
| but don't you see" instead of just building the easy on-
| ramp that every other field on earth has built
|
| .
|
| > > You might be tired of wandering into someone else's
| area of expertise and telling them: > > You must change!
| You must make it more accessible!
|
| No, we generally just fix the problem. If people are
| saying "this isn't accessible enough," we just work on
| it.
|
| I would like for you personally to be aware of Bret
| Victor's work. He's incredibly potent and clear on these
| topics.
|
| Programmers work _really_ _really_ hard on learnability
| and understandability. This is a big deal to us. That 's
| why we can't understand why it's not a big deal to you.
|
| http://worrydream.com/LearnableProgramming/
|
| We have, in fact, mostly given up on waiting for you, and
| started to make our own tooling to understand your work,
| using obvious principles like live editors and
| witnessable effects.
|
| http://worrydream.com/MediaForThinkingTheUnthinkable/
|
| Edit: those are the talk notes. Wrong link, sorry. I
| should have used this instead: https://vimeo.com/67076984
|
| This is a big part of how we criticize ourselves, is for
| failing to provide the tooling to allow new modes of
| approach.
|
| https://www.youtube.com/watch?v=PUv66718DII
|
| We frequently think of our programming languages as new
| modes for thought. This line of discussion is
| particularly popular in the Lisp, Haskell, and Forth
| communities, though it crops up at some level everywhere.
|
| We frequently think that the more opaque the language,
| the less useful it is in this way.
|
| That's why programming languages, which are arguably 70
| years old as a field, have so much more powerful tools
| for teaching and explanation than math, which is
| literally older than spoken language
|
| You guys don't even have documentation extraction going
| yet. We have documentation where you have a little code
| box and you can type things and try it. You can screw
| with it. You can see what happens.
|
| This is why we care about things like Active Reading and
| explorable explanations.
|
| http://worrydream.com/ExplorableExplanations/
|
| This is why we care about things like live reactive
| documents. It _really_ changes your ability to
| intuitively understand things.
|
| http://worrydream.com/Tangle/
|
| Math hasn't grokked non-symbolic communication since
| Archimedes, that's why it took nearly two thousand years
| to catch up with him.
|
| We are asking you to come into step with the didactic
| tools of the modern world. It's not the 1850s anymore. We
| have better stuff than blackboards.
|
| Are these flat symbolic equations cutting it for you guys
| to communicate with one another? Sure.
|
| Are they cutting it for you guys to onboard new talent,
| or make your wealth available to the outside? No. (Do you
| realize that there is an outside to you, which isn't true
| of most technical fields anymore?)
|
| These problems are not unique to mathematics, of course.
| Formal logic is similar. Within my own field of
| programming, the AI field is similar, as is control
| theory, as tends to be database work. They don't want to
| open the doors. You have to spend six years earning it.
|
| But the hard truth is there are more difficult fields
| than mathematics that have managed to surmount these
| problems, such as physics (which no, is not applied
| mathematics,) and I think it might be time to stop
| protesting and start asking yourself "am I failing the
| next generation of mathematicians?"
|
| An example of who I believe to be genuinely good math
| communicators in the modern era are Three Blue One Brown.
|
| .
|
| > > Believe me, mathematicians are tired of non-
| mathematicians wandering up and saying: > > Look!
| Computer programs are easy and intuitive and everyone can
| understand them, even without training! Make math like
| that!
|
| Then fix the problem.
|
| It _IS_ fixable.
|
| .
|
| > Do you really believe that math notation is
| deliberately designed to make it hard for people
| untrained in math to learn how to use it?
|
| Given the way you guys push back on being asked to write
| simple reference material?
|
| No, but I understand why they do.
|
| .
|
| > Do you really believe that no one has tried to make it
| more accessible?
|
| No. Instead, I believe that nobody has succeeded.
|
| Try to calm down a bit, won't you? People tried to
| explain Berkeley sockets in a simple way for 12 years
| before Beej showed up and succeeded. The Little Schemer
| was 16 years after Lisp.
|
| Explaining is one of the very hardest things that exists.
|
| We're not saying you didn't try! The battlefield is
| littered with the corpses of attempts to get past
| Flatland.
|
| We're just saying "you haven't succeeded yet and this is
| important. Keep trying."
|
| .
|
| > Do you really believe you know more about why math
| notation is what it is than mathematicians and trained
| mathematics educators do?
|
| No. The literal ask is for you to repair that. Crimeny.
| Jensson wrote:
| > Surely you must realize that you're protesting this
| because it has this reputation, though?
|
| I've never heard anyone make this accusation until I read
| it here on HN today. The reputation doesn't seem to be
| widespread.
|
| > Programmers work really really hard on learnability and
| understandability. This is a big deal to us. That's why
| we can't understand why it's not a big deal to you.
|
| How to better teach math is like one of the most studied
| topics in education since it is so extremely important
| for so many outcomes. People learn programming faster
| since programming is simply easier, not because more
| effort has been done to make programming easy. There
| hasn't, way more effort has been put into making math
| easy and the math we have is the results of all that
| work.
|
| https://en.wikipedia.org/wiki/List_of_mathematics_educati
| on_...
|
| > Given the way you guys push back on being asked to
| write simple reference material?
|
| Nobody pushes back on writing simple reference manuals.
| There are tons of simple reference manuals for math
| everywhere on the internet, in most math papers, in most
| math books, everywhere! Yet still people fail to
| understand it. Many billions has been put trying to
| improve math education, trying to find shortcuts, trying
| to do anything at all. You are simply ignorant thinking
| that there are some quick fix super easy to implement
| things that would magically make people understand math.
| There isn't. It is possible that math education could get
| improved, but it wont be a simple thing.
| [deleted]
| wirrbel wrote:
| I graduated in physics so I am no stranger to math notation
| quirks and I think I also do understand their usefulness at
| times (conciseness in notation, etc). And it can be
| dangerous, too. As soon as the notation lures you into
| doing transformations that are invalid.
|
| Doesn't help that then notation is often poorly defined,
| and sometimes a weird mix of notations is presented.
|
| Overall the situation is also not pleasant for math people
| changing topics, or physicists reading papers from physical
| chemistry professors who 'grew up' in mathematical
| chemistry.
| milesvp wrote:
| You may not realize that in a given field, the same
| variable that represents the same basic thing may be
| negated depending on the part of the world the paper is
| published from. This can be fine, if it's your subfield,
| you happen to know to be careful with said variable. I
| don't personally dig into a lot of disperate maths in
| different papers very often, but this is the single biggest
| complaint my polyglot friend talks about. The second
| biggest is when he has to read and parse the math from a
| dozen unrelated papers in a field to find out what some
| random undefined variable means in the actual paper he
| cares about.
| JohnHaugeland wrote:
| No.
|
| Leave math notation whatever it is
|
| Just do a better job of documenting it
| f38zf5vdt wrote:
| I'm glad I'm not the only person like this. I've never liked
| tradition math notation and found it about as useful as
| traditional musical notation, that is, hard to read for the
| layman and for no other reason than "this is how people have
| been doing it for a long time". Maybe I'm the minority, but
| when I read a CS paper I mostly ignore the maths and then go
| to the source code or pseudocode to see how the algorithm was
| implemented.
| ColinWright wrote:
| > _...for no other reason than "this is how people have
| been doing it for a long time"._
|
| I disagree. Math notation has evolved to be as it is
| because it is useful for the purpose of doing math. If
| there were some way of doing it better, people would be
| evolving to be doing so.
|
| In some ways they are ... people are using computer algebra
| packages more for a lot of the grunt work, and are using
| proof assistants to verify some things, but there's a lot
| of math that's still done by sketching why something is
| true and letting the reader work through it. Math notation
| isn't about executing algorithms, it's about communicating
| what the result is, and why it works.
|
| "Doing Math" is not "Writing Programs", so math notation is
| different.
| KptMarchewa wrote:
| > If there were some way of doing it better, people would
| be evolving to be doing so.
|
| I don't see why wouldn't it be some kind of local
| maximum. Maybe there are better ways, but they are
| sufficiently far away from current notation, that they
| aren't even thought about.
| marcosdumay wrote:
| Hum... Apart from math, music notation maps linearly from
| the symbols to the instrument positions and time.
|
| It's absolutely not used only because "this is how people
| have been doing it for a long time". It's a very efficient
| notation to decode.
| randomNumber7 wrote:
| 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...
| KptMarchewa wrote:
| People sometimes do that when their code is reasonably
| abstract. For example, usual convention for generic type in
| Java is something like T.
|
| https://docs.oracle.com/javase/tutorial/java/generics/types
| ....
| drdec wrote:
| As someone trained in mathematics, I can tell you that
| using single character variables allows one to focus better
| on the concepts abstractly which is one of the goals of
| mathematics. That is to say, it is a practice well-suited
| to mathematics.
|
| It doesn't carry over to programming where explicit
| variables are better suited. In mathematics one is dealing
| with relatively few concepts compared to a typical program
| so assigning a single letter (applied consistently) to each
| is not a problem. This is not so in programming, except for
| a few cases like using i and j for loop variables (back
| when programs had explicit loops).
| colechristensen wrote:
| 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
| randomNumber7 wrote:
| 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....
| colechristensen wrote:
| It's not horrible, it's different, has different goals
| and different audiences. Context is king, and the bulk of
| professional programmers criticizing scientist code is
| just lack of context and a different set of priorities.
|
| From a more science based background i often think
| programmers write horrible code as i search in vain for
| where anything actually happens in a sea of abstractions.
| [deleted]
| Jtsummers wrote:
| As far as programmers, forget about the names. Does every C
| source file that uses pointer arithmetic include an
| explanation of how it works? Nope. They just use it and
| assume the reader understands it or is clever enough to ask
| for help or read up on the language.
|
| Mathematical writing is similar. At some point you have to
| assume an audience, which may be more or less
| mathematically literate. If you're writing for graduate
| students or experts in a domain, you don't include a
| tutorial and description of literally every term, you can
| assume they're familiar with the domain jargon (just like C
| programmers can assume that others who read their program
| understand pointers and other program elements). Whenever
| something is being used that is unique to the context, a
| definition is typically provided, at least if the writer is
| halfway decent.
|
| If the audience is assumed to be less mathematically
| literate (like a Calculus course textbook audience), then
| more terms will be defined (chapter 1 of most Calculus
| books include a definition of "function"). But a paper on
| some Calculus topic shouldn't have to define the integral,
| it should be able to use it because the audience will be
| expected to understand Calculus.
| Jensson wrote:
| > there is one true noation (or rather, a separate one for
| each language) that is unambiguous and clearly defined.
|
| This is such a disingenuous take. How many of the source code
| files you write are 100% self contained and well defined? I'd
| bet not a single one of them are. You reference libraries,
| you depend on specific compiler/runtime/OS versions, you
| reference other files etc. If you take a look at any of these
| scientific papers you call "badly defined", did you really go
| through all of the referenced papers and look if they defined
| the things you didn't get? If not then you can't be sure that
| the paper uses undefined notation. If you argue that it is
| too much work to go through that many references, well that
| is what you would have to do to understand one of your
| program files.
| throwaway31338 wrote:
| 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 about an ISA like RISC-V,
| though, specified at a level sufficient to go down to the
| gates. You might be talking about an ISA like 6502 where
| the gate-level implementations have been reverse-
| engineered.
|
| You can take programming all the way down boolean logic if
| you need to and the tools are readily available. They don't
| rely on you "just knowing" something.
| pjc50 wrote:
| > 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.
| throwaway31338 wrote:
| 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.
| Jensson wrote:
| > All behavior in an implementation can be teased-out if
| given sufficient time.
|
| Can it? Given what? You would need to understand how the
| CPU is supposed to execute the compiled code to do that.
| In order to understand the CPU you would need to read the
| manual for its instruction set, which is written in human
| language and hence not any better defined than math. At
| best you get the same level of strictness as math.
|
| If you assume you already have a perfect knowledge of the
| CPU workings, then I can just assume that you already
| have perfect knowledge of the relevant math topic and
| hence don't even need to read the paper to understand the
| paper. Human knowledge needs to come from somewhere. If
| you can read a programming language manual then you can
| read math. Every math paper is its own DSL in this
| context with its own small explanations for how it does
| things.
| webmaven wrote:
| _> Every math paper is its own DSL in this context with
| its own small explanations for how it does things._
|
| That's really the point though: not _every_ piece of
| software defines it 's own DSL, nor does it necessarily
| incorporate a DSL from some library or framework (which
| in turn may or may not borrow from other DSLs, etc.). It
| is also impossible to incorporate something from other
| software without actually referencing it explicitly.
|
| Math, though, is more like prose in this respect - while
| any given novel probably has a lot of structure,
| terminology, and notation in common with other works in
| its genre, unless it is extremely derivative it almost
| certainly has a few quirks and innovations specific to
| the author or even unique to that particular work that
| you can absorb while reading or puzzle out due to
| context, as long as you accept that the context is quite
| a lot of other works in the genre (this is more true of
| some genres/subfields than others). Unlike novels, at
| least in math papers (but not necessarily books) you get
| explicit references to the other works that the author
| considered most relevant, but those references are not
| usually sufficient on their own, nor necessarily
| complete, and you have to do more spelunking or happen to
| have done it already.
|
| Finally, like prose, with math you have to rely on other
| (subsequent) sources to point out deficiencies in the
| work, or figure them out on your own. Math papers, once
| published, don't usually get bug fixes and new releases,
| you're expected to be aware (from the context that has
| grown around the paper post-publication) what the
| problems are. Which means reading citations forward in
| time as well as backward for each referenced paper. The
| combinatorial explosion is ridiculous.
|
| It would be great if there were something like tour
| guides published that _just_ marked out the branching
| garden paths of concepts and notation borrowed and
| adapted between publications, but textbooks tend to focus
| on teaching one particular garden path.
| Jensson wrote:
| > It is also impossible to incorporate something from
| other software without actually referencing it
| explicitly.
|
| No, some programming languages just injects symbols based
| on context. You'd have to compile it with the right
| dependencies for it to work, so it is impossible to know
| what it is supposed to be.
|
| And even if they reference some other file, that file
| might not even be present in the codebase, instead some
| framework says "fetch this file from some remote
| repository at this URL on the internet" and then it
| fetches some file from the node repository, which can be
| another file tomorrow for all we know. This sort of time
| variance is non-existent in math, so to me math is way
| more readable than most code.
|
| And you have probably seen a programming tutorial or
| similar which uses library functions that no longer
| exists in modern versions, tells you to call a function
| but the function was found in a library the tutorial
| forgot to tell you about, or many of the other things
| that can go wrong.
| tedunangst wrote:
| All meaning of math notation can be teased out if given
| sufficient time.
| Jensson wrote:
| > 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.
|
| I doubt you actually can do that and understand it all. A
| computer can do it, but I doubt you the human can do that
| and get a perfect picture of any non trivial program
| without making errors. Human math is a human language
| first and foremost, its grammar is human language which
| is used to define things and symbols. This lets us write
| things that humans can actually read and understand the
| entirety of, unlike a million lines of code or cpu
| instructions.
|
| Show me a program written by 10 programmers over 10 years
| and I doubt anyone really understands all of it. But we
| have mathematical fields that hundreds of mathematicians
| have written over centuries, and people still are able to
| understand it all perfectly. It is true that a computer
| can easily read a computer program, but since we are
| arguing about teaching humans you would need to show
| evidence that humans can actually read and understand
| complex code well.
| readme wrote:
| came here to say this
|
| some people literally make the notation up as they go along
| User23 wrote:
| Edsger Dijkstra, who was a mathematician by training, wrote a
| wonderful little monograph on this subject called _The
| notational conventions I adopted, and why_ [1]. I am
| particularly fond of his commented equational proof format.
|
| [1]
| https://www.cs.utexas.edu/users/EWD/transcriptions/EWD13xx/E...
| klibertp wrote:
| Why are you telling OP what his problem is? Shouldn't you
| address _his_ pain points, not _your rationalization_ of them?
|
| I wrote it many times already and am bit tired of it, so just a
| quick summary:
|
| - programmers[1] also use cryptic notation and tend to think in
| concepts rather than syntax
|
| - nevertheless, programmers spend a lot of time commenting the
| code, documenting it, specifying it, and so on.
|
| - why can't mathematicians emulate it? What is so wrong about
| attaching additional few pages to every paper that nobody wants
| to do it? Pages with explanation of the syntax used, even the
| common bits. And you know what they could also do? Link to
| external resources with explanations! But no. This is not
| happening. Do their PDFs have a size limit or something? Is
| inserting a link into a paper considered some kind of
| blasphemy?
|
| I don't know the reason, but in all the discussions on this
| topic mathematicians almost always _underestimate the
| importance of knowing the syntax_. It 's much more important
| for comprehension than they tend to admit. And in the end they
| do exactly nothing to make the syntax more approachable for
| newcomers. And then newcomers are out-goers in a heartbeat.
| It's so obvious that I can't help thinking it's premeditated...
|
| EDIT: [1] Among many others, of course.
| TheTrotters wrote:
| Note that the OP is asking about college-level math, not
| cutting-edge papers.
|
| Textbooks routinely have a list of symbols and their
| definitions.
|
| But, from my experience, notation is rarely the problem. I'd
| bet that the root cause of OP's frustration is lack of
| understanding of concepts, not notation. (But, of course,
| it's hard to say more without specific examples).
| Jtsummers wrote:
| Mathematicians _do_ document and comment, that 's what papers
| and textbooks are: commentary on the math. They don't throw
| out formulae and equations and call it a day. Attaching a
| full tutorial for _every_ level of reader is tantamount to
| attaching Stroustrup 's C++ books to every C++ program, or
| K&R to every C program. You wouldn't do that, you'd expect
| the reader to ask you for references or to seek them out
| themselves.
| klibertp wrote:
| > or K&R to every C program.
|
| That's actually doable... ;) K&R is rather terse, what, 1/5
| of Stroustrup or something like that. But I digress.
|
| More on topic: there's also a class of programs that DO
| come with a book attached - or rather, multiple books, for
| every level; if not included outright in the distribution
| then at least linked to in the "learn" tab on a homepage.
| They're called programming languages. So, it can be done.
| That's all I want to say.
| Jtsummers wrote:
| > What is so wrong about attaching additional few pages
| to every paper that nobody wants to do it? Pages with
| explanation of the syntax used, even the common bits.
|
| Programs don't do this, why do you expect every math
| paper to do it?
|
| > Link to external resources with explanations!
|
| This is called a bibliography, every book that isn't so
| old that it _is_ the definition and paper includes one.
| In many textbooks there are also appendices which cover
| (some of) the foundational material. And most include
| sections (often in the front and back covers) that show
| the symbols and their names, if not their definitions.
| klibertp wrote:
| > Programs don't do this, why do you expect every math
| paper to do it?
|
| Well, I don't. It was you moving the goalpost. I talked
| about "a few pages", and you made "a book" out of it. I
| simply don't agree with you here and so I have very
| little to add at this point, sorry.
|
| > This is called a bibliography, every book that isn't so
| old that it is the definition and paper includes one.
|
| No. Bibliography is like a list of libraries you depend
| on. It has literally nothing to do with explaining the
| syntax close to where it's used.
|
| > appendices which cover (some of) the foundational
| material.
|
| Ha, ha, ha. No. If it's not front and center, then it
| doesn't count. I'm sorry, but I'm really tired of this
| subject. I would be willing to compromise more if that
| wasn't the case, believe me.
|
| > show the symbols and their names, if not their
| definitions.
|
| Ok. Putting that on the cover is a bit strange, but ok.
| That's a nice, but very small, step in the right
| direction. Please iterate and improve upon it!
|
| EDIT: again, because I missed it at first:
|
| > Programs don't do this, why do you expect every math
| paper to do it?
|
| Programs do come with man pages! And tutorials,
| interactive tours, contextual help, and more. Emacs comes
| with 3 books, and a tutorial. (GNU) libc has a book to
| it. Firefox has a whole portal (MDN) as its
| documentation. Visual Studio comes with MSDN and a huge
| amount of explanatory material. And when it comes down to
| code, you have auto-completion, go to definition, search
| for callers; you can hover over a symbol and you get a
| popup with documentation and types; you can also trace
| execution, stop the execution, rewind the execution (if
| you have good debugger), experiment with various
| expressions evaluated at different points.
|
| The most important difference between math and
| programming (or CS)is that programmers can (and do) build
| automated tools that help the next generation of newbies
| get into programming, while mathematicians can't. It's
| just that they don't want to admit this is a weakness,
| and only fortify more in their ivory towers.
|
| TLDR: I just can't see how you can even put math papers
| and programs on the same scale in terms of accessibility!
| Jensson wrote:
| > Programs do come with man pages! And tutorials,
| interactive tours, contextual help, and more. Emacs comes
| with 3 books, and a tutorial. (GNU) libc has a book to
| it. Firefox has a whole portal (MDN) as its
| documentation. Visual Studio comes with MSDN and a huge
| amount of explanatory material. And when it comes down to
| code, you have auto-completion, go to definition, search
| for callers; you can hover over a symbol and you get a
| popup with documentation and types. I just can't see how
| you can even put math papers and programs on the same
| scale in terms of accessibility!
|
| You are comparing big teams and products to a single guy
| writing a paper intended for a niche audience and to be
| read maybe a few hundred times if he is lucky. People
| makes mistakes and sometimes forget to document
| everything, they try to document everything though as can
| be seen in their papers where most things are documented
| well, but sometimes they miss things and unlike code you
| don't have compiler warnings telling you about it. And
| given how few people read those papers it isn't worth
| investing in a team to go through and update all of those
| papers to properly add definitions for everything they
| missed.
|
| The equivalent to those programs in math would be high
| school textbooks, and they are extremely well documented
| and easy to read in most cases.
| klibertp wrote:
| > it _isn 't worth investing_ in a team to go through and
| update all of those papers to properly add definitions
| for everything they missed.
|
| Thank you. There's nothing else left to discuss.
| Jensson wrote:
| Thanks for understanding, math is a small field without
| money for things like this, there is no way anyone should
| expect those niche papers to be as well documented as big
| programming projects used by millions.
|
| If you still think that is a problem then start some open
| source organization to fix that. Nobody has done that yet
| though since so few people care about math papers, but
| since you feel so strongly about this you could do it,
| someone has to be the one to start it.
| klibertp wrote:
| No, I mean, well, it's very understandable when you
| describe it that way. Actually, I think your post here
| changed my perception of the problem the most out of all
| discussions I had on the subject. It made me think about
| _people_ who are behind the papers. I somehow missed it.
| Thank you.
|
| (And, sorry for being a jerk in this thread. I said too
| much in a few places, exactly because I didn't think of
| innocent mathematicians who might read it. I'm still
| convinced that there is a lot that math can borrow from
| CS and SE, but I'm definitely going to argue this
| differently.)
| Jensson wrote:
| I wrote one math paper before I went into programming. It
| is a lot of work, like code reviewing but much much
| longer. It isn't fun. A big reason I got into programming
| is because that process is so much work. Of course I, the
| professor who reviewed it and the professors who looked
| at it afterwards understood it, but I can't guarantee
| that someone who hasn't read a lot about research level
| topology or combinatorics will easily understand much at
| all. However I doubt that anyone who didn't do those
| things will ever read it since it is an uninteresting
| niche topic. I'd be surprised if even 10 people read it
| fully.
| klibertp wrote:
| Yeah, I didn't think about it at all - I didn't realize
| that what I'm saying is basically demanding people to
| work for free (and on things that won't be useful to
| anyone in 99% of cases), and that's on top of already
| huge effort that is writing the paper in the first place.
| Honestly, I was behaving like people who open tickets in
| an open source project just to _demand_ that someone
| implements a particular feature, just for them, and right
| now. I dislike such behavior, and realizing that I 'm
| doing the same hit me hard :)
| jonnybgood wrote:
| I think the GP post is criticizing the lack of documenting
| syntax. Math papers tend to document semantics, whereas the
| understanding of the syntax by the reader is presumed.
| kazinator wrote:
| > _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._
|
| No, that belief isn't the problem; that actual _status quo_
| itself is obviously the problem. There are numerous notations
| and authors don 't explain what they are using, assuming
| everyone has recursively read all of their references depth-
| first before reading their paper.
| hansvm wrote:
| Why does RTFM not apply to mathematics dependencies?
| layer8 wrote:
| Because there usually isn't anything resembling an actual
| manual.
| Jensson wrote:
| Sure there is, read these and you will understand most
| math papers people here struggle with:
| https://mathblog.com/mathematics-books/
|
| Of course you still wont be able to understand most math
| papers written by pure mathematicians, but it should be
| fine for whatever you need in CS. I know all the topics
| on that page, it is just a very fleshed out math major.
| rullopat wrote:
| 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.
| Jensson wrote:
| That is a physics based paper, it used physics notation. Can
| find common ones here (including that triangle):
|
| https://en.wikipedia.org/wiki/List_of_common_physics_notatio.
| ..
| NamTaf wrote:
| What you're looking at is calculus, specifically
| differentiation. This is pretty core to understanding
| physics, because so much of physics depends on the time-
| evolving state of things. That's fundamentally what's
| happening here.
|
| The triangle, for example, is the upper-case greek letter
| delta, which in calculus represents 'change of'. You might
| have heard of 'delta-T' with respect to 'change of time'.
|
| In calculus, upper-case delta means 'change over a finite
| time' vs lower-case delta meaning 'instantaneous change'. The
| practical upshot, for example, is that the lower-case is the
| instantaneous rate-of-change at an instant in time, whereas
| the upper-case is the change over a whole time (e.g. the
| average rate of change per second for time = 0 seconds to
| time = 3 seconds).
|
| If you are trying to grok this, I would suggest an
| introductory calculus or pre-calculus resource. It doesn't
| have to be a uni textbook - higher-level high school maths
| usually teaches this. In this particular case, the Khan
| Academy would be my recommendation because it is about the
| right level (we're not talking esoteric higher-level
| university knowledge here) and it is eminently accessable.
| For example, this link may be a good starter in this
| instance:
|
| https://www.youtube.com/watch?v=MeU-KzdCBps
| forgetfulness wrote:
| Looks like you need to grind through an elementary calculus
| book. With the exercises, you may think you build intuition
| by reading just the definitions, but half of the
| understanding is tacit and you get through the exercises.
|
| If you're trying to get into signal processing, it'll involve
| calculus in complex numbers, and knowledge of that is often
| gained through plodding through proofs and exercises over and
| over.
| kens wrote:
| Everyone is talking about the D symbol, but the real problem
| that you'll encounter will be later in the paper where they
| start talking about H(o), 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
| https://en.wikipedia.org/wiki/Impulse_response
| forgetfulness wrote:
| Wikipedia is truly atrocious for learning math, the
| articles are like man pages in that they precisely describe
| the concepts in terms that will only make sense if you
| already know the thing. They just aren't written for
| pedagogy.
|
| Like in 300 BC today, there's no royal road to geometry.
| kergonath wrote:
| Just like with programming, knowing the meaning of keywords
| is not enough to understand something.
| nickcw wrote:
| The triangle is a capital Greek Delta and is usually used to
| indicate a change in something. So DT means some change in T.
| ColinWright wrote:
| 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've used is where the triangle appears in
| the formula.
|
| But that can't be it, because just two lines above it we
| have:
|
| "For a pulse of width Dt, the critical time ..."
|
| So that's stating that "Dt" is the width of the pulse, and
| should be thought of as a single term.
|
| So maybe that's the wrong formula, or maybe it was just a bad
| example. So trying to be more helpful, the "triangle" is a
| Greek capital delta and means different things in different
| places. However, it is often used to mean "a small change
| in".
|
| https://en.wikipedia.org/wiki/%CE%94T
|
| FWIW ... at a glance I can't see where that result is
| derived, it appears simply to be stated without explanation.
| I might be wrong, I've not read the rest of the paper.
| tomerv wrote:
| 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 here is that "For a pulse of width Dt"
| is the definition of this variable, but this can be easily
| missed if you're not used to this naming convention.
| enriquto wrote:
| > it's just a single variable with a named made of 2
| characters.
|
| I have this same problem with programming, when I have to
| deal with code written by non-mathematicians. They tend
| to use all these stupid variables with more than one
| letter and that confuses the heck out of me.
| kergonath wrote:
| That's because "D" means "a change of" or "an interval
| of". So, Dt 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 2pth instead of sin(2 p th) ).
| rullopat wrote:
| Sorry I didn't mean to make you work for me, but it's a PDF
| and I didn't know how to explain better the position (maybe
| I should have told you the first formula on page X).
|
| For you it was a D, for me it was a triangle and I didn't
| get the meaning of that Dt. Maybe it's just a too advanced
| paper for my knowledge.
| gspr wrote:
| I'll use this as an example for the point I'm trying to
| make in my comment
| https://news.ycombinator.com/item?id=29341727
|
| Please don't take this the wrong way. It is not meant to
| be demeaning, and it is not meant to be gatekeeping
| (quite the contrary!). But: If you do not know what a
| derivative is, then learning that that symbol means
| derivative (assuming that it does, I have not actually
| looked at what you link to) will help you next to
| nothing. OK, you'll have something to google, but if you
| don't already have some idea what that is, there is no
| way you will get through the paper that way.
|
| I hope you take this as motivation to take the time to
| properly learn the fundamentals of mathematics (such as
| for example calculus for the topic of derivatives).
| [deleted]
| ColinWright wrote:
| I'm just saying "D" because I can't immediately type the
| symbol here and it was easier just to use that. Not
| least, I didn't know if that was the formula you meant.
|
| But as I say, immediately above the formula it says:
|
| "For a pulse of width [?]t, the critical time ..."
|
| So that really is saying exactly what that cluster of
| symbols means. There will be things like this
| _everywhere_ as you read stuff. Things are rarely
| completely undefined, but you are expected to be reading
| along.
|
| And you need to work. I just typed this into DDG:
|
| "What does [?]t mean?"
|
| The very first hit is this:
|
| https://en.wikipedia.org/wiki/Delta_%28letter%29
|
| That gives you a _lot_ of context for what the symbol
| means, and this is the sort of thing you 'll need to do.
| You need to stop, look at the thing you don't understand,
| read around in the nearby text, then type a question (or
| two, or three) into a search engine.
| wrycoder wrote:
| The triangle, or "delta", is used to indicate a tiny
| change in the following variable.
|
| Let's say you go on a journey, and the distance you've
| travelled so far is "x" and the time so far is "t".
|
| Then your average velocity since the beginning is x / t .
|
| But, if you want to know your _current_ velocity, that
| would be delta x divided by delta t .
|
| The delta is usually used in a "limiting" sense - you can
| get a more accurate measurement of your velocity by
| measuring the change in x during a tiny time interval.
| The tinier the interval, the more accurate the estimate
| of current velocity.
|
| What I'm talking about here is the first steps in
| learning differential calculus. You could look for that
| at kahnacademy.com. You might also benefit by looking at
| their "precalculus" courses.
|
| Just keep plugging away at it, the concepts take awhile
| to seep in. Attaining mathematical maturity takes years.
| simiones wrote:
| I would say that delta is typically used to mean
| difference of any magnitude (i.e. Dt could mean 2 hours).
| wrycoder wrote:
| True. To continue my example, the average velocity in the
| last hour would be taken with a [?]t of 60 minutes.
| pbhjpbhj wrote:
| Yes, small changes usually use lowercase delta, e.g. dt.
| Not to be confused with the derivative symbol dt, nor
| with the partial derivative symbol [?]t !
|
| Before I continued my maths learning after highschool (ie
| before UK A-levels) I learnt the Greek alphabet to make
| it easier to understand maths notations as I could
| 'voice' (internally) all the funny glyphs adopted from
| Greek.
|
| At uni I learnt how to properly write an ampersand (for
| logic classes) and how to write Aleph and Beth (for pure
| maths, particularly transcendental numbers).
|
| Some professors have a fondness for the more confusing
| Greek letters (lowercase xi, lowercase eta) ... is it n
| or eta, epsilon or xi, ...
| wrycoder wrote:
| https://en.wikipedia.org/wiki/Riemann_sum usually uses
| [?]. See also the explanation of [?] at
| https://en.wikipedia.org/wiki/Derivative#Explanations .
|
| That's the sense in which I'm using [?], which is common,
| at least in the US.
| Jensson wrote:
| But this is a physics paper, that isn't how you use
| uppercase delta in physics. It is just a range. In
| physics however you do a ton of approximations all the
| time in ways mathematicians hate (you don't care about
| errors smaller than you can measure), so uppercase delta
| is often approximated with derivatives etc, but it isn't
| a derivative. Math in physics is way more practical and
| uses very different techniques than math in math, often
| because physicists invented the math first and
| mathematicians later went and formalized it.
| ColinWright wrote:
| BTW ... you say:
|
| > _Maybe it 's just a too advanced paper for my
| knowledge._
|
| Maybe it is _for now_ ... the point being that if you
| start at the beginning, chip away at it, search for terms
| on the 'net, read multiple times, try to work through
| it, and then ask people when you're really stuck, that's
| one way of making progress.
|
| You can, instead, enroll in an on-line course, or night-
| school, and learn all this stuff from the ground up, but
| it will almost certainly take longer. Your knowledge
| would be better grounded and more secure, but learning
| how to read, investigate, search, work, then ask, is a
| _far_ greater skill that "taking a course".
|
| Others have answered your specific question about the
| delta symbol, but there are deeper
| processes/problems/questions here:
|
| * Not all concepts or values or represented by a single
| glyph, sometimes there are multi-glyph "symbols", such as
| "Dt" in your example.
|
| * When you see a symbol you don't recognise, read the
| surrounding text. The symbol will almost always be
| referenced or described.
|
| * The notation isn't universal. Often it's an aid to your
| memory, to write in a succinct form the thing that has
| been described elsewhere.
|
| * In these senses, it's very much a language more akin to
| natural languages than computer languages. The formulas
| are things used to express a meaning, not things to be
| executed.
|
| * Specific questions about specific notation can be
| answered more directly, but to really get along with
| mathematical notation you need to "read like math" and
| not "read like a novel".
|
| * None of this is correct, all of it is intended to give
| you a sense of how to make progress.
| NamTaf wrote:
| I feel you're coming at this without appreciating your body
| of prior knowledge. Intended or not, your statment "But
| that can't be it, because just two lines above it we
| have..." assumes a whole lot of knowledge.
|
| You and I both know that it reads as one term, but for
| someone unfamiliar with calculus but exposed to algebra
| they are drilled to understand separate graphemes as
| _separate_ items, because the algebraic 'multiply' is so
| often implied, e.g. 3x = 3 * x as two individual 'things'.
|
| I think there's merit in explaining the concept of delta
| representing change, because it's not obvious. For example,
| when I was taught the concept in school, my teacher
| explicitly started with doing a finite change with numbers,
| then representing it in terms of 'x' and 'y', then merged
| them into the delta symbol. That's a substantial intuitive
| stepping stone and I think it's pretty reasonable that
| someone may not find this immediately apparent.
| Someone wrote:
| > with calculus but exposed to algebra they are drilled
| to understand separate graphemes as separate items
|
| But most will already be familiar with the family of
| goniometric functions such as _sin_ and _cos_ , there's
| _log_ and possibly _exp_ and _sqrt_. There's _min_ and
| _max_ ; advanced math has _inf_ and _sup_.
| ColinWright wrote:
| I agree completely that I'm coming at this with a lot of
| background knowledge, but if I'm reading in an unfamiliar
| field and I see a symbol I don't recognise, I look in the
| surrounding text to see if the symbol appears nearby. As
| I say, "Dt" appears immediately above ... that's a clue.
| As you say, it's drilled in at school that everything is
| represented by a single glyph, and if these are
| juxtaposed then it means multiplication, and that is
| another thing to unlearn.
|
| But I think the problem isn't the specifics of the "D",
| it's the meta-problem of believing that symbols have a
| "one true meaning" instead of being defined by the scope.
|
| I agree that explaining the delta notation would be
| helpful, but that's like giving someone a fish, or making
| them a fire. They are fed for one day, or warm for one
| night, it's the underlying misconceptions that need
| addressing so they can learn to fish and be fed, or set
| on fire and be warm, for the remainder of their life.
| NamTaf wrote:
| I absolutely agree with your comments regarding teaching
| the underlying approach to digesting a paper. You
| definitely raise good points, especially the 'one true
| meaning' comment. I should state that I'm not discounting
| the value of your point, especially given this
| clarification, however I guess that when I reflect on my
| experience in my time learning this, the time I best
| learnt was via initial expalnation, then worked example,
| then customary warning of corner-cases and here-be-
| dragons.
|
| e: I also think, on reflection, that a signfigicant part
| of your ability to grok a new paper per your comments is
| your comfort in approaching these concepts due to your
| familiarity. Think of learning a new language - once you
| have a feel for it, you're likely more comfortable
| exploring new concepts within it, however when you're
| faced with it from the start you probably feel very lost
| and apprehensive.
|
| I feel that understanding calculus is a fairly
| fundamental step in the 'language of maths', teaching
| that symbols don't necessarily represent numbers but can
| represent concepts (e.g. delta being change). This isn't
| something you encounter until then, but once you do you
| begin to understand the characters associated iwth
| integrals, matricies, etc. in a way that you may not have
| previously with algebra alone.
| ColinWright wrote:
| Agreed ... I think we're pulling in the same direction
| ...
| erichocean wrote:
| > _a symbol_
|
| That's literally the whole problem: he sees two symbols.
| BeetleB wrote:
| A few points:
|
| 1. You're reading a journal article. They will assume you
| know the notation not just of the broader discipline (e.g.
| physics/electrical engineering), but of the _subdiscipline_
| and at times the _subsubdiscipline_. Journal papers are
| explicitly written not to be easy to comprehend by
| beginners.[1] Notation will be only one problem you 'll face.
|
| 2. As has been pointed out, this is not a mathematics paper.
| Mathematicians have their own notation, as do physicists and
| engineers. As I mentioned in the above bullet, they can have
| their own notation even in subdisciplines (e.g. circuit folks
| use "j" for the imaginary number, and semiconductor folks use
| "i"). There is a lot of overlap in notation amongst these
| parties, but you should never assume because you know one
| notation that you'll easily understand the math written by
| other fields.
|
| 3. Most introductory textbooks will explain the basic
| notation. Unfortunately, I often do find gaps where you go to
| higher level textbooks and they use notation that they don't
| explain (i.e. they assume you've seen it before), but is not
| covered in the prior textbooks.
|
| 4. Finally, sorry to say this, but "delta" (the triangle) for
| representing change is used in almost all sciences and
| engineering. It was heavily used in my high school as well.
| If you're struggling with this you really need to read some
| introductory textbooks in, say, physics.
|
| [1] I'm not kidding. I've spent time in academia and I've
| complained how obtuse some articles are, and almost
| universally the response is "We write for other experts, not
| for new graduate students". One professor took pride at the
| fact that in his field, one can comprehend only about one
| page of a paper per day - and this coming from someone who is
| an _expert_. These people have issues.
| pfortuny wrote:
| You probably want to try and read Courant-Johns Calculus (I
| forget the exact title).
| motohagiography wrote:
| I sometimes think math notation is a conspiracy against the
| clever but lazy. Being able to pronounce the greek alphabet is a
| start, as you can use your ear and literary mind once you have
| that, but when you encounter <...>, as in an unpronouncable
| symbol, the meaningless abstraction becomes a black box and
| destroys information for you.
|
| Smart people often don't know the difference between an elegant
| abstraction that conveys a concept and a black box shorthand for
| signalling pre-shared knowledge to others. It's the difference
| between compressing ideas into essential relationships, and using
| an exclusive code word.
|
| This fellow does a brilliant job at explaining the origin of a
| constant by taking you along the path of discovery with him,
| whereas many "teachers" would start with a definition like
| "Feigenbaum means 4.669," which is the least meaningful aspect to
| someone who doesn't know why.
| https://www.veritasium.com/videos/2020/1/29/this-equation-wi...
|
| It wasn't until decades after school that it clicked for me that
| a lot of concepts in math aren't numbers at all, but refer to
| relationships and relative proporitons and the interactions of
| different types of things, which are in effect just _shapes_ ,
| but ones we can't draw simply, and so we can only specify them
| using notations with numbers. I think most brains have some low
| level of natural synesthesia, and the way we approach math in
| high school has been by imposing a three legged race on anyone
| who tries it instead.
|
| Pi is a great example, as it's a proportion in a relationship
| between a regular line you can imagine, and the circle made from
| it. There isn't much else important about it othat than it
| applies to everything, and it's the first irrational number we
| found. You can speculate that a line is just a stick some
| ancients found on the ground and so its unit is "1 stick" long,
| which makes it an integer, but when you rotate the stick around
| one end, the circular path it traces has a constant proportion to
| its length, because it's the stick and there is nothing else
| acting on it, but amazingly that proportion that describes that
| relationship pops out of the single integer dimension and yields
| a whole new type of unique number that is no longer an integer.
| The least interesting or meaningful thing about pi is that it is
| 3.141 etc. High school math teaching conflates computation and
| reasoning, and invents gumption traps by going depth first into
| ideas that make much more sense in their breadth-first contexts
| and relationships to other things, which also seems like a
| conspiracy to keep people ignorant.
|
| Just yesterday I floated the idea of a book club salon idea for
| "Content, Methods, and Meaning," where starting from any level,
| each session 2-3 participants pick and learn the same chapter
| separately and do their best to give a 15 minute explanation of
| it to the rest of the group. It's on the first year syllabus of a
| few universities, and it's a breadth-first approach to a lot of
| the important foundational ideas.
|
| The intent is I think we only know anything as well as we can
| teach it, so the challenge is to learn by teaching, and you have
| to teach it to someone smart but without the background. Long
| comment, but keep at it, dumber people than you have got further
| with mere persistance.
| swframe2 wrote:
| Naively, I would say the following:
|
| 1) Search youtube for multiple videos by different people on the
| topic you want to learn. Watch them without expecting to
| understand them at first. There is a delayed effect. Each content
| creator will explain it slightly differently and you will find
| that it will make sense once you've heard it explained several
| different times and ways.
|
| I will read the chapter summary for a 1k page math book
| repeatedly until I understand the big picture. Then I will
| repeated skim the chapters I least understand until I understand
| its big picture. I need to know the terms and concepts before I
| try to understand the formulas. I will do this until I get too
| confused to read more then I will take a break for a few
| hours/days and start again.
|
| 2) You have to rewrite the formulas in your own language. At
| first you will use a lot of long descriptions but quickly you
| will get tired and you will start to abbreviate. Eventually, you
| get the point where you will prefer the terse math notation
| because it is just too tedious to write it out in longer words.
|
| 3) You might have to pause the current topic you are struggling
| with and learn the math that underlies it. This means a topic
| that should take 1 month to learn might actually take 1 year
| because you need to understand all that it is based on.
|
| 4) Try to find an applied implementation. For example
| photogrammetry applies a lot of linear algebra. It is easer to
| learn linear algebra if you find an implementation of
| photogrammetry and try to rewrite it. This forces you to
| completely understand how the math works. You should read the
| parts of the math books that you need.
| gspr wrote:
| I hear this question asked quite often, particularly on HN. I
| think the question is quite backwards. There is little value
| alone in learning "math notation", even ignoring what many people
| point out (there is no one "math notation"). "Math notation", at
| best, translates into mathematical concepts. Words, if you will,
| but words with very specific meaning. Understanding those
| concepts is the crux of the matter! That is what takes effort -
| and the effort needed is that of learning mathematics. After
| that, one may still struggle with bad (or "original", or
| "different", or "overloaded", or "idiotic", or...) notation, of
| course, but there is little use in learning said notation(s) on
| their own.
|
| I've been repeatedly called a gatekeeper for this stance here on
| HN, but really: notation is a red herring. To understand math
| written in "math notation", you first have to understand the math
| at hand. After that, notation is less of an issue (even though it
| may still be present). Of course the same applies to other
| fields, but I suspect that the question crops up more often
| regarding mathematics because it has a level of precision not
| seen in any other field. Therefore a lot more precision tends to
| hide behind each symbol than the casual observer may be aware of.
| thuccess129 wrote:
| Look for an etymology dictionary on math notation? The
| biographical sketch of the person who introduced the equal sign
| is an interesting read.
| excitednumber wrote:
| Read calculus made easy. It won't solve all of your problems but
| damn is this book good.
|
| https://www.gutenberg.org/ebooks/33283
| amitkgupta84 wrote:
| First, just to state the obvious, if you can accurately describe
| a notation in words, you can do an Internet search for it.
|
| When that fails, math.stackexchange.com is a very active and
| helpful resource. You can ask what certain notation means, and
| upload a screenshot since it's not always easy to describe math
| notation in words.
|
| If you don't want to wait for a human response, Detexify
| (https://detexify.kirelabs.org/classify.html) is an awesome site
| where you can hand draw math notation and it'll tell you the
| LaTeX code for it. That often gives a better clue for what to
| search for.
|
| For example you could draw an upside down triangle, and see that
| one of the ways to express this in LaTeX is \nabla. Then you can
| look up the Wikipedia article on the Nabla symbol. (Of course in
| this case you could easily have just searched "math upside down
| triangle symbol" and the first result is a Math Stackechange
| thread answering this).
| pgcj_poster wrote:
| If you haven't already, I would start by learning the Greek
| alphabet and the sounds that the letters make. Conventions like S
| for sum and D for difference seem much less strange when you
| realize that they're basically just S and D.
| thorin wrote:
| You might be better picking an area, and trying to work out the
| notation relating to that area e.g. vectors / matrices / calculus
| etc. As Colin says below there are often multiple equivalent ways
| of representing things across different fields and timeframes. I
| seem to remember maths I studies in Elec Eng looking different
| but equivalent to the way it was represented in other disciplines
| whatsakandr wrote:
| I have a masters in engineering, but there was a lot of pure math
| things that I never understood until recently. I found the same
| approach to learning software concepts and APIs. Just start at
| the one you don't know and recursively explore the concepts until
| you find stuff you do know.
| lukeplato wrote:
| I found that there is a physicality/motion to the progression of
| notation that you learn by solving a lot of problems, especially
| solving them quickly during tests
| solmag wrote:
| Practice, just like you learned programming. "The Context" gives
| you the meaning for the notation, sadly. You have to kind of know
| it to understand the notation properly.
| solmag wrote:
| You can also get sufficiently angry and just write out linear
| algebra books and what not in Agda / Coq / Lean if it pisses
| you off so much (I've done a bunch of exercises in Coq)
| db48x wrote:
| I like the approach they took in Structure and Interpretation
| of Classical Mechanics, where the whole book is done in
| Scheme: (define ((Lagrange-equations
| Lagrangian) q) (- (D (compose ((partial 2)
| Lagrangian) (Gamma q))) (compose ((partial 1)
| Lagrangian) (Gamma q))))
| kergonath wrote:
| I have no idea what the hell that means, and I am quite
| familiar with Lagrangian mechanics.
| db48x wrote:
| Compare it to D([?]2L[?]G[q]) - [?]1L[?]G[q] = 0.
|
| Of course, even that isn't quite the standard notation;
| it's using a less ambiguous notation which they invented
| for the book. From the preface (https://mitpress.mit.edu/
| sites/default/files/titles/content/...):
|
| ---
|
| Classical mechanics is deceptively simple. It is
| surprisingly easy to get the right answer with fallacious
| reasoning or without real understanding. Traditional
| mathematical notation contributes to this problem.
| Symbols have ambiguous meanings that depend on context,
| and often even change within a given context.1 For
| example, a fundamental result of mechanics is the
| Lagrange equations. In traditional notation the Lagrange
| equations are written
|
| d/dt [?]L/[?]q - [?]L/[?]q = 0.
|
| The Lagrangian L must be interpreted as a function of the
| position and velocity components q and q, so that the
| partial derivatives make sense, but then in order for the
| time derivative d/dt to make sense solution paths must
| have been inserted into the partial derivatives of the
| Lagrangian to make functions of time. The traditional use
| of ambiguous notation is convenient in simple situations,
| but in more complicated situations it can be a serious
| handicap to clear reasoning. In order that the reasoning
| be clear and unambiguous, we have adopted a more precise
| mathematical notation. Our notation is functional and
| follows that of modern mathematical presentations.2 An
| introduction to our functional notation is in an
| appendix.
|
| Computation also enters into the presentation of the
| mathematical ideas underlying mechanics. We require that
| our mathematical notations be explicit and precise enough
| that they can be interpreted automatically, as by a
| computer. As a consequence of this requirement the
| formulas and equations that appear in the text stand on
| their own. They have clear meaning, independent of the
| informal context. For example, we write Lagrange's
| equations in functional notation as follows:3
|
| D([?]2L [?] G[q]) - [?]1L [?] G[q] = 0.
|
| The Lagrangian L is a real-valued function of time t,
| coordinates x, and velocities v; the value is L(t, x, v).
| Partial derivatives are indicated as derivatives of
| functions with respect to particular argument positions;
| [?]2L indicates the function obtained by taking the
| partial derivative of the Lagrangian function L with
| respect to the velocity argument position. The
| traditional partial derivative notation, which employs a
| derivative with respect to a "variable," depends on
| context and can lead to ambiguity.4 The partial
| derivatives of the Lagrangian are then explicitly
| evaluated along a path function q. The time derivative is
| taken and the Lagrange equations formed. Each step is
| explicit; there are no implicit substitutions.
|
| --- (define ((Lagrange-equations
| Lagrangian) q) (- (D (compose ((partial 2)
| Lagrangian) (Gamma q))) (compose ((partial
| 1) Lagrangian) (Gamma q))))
|
| I think you can see that the Scheme code is a direct and
| very simple translation of the equation.
|
| And it has the advantage that you can run it immediately
| after typing it in, assuming you have a coordinate path
| to pass to it. They immediately go to a concrete example:
| (define ((L-free-particle mass) local) (let ((v
| (velocity local))) (* 1/2 mass (dot-product v
| v)))) (define (test-path t) (up (+
| (* 'a t) 'a0) (+ (* 'b t) 'b0)
| (+ (* 'c t) 'c0))) (((Lagrange-equations
| (L-free-particle 'm)) test-path) 't)
| = (down 0 0 0)
|
| As the book says, "That the residuals are zero indicates
| that the test path satisfies the Lagrange equations."
|
| They then give another example, symbolic this time:
| (show-expression (((Lagrange-equations (L-free-
| particle 'm)) (literal-function 'x))
| 't)) = (* (((expt D 2) x) t) m)
|
| Quoted from https://mitpress.mit.edu/sites/default/files/
| titles/content/...
| Jtsummers wrote:
| https://mitpress.mit.edu/sites/default/files/titles/conte
| nt/...
|
| It's formula 1.12 at the start of section 1.5 on this
| page converted into a Scheme representation, in section
| 1.5.2.
| kergonath wrote:
| Thanks! I am not sure I like the Scheme-like notation,
| but the effort is interesting.
| Jtsummers wrote:
| It's actually executable, which is part of why they wrote
| this particular book. The intent was to have a more
| uniform syntax for presenting the math and being able to
| (programmatically) use it.
| solmag wrote:
| I should really pick that one up some day. It had an
| inspiring story, I believe the author wanted to understand
| the classical mechanics and just wrote them out in Scheme.
| db48x wrote:
| Pretty much, yea. And because they are literally a 100x
| programmer, they also extended Scheme to support stuff
| you usually use a computer algebra system for at the same
| time. After all, if your CAS can take the derivative of a
| function, why can't your programming language?
| dqpb wrote:
| Ultimately I think this is the right answer.
| ReleaseCandidat wrote:
| Well, the real fun is deciphering a lower case xi - x - when
| written on the blackboard (or whiteboard), specially compared to
| a lower case zeta - z (fortunately way less commonly used).
|
| As all the others already told you. you don't learn by reading
| alone.
| pbhjpbhj wrote:
| xi vs epsilon vs zeta when chalked on a blackboard at pace and
| read from 30m away!
|
| Learning the Greek alphabet pays off.
| contravariant wrote:
| Ah, yes. I remember the time when I saw someone write something
| vaguely like the following
|
| [0,x[={x|0<=x<x}
|
| Which was fun trying to figure out when written in handwriting
| where x,{,} all look the same.
|
| If you can't figure out what it's supposed to be, this equation
| starts with a half-open interval denoted: [x,0[. This notation
| has some advantages but can be make things hard to read.
| caffeine wrote:
| Khan academy would be a great place - generally in high school
| you learn enough to get through basic notation.
| pmontra wrote:
| School, one year after another. Delta t stuff were probably in
| the last year before college.
| housu wrote:
| I got a bachelor's degree in math
| 734129837261 wrote:
| If math was a programming language, all mathematicians would be
| fired for terrible naming conventions and horrible misuse of
| syntax freedom.
|
| Honestly, most math formulas can be turned into something that
| looks like C/C++/C#/Java/JavaScript/TypeScript code and become
| infinitely more readable and understandable.
|
| Sadly, TypeScript is one of the languages that is attempting to
| move back to idiocy by having generics named a single letter.
| Bastards.
| kaetemi wrote:
| Math notation feels like a write-only language somehow.
|
| I can read and understand undocumented code with relative ease.
| Reading math notation without any documentation seems pretty much
| impossible, otoh.
| CogitoCogito wrote:
| You get better at it the more you do. A tip is also to actually
| change a mathematical exposition into a form you better
| understand (e.g. by writing it in a different notation and/or
| expanding it out in words to make the existing notation less
| dense). Basically convert the presentation into the way you
| would personally like to see it.
|
| If you do this enough, the process becomes easier and the
| original notation becomes easier to understand. But it takes a
| lot of time and patience (as I'm sure it took for you
| understand undocumented code did as well).
| xemdetia wrote:
| One of the best things I figured out that at least in the last 70
| years or so ago it's pretty easy to find the "first" or
| foundational paper for a particular construct where they have to
| explain their notation for the first reader or they have the vibe
| of working with the new idea in the raw rather than 40 years
| later where it is matured. One example I use for this is hamming
| codes where some of the recent examples or explanations don't
| build it from first principles, but the original articles do
| explain it very clearly.
| the__alchemist wrote:
| It can be quite provincial. Could you please post a link to a
| paper or website that has notation you'd like to understand?
| Which domains are you interested in particularly?
| [deleted]
| tgflynn wrote:
| It sounds like you're trying to read papers that assume a certain
| level of mathematical sophistication without having reached that
| level. Typical engineering papers will assume at least what's
| taught in 2 years of college level mathematics, mainly calculus
| and linear algebra, and no they aren't going to be explaining
| notation used at that level.
|
| But it isn't just about the notation. You also need to understand
| the concepts the notation represents, and there aren't really any
| shortcuts to that.
|
| These days there are online courses (many freely available) in
| just about every area of mathematics from pre-high school to
| intro graduate level.
|
| It's possible for a sufficiently motivated person to learn all of
| that mathematics on their own from online resources and books,
| but it isn't going to be an easy task or one that you can
| complete in a few weeks/months.
| arcbyte wrote:
| The author explained his problem and asked for resource
| recommendations.
|
| Your response is to scold him for having the problem he already
| said he had and instead of recommending resources you told him
| to go look on the internet.
|
| And you implied he doesn't have motivation.
| rackjack wrote:
| All math notation was created by mathematicians who wanted to
| quickly represent something, either to:
|
| - better see the structure of the problem; or
|
| - reduce the amount of ink they need to write the problem
|
| Very similar to how programmers use functions, in fact.
|
| To this end, mathematicians in different fields have different
| notation, and often this notation overlaps with different
| meaning. Think how Chinese and Japanese have overlapping
| characters with different meanings.
|
| As others have stated, there is no "one true notation" -- all
| notation is basically a DSL for that math field.
|
| Instead, choose a topic you are interested in, find an
| introductory text, and start reading. They will almost certainly
| explain the notation. Unfortunately, even within a field,
| notation can vary, but once you have a grasp of one you will
| probably grasp the rest quick enough.
|
| I will mention, though, that some notation is "mostly" universal.
| Integrals, partial derivatives, and more that I can't recall
| right now all use basically the same notation everywhere, since
| they underlie a lot of other math fields.
| amelius wrote:
| I have a notation problem. I want to write "approximately 24
| volt" on my printed circuit board, but I have little space. I
| could write "[?]24V", but the wavy symbol makes it look like it
| is AC instead of DC. How to solve this without adding more
| characters or changing my circuit?
| [deleted]
| pbhjpbhj wrote:
| Use =c.24V (read as 'equals circa 24 volts', _circa_ is Latin
| for 'about').
|
| Use the 3 line version of approximately equal (looks like tilde
| above an equal sign, [?]).
| srcreigh wrote:
| First math course at university of Waterloo:
| https://cs.uwaterloo.ca/~cbruni/Math135Resources/courseNotes...
|
| Learning everything about math is nearly impossible like knowing
| everything about all code that exists.
|
| That course should teach some basics for proof strategies. Ex
| here on page 2, there are definitions with examples:
| https://cs.uwaterloo.ca/~cbruni/pdfs/Math135SeptDec2015/Lect...
|
| Specialized math tends to have specialized notation. For ex
| Linear Algebra, Calculus, Combinatorics. Any decent textbook will
| have an appendix or table with what the notation means.
| Zolomon wrote:
| I asked this on Mathematics StackExchange some time ago and got
| good responses:
|
| https://math.stackexchange.com/a/13281
| sealeck wrote:
| Most textbooks come with a list of definitions.
|
| Try to read it aloud.
|
| "The Probability Lifesaver" has a lot of good mathematics tips
| (which are not even mathematics related) most of which are not
| probability-specific. It's a goldmine.
| yongjik wrote:
| Try reading a good undergraduate calculus textbook. It would be
| hefty and a bit wordy, and it may take a few months to go
| through, but calculus requires surprisingly little amount of
| prior knowledge - even the concept of limit should be defined in
| the textbook (the famous epsilon-delta).
|
| Also remember that math notations are meant for people. If you
| learn the sigma summation notation, and if you wonder "So I
| understand what is \Sigma_{i=0}^{10}, but what is
| \Sigma_{i=0}^{-1}?" then you're wondering irrelevant stuff. If a
| math notation is confusing to use, good mathematicians will
| simply not use it and devise an alternative way to express it (or
| re-define it more clearly for their purpose).
|
| Also, don't skip exercises. Try to solve at least 1/3 of them
| after each chapter. Exercises are the "actually riding a bike"
| part of learning how to ride a bike.
| aabaker99 wrote:
| Is there any particular topic? I agree with other posters though
| that the notation is a short hand for the concepts and you need
| the concepts, not the notation.
| bradlys wrote:
| https://www.amazon.com/Introduction-Mathematical-Reasoning-N...
|
| I'd highly recommend this book. It's what I had for my intro to
| proofs class in college and it was the best book I found for
| understanding. I found many other books on this topic to be kinda
| garbage but this one was amazing.
| conjectures wrote:
| Khan academy and Schaum's Outlines are your friends.
|
| Then some textbooks with exercises (e.g. Axler on lin alg).
|
| The notation is usually an expression of a mental model, so just
| approaching via notation may cause some degree of confusion.
| readme wrote:
| the notation you need to know _should_ be defined somewhere in
| the book or paper you 're reading
|
| if it's not, try intuition
|
| if that fails, email your mathematician friend and ask
|
| don't have a mathematician friend? there's your next goal, go
| make one.
| kergonath wrote:
| > if it's not, try intuition
|
| If it's not, the book is badly written. Most of the time, you
| can't rely on a specific bit of notation to be consistent
| across books or articles. Smart arses who try to impress the
| readers with their fancy unique notations are the bane of
| scientists doing literature reviews.
|
| 90% of the time, there needs to be a keyword when a symbol is
| introduced, e.g. "where L is the time-dependent foo operator"
| so you can get a textbook to find what the fuck a "foo
| operator" is. Then, the first time you spend a day learning
| what it is, and the next million times you mumble "what a
| stupid notation for such a straightforward concept".
| anter wrote:
| Related question, does anyone know of any websites/books that
| have mathematical notation vs the computer code representing the
| same formula side by side? I find that seeing it in code helps me
| grasp it very quickly.
| hdinh wrote:
| https://github.com/Jam3/math-as-code
| OneTimePetes wrote:
| 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.
| b20000 wrote:
| get some english math textbooks used in high schools across
| europe.
| hatmatrix wrote:
| I think the problem is that there is no authoritative text, that
| I know of, and as ColinWright says, the same ideas can be notated
| differently by different fields or sometimes by different authors
| in the same field (though often they converge if they are in the
| same community).
|
| Wikipedia has been helpful sometimes but otherwise I have found
| reading a lot of papers on the same topic has been useful.
| However, this is kind of an "organic" and slow way of learning
| notation common to a specific field.
| smitty1e wrote:
| The Greek alphabet would like to thank all the scholars for the
| centuries of overloading and offer a "tee hee hee" to all of
| the students tormented by attendant ambiguities.
|
| Tough love, kids.
| rightly wrote:
| 99% of the time it's not needed in software.
| janeroe wrote:
| > I find it really hard to read anything because of the math
| notations and zero explanation of it in the context.
|
| So many answers and no correct one yet. Read and solve "How to
| Prove It: A Structured Approach", Velleman. This is the best
| introduction I've seen so far. After finishing you'll have enough
| maturity to read pretty much any math book.
| slipmasterflex wrote:
| Great recommendation!
| CogitoCogito wrote:
| My advisor's advise was basically "find a notation that you
| yourself like and understand well" and stick consistently to it.
| He said this in a context of having seen many standard notations
| before (so he's not saying to re-invent the wheel), but his point
| was just that notations and ways of thinking are personal. Try to
| be clear and precise (for yourself and others), but realize that
| you are crafting something that reflects you and your way of
| thinking.
|
| It's kind of a cop-out, but to be fair it's basically what I
| would say for programming as well. Try to simultaneously write
| code that clear to yourself and clear to others. There's no
| perfect method. Just constantly self-critique and try to improve.
| merlinran wrote:
| Had been in the same situation for years. Read a paper, encounter
| the first equation, scratch my head and search around trying to
| understand it, give up. That changed half a month ago, after
| watching the Linear Algebra and Calculus course at
| https://www.youtube.com/c/3blue1brown/playlists?view=50&sort....
|
| Let me explain a little bit. Just like a foreign language you
| stopped learning and using after high school, what prevents you
| from using it fluently is not just the vocabulary and grammar,
| but also the intuition and the understanding of the language as a
| whole. Luckily, math is a human designed language, with linear
| algebra and calculus being the fundamentals. And again, learning
| them is about building intuition on why and how they are used, so
| whenever you encounter transformation, you think in terms of
| vectors and matrices, and derivative for anything relevant to
| rate of change. By using carefully designed examples and visual
| representation, Grant Sanderson greatly smoothed the learning
| curve in the video courses. Try it out and you'll see.
|
| Beyond that, different fields do have slightly different
| notation. When you first encounter them, just grab some
| introduction books or online courses and skim over the very first
| chapters.
| cjfd wrote:
| Could it be that you are trying to read things that are a bit too
| advanced? Maybe look for some first year university lecture
| notes? In general, if you cannot follow something, try to find
| some other materials on the same subject, preferably more basic
| ones.
| erichocean wrote:
| Math papers can be pretty sloppy, and you don't realize this
| until you start working with formal mathematics--then it's
| obvious.
|
| Almost all hand "proofs" in math papers have minor bugs, even if
| they're mostly correct in the big picture sense.
|
| Even math designed to support programming (e.g. in computer
| graphics) is almost always incomplete/outright wrong in some
| meaningful way.*
|
| But with a struggle, it's still largely usable/useful.
|
| I've used advanced mathematics most of my career to do work (i.e.
| read a paper, implement it), but the ability to actually use math
| to do _new_ things in computer science that mattered only to me
| only happened after I learned TLA+, which took a few weeks of
| solid study to click. Since then, it 's been a pleasure. My specs
| have never been this good!
|
| Lamport's video course on TLA+ is pretty good, but honestly I've
| read everything I can find on the topic so it's difficult to know
| what helped me the most.
|
| *I think this is because, short of doing formal mathematics,
| there's no way to "test" your math. It's the equivalent of
| expecting programmers to write correct code the first time with
| no tests, and without even running the code.
| [deleted]
| mixmastamyk wrote:
| Community college is a good way, low commitment as well.
| dwheeler wrote:
| There is no single authoritative source for mathematical
| notation. That said, there are a lot of common conventions. You
| could do worse than this NIST document if it's just a notation
| question:
|
| https://dlmf.nist.gov/front/introduction
|
| Of course, if the real problem is that you need to learn some
| mathematical constructs, that is a different problem. The good
| news is that there's a lot of material online, the bad news is
| that not all of it is good... I often like Khan Academy when it
| covers the topic.
|
| I wish you luck!
| Tycho wrote:
| I think the _Princeton Companion to Mathematics_ covers a lot of
| it at the start of the book.
| tclancy wrote:
| I've run into this problem as well and it's put me off learning
| TLA+ and information theory, which bums me out. I assume there's
| a Khan Academy class that would help but it's hard to find.
| macrowhat wrote:
| My pre-calc teacher would regularly use smiley faces and
| christmas trees in place of common symbols. After a while you
| start to start to see past the funny symbols and look at it as
| pure symbol manipulation. Very interesting approach, and I went
| on to get (literally) a 100 in calc 3--with the curve ;)
| anthomtb wrote:
| 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.
| analog31 wrote:
| Maybe a problem is trying to learn it by reading it.
|
| I was a college math major, and I admit that I might have flunked
| out had I been told to learn my math subjects by reading them
| from the textbooks without the support of the classroom
| environment. It may be that the books are "easy to read if a
| teacher is teaching them to you."
|
| Talking and writing math also helped me. Maybe it's easier to
| learn a "language" if it's a two way street and involves more of
| the senses.
|
| Perhaps a substitute to reading the stuff straight from a book
| might be to find some good video lectures. Also, work the chapter
| problems, which will get your brain and hands involved in a more
| active way.
|
| As others might have mentioned, there's no strict formal math
| notation. It's the opposite of a compiled programming language.
| In fact, math people who learn programming are first told: "The
| computer is stupid, it only understands exactly what you write."
| In math, you're expected to read past and gloss over the slight
| irregularities of the language and fill in gaps or react to
| sudden introduction of a new symbol or notational form by just
| rolling with it.
| zwerdlds wrote:
| My suggestion to you is going to sound pithy, but its what worked
| for me: do problems. Lots and lots of problems.
|
| Pick a direction (maybe discrete math, if you're trying to do CS)
| and get a book (I like EPP, as it is super accessible) and go, in
| order, through each chapter. Read, do the example problems, and
| do EVERY SINGLE PROBLEM in the (sub)chapter section enders.
|
| Its a time commitment, but if you really want to learn it, this
| is one way to do so. IMO finding the right textbook is key.
| klodolph wrote:
| > [...] I find it really hard to read anything because of the
| math notations and zero explanation of it in the context.
|
| I suggest finding contexts first, and exploring math within those
| contexts. Different subfields have their own conventions and
| notation.
|
| For example, you might be working in category theory, and see an
| arrow labeled "p". When I see that, I think, "Ah, that's probably
| a projection! That's what p stands for!"
|
| Or you might be in number theory, and see something like p(x).
| When I see that, I think, "Ah, that's the prime number counting
| function! That's what p stands for, 'prime'!"
|
| Or you might be in statistics, and see 1/2[?]p e^(-1/2 x^2). When
| I see that, I think, "Ah, that's the number p! It's about 3.14"
|
| Or you might see a big [?] which stands for "product".
|
| The fact that such a common symbol, p, stands for four different
| things in four different contexts can be a bit confusing. So if
| you want to learn mathematical notation, pick a context that you
| want to study (like linear algebra), and look for accessible
| books and videos in that subfield. The trick is finding stuff
| that is advanced enough that you're getting challenged, but not
| so advanced that it's incomprehensible. A bit of a razor's edge
| sometimes, which is unfortunate.
| todd8 wrote:
| For about $5 you can find an old (around 1960-1969) edition of
| the "CRC Handbook of Standard Mathematical Tables. I've owned two
| of the 17th edition published in 1969, because back then hand
| calculators didn't exist and many of the functions used in
| mathematics had to be looked up in books, like what is the square
| root of 217. Engineers used these handbooks extensively back
| then.
|
| Now, of course, you have the internet and it can tell you what
| the square root of 217 is. Consequently, the value of these used
| CRC handbooks is low and many are available on eBay for a few
| dollars. Pick up a cheap one and in it you will find many useless
| pages of tables covering square roots and trigonometry, but you
| will also find pages of formulas and explanations of mathematical
| terms and symbols.
|
| Don't pay too much for these books because the internet and
| handheld calculators have pretty much removed the need from them,
| but that is how I first learned the meanings of many mathematical
| symbols and formulas.
|
| You might also look for books of "mathematical formulas" in you
| local bookstores. Math is an old field and the notations you are
| stumbling over have likely been used for 100 years, like the
| triangle you were wondering about. (Actually the triangle is the
| upper case greek letter delta. Delta T refers to an amount of
| time, usually called an interval of time.)
|
| Unfortunately, because math is an old subject it is a big
| subject. So big that no one person is expert in every part of
| math. The math covered in high school is kind of the starting
| point. All branches of mathematics basically start from there and
| spread out. If you feel you are rusty on your high school math,
| start there and look for a review book or study guide in those
| subjects, usually called Algebra 1 and Algebra 2. If you recall
| your Algebra 1 and 2, take a look at the books on pre-calculus.
| The normal progression is one year for each of the following
| courses in order, Algebra 1, Geometry, Algebra 2, Pre-Calculus,
| and Calculus. This is just the beginning of math proficiency, but
| by the time you get through Calculus you will be able to read the
| paper you referenced.
|
| Is it really a year for each of those subjects? It can be done
| faster but math proficiency is a lot of work. Like learning to be
| a good golfer, it would be unusual to become a 10 handicap in
| less than 5 years of doing hours of golf each and every week.
|
| Calculus is kind of the dividing line between high-school math
| and college level math. Calculus is the prerequisite for almost
| all other higher level math. With an understanding of Calculus
| one can go on to look into a wide range of mathematical subjects.
|
| Some math is focused on its use to solve problems in specific
| areas; this is called _applied math_. In applied math there are
| subjects like Differential Equations, Linear Algebra, Probability
| and Statistics, Theory of Computation, Information & Coding
| Theory, and Operations Research.
|
| Alternatively, there are areas of math that are studied because
| they have wider implications but not because they are trying to
| solve a specific kind of problem; this is called _pure math_. In
| pure math there are subjects like Number Theory, Abstract
| Algebra, Analysis, Topology & Geometry, Logic, and
| Combinatorics.
|
| All of these areas start off easy and keep getting harder and
| harder. So you can take a peek at any of them, once you are
| through Calculus, and decide what to study next.
| specproc wrote:
| As someone else who'd like to have a better understanding of
| formal notation, I think that's a great answer. Thanks for
| taking the time.
| zoomablemind wrote:
| >... I'd really like to learn "higher level than highschool"
| math...
|
| This sounds somewhat abstract, as the math field is vast. If you
| consider the next level from where you believe your present
| standing is, I would try to revisit the college-level math which
| you probaby experienced back in time.
|
| Generally, the textbooks rely on previous knowledge and gradually
| feed the new concepts, including the math notation as needed in
| the new scope.
|
| I find it easier to get the feel for the notation by actually
| writing it by hand. Indeed it's just an expression tool. Also,
| you may develop your own way of making notes, as you go on
| dealing with math-related problems.
|
| But in the core of this you are learning the concepts and an
| approach to reasoning. Of course, for this path to have any
| practical effect, you would need to memorize quite a bit, some
| theorems, some methods, some formulas, some applications.
| Internalizing the notation will help you condense all of that new
| knowledge.
|
| Picking a textbook for your level is all that is needed to
| continue the journey!
| Grustaf wrote:
| If you don't remember notation, surely you don't remember the
| material either, so why not just skim through the basic
| textbooks?
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