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Ask HN: How did you learn math notation?

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

#61

Post by Jensson »

throwaway31338 wrote: post #56
Jensson wrote: post #51

Earlier quoted context omitted.

> 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 sci…

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

> 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.

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

#65

Post by janeroe »

> 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.

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

#66

Post by kens »

rullopat wrote: post #11
ColinWright wrote: post #2

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

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

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

One place to start is https://en.wikipedia.org/wiki/Impulse_response

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

#67

Post by CogitoCogito »

throwaway31338 wrote: post #54
cabalamat wrote: post #50

Earlier quoted context omitted.

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

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…

> 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.

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

#68

Post by slipmasterflex »

janeroe wrote: post #65

> 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.

Great recommendation!

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

#69

Post by ColinWright »

throwaway31338 wrote: post #54
cabalamat wrote: post #50

Earlier quoted context omitted.

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

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…

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".

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

#70

Post by Jtsummers »

randomNumber7 wrote: post #60
cabalamat wrote: post #50

Earlier quoted context omitted.

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

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...

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.

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