The easy way of seeing the first part is to do the prime factorization. The 7 doesn't matter since it's prime. If n has a 2 in its factorization it now has 2^3. But if it doesn't have a 2 it won't suddenly acquire one.
All the symbol soup proofs aren't wrong but I don't think they satisfyingly explain the why.
If n is even, we can choose some m such that n = 2m, and p(n) = p(2m) = 7 * 8m^3 + 2m = 2 * (7 * 4m^3 + m), which is divisible by 2 since we could factor out the 2 at the start.
If n is odd, similarly we can say n = 2m + 1. p(2m) = 7 * (2m + 1)^3 + (2m + 1) = 56m^3 + 84m^2 + 44m + 8 = 2 * (28m^3 + 42m^2 + 22m + 4), which is also divisible by 2 per the 2 at the start.
You're the second commenter, so far, to mention exclamation marks. What do they mean to you that would bother you so much to point it out, or anyone for that matter? I haven't even noticed them until I read the comments here on hn.
Not gp, but I feel similarly. For me, I can't help read it with emphasis. As in, the voice in my head gets all fancy in an annoying way. If you imagine someone in person reading it out-loud with exaggerated emphasis, that's what it feels like. Same thing with comic books for me, the sprinkled bolded words in dialog are really grating.
To me it's fairly similar to someone making excessive use of CAPS LOCK. It can be used as a stylistic choice at times, but use it TOO MUCH and it just becomes DISTRACTING.
I DON'T SEE A PROBLEM WITH THIS EITHER! BUT I EMPATHIZE! I GET COMMENTS FROM PEOPLE SAYING THAT I'M SOMEHOW YELLING AT THEM ALL THE TIME BUT I'M ACTUALLY SITTING IN SILENCE, TYPING QUIETLY ON A MEMBRANE KEYBOARD! LOL???
Oh this is nothing. One of my colleagues does that and adds random colour changes, underlines and font face changes. It's like working with a serial killer.
Ahh. I honestly miss that amount of self-expression, garish as it was. Or rather, I intensely dislike the mono-culture where every vertical video with one-word subtitles looks the same.
Interesting topic, but the writing style is so tedious that it really subtracts from the content as a whole. There's on average one exclamation mark per paragraph. Surely every single thought you write down isn't groundbreaking enough to warrant that.
I beg to differ, I think the writing conveys beautifully, the deeper abstract ideas embedded in what appears to be a simple problem - hence, it captures the essential spirit of what math is about
I like when the author’s personality shines through, and frankly I can’t imagine finding occasional exclamation marks _tedious_ of all things. I just don’t take things so seriously, I suppose.
Just out of curiosity, why do you find the code horrid? There are really only 9 lines of code that aren't glue code, and apart from the error prints (which are really irrelevant for these demonstration purposes), the code looks basically fine to me.
EDIT:
seen = []
for b in boards:
if set(seen).isdisjoint(orbit(b)):
seen.append(b)
Sage is a Python (and the snippet you pasted works fine in Python). And I'm also curious what would qualify that code as "horrid." I'd make light suggestion to improve performance by making `seen` a set right off the bat, but for this size of problem, that sort of detail is unimportant. Calling somebody's code "horrid" without even understanding what it's doing isn't a productive approach to, well; anything, really.
Math pages on Wikipedia have this bad. I don’t know whether the programming concepts pages are more sanely written or I already understand the circular reasoning. But it feels like they’re more approachable.
The math pages on Wikipedia value correctness above lay comprehensibility. It's easy to understand how this happens: a learned mathematician points out that a simplification for the purposes of making an explanation more approachable is not actually correct, so the explanation gets desimplified... and repeat ad absurdum until most math pages on Wikipedia cater primarily to experts and are too advanced for a good fraction of the audience.
I love the math pages on Wikipedia but I have a math degree. They are written (written clearly and concisely) for mathematicians.
If you're a programmer (but don't have a math degree) then I would offer up API docs as a comparison. They are written for you, the user of the API, to be as concise and straightforward as possible so that you can get up and running with the API. API docs are definitely not written for beginners who have never written a line of code (or a line of code in the language the API is written in) before.
If there's one complaint I may entertain, it's that Wikipedia isn't supposed to be a resource for specialists. It's intended to be an encyclopedia for a general audience. But then by that reasoning, many of these math pages on Wikipedia probably ought to be deleted outright because they're simply too specialized in the first place. So we're left with the dilemma: do we keep these articles as-is (and keep mathematicians happy) or do we delete them outright because they're too specialized?
The third option, rewriting them for a general audience, is likely to run afoul of Aesop's fable #721 "The miller, his son, and the donkey" [1]. You'll get a highly technical and complex article that explains far too much and buries its insights in overly verbose and cumbersome prose (which cannot assume any prerequisite mathematical knowledge). It'll please neither the mathematician nor the general audience member.
It's not. The language of mathematics is prose, usually written in English. The formulae are meant to illustrate the relationships in a very concise way but they're meaningless without the accompanying prose.
You surely know the syntax for multiplication, addition, and exponentiation. So you understand the polynomial in the opening paragraph. I'm sure you know function notation and division, so you understand that P(n)/k is always an integer.
You probably have seen the sigma notation before because that's usually taught in high schools around the world, so you know that the lemma which is not Burnside's is about adding and dividing. You probably have seen the notation |X| to mean absolute value, so perhaps we finally are reaching the limits of your knowledge if it has been a while since you've seen the notation |X| can also mean the size of a set (which is, in a way, a sort of absolute value, mathematicians love type-punning).
I'm sure you've seen function notation f: X -> Y to indicate that f is a function from the set X to the set Y, but I'll believe if you didn't know that [n] is the set {1, 2, 3, 4, ..., n}.
I haven't done a careful calculation here, but I believe as a moderate estimate that this already covers at least 30% of the mathematical symbols in this page.
My point is: the notation isn't probably the problem. You surely have seen these symbols before or can figure out what the individual symbols mean. I daresay the most esoteric symbol here is the Fraktur S for the symmetric group (here for the symmetric group of 6 symbols),
but I assume that if you didn't know what the symmetric group is, the more common notation of S_6 would probably not have helped you much.
So if the notation isn't the problem, I wager that the concepts and the difficulty of absorbing these ideas is the problem. It requires you to compile stuff yourself in your own head.
With programming we are used to a machine doing this for us. We write the code, give it to a machine, the machine basically tells us if we're right or not. With mathematics we don't generally have that machine for all cases. You have to do the work in your own head. The problem isn't the symbols. The problem is that you have to think about them harder, work out what is being multiplied, what are the sets in question, what are the operations, and put them together yourself. You have to read something like not Burnside's lemma and understand what it's saying about permutations and sets and grouping and counting.
Reading mathematics is a special skill. It's slow. It requires you to take out pen and paper and work it out yourself. Yes, like that. You have to do input and output on your own as you do mathematics. It's a unique skill that sadly isn't usually independently taught except at the university level.
The concepts and the work required to understand them are the problem. Not the symbols. The symbols are superficial and easily dispensed with.
More on integer valued polynomials: https://cameroncounts.wordpress.com/2017/01/31/polynomials-t...
Well I was curious, but there's a lot there I didn't understand. Apparently I'm good enough at math to do the proofs, but not to write the exercises.
Exercise left to the reader:
Prove 7*n^3 + n is divisible by 2
7*n^3 is even when n is even and odd otherwise.
odd + odd is even, as is even + even.
The easy way of seeing the first part is to do the prime factorization. The 7 doesn't matter since it's prime. If n has a 2 in its factorization it now has 2^3. But if it doesn't have a 2 it won't suddenly acquire one.
All the symbol soup proofs aren't wrong but I don't think they satisfyingly explain the why.
7n^3 +n (mod 2) = 1 n^3 + n = n + n = 2*n = 0*n = 0
Given p(n) = 7n^3 + n:
If n is even, we can choose some m such that n = 2m, and p(n) = p(2m) = 7 * 8m^3 + 2m = 2 * (7 * 4m^3 + m), which is divisible by 2 since we could factor out the 2 at the start.
If n is odd, similarly we can say n = 2m + 1. p(2m) = 7 * (2m + 1)^3 + (2m + 1) = 56m^3 + 84m^2 + 44m + 8 = 2 * (28m^3 + 42m^2 + 22m + 4), which is also divisible by 2 per the 2 at the start.
[stub for offtopicness]
I feel silly saying this, but I wish the author would use more periods and fewer exclamation marks.
Elaine Benes would be proud of their writing...
I was in the pool!
You're the second commenter, so far, to mention exclamation marks. What do they mean to you that would bother you so much to point it out, or anyone for that matter? I haven't even noticed them until I read the comments here on hn.
Not gp, but I feel similarly. For me, I can't help read it with emphasis. As in, the voice in my head gets all fancy in an annoying way. If you imagine someone in person reading it out-loud with exaggerated emphasis, that's what it feels like. Same thing with comic books for me, the sprinkled bolded words in dialog are really grating.
To me it's fairly similar to someone making excessive use of CAPS LOCK. It can be used as a stylistic choice at times, but use it TOO MUCH and it just becomes DISTRACTING.
I DON'T SEE A PROBLEM WITH THIS EITHER! BUT I EMPATHIZE! I GET COMMENTS FROM PEOPLE SAYING THAT I'M SOMEHOW YELLING AT THEM ALL THE TIME BUT I'M ACTUALLY SITTING IN SILENCE, TYPING QUIETLY ON A MEMBRANE KEYBOARD! LOL???
Oh this is nothing. One of my colleagues does that and adds random colour changes, underlines and font face changes. It's like working with a serial killer.
Maybe he was a teenager on IRC in the late 90s or early 00s and decided to never change
Thinking about it I guess MSN messenger and My Space also allowed/encouraged font shenanigans? My memory falters
Ahh. I honestly miss that amount of self-expression, garish as it was. Or rather, I intensely dislike the mono-culture where every vertical video with one-word subtitles looks the same.
Interesting topic, but the writing style is so tedious that it really subtracts from the content as a whole. There's on average one exclamation mark per paragraph. Surely every single thought you write down isn't groundbreaking enough to warrant that.
I beg to differ, I think the writing conveys beautifully, the deeper abstract ideas embedded in what appears to be a simple problem - hence, it captures the essential spirit of what math is about
I like when the author’s personality shines through, and frankly I can’t imagine finding occasional exclamation marks _tedious_ of all things. I just don’t take things so seriously, I suppose.
[flagged]
> That python code is horrid.
Just out of curiosity, why do you find the code horrid? There are really only 9 lines of code that aren't glue code, and apart from the error prints (which are really irrelevant for these demonstration purposes), the code looks basically fine to me.
EDIT:
Ah, well...EDIT 2: also, see comment below, it's not Python
Sage is a Python (and the snippet you pasted works fine in Python). And I'm also curious what would qualify that code as "horrid." I'd make light suggestion to improve performance by making `seen` a set right off the bat, but for this size of problem, that sort of detail is unimportant. Calling somebody's code "horrid" without even understanding what it's doing isn't a productive approach to, well; anything, really.
edit: nevermind, it's sage code not python code
It's not python, it's sage, so those actually work.
You perfectly summarized the reaction of any programmer looking at the work of a mathematician.
Math pages on Wikipedia have this bad. I don’t know whether the programming concepts pages are more sanely written or I already understand the circular reasoning. But it feels like they’re more approachable.
The math pages on Wikipedia value correctness above lay comprehensibility. It's easy to understand how this happens: a learned mathematician points out that a simplification for the purposes of making an explanation more approachable is not actually correct, so the explanation gets desimplified... and repeat ad absurdum until most math pages on Wikipedia cater primarily to experts and are too advanced for a good fraction of the audience.
I love the math pages on Wikipedia but I have a math degree. They are written (written clearly and concisely) for mathematicians.
If you're a programmer (but don't have a math degree) then I would offer up API docs as a comparison. They are written for you, the user of the API, to be as concise and straightforward as possible so that you can get up and running with the API. API docs are definitely not written for beginners who have never written a line of code (or a line of code in the language the API is written in) before.
If there's one complaint I may entertain, it's that Wikipedia isn't supposed to be a resource for specialists. It's intended to be an encyclopedia for a general audience. But then by that reasoning, many of these math pages on Wikipedia probably ought to be deleted outright because they're simply too specialized in the first place. So we're left with the dilemma: do we keep these articles as-is (and keep mathematicians happy) or do we delete them outright because they're too specialized?
The third option, rewriting them for a general audience, is likely to run afoul of Aesop's fable #721 "The miller, his son, and the donkey" [1]. You'll get a highly technical and complex article that explains far too much and buries its insights in overly verbose and cumbersome prose (which cannot assume any prerequisite mathematical knowledge). It'll please neither the mathematician nor the general audience member.
[1] https://en.wikipedia.org/wiki/The_miller,_his_son_and_the_do...
The programming pages use symbols that exist on your keyboard and deconstruct their process.
The math pages look like they’re trying to be Perl one-liners. Why is everything so jammed up, Mathematics?
Why is everything so jammed up, Mathematics?
It's not. The language of mathematics is prose, usually written in English. The formulae are meant to illustrate the relationships in a very concise way but they're meaningless without the accompanying prose.
One of my preferred example:
https://en.wikipedia.org/wiki/Bremsstrahlung#Quantum_mechani...
It is actually physics, but for all I know it could just as well summon Cthulhu.
I don't believe your 99.9% figure.
You surely know the syntax for multiplication, addition, and exponentiation. So you understand the polynomial in the opening paragraph. I'm sure you know function notation and division, so you understand that P(n)/k is always an integer.
You probably have seen the sigma notation before because that's usually taught in high schools around the world, so you know that the lemma which is not Burnside's is about adding and dividing. You probably have seen the notation |X| to mean absolute value, so perhaps we finally are reaching the limits of your knowledge if it has been a while since you've seen the notation |X| can also mean the size of a set (which is, in a way, a sort of absolute value, mathematicians love type-punning).
I'm sure you've seen function notation f: X -> Y to indicate that f is a function from the set X to the set Y, but I'll believe if you didn't know that [n] is the set {1, 2, 3, 4, ..., n}.
I haven't done a careful calculation here, but I believe as a moderate estimate that this already covers at least 30% of the mathematical symbols in this page.
My point is: the notation isn't probably the problem. You surely have seen these symbols before or can figure out what the individual symbols mean. I daresay the most esoteric symbol here is the Fraktur S for the symmetric group (here for the symmetric group of 6 symbols),
https://en.wikipedia.org/wiki/Fraktur#Unicode
but I assume that if you didn't know what the symmetric group is, the more common notation of S_6 would probably not have helped you much.
So if the notation isn't the problem, I wager that the concepts and the difficulty of absorbing these ideas is the problem. It requires you to compile stuff yourself in your own head.
With programming we are used to a machine doing this for us. We write the code, give it to a machine, the machine basically tells us if we're right or not. With mathematics we don't generally have that machine for all cases. You have to do the work in your own head. The problem isn't the symbols. The problem is that you have to think about them harder, work out what is being multiplied, what are the sets in question, what are the operations, and put them together yourself. You have to read something like not Burnside's lemma and understand what it's saying about permutations and sets and grouping and counting.
Reading mathematics is a special skill. It's slow. It requires you to take out pen and paper and work it out yourself. Yes, like that. You have to do input and output on your own as you do mathematics. It's a unique skill that sadly isn't usually independently taught except at the university level.
The concepts and the work required to understand them are the problem. Not the symbols. The symbols are superficial and easily dispensed with.