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It's kind of humbling to know that I can't understand any of that. I barely passed mathematics in British high school - but for some reason took to quadratic eq

It's kind of humbling to know that I can't understand any of that. I barely passed mathematics in British high school - but for some reason took to quadratic equations and other slightly more advanced maths in college, I enjoyed it back then - but simply cannot read nor understand just about all of the topics in that thread. I'd have to look up what the "/" in "M/L" means in the 5th reply (I didn't) There are so many _symbols_ used in mathematics. It's all as if they're spells written by wizened greybeard wizards and passed down to apprentices throughout the centuries.

_lvym 5y ago
Quadratic equations are now a standard part of British secondary school mathematics.
- scarygliders 5y ago
  "Back in mah day" - they weren't :)
  - aardvark179 5y ago
    I think you may have wiped them from your memory. :-)
qcoh 5y ago
> I'd have to look up what the "/" in "M/L" means in the 5th reply (I didn't) M/L means that M is a field-extension over L. A concrete example would be C/R (the complex numbers over the reals). Algebraic number theory, in particular Galois theory, studies field extensions by looking at the group of symmetries: the field automorphisms of the larger field that fix the smaller field. For the concrete example above, the Galois group is a group with two elements: the identity function and the function that maps i to -i and keeps real numbers fixed. It's not a coincidence that the dimension of C as an R-vector space is the same as the size of this group (or that the degree of the polynomial that has i (and -i) as roots has degree 2).
- skulk 5y ago
  If anyone is wondering why field extensions aren't transitive, you can watch Fields medalist Dr. Richard Borcherds explain it: https://youtu.be/zAqVrdNE81g?t=798 https://youtu.be/zAqVrdNE81g?t=798
openfuture 5y ago
> There are so many _symbols_ used in mathematics. It's all as if they're spells written by wizened greybeard wizards and passed down to apprentices throughout the centuries. That's exactly what they are!
thomasahle 5y ago
> There are so many _symbols_ used in mathematics. It's all as if they're spells written by wizened greybeard wizards and passed down to apprentices throughout the centuries. Think of it as jumping into a new codebase you have never worked with before. It is full of strange functions calling each other for unknown reasons. It takes a while to get a good intuition for why. M/L could be written Field-extension(M, L). That might make it easier to Google what's going on, rather than having to pick up a group theory book. And indeed in this case it would be extra helpful, because M/L could also mean "quotient group" or even "division" depending on the type of the objects M and L. I personally prefer M:L for field extensions. But Googling individual concepts is hardly a good way to learn group theory. So I don't think it matters too much. For anyone that spends enough time working in the given field, the notation is by far the easiest thing to figure out. In general it is nice that mathematics - still being largely a handwritten endeavor - has the freedom to introduce new notation where it makes sense. See https://mathoverflow.net/questions/42929/suggestions-for-good-notation https://mathoverflow.net/questions/42929/suggestions-for-goo... for cool examples of new notation improving the lives of everyone.
- dwohnitmok 5y ago
  > For anyone that spends enough time working in the given field, the notation is by far the easiest thing to figure out. I want to emphasize this because the weirdness of math notation comes up often. The difficulty of grasping mathematical concepts far outweighs notational concerns. Getting used to notation in a specific field is an absolute drop in the bucket compared to understanding the theorems and proofs, and often the reason notation is so idiosyncratic and inconsistent is because it's meant to try to reflect human intuition, which is idiosyncratic and inconsistent in comparison to the mathematics from which it is derived. Once you really get into mathematics, notation is so far down the list of things that make learning mathematics hard that I don't think it's worth the associated cost of trying to change it on a large scale.
  - t-3 5y ago
    I have to vehemently disagree. Notation does more to keep mathematics the domain of nerds than anything else. It's unintuitive, uninformative, excessively complicated jargon which exists only to gatekeep and save paper. It's simply backwards to continue to use such an error-prone system in times of modern technology.
chaxor 5y ago
To be fair, I have a graduate degree in engineering and working on a different PhD and there were many things in that these I didn't understand. It's really incredible to me how quickly everything just became category theory in literature - like overnight. And stuff that used to be simple algebra is now explained as something like 'Abelian group linear transformations'. I haven't heard of "smash products on pointed spaces" for example, and idk why I would think them to be associative. Fun language though.
- whythatis 5y ago
  From what little I scrolled, I didn't see any category theory. Just some elementary group theory, point-set topology, real analysis, linear algebra etc Most of it standard material for undergrad math majors. > And stuff that used to be simple algebra is now explained as something like 'Abelian group linear transformations'. "Simple algebra" means different things in different professions? For example, Dummit&Foote's Abstract Algebra[0] would be considered "elementary algebra" by most mathematicians. For another example, Principles of Mathematical Analysis by Walter Rudin[1] is "elementary analysis". [0] https://www.amazon.com/Abstract-Algebra-3rd-David-Dummit/dp/0471433349 https://www.amazon.com/Abstract-Algebra-3rd-David-Dummit/dp/... [1] https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/0070856133/ref=pd_lpo_1?pd_rd_i=0070856133&psc=1 https://www.amazon.com/Principles-Mathematical-Analysis-Inte...
kzrdude 5y ago
I'm not a mathematician so I don't know if my reply here has any impact either. I don't understand the + in the first reply! It's an addition of vector spaces, but I'd have to go look up into definitions or subject material to understand what it means.. On the note of notation, mathematics repurposes notation all the time, hopefully in a way that helps the person reading to understand by analogy or at least not be mislead by the ananlogy. You said M/L and the / I think of then is quotient group, but that's apparently not what it means (I wouldn't know without reading it in context).
- Koshkin 5y ago
  > I don't understand the + To add to the confusion, in some contexts the sum is the same as ("isomorphic to") the product!
- enriquto 5y ago
  > I don't understand the + in the first reply! If X and Y are vector spaces (e.g., a line or a plane through the origin), then X+Y is the sum of all elements of each. If you think about this example (dim(X)=1 and dim(Y)=2) you can easily recover the first formula intutively. There are two cases for the formula dim(X+Y) = dim(X) + dim(Y) - dim(X ∩ Y) Case 1: X is a line inside the plane Y. In that case X+Y=Y and X∩Y=X, and the formula of the dimensions becomes 2=1+2-1 Case 2: X intersects Y at the origin. In that case X+Y is a three-dimensional space and X∩Y is 0, thus the formula says 3=1+2-0.
  - kzrdude 5y ago
    And you need some rule for what summing the elements means, where does it come from, I'm curious? If it's just a formal sum then the elements are incomparable between different spaces - and we can't say that one contains the other. I think I must understand when you say "X + Y" is the sum of elements that you're not using a formal sum {x} + {y} of each element x, and y but you mean sort of like the union of elements in X and Y?
    - enriquto 5y ago
Cthulhu_ 5y ago
It's not a race, but I'm even worse; I just tune out and shut down completely in the face of mathematical formulas on webpages. It's a really visceral reaction. I've done mathematics in secondary school (coordinate systems, Pythagorean wotsit), but I can still visualize those and put them into practical use. Later on I had statistics and linear algebra - that last one I had to redo and really work on, it was only after I did a minor in game design that I could finally map that math to a practical application. But I just don't have the background - or interest! - in anything else related to math. And I don't miss it either, else I would force myself to learn about it.
- actually_a_dog 5y ago
  It's not just you. Stephen Hawking said that for each equation he included in A Brief History of Time, he would cut his readership in half.
- Tarq0n 5y ago
  Do you have the same reaction to math in code form? I think I often don’t have the patience to expand things like symbols and ranges in my head in the case of mathematical notation, but math-as-code doesn’t require the same effort.
  - Fnoord 5y ago
    I have the same with code. I only read or write simple stuff. I used to like mathematics on school, until I just couldn't follow it anymore. It got me hot (physically), with headache akin to migraine. Which gotten worse throughout the years I studied. Now I tend to avoid such state. Its not worth the hassle. There's a couple of professions/qualities I have a lot of respect for, mathematicians is one of them. I appreciate them for three reasons: their ability to deal with the sheer complexity correctly and calmly, the fact it is unbiased, like science used to be (or rather: a couple of sciences aren't). The third reason is the one outlined above, generally: its a quality I've been unable to possess.
  - dekhn 5y ago
    I struggle with math notation but have no trouble with math written as code (C, python functions....). It took me years to be able to see a summation as a for loop.
gspr 5y ago
> There are so many _symbols_ used in mathematics. It's all as if they're spells written by wizened greybeard wizards and passed down to apprentices throughout the centuries. It's a misconception commonly held by non-mathematicians that the symbols are somehow the key to mathematics. I often see complaints, particularly on HN, of it being "impossible to read" research mathematics due to "all the dense symbols". This isn't how it works. You learn the mathematics. That's hard. As you go along, you realize that some things that require a lot of words can be condensed. So you make up symbols, or, more likely, the source that you're using teaches you some commonly-agreed-upon symbols. Now, once you've learned the math at hand, the symbol isn't the issue anymore. (Sure, it's a bit of a pain when people disagree about symbols for concepts that everyone agrees should be abbreviated, but in the big picture it's minor)
- genewitch 5y ago
  Japanese is easy, you just learn Japanese and the symbols that are commonly agreed upon to represent it.
  - Koshkin 5y ago
    > That's hard.
  - injb 5y ago
    This isn't a good analogy, because if you have a human brain then you already understand all the concepts that Japanese is used to represent (except maybe some cultural things that are hard to translate). In contrast, the language of mathematical notation is used for expressing concepts that are utterly unfamiliar and unintuitive. As a language, mathematical notation is nowhere near the complexity of a full human language, or even a programming language. It's much simpler than that. But the concepts behind it don't come naturally and have to be studied carefully.
    - samatman 5y ago
      While this is true, it doesn't help that postwar mathematics seems to have run out of useful names for things. Understanding what an algebraic group "really is" (with variations and extensions e.g. Lie groups) is more than one lifetime's work, but if you know what "group" means in English, well, you won't be disappointed. "Ring", even "field" kind of work this way. If I ever really develop an intuition for what a magma is, and I doubt it, this will not resemble the molten core of the earth. Or will it?
PebblesRox 5y ago
Reading about unfamiliar mathematics always gives me empathy for my 7yo's struggles to remember and work with basic arithmetic notation. It's only obvious once you've internalized it! Until then it's completely arbitrary and you constantly need to remind yourself what all the squiggles on the page really mean, which is hard when you're still trying to wrap your minds around the concepts themselves.

_lvym 5y ago

Quadratic equations are now a standard part of British secondary school mathematics.

scarygliders 5y ago
"Back in mah day" - they weren't :)
- aardvark179 5y ago
  I think you may have wiped them from your memory. :-)

qcoh 5y ago

> I'd have to look up what the "/" in "M/L" means in the 5th reply (I didn't) M/L means that M is a field-extension over L. A concrete example would be C/R (the complex numbers over the reals). Algebraic number theory, in particular Galois theory, studies field extensions by looking at the group of symmetries: the field automorphisms of the larger field that fix the smaller field. For the concrete example above, the Galois group is a group with two elements: the identity function and the function that maps i to -i and keeps real numbers fixed. It's not a coincidence that the dimension of C as an R-vector space is the same as the size of this group (or that the degree of the polynomial that has i (and -i) as roots has degree 2).

skulk 5y ago
If anyone is wondering why field extensions aren't transitive, you can watch Fields medalist Dr. Richard Borcherds explain it: https://youtu.be/zAqVrdNE81g?t=798 https://youtu.be/zAqVrdNE81g?t=798

openfuture 5y ago

> There are so many _symbols_ used in mathematics. It's all as if they're spells written by wizened greybeard wizards and passed down to apprentices throughout the centuries. That's exactly what they are!

thomasahle 5y ago

> There are so many _symbols_ used in mathematics. It's all as if they're spells written by wizened greybeard wizards and passed down to apprentices throughout the centuries. Think of it as jumping into a new codebase you have never worked with before. It is full of strange functions calling each other for unknown reasons. It takes a while to get a good intuition for why. M/L could be written Field-extension(M, L). That might make it easier to Google what's going on, rather than having to pick up a group theory book. And indeed in this case it would be extra helpful, because M/L could also mean "quotient group" or even "division" depending on the type of the objects M and L. I personally prefer M:L for field extensions. But Googling individual concepts is hardly a good way to learn group theory. So I don't think it matters too much. For anyone that spends enough time working in the given field, the notation is by far the easiest thing to figure out. In general it is nice that mathematics - still being largely a handwritten endeavor - has the freedom to introduce new notation where it makes sense. See https://mathoverflow.net/questions/42929/suggestions-for-good-notation https://mathoverflow.net/questions/42929/suggestions-for-goo... for cool examples of new notation improving the lives of everyone.

dwohnitmok 5y ago
> For anyone that spends enough time working in the given field, the notation is by far the easiest thing to figure out. I want to emphasize this because the weirdness of math notation comes up often. The difficulty of grasping mathematical concepts far outweighs notational concerns. Getting used to notation in a specific field is an absolute drop in the bucket compared to understanding the theorems and proofs, and often the reason notation is so idiosyncratic and inconsistent is because it's meant to try to reflect human intuition, which is idiosyncratic and inconsistent in comparison to the mathematics from which it is derived. Once you really get into mathematics, notation is so far down the list of things that make learning mathematics hard that I don't think it's worth the associated cost of trying to change it on a large scale.
- t-3 5y ago
  I have to vehemently disagree. Notation does more to keep mathematics the domain of nerds than anything else. It's unintuitive, uninformative, excessively complicated jargon which exists only to gatekeep and save paper. It's simply backwards to continue to use such an error-prone system in times of modern technology.

chaxor 5y ago

To be fair, I have a graduate degree in engineering and working on a different PhD and there were many things in that these I didn't understand. It's really incredible to me how quickly everything just became category theory in literature - like overnight. And stuff that used to be simple algebra is now explained as something like 'Abelian group linear transformations'. I haven't heard of "smash products on pointed spaces" for example, and idk why I would think them to be associative. Fun language though.

whythatis 5y ago
From what little I scrolled, I didn't see any category theory. Just some elementary group theory, point-set topology, real analysis, linear algebra etc Most of it standard material for undergrad math majors. > And stuff that used to be simple algebra is now explained as something like 'Abelian group linear transformations'. "Simple algebra" means different things in different professions? For example, Dummit&Foote's Abstract Algebra[0] would be considered "elementary algebra" by most mathematicians. For another example, Principles of Mathematical Analysis by Walter Rudin[1] is "elementary analysis". [0] https://www.amazon.com/Abstract-Algebra-3rd-David-Dummit/dp/0471433349 https://www.amazon.com/Abstract-Algebra-3rd-David-Dummit/dp/... [1] https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/0070856133/ref=pd_lpo_1?pd_rd_i=0070856133&psc=1 https://www.amazon.com/Principles-Mathematical-Analysis-Inte...

kzrdude 5y ago

I'm not a mathematician so I don't know if my reply here has any impact either. I don't understand the + in the first reply! It's an addition of vector spaces, but I'd have to go look up into definitions or subject material to understand what it means.. On the note of notation, mathematics repurposes notation all the time, hopefully in a way that helps the person reading to understand by analogy or at least not be mislead by the ananlogy. You said M/L and the / I think of then is quotient group, but that's apparently not what it means (I wouldn't know without reading it in context).

Koshkin 5y ago
> I don't understand the + To add to the confusion, in some contexts the sum is the same as ("isomorphic to") the product!
enriquto 5y ago
> I don't understand the + in the first reply! If X and Y are vector spaces (e.g., a line or a plane through the origin), then X+Y is the sum of all elements of each. If you think about this example (dim(X)=1 and dim(Y)=2) you can easily recover the first formula intutively. There are two cases for the formula dim(X+Y) = dim(X) + dim(Y) - dim(X ∩ Y) Case 1: X is a line inside the plane Y. In that case X+Y=Y and X∩Y=X, and the formula of the dimensions becomes 2=1+2-1 Case 2: X intersects Y at the origin. In that case X+Y is a three-dimensional space and X∩Y is 0, thus the formula says 3=1+2-0.
- kzrdude 5y ago
  And you need some rule for what summing the elements means, where does it come from, I'm curious? If it's just a formal sum then the elements are incomparable between different spaces - and we can't say that one contains the other. I think I must understand when you say "X + Y" is the sum of elements that you're not using a formal sum {x} + {y} of each element x, and y but you mean sort of like the union of elements in X and Y?
  - enriquto 5y ago

Cthulhu_ 5y ago

It's not a race, but I'm even worse; I just tune out and shut down completely in the face of mathematical formulas on webpages. It's a really visceral reaction. I've done mathematics in secondary school (coordinate systems, Pythagorean wotsit), but I can still visualize those and put them into practical use. Later on I had statistics and linear algebra - that last one I had to redo and really work on, it was only after I did a minor in game design that I could finally map that math to a practical application. But I just don't have the background - or interest! - in anything else related to math. And I don't miss it either, else I would force myself to learn about it.

actually_a_dog 5y ago
It's not just you. Stephen Hawking said that for each equation he included in A Brief History of Time, he would cut his readership in half.
Tarq0n 5y ago
Do you have the same reaction to math in code form? I think I often don’t have the patience to expand things like symbols and ranges in my head in the case of mathematical notation, but math-as-code doesn’t require the same effort.
- Fnoord 5y ago
  I have the same with code. I only read or write simple stuff. I used to like mathematics on school, until I just couldn't follow it anymore. It got me hot (physically), with headache akin to migraine. Which gotten worse throughout the years I studied. Now I tend to avoid such state. Its not worth the hassle. There's a couple of professions/qualities I have a lot of respect for, mathematicians is one of them. I appreciate them for three reasons: their ability to deal with the sheer complexity correctly and calmly, the fact it is unbiased, like science used to be (or rather: a couple of sciences aren't). The third reason is the one outlined above, generally: its a quality I've been unable to possess.
- dekhn 5y ago
  I struggle with math notation but have no trouble with math written as code (C, python functions....). It took me years to be able to see a summation as a for loop.

gspr 5y ago

> There are so many _symbols_ used in mathematics. It's all as if they're spells written by wizened greybeard wizards and passed down to apprentices throughout the centuries. It's a misconception commonly held by non-mathematicians that the symbols are somehow the key to mathematics. I often see complaints, particularly on HN, of it being "impossible to read" research mathematics due to "all the dense symbols". This isn't how it works. You learn the mathematics. That's hard. As you go along, you realize that some things that require a lot of words can be condensed. So you make up symbols, or, more likely, the source that you're using teaches you some commonly-agreed-upon symbols. Now, once you've learned the math at hand, the symbol isn't the issue anymore. (Sure, it's a bit of a pain when people disagree about symbols for concepts that everyone agrees should be abbreviated, but in the big picture it's minor)

genewitch 5y ago
Japanese is easy, you just learn Japanese and the symbols that are commonly agreed upon to represent it.
- Koshkin 5y ago
  > That's hard.
- injb 5y ago
  This isn't a good analogy, because if you have a human brain then you already understand all the concepts that Japanese is used to represent (except maybe some cultural things that are hard to translate). In contrast, the language of mathematical notation is used for expressing concepts that are utterly unfamiliar and unintuitive. As a language, mathematical notation is nowhere near the complexity of a full human language, or even a programming language. It's much simpler than that. But the concepts behind it don't come naturally and have to be studied carefully.
  - samatman 5y ago
    While this is true, it doesn't help that postwar mathematics seems to have run out of useful names for things. Understanding what an algebraic group "really is" (with variations and extensions e.g. Lie groups) is more than one lifetime's work, but if you know what "group" means in English, well, you won't be disappointed. "Ring", even "field" kind of work this way. If I ever really develop an intuition for what a magma is, and I doubt it, this will not resemble the molten core of the earth. Or will it?

PebblesRox 5y ago

Reading about unfamiliar mathematics always gives me empathy for my 7yo's struggles to remember and work with basic arithmetic notation. It's only obvious once you've internalized it! Until then it's completely arbitrary and you constantly need to remind yourself what all the squiggles on the page really mean, which is hard when you're still trying to wrap your minds around the concepts themselves.