Which order to learn math topics in?

For the discussion of math. Duh.

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DMalik
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Which order to learn math topics in?

Postby DMalik » Fri Apr 01, 2011 4:24 pm UTC

I've been teaching myself mathematics from textbooks and webpages for a while. I have never had formal math education, though, so I am unfamiliar with how this is usually done.

An example: I was reading a graph theory textbook when, about a tenth of the way in, the author assumes knowledge of linear algebra for the rest of the book. This annoys me because I'd prefer not to stop-and-start textbooks.

So, in what order are you supposed to learn things? I'm looking for a sort of 'technology tree' of math; what are each subfield's dependencies? What should I know before getting a textbook about abstract algebra or topology?

I am currently pretty much up to speed with high-school math; I am more-or-less familiar with calculus, I know (extremely) basic number theory, but beyond that I know nothing. I am interested in basically all math, maybe slightly biased to the discrete side. I'm learning this for fun.

Thanks for any help (and links to good help) :)

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Re: Which order to learn math topics in?

Postby Meem1029 » Sun Apr 03, 2011 2:41 am UTC

I'm not sure as I'm currently a freshman math major myself, but Real Analysis is one thing that seems to be foundational to a lot of fields. At least the concepts presented in an analysis class are. Are you familiar with rigorous proofs (I'm not talking the kind you did in geometry in high school, but there are similarities)? If not, I'd recommend getting familiar with them, as that is how a lot of math is presented. If you're just interested in learning the concepts, I'm not sure how much you'd need to know how to prove stuff though. I'm sure other people will be along shortly with specific books that are good.
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Re: Which order to learn math topics in?

Postby Dopefish » Sun Apr 03, 2011 3:15 am UTC

Linear algebra/Matrix theory (often they're taught together, although some books/courses will focus more on the abstract side of lin alg, but most I've seen are all about matrices) is fairly useful stuff that shows up all over the place, and would be low on the 'tech tree', so as long as your fine with basic algebra and calculus that should be safe for you to tackle. This is probably the most important thing that'll pop up as a pre-req all over the place, since it's so useful, and the abstract side of it can lead to some deep magics.

Multivariable calculus is handy too, and reasonably just builds on your normal one dimensional calculus, so is another thing you could probably safely look into.

One thing I might suggest to work out a 'tech tree' is to just check out the pre-reqs for various math courses at university websites to give you an idea of what math courses you're expected to have covered by then. If you happen to notice theres topology courses being offered that don't list linear algebra as a pre-req for example, you can probably safely pursue it.

Of course, many courses/books on subjects can be made assuming differing levels of background, and with math in particular some 'simple' math can be presented in a sufficiently abstract way that it can look scary and impossible from one source, but another source presents it at a very basic level that doesn't require much background, so it's good to check around abit if you find one resource seems over your head. I'm not a fan of most Wiki math pages for this reason, as many things are presented alongside some very rigourous/abstract form that is probably correct, but can work to scare off people who have no idea what it's even saying.

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Re: Which order to learn math topics in?

Postby lu6cifer » Sun Apr 03, 2011 4:36 am UTC

Well, in high school, it's usually

Alg2
Geometry
Pre-Calc
Calculus

In college, you'll usually have to take Calcs I, II, III, lin algebra/matrices, diffeqs, and so on, but sometimes differential equations is doable with only Calcs I and II. Analysis is fundamental, but it's really considered a high level course.

On the discrete side of things, graph theory can be learned without incorporating matrices/lin alg, and since discrete math is...well, discrete, you can learn a bunch of different topics without needing calculus (although calculus can help), like abstract algebra, graph theory, elementary number theory, set theory, formal logic, combinatorics, etc...
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Re: Which order to learn math topics in?

Postby Ubercomrade » Sun Apr 03, 2011 7:51 pm UTC

DMalik wrote:So, in what order are you supposed to learn things? I'm looking for a sort of 'technology tree' of math; what are each subfield's dependencies? What should I know before getting a textbook about abstract algebra or topology?


Dopefish wrote:One thing I might suggest to work out a 'tech tree' is to just check out the pre-reqs for various math courses at university websites to give you an idea of what math courses you're expected to have covered by then. If you happen to notice theres topology courses being offered that don't list linear algebra as a pre-req for example, you can probably safely pursue it.


I'm about to hijack the topic, but please bear with me. I was planning to post this on xkcd forums (first) precisely because of the sort of people who post and lurk here. I hope to make this post both relevant and useful.

Going through college, I have been seeing the need for a 'tech tree' since sophomore year. The way we approach education (formal and otherwise) these days is almost like trying to look things up in an encyclopedia without alphabetical order, table of contents, or any other means to optimize the search. 'Almost' because institutions of formal education already arrange courses in order of loosely increasing complexity, which causes most students to climb the tech tree without the ability to see the big picture. Concepts are lumped into courses, which reduces the resolution with which one can see a conceptual tech tree by looking at course prerequisites. This system is far from optimal: even the professors who design the curriculum can only guess about how much the students know; if they guess incorrectly (which is not uncommon) some conceptual material either gets taught twice, or omitted all-together. As for non-formal education, I can relate to your problem, too: I have had electronics as a hobby since elementary school, but I just barely knew my anode from my cathode until I went to college - because I didn't know my way up the tech tree.

Like adding a table of contents to a textbook or alphabetical order to an encyclopedia, an explicitly-defined conceptual tech tree would effectively improve our capacity for education. I am sure everyone can see why this is important, but I will explain anyway. In the United States alone, we have employers and politicians raving about the inadequacy of the greater part of the workforce. We know that not everyone can afford formal education. At the same time, the Internet is full of knowledge - free knowledge - for those who know what to look for. We have a problem and a solution, but we don't have anything that would connect the two.

Here's what I propose, although I can't really claim it's my idea anymore:

We, the xkcd fans, build the first tech tree. We start with the purest academic subject: mathematics. We will need the knowledge of the mathematically-skilled people on these forums to define the dependency chains and break blobs of lumped concepts into indivisible atoms of knowledge. We will need the skills of the computer science people to bring the tech tree to life (possibly as a java-based browser applet.) We will need the collective ingenuity of all members to design and refine the underlying principles of the tech tree, and we will need the diversity of academic knowledge of all xkcd forum members when the tech tree is expanded to other intellectual subjects.

Ultimately, what we want from the tech tree is the most intuitive way to browse the map of human knowledge. By breaking intellectual subjects down into bite-size concepts, we can divide and conquer them more effectively. By defining conceptual dependencies for all to see, we can quickly find the shortest path from what we know to what we want to know. By creating a map of the path (which is unfortunately not as often traveled as we want,) we would show the way to those who would otherwise not find it.

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z4lis
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Re: Which order to learn math topics in?

Postby z4lis » Sun Apr 03, 2011 8:27 pm UTC

I believe the Khan Academy (http://www.khanacademy.org/#browse) has already started doing this. However, must of the curriculum is what I would place at high-school or lower-level university courses. What you're describing (a tech tree of concepts) is what I so eagerly wanted in high school to see. I needed a well-planned set of books or lectures I could read that were arranged in such a way that I could move from one to the next in a smooth fashion. If such a project took off in the hands of some members of this forum, I'd be happy to help out.

There is, however, one thing I would request be made very clear to anybody wanting to learn math via online lectures and the sort, and it's that just reading through things is not enough to understand the topics. I've tried to just sit down and read a book on math. It's very difficult, and I got little out of it. One has to spend time pouring over the concepts, linking them up with other things, and trying to maintain the big picture whilst sifting through details of a proof. Also, exercises must, must be included. In fact, perhaps it might even be better to have the "tech tree" be exercise-based, rather than concept based. The definitions and concepts are taught as exercises and problem sets demand.

To the OP: You'll find that lots of math are interlinked and intertwined. As far as a basic set, I would start out with anything that teaches basic logic and set theory, just to become familiar with the notation frequently used. Concepts like induction need to be second nature. Many introductory texts have a chapter devoted to this stuff. After you've done that, I would suggest getting a solid understanding of linear algebra. Analysis requires it for differentiation. Algebra makes heavy use of it. As you saw, graph theory needs it. Linear algebra such a fundamental field, I would make that my first stop if I were going to study mathematics, as it's so ubiquitous. Sadly, I'm not really sure what would be the best book to really learn it from. There are tons of books about it, of course, but I have hunch that many of them don't really discuss on a level above simple matrix manipulation, which is certainly not what you're after.

Once you've managed to get linear algebra down, you have two options: you can pursue either analysis or algebra. If you really want to go far, you must learn a great deal of both. A note on topology: you could technically go after this field without having any prerequisites than set theory. However, some of the magic behind the field, not to mention motivation, might be lost.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

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Re: Which order to learn math topics in?

Postby ++$_ » Mon Apr 04, 2011 1:19 am UTC

Here's my very rough gesture at a tech tree for math. All requirements are transitive. Note that double-headed arrows normally mean that you can start learning either of the two subjects before the other, except for the single red arrow, which indicates that you should learn them both at basically the same time (at least in my opinion). Also, I have left out subjects that I don't know much about.
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math tech tree.png

mark999
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Re: Which order to learn math topics in?

Postby mark999 » Mon Apr 04, 2011 2:17 am UTC

++$_ wrote:Note that double-headed arrows normally mean that you can start learning either of the two subjects before the other, except for the single red arrow, which indicates that you should learn them both at basically the same time (at least in my opinion).


Could you explain your reasoning behind the red arrow? Did you learn single-variable calculus and real analysis at the same time? That seems strange to me.

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Re: Which order to learn math topics in?

Postby z4lis » Mon Apr 04, 2011 2:39 am UTC

I'll add a few stray fields to that tree, but in text form:

group theory, basic point set topology => algebraic topology
ring/field theory => module theory
measure theory, advanced linear algebra, point set topology => functional analysis
??? => metric spaces (point set topology? not really... Euclidean geometry?)
??? => uniform spaces

Also, many of these fields interact as you dive deeper into them. Topology has things to say about tons of things, even algebra. PDEs and differential geometry interact. Functional analysis (and so measure theory) are important for PDEs, as well. Also, I don't think that basic complex analysis really requires multivariable. In fact, I've found thinking about complex functions as "just maps from R^2" to be a confusing viewpoint in some respects.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

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Re: Which order to learn math topics in?

Postby Ubercomrade » Mon Apr 04, 2011 5:09 am UTC

z4lis wrote:I believe the Khan Academy (http://www.khanacademy.org/#browse) has already started doing this. However, must of the curriculum is what I would place at high-school or lower-level university courses. What you're describing (a tech tree of concepts) is what I so eagerly wanted in high school to see. I needed a well-planned set of books or lectures I could read that were arranged in such a way that I could move from one to the next in a smooth fashion. If such a project took off in the hands of some members of this forum, I'd be happy to help out.


Yes. I imagined the tech tree to be something like Wikipedia, in the sense of being interlinked, but in a more intuitive - and this is key - interactive graphical interface form rather than text. It's difficult enough to draw even a partial math technology tree without having to drag-and-drop labels and re-draw all the arrows. To really get started, we need someone who can program a java/flash based applet, and some place to host it.

I see ++$_ already drew a tech tree - so we already have a model to start from. It consists of four types of elements: nodes, prerequisite arrows, and co-requisite arrows. In terms of objects, the tech tree is a list of nodes; each node would have a list of pre-requisite pointers, a list of post-requisite pointers, and a list of co-requisite pointers. Each pointer in a node corresponds to another pointer in some other node: pre-requisite pointers in one node correspond to post-requisite pointers in the other (pre-requisite) node, and vice-versa. Co-requisite pointers in one node correspond to other co-requisite pointers in other node(s). All the applet would do, given the tech tree data, is parsing the list and displaying it as an intuitive, interactive GUI. For the sake of usefulness, I guess each node on the interactive tech tree should also have links to the relevant article(s) or video(s) or something.
Last edited by Ubercomrade on Thu Aug 04, 2011 4:51 pm UTC, edited 2 times in total.

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Re: Which order to learn math topics in?

Postby ++$_ » Mon Apr 04, 2011 5:38 am UTC

mark999 wrote:Could you explain your reasoning behind the red arrow? Did you learn single-variable calculus and real analysis at the same time? That seems strange to me.
Yeah, basically. It's certainly possible to learn calculus before learning analysis, but this involves learning methods for differentiation and integration without much understanding of what is going on. I think this is a bad idea.

I see real analysis and calculus as basically the same subject called by two different names. Calculus is the name it is given in applied contexts, while analysis is its name in pure contexts.
z4lis wrote:??? => metric spaces (point set topology? not really... Euclidean geometry?)
I was thinking of that as just part of real analysis.
z4lis wrote:Also, I don't think that basic complex analysis really requires multivariable. In fact, I've found thinking about complex functions as "just maps from R^2" to be a confusing viewpoint in some respects.
Yeah, I guess you could do it without knowing any multivariable calculus, but I think it would be more difficult.

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Re: Which order to learn math topics in?

Postby Qaanol » Mon Apr 04, 2011 5:50 am UTC

I like this idea.

I’m picturing it as a tree with names of fields, much like ++$ made, so that someone who doesn’t know about the subjects ahead of them will at least now know what those subjects are called and will be able to look them up on Wikipedia. Also, it might be helpful to have the individual items be rather more specific concepts, and enclose them (Venn-style) in ovals labeled by broader subject area. If this does become more complicated and interactive, I think it is important to still be able to view the entire tree in one image, with all its interconnectedness.

As far as exercises go, I’d leave them out, but perhaps put in a brief summary of the types of problems the subject is designed to solve. As in, very brief, maybe just a couple sentences. Once somebody knows the name of the subject and vaguely what it’s about, they can go look it up on Wikipedia. No sense in our trying to duplicate what’s already there. Perhaps a single archetypal example problem could be included for each concept, but not worked out or solved, just described in a “Here’s a typical problem that can be solved easily with the tools of this subject” sort of way.
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Re: Which order to learn math topics in?

Postby Quaternia » Mon Apr 04, 2011 1:28 pm UTC

Just something that might interest you, in a lot of math books at the university level they have a tech tree for the material presented, especially when it's a survey book about a new field or something that bridges a couple of fields. So really if Khan Academy has the high school tech tree covered, the only missing link is the undergraduate stuff :D
(I know there's a tech tree for category theory out there, with associated youtube lectures. Haven't watched many, but yeah.)

The concept is cool though, this would save SO MUCH TIME during course advising.
Instead of "oh Bobby you have like, 104 credits left to go, and 24.5 have to be in category OUQTJG, the rest being in SAGIU with the exception of another 12 in AGJOJA such that these 12 credits are not cross-referenced with any in SAGIU but are all cross-referenced with OUQTJG..."
It would be like "oh Bobby put 100 more points in your Major Tree, then level up your Distribution Requirements"
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Re: Which order to learn math topics in?

Postby DMalik » Wed Apr 06, 2011 5:16 pm UTC

Thanks for all the responses. I've decided to learn linear algebra and then decide what to do next (graph theory, number theory, more calculus, etc.).

Making a tech-tree sounds good. I also found http://hbpms.blogspot.com/ which looks like it would be helpful for math self-education.

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Re: Which order to learn math topics in?

Postby uukgoblin » Tue Jan 06, 2015 9:32 pm UTC

I've re-written the graph in dot, the source is at github in goblin/mathdeps.
I'd link the resulting png, but the message gets flagged as spam (hooray for human/spam-filters!). You can see it there on github too.

Please feel free to fork and correct :-)

We could use sub-graphs to categorize subjects. Dot's subgraphs are demonstrated on page 27 on the graphviz/dot guide. I'd link it too, but hey, spam!

One question: where do Fourier series and transforms fit into this?

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Re: Which order to learn math topics in?

Postby Qaanol » Wed Jan 07, 2015 6:04 am UTC

For the links: uuk’s GitHub project, and the corresponding png image.

Maybe it’s just because I’m posting at 1:00am, but I don’t immediately grok why there are arrows from calculus to combinatorics, and from real analysis to PDEs.

Things that jump out as absent:
Functional analysis
Topology
Algebraic topology
Algebraic geometry

Also, exactly three of the arrows are bidirectional, and one of them is red, but I do not see any documentation explaining what those things mean.
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Re: Which order to learn math topics in?

Postby Forest Goose » Wed Jan 07, 2015 6:14 pm UTC

I'm not sure how useful such a graph is, it makes a lot of assumptions about what you will study at what level - I'll speak to what I know.

After the most basic of basic set theory, you definitely need to have an awareness of real analysis, measures, probability, etc.
Logic and Set Theory include Model Theory, so you definitely need some algebra in there, if not more than some -- Category Theory fits in that cluster too, so that's something else that requires algebra, topology to really get. There's more along the same lines - combinatorics and infinite Ramsey stuff, various axioms, etc.

Unless, of course, you mean really basic logic and really basic set theory, say, a quick undergrad course in the material. However, if we are looking at things that way, then this graph is less "What you need to know" and more "What you need to know, in basic form, to follow where it leads", but who says we should? I like the idea, but the problem is that if you want to even touch any of these topics beyond what is covered in an introductory chapter, or two, you'll probably need stuff from various other branches.

I would suggest less following some specific ordering, nothing this specific - take it from someone who's tried in the past... - and, instead, sample various books till you find ones of interest that you can follow, look up results you don't fully understand, read up on them as encountered - and if something is wanted to be learned, but too advanced, then read the immediate prereqs. for it. If you do this, loosely, and follow your gut, as long as you are doing every possible exercise and attempting to prove the theorems before reading the book proofs, then you should keep advancing, advancing towards what you like via what you like - and that will keep you motivated and going. If you feel you missed a subject along the way, return back to it once you've built up your maturity, it will be a breeze, then, even if new.

Just my 2 cents.
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Re: Which order to learn math topics in?

Postby f5r5e5d » Wed Jan 07, 2015 6:58 pm UTC

Which order to learn math topics in?
what do you want to do with "math"? - ultimately math can be viewed as abstract "games with symbols" - and way "meta" about what types games are possible

but as an engineer I can assure you that lots of highly technical professions don't use most of what excites graduate level math types here, most of us using math in the real world only learn what is needed by the engineering/technical or scientific problems in front of us

undergrad technical programs do try to put basic tool kits in place and few insist you can't use Calculus, Fourier, Laplace Transforms without solid understanding of Topology, Analysis, Measure Theory... even if a math person would insist you really don't know how Calculus works without

a tree of "math" may not be a useful view, subject dependency graphs are very conditional on the level, application domain - very sophisticated tools can be used without complete knowledge to derive them on your own

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Re: Which order to learn math topics in?

Postby uukgoblin » Wed Jan 14, 2015 5:38 pm UTC

Thanks for the links Qaanol, maybe when I grow up I'll be able to include them too ;-)

Qaanol wrote:Also, exactly three of the arrows are bidirectional, and one of them is red, but I do not see any documentation explaining what those things mean.

I've copy&pasted the explanation of what the arrows mean into the README.

Qaanol wrote:I don’t immediately grok why there are arrows from calculus to combinatorics, and from real analysis to PDEs.

I don't know much about combinatorics so can't answer this one, but real analysis to PDE is indeed weird. If anything, I think it should go to ODE first. It probably is a requirement for PDEs, but not an immediate one. I'm not sure about the difference between analysis and calculus though.

Forest Goose wrote:I'm not sure how useful such a graph is, it makes a lot of assumptions about what you will study at what level

Well the usefulness is to make sense of the dependency hierarchy. IMO it shouldn't make assumptions about the level of study. It should group topics from a particular area (let's say topics would be like "addition", "subtraction" and the area "elementary arithmetic"), and then create dependencies based on those.

The graph is not meant to guess what's "useful" and what isn't. It should provide quick reference and prioritisation of subjects. For instance, let's say I want to understand Shinichi Mochizuki's abc conjecture proof. I already know some basic arithmetics and elementary algebra, and I'd like to know which direction to proceed with my self-study in order to grok that. Once I see how overwhelming all that is, it would be my decision whether to give up right away or not ;-)

Another example would be if someone wanted to understand maybe a non-math subject like antenna design. Physics (ab)uses maths a lot, and maths is often taught alongside physics, with a lot of details neglected or omitted. I'd rather learn the required maths properly first, and only then proceed to applications in physics. With a dependency graph, a physics teacher could quickly pick maths areas that are required or useful for a particular subject and point it out to students to learn before taking the course.

f5r5e5d wrote:few insist you can't use Calculus, Fourier, Laplace Transforms without solid understanding of Topology, Analysis, Measure Theory

That's an interesting remark. I can see how knowing more stuff can boost your understanding of some more basic thing. In this case, perhaps a gray arrow to mean "recommends" would be useful? And maybe the shade of gray to mean the strongness of the recommendation? Also, more granularity would improve things, I think. It'd perhaps make the graph uglier, but maybe we could have multiple views or a zoomable map.

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Re: Which order to learn math topics in?

Postby Forest Goose » Wed Jan 14, 2015 7:40 pm UTC

[quote="uukgoblin"
The graph is not meant to guess what's "useful" and what isn't. It should provide quick reference and prioritisation of subjects. For instance, let's say I want to understand Shinichi Mochizuki's abc conjecture proof. I already know some basic arithmetics and elementary algebra, and I'd like to know which direction to proceed with my self-study in order to grok that. Once I see how overwhelming all that is, it would be my decision whether to give up right away or not ;-)

Another example would be if someone wanted to understand maybe a non-math subject like antenna design. Physics (ab)uses maths a lot, and maths is often taught alongside physics, with a lot of details neglected or omitted. I'd rather learn the required maths properly first, and only then proceed to applications in physics. With a dependency graph, a physics teacher could quickly pick maths areas that are required or useful for a particular subject and point it out to students to learn before taking the course.
[/quote]

It most certainly does make a number of assumptions. Descriptive Set Theory, and loads of other set theory, has to do with the real line, yet Set Theory points to Real Analysis. Model Theory belongs to Logic, as does Universal Algebra, (and, arguably, Category Theory), yet nothing from algebra points to Logic.

So, a big assumption being made is that "Set Theory" doesn't cover everything in the subject usually called that, the same for Logic - indeed, going off the dependencies of the graph, a lot of things in an undergrad/beginning grad text on sets and logic wouldn't be covered under the graph's "Set Theory" since those things depend on all sorts of other things, ultimately. So, yes, there is a big assumption in there. -- And how is it that Real Analysis requires Logic and Set Theory, yet the algebra line doesn't? That's really bizarre.

There's also a lot of stuff that is missing, so it's assuming a bunch right there - any listing of subjects that includes "Representation Theory", but can't be bothered algebraic topology, algebraic geometry, and homological algebra, module theory, etc...isn't omitting them because they are advanced, so why? Or, is there an assumption that we can safely shuffle those under the rug of something listed, but calculus (both SV and MV) and analysis (real and complex - and ODE and PDE) are all worthy of their own specific category?

This can go on and on and on.

What it looks like, to me, is a dependency graph and prioritization specifically geared towards a certain type of mathematics - one that would be useful to applications, physics etc. Which, definitely, is important, but is, definitely, assumptive and biased.

-- I'm not sure you can make a graph that doesn't make assumptions about level of study, yet highlights dependencies...math isn't really a progression of subfields, but an interplay of them, and building ideas upon ideas. Again: you need set theoretical notions to do real analysis, you need real analysis to do descriptive set theory. So, what do we do, then? Further subdivide them...that doesn't sound useful, since we're going to be subdividing till eternity and end up with bidirectional arrows pointing all over the place. In the end, the dependencies are a matter of where your interests are. (what you need from real analysis to do set theory, algebraic geometry, Iwasawa theory, or complex analysis are going to be really really really different, there's no sense pretending like there is some sort of generic way to capture all of that.) --
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Re: Which order to learn math topics in?

Postby uukgoblin » Wed Jan 14, 2015 8:36 pm UTC

Right, I wasn't saying that the graph in its current form is free of assumptions. I didn't author it, I just copied it from ++$_'s post, and I have little knowledge about most of its nodes. The only thing I'm sure about it is that it's incomplete ;-)

Forest Goose wrote:math isn't really a progression of subfields, but an interplay of them, and building ideas upon ideas. Again: you need set theoretical notions to do real analysis, you need real analysis to do descriptive set theory. So, what do we do, then?


Well, this part would look like:

set theoretical notions -> real analysis -> descriptive set theory

I don't see a problem there (although I'm not sure what those represent exactly).

Forest Goose wrote:In the end, the dependencies are a matter of where your interests are.


If there are different types of categorisation, or different types of interests, we could have multiple graphs, one for each type...

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Re: Which order to learn math topics in?

Postby Forest Goose » Thu Jan 15, 2015 9:39 am UTC

tl;dr
Using graphs as a means of what to study is like telling a budding musician to follow a graph of the ridiculous number of genres of music, with the addition that they aren't allowed to hear any full songs from a genre till they're ready to learn to play them...

uukgoblin wrote:Well, this part would look like:

set theoretical notions -> real analysis -> descriptive set theory


That doesn't really work, though. What happens when you have homological algebra pointing at algebraic geometry and homotopy theory, then both of those pointing at hom. alg., then hom. alg. and hom. the. pointing at alg. geo.? Any configuration of arrows, honestly, for those three could work. --And just how much of algebra and topology, and cat theory, do you need to deal with model categories and all of that stuff?

What about topological logic and it's relation to intuitionistic logic? Where do sets, cats, first order logic, model theory, topology, and etc. point? Bet they don't point the same way if you're interested in banach space geometry.

Even with the, simpler, example above - what if your study of real analysis requires descriptive set theory, do we reverse it?

My point: any graph is going to be really really cluttered, full of ridiculously narrow subdomains and levels of depth into them, and have tons and tons of arrows...it would, probably, take a longer investment to appreciate and intuit just what the hell was going on than it would to learn a decent bulk of undergrad mathematics to begin with!

If there are different types of categorisation, or different types of interests, we could have multiple graphs, one for each type...


We could, but they aren't going to be that much simpler - or, they're really really specific, meaning the complexity gets shuffled off to the number of graphs...and that's just as bad.

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But, really, while it sounds like a useful thing, how useful are these graphs, really? If you can't read a few paragraphs of something and determine what you need to get there, then you aren't anywhere near getting there. In other words, if you can't formulate a realistic idea of prereqs. on your own, then you aren't really going to understand the what's and why's of any graph pointing there.

If you can't make heads or tails of the Hodge Conjecture, it's all foreign sounding to you, then does it help you to know you need to learn algebraic topology, algebraic geometry, diff. geo., and homo. algebra - not what you need to know, just that you need those subjects? Not really, each of those areas is vast, so it's fairly uninformative - for example, does reading about homotopy groups of spheres and the geometry of 3-manifolds get you closer? The student won't know, so they aren't closer to knowing what to read about to read about to read about to, finally, read about the Hodge Conjecture...nope, they just no they can't do it yet, the same as when they started.

The better method is to follow your interest and set aside things that seem like they'd be interesting, but can't be approached yet, then return back when they seem more tractable. In the end, you end up finding your interest, instead of following paths to something you may, actually, not even find that interesting.*

*I used to think that cohomology theories, and such, seemed really really cool...then, I found out, once I could follow them and work with them, that I found it all kind of unpleasant. On the flip side, I used to find logic and sets (especially ordinals and cardinals) really really tedious, then, after finally really grasping them, I could see how beautiful they were, and I fell in love with them. Following a graph, which I've tried, would've taken me somewhere very very different and less my cup of tea, had I not thrown it in the waste bin at one point.

Math is beautiful, follow what strikes and moves you, not a path of dependencies leading somewhere you can't even understand (and, hence, can't "really" desire to get to).
Forest Goose: A rare, but wily, form of goose; best known for dropping on unsuspecting hikers, from trees, to steal sweets.

uukgoblin
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Re: Which order to learn math topics in?

Postby uukgoblin » Thu Jan 15, 2015 3:00 pm UTC

Thanks for your great answer!

Now I'm sad :-( the idea sounded so good... :-(

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Dopefish
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Re: Which order to learn math topics in?

Postby Dopefish » Thu Jan 15, 2015 7:13 pm UTC

I wouldn't be sad, it is something that sounds like a good idea, and many of us have experience with 'tech trees' of research in video games and it's not unreasonable to want something similar in real life. It just turns out that real life is super complicated and massively interconnected so trying to simplify it (or even the relatively narrow math portion of it [which as you see isn't even that narrow]) down to a graph isn't practical.

Tirian
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Re: Which order to learn math topics in?

Postby Tirian » Thu Jan 22, 2015 1:09 pm UTC

In addition, the assumed dependence of one field of mathematics on understanding another can often be an artifact of the textbook author's purpose or bias. I would reference back to the OP, which described a graph theory textbook that required an understanding of linear algebra throughout it. I don't deny that there are some results both elementary (like calculating the number of walks of length n between two given vertices given its adjacency matrix) and profound (like calculating the number of spanning trees of an arbitrary graph) and curious (like showing that if the eigenvalues of the adjacency matrix of a simple graph are distinct, then the automorphism group of the graph is abelian), and obviously things depend of whether the purpose of the text was to write computer programs to implement all the various algorithms. But the rest of the field as an abstract strikes me as delightfully independent of prerequisites beyond naive set theory and is an excellent framework for practicing how to structure and write a valid mathematical argument. A vital part of my mathematical development was that I was able to take my college's course in graph theory in my freshman year instead of having to wait to be a senior (and lucky to be at a school where it wasn't a graduate-level course).


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