Obscure branches of math?

[lu6cifer]

I've been looking into spherical trigonometry lately, a pretty interesting branch of trig that seems to have slipped out of the curriculum. I was just wondering, what other obscure, not regularly taught fields of math are there? [Preferably pre-calc, and I already know of linear algebra]

[TheQntty]

What curriculum? High School? University? Or branches of math that are considered 'serious' branches by mathematicians?

[achan1058]

I would say the whole of discrete math, if you are talking about high school. Though it is regularly taught in university (with dozens of different branches and courses), high school never seems to touch on it. Furthermore, it's as real math as it gets, instead of telling the monkeys (I mean students) to run algorithms by hand.

[doogly]

Continued fractions are largely neglected. Lots of early geometry topics, like Italian style algebraic geometry, or Lobachevsky's hyperbolic geometry. Now these things are all done with fancier tools that are quite powerful, but obscure a lot of the history and motivation.

And if you are digging spherical trig, you could take a look at celestial navigation. If you acquire practical astrolabe skills I would offer you Mad Props.

Edit: Also, there are plenty of things that are accessible to someone without calc, but are neglected in the course of a math education until you have it. Things like set theory, number theory and group theory. Not at all obscure among mathematicians, but you could easily start dipping your toe into these areas years before your education would present them.

[t0rajir0u]

As far as the usual high school curriculum goes, discrete math is the biggie. This includes, as subfields: elementary number theory (which itself includes continued fractions, basic Diophantine equations, ...), graph theory (which includes networks, circuits, ...), poset theory, error-correcting codes, finite geometry, elementary combinatorics (no, you don't ever actually learn combinatorics in high school; maybe you learn "probability," but even then, not really), ...

Most of the reason for this is likely to be historical. The standard American curriculum was largely set in stone long before people realized that discrete math was important (i.e. before computers existed) and before a lot of it existed, so you're not likely to learn anything that people didn't already know hundreds of years ago, even though newer math doesn't necessarily mean harder math. Poset theory, in particular, I think is a relatively recent development even though it's both extremely intuitive and very useful in other branches of mathematics. (For example, the open sets of a topology form a poset under inclusion, so poset theory is relevant to topology. The natural numbers form a poset under divisibility, so poset theory is relevant to number theory. And so forth.)

Group theory doesn't really fall under "pre-calc" for me; while it doesn't require any calculus to understand, it does require some mathematical maturity to appreciate. You should ideally have a pretty strong intuition for what a homomorphism is (even if you don't know what the term means) and should be comfortable with the notion of proof. Clearly there are plenty of bright students who haven't learned calculus who would do just fine in a group theory course, but all the same I take your question to mean "what branches of mathematics could easily be taught in high schools but aren't?"

Edit: And set theory is in its own special category. Most mathematicians would argue that you don't really understand mathematics until you understand set theory, since that's how most of mathematics is axiomatically built up these days. (I say "most of" because there are interesting exceptions, but they would take awhile to discuss.) So depending on mathematical maturity this should either be at the top of your list for high school or the first thing you learn in college.

[btgreat]

For anyone interested in math (even a high schooler), and willing to sit down and go through a good set of exercises, I think elementary number theory, set theory, and basic algebra are all possible, out of the ordinary (at least for high schoolers) branches of math that could be explored by someone before seeing calculus. Ultimately, a good introduction to proofs is much more important than an understanding of limits, etc. for nearly any non-continuous math (i.e. anything but analysis) - calculus really won't help you learn much of the interesting math that is out there.

A couple years ago, before I had seen much in the way of higher level math, I grabbed a book called How to Prove It, by Daniel J. Velleman. The book uses elementary set theory to teach the reader about different proof techniques, and gives a good introduction to the common language and symbols thrown around in typical math proofs (or texts that assume knowledge of proofs). I found the book to be excellent preparation for the first largely proof-based math class I took. A typical high school student could almost certainly use the introduction to rigorous proof.

I disagree about group theory not being a precalculus class - I don't think it is inaccessible to someone with less mathematical maturity, it just requires a more gentle introduction (e.g. stay away from texts intended for graduate students). Because the ideas of homomorphism and isomorphism are so important in math, it is (in my opinion) very important that someone see these ideas as early as possible. I don't know if you are interested in studying the fields you are asking about, but if you are, I think linear algebra would be the best thing you could study. It's applications come up everywhere in later courses in algebra and number theory, and in examples from nearly every branch of mathematics. It also has many real world applications if you care for that too.

[vilidice]

I think a lot of functional analysis (which is more of a subset of set theory and linear algebra) is lacking in many areas, as I never was introduced to the concept of a function in its real form until calculus, and it's a hugely powerful mathematical construct that current algebra and even a lot of pre-calculus courses really don't get into the level of detail that they could, and possibly should (especially if we want more scientists and electrical engineers), and it really takes no calculus to understand, but would help many people understand calculus better, however to cover everything in the subject to its fullest extent does require a lot of math theory usually reserved for algebraic structures courses in college, however the direct applications of the concepts in this type of mathematics don't fully come out until higher level differential equations and signal analysis (however anything you've done in physics dealing with harmonics was derived in some way using this).

As said before learning types of proofs is essential for mathematics and helps to understand almost everything in higher mathematics, calculus based or not, one thing you'll probably notice as you continue further into theoretical mathematics is that actual numbers become less involved in most cases, and proofs dominate everything, so having that as a background would also be a good place to start.

also Euler's work and manipulations of the natural exponential give a good introduction to complex analysis which doesn't require much calculus, mostly trig functions and their exponential equivalents, also graph theory, and many mechanics problems, which through some amount of derivation come out to some very elegant, calculus-free solutions.

In general though most higher mathematics come from Set theory, algebraic structures, proofs, and vector spaces (which really aren't covered that well in most introductory linear texts because it gets lost in the general linear methods and the amount of material they need to cover for courses so I'd recommend getting a book more focused on that area once you understand the general concepts and methods of linear algebra if you intend to pursue it.), so getting a solid understanding of those concepts will probably make venturing into calculus based mathematics much easier, as well as really understanding the intricacies of any numerical theory and spacial geometry.

[Incompetent]

At school (in the UK at least)? All of pure mathematics is ignored. The kids only get taught mathematical methods, the kind of thing you'd learn in training to be an economist, physicist or engineer.

[the tree]

Lies! There is like 3 marks on the GCSE paper for being able to correctly classify a number as rational or not, that is slightly pure maths.

[TheQntty]

Do you have to provide of proof of irrationality/rationality?

[Zamfir]

At school (in the UK at least)? All of pure mathematics is ignored. The kids only get taught mathematical methods, the kind of thing you'd learn in training to be an economist, physicist or engineer.

If you compare the number of economists, physicists and engineers the world has or needs, and compare it to the number of pure mathematicians it needs, and this suddenly sounds like a very wise policy.

[Talith]

I assume you're refering to the GCSE curriculum rather than the A level one. Because Maths at A level is much more proof orientated.

[Incompetent]

I assume you're refering to the GCSE curriculum rather than the A level one. Because Maths at A level is much more proof orientated.

No, I'm not. I did A-level Maths and Further Maths, and unless they've changed radically in favour of pure maths in the past 7 years (and I doubt it, given the impressions some of our current crop of undergrads have of 'proof'), I stand by my claim that they have almost nothing in terms of pure maths content - practically everything was (is?) presented as specific methods to solve specific problems.

@Zamfir: I know why it's done. (I happen to think though that engineers, economists and the like would learn mathematical methods specific to their trade better at university, so they'd also benefit from a more general grounding in maths at A-level. The A-level system as it is seems to be designed for people who finish their education at 18.) I just wish they wouldn't lie when they call A-level maths modules 'pure'.

[Zamfir]

I happen to think though that engineers, economists and the like would learn mathematical methods specific to their trade better at university

But the stuff needed by them is very uniform: good algebraic manipulation skills and basic calculus are needed everywhere, not in one specific field. The more general grounding is, perhaps paradoxically, needed by far fewer people. Computer science has probably created a trade that needs different kinds of basic math, and perhaps high school curricula should faster adapt to that. On the other hand, it seems programmers rely less on discrete math in their day-to-day work, compared to the use of calculus by engineers. But I could be mistaken there.

[achan1058]

But the stuff needed by them is very uniform: good algebraic manipulation skills and basic calculus are needed everywhere, not in one specific field. The more general grounding is, perhaps paradoxically, needed by far fewer people. Computer science has probably created a trade that needs different kinds of basic math, and perhaps high school curricula should faster adapt to that. On the other hand, it seems programmers rely less on discrete math in their day-to-day work, compared to the use of calculus by engineers. But I could be mistaken there.

Loops are induction, and ifs are logic. Programmers use these everyday.

[Zamfir]

Loops are induction, and ifs are logic. Programmers use these everyday.

Yes, that;s very true, and the I guess abstracted logic is a prime candidate to teach more in high school, now that it has moved from a philosopher's tool to an everyday practicality.

[diotimajsh]

Yes, that;s very true, and the I guess abstracted logic is a prime candidate to teach more in high school, now that it has moved from a philosopher's tool to an everyday practicality.

I am highly in favor of logic being taught as a mandatory part of high school education. I guess if we're going to argue that it's a useful tool for the layman, though, we should actually get some empirical studies going to show how it impacts other areas of life.

[t0rajir0u]

I disagree. The way things are usually taught, logic would go the same way as everything else - memorize some rules (de Morgan's laws, for example) and apply them without ever understanding them. There's a big difference between teaching logic in the abstract and teaching people how to think.

(For example, ideally the distinction between converses and contrapositives should be intuitive. You shouldn't have to be taught the words to understand the concepts if your thinking is solid.)

[Zamfir]

Yes, that;s very true, and the I guess abstracted logic is a prime candidate to teach more in high school, now that it has moved from a philosopher's tool to an everyday practicality.

I am highly in favor of logic being taught as a mandatory part of high school education. I guess if we're going to argue that it's a useful tool for the layman, though, we should actually get some empirical studies going to show how it impacts other areas of life.

We teach other things in high school that have little use to laymen. Calculus would be the prime example, and the reason to teach it anyway is to some extent that the fraction of people who will use it is relatively high. Logic, of a CS type, might fall in the same category. On the other hand, I think there is, or should be, a very valid debate whether calculus should be taught in high school. For many people it is something they will never ever see again, more than almost any other high school subject I can think of.

I disagree. The way things are usually taught, logic would go the same way as everything else - memorize some rules (de Morgan's laws, for example) and apply them without ever understanding them. There's a big difference between teaching logic in the abstract and teaching people how to think.

There are two related, but separate reasons to teach logic, either from a rethorical/philosophical side about proper arguments, or the programming application, for designing effective IF conditions etc. Both use more or less the same concepts, and both are relevant today, but the way to teach them is very different. XORs are the great example difference: Completely irrelevant for the first, automatically included in the second.

[t0rajir0u]

I think there is, or should be, a very valid debate whether calculus should be taught in high school. For many people it is something they will never ever see again, more than almost any other high school subject I can think of.

Agreed. If you're planning to go into the soft sciences, statistics should be stressed over calculus, and if you're planning to go into the hard sciences, you'd be better off learning calculus in college in the context of whatever else it is you're studying (biology, chemistry, pure math). Most people don't gain anything from being taught calculus, and math types would gain more from a thorough grounding in set theory and proof.

While we're on the subject, what do you think is the most useful high school subject? I think, for those who master it, it's learning how to write properly.

There are two related, but separate reasons to teach logic, either from a rethorical/philosophical side about proper arguments, or the programming application, for designing effective IF conditions etc. Both use more or less the same concepts, and both are relevant today, but the way to teach them is very different. XORs are the great example difference: Completely irrelevant for the first, automatically included in the second.

Mathematical logic is what I was thinking of - first-order propositional calculus, etc. It happens to tie into a lot of other pure math ideas in a very interesting way - XOR, for example, is the same thing as addition in a vector space over F_2, so any logic problem phrased in terms of XOR alone can be solved using linear algebra. (ANDs are harder.) Pretty much irrelevant to most people. But I think philosophy would be a good alternative.

[Incompetent]

One could equally well argue that it's useless to teach kids history, or art, or literature. The reason we do is that these are seen as important to our culture and hence to education in the broader sense (as opposed to just training). Why are maths and philosophy not seen in the same way? Sure, maths is useful as well, but I think someone who knows nothing of maths is culturally impoverished as much as they are materially impoverished.

[crazymike811]

so you say you know of linear algebra, have you explored it at all?
that's got to be one of my favorites so far, once you get past the foundations and see how many actual real life applications are available, it's really astounding, I've only taken one course in it and it freakin rocks! I'd also have to agree that a strong foundation in proofs, with a serious emphasis on what it means to be grammatically correct when writing mathematically is crucial. Linear Algebra I and proofs can easily be approached without a firm basis in calculus, although any understanding of calculus can't really hurt.

[achan1058]

I agree that basic logic, statistics, and probability should be taught more. In particular, logic should be taught like in philosophy (which involves analyzing word arguments). The reason for statistics and probability is simple, to (hopefully) steer people away from compulsive gambling. The other reason is that probability is very counterintuitive, so it is best to have some real knowledge into it so that you can judge things properly.

[Zamfir]

While we're on the subject, what do you think is the most useful high school subject? I think, for those who master it, it's learning how to write properly.

I think that is a good point, especially because it can be taught at a wide range of levels. No matter how smart or experienced you are, you can always be better at the exposition of your ideas, and it is always valuable. And it also works in passive mode, since being better in writing clear, convincing texts helps when reading and judging other people's texts.

For mathematical subjects I would move away from subjects that are useful for active practioners.Calculus and formal logic are topics that are mostly encountered by people who use them theirselves, which means a minority of people, and other people will gain very little from their high school experience. Statistics on the other hand has become so important for all kinds of decisions. Passive users who only read the results of statistical analyses should still have had some active experience. It would be great if a lot of newspaper-reading people had done some linear regressions in school, to get a tiny feel for the strengths and the weaknesses of such approaches.

For similar reasons I would like to see more computer programming in high schools, just so that people have an idea what happens inside and how hard or easy it is to do certain things. But I think I underestimate how hard these things, both statistics and programming, can be for people with little feel for them.

[MHD]

Meta-Meta-mathmatics.

Now you probably know math: Numbers, operators, sets, calculus, etc.

Now you probably know meta-math too: Algebra, function operations, lambda calculus, etc.

Now you probably don't know meta-meta-math:
The theories on how our mathematical system is put together, how it is even possible to define a function, how to conceive algebra, how functions moves numbers between limited yet infinite parralell planes, etc.

[t0rajir0u]

how functions moves numbers between limited yet infinite parralell planes, etc.

I don't know what you mean by this. It sounds like by meta-meta-mathematics you mean set theory, which a lot of people would just consider mathematics, whereas a mathematician would probably call meta-mathematics something more along the lines of model theory and reverse mathematics and quite arguably the term "meta-meta-mathematics" is either quite obscure or quite useless.

[the tree]

I just wish they wouldn't lie when they call A-level maths modules 'pure'.

The 'core' modules stopped being called 'pure' shortly after the end of the last ice age. The 'further pure' kept the same name because in the grand scheme of things, no-one takes those modules anyway.

One could equally well argue that it's useless to teach kids history, or art, or literature. The reason we do is that these are seen as important to our culture and hence to education in the broader sense (as opposed to just training). Why are maths and philosophy not seen in the same way?

Very simple. No-one believes you when you start telling them about maths.

Now you probably don't know meta-meta-math:
The theories on how our mathematical system is put together, how it is even possible to define a function, how to conceive algebra, how functions moves numbers between limited yet infinite parralell planes, etc.

Yeah, all of that is just regular maths.

[MHD]

how functions moves numbers between limited yet infinite parralell planes, etc.

I don't know what you mean by this. It sounds like by meta-meta-mathematics you mean set theory, which a lot of people would just consider mathematics, whereas a mathematician would probably call meta-mathematics something more along the lines of model theory and reverse mathematics and quite arguably the term "meta-meta-mathematics" is either quite obscure or quite useless.

It is kindof hard to descibe, but I am definitely sure it's not set theory (I undestand set theory, sort of).

It is currently a research matter and not yet in publicity.

[doogly]

Category theory? That's fairly meta, but of course still math.

[Billy]

I would recommend:

• Hyperexponentiation, such as tetration or Knuth's up arrow notation.
  
• Digital signal processing: stuff like dicrete convolution, discrete fourier transforms, etc.
  
• One cool thing to know is how to take the cross product of two seven dimensional vectors, or the n-1ary cross product of n-1 n dimensional vectors.
  
• Quantum theory: kronecker products, bra-ket notation, inner and outer products, unitary matrices, etc.
  
• While not too obscure, big O notation and its relatives are good to know.
  
• Game theory. Again, not too obscure, but good.

Hope this helps!

[tomtom2357]

I would recommend:
[list]
[*] One cool thing to know is how to take the cross product of two seven dimensional vectors, or the n-1ary cross product of n-1 n dimensional vectors.

Wait, I thought that you could only take the cross product of two 3-dimensional vectors.

[doogly]

In 7 dimensions you can do something kind of like the cross product, but it isn't a unique operation.

[jestingrabbit]

but it isn't a unique operation.

Neither is the usual regular cross product (right handed vs left handed).

[doogly]

Unless you include right handedness in the definition. Which probably requires some extra input information, like a notion of coordinates on the space. Hmm, now I already feel dirty this early in the morning.

[mike-l]

Unless you include right handedness in the definition. Which probably requires some extra input information, like a notion of coordinates on the space. Hmm, now I already feel dirty this early in the morning.

You don't need coordinates, just an orientation. Which isn't much of a stretch, as you already require a notion of angle and size.

[doogly]

How do you practically set up an orientation without coordinates? All changes of basis from one set of coordinates which preserve the orientation form a neat little family. If you have R^3, and you want to say "this family of coordinates are the right handers, I want these," is there a way to point out which family you are talking about without giving one of the coordinate systems explicitly?

[mike-l]

Well, if you're doing any calculations at all, I assume you have something you're working with, and I can just tell you the orientation of one item you're working with, and that gives you everything. If you aren't doing the actual calculations, I'm handing you the whole damn function {Bases} -> {+/- 1}. I can give you the orientation as coordinate free as what you need it for.

[doogly]

I guess this becomes a sort of weird question. Let me pull it back.

If you *can* just pick a basis and say that is the one that goes to +1, but you don't want to ever actually explicitly give any, are you now stuck using the axiom of choice, or something like this? When you, for the sake of abstraction, decline to be explicit, but very easily could, what goes on?

Cause it looks to me like you can easily say there is a map
f: {bases} -> {-1,+1}
but you need to sort of "anchor" this map with one specific basis, or else if I have a map g that purports to do the same thing, we won't have any way of knowing if our f and g are the same or opposites.

[mike-l]

We certainly don't need the axiom of choice. You only need that when you want to do infinitely many things and don't have a way of doing it easily. Eg, you don't need the axiom of choice to pick one of each of infinitely many shoes, as you can always pick the left one. But you do need the AoC to pick one of each of infinitely many socks, as there's no way to tell one sock from the other without arbitrarily picking one each and every time. (To further clarify, you need only make one arbitrary choice with the shoes, left or right. You need to make infinitely many arbitrary choices with the socks. AoC says we can make infinitely many arbitrary choices. We can always make finitely many)

In our case, we're like the shoes, I can tell if 2 bases are the same orientation by taking the determinant of the transformation between the two.

In terms of anchoring it, it's not that I'm not letting you evaluate it anywhere, it's that I'm thinking of it as a whole with no 'special' basis. When you actually want to 'use' it, one of two things will happen. Either you'll be working with coordinates, in which case you evaluate it once and be done with it, or you aren't, and it's good enough in function form.

[doogly]

So how are you getting these determinants?
This sounds like I should just look at linear algebra done right, the Axler book. It seems like the coordinate free thing in the case of orientation is more like not admitting to using coordinates, but having them in the back of your head or up your sleeve the whole time.