The problem is that high school math and basic calculus is not much mathematics at all, and it still has the problems of foreign notation and ambiguity.
HS math and calc a is some math, definitely not all, but a good chunk of what I would call "applied math"---the math that links directly to the real-world, meaning the concepts can be understood intuitively, e.g., the connections to physics.
The trick is to covertly throw in some formal proofs to prepare the reader for the more abstract stuff. Going directly into proofs might be too much of a jump for some people, though I agree that, ultimately, this is what mathematics is about.
> HS math and calc a is some math, definitely not all, but a good chunk of what I would call "applied math"---the math that links directly to the real-world, meaning the concepts can be understood intuitively, e.g., the connections to physics.
IMHO there's little mathematics that can't be applied to the real world rather directly. I even heard mathematicians say that the distinction between pure and applied mathematics is mostly for historical reasons. Especially if you consider connections to physics as intuitive understandability - for example in string theory or quantum field theory you'll find lots of highly complicated mathematics.
I even heard mathematicians say that the distinction between pure and applied mathematics is mostly for historical reasons.
It must have been applied mathematician, for I cannot even imagine pure mathematician uttering anything like this.
I mean, seriously, I know quite a lot of modern pure mathematics, and for most of it I cannot picture even far-fetched connection with anything existing in real world, not to mention an actual application in solving some problem that wasn't created only for this application.
Most of the time, for almost every field of mathematics, the way it works is that for an enormous amount of knowledge, and enormous amount of research happening and results and papers being published, only very, very small amount will actually get applied any time soon (soon as in next 200 years). For some fields, like calculus, or probability, or partial differential equations, amount of applicable stuff is larger (mostly it's just old, one or two centuries old stuff anyway), and for other fields, like say homological algebra, or algebraic topology, or descriptive set theory, the applicable stuff is almost nonexistent - I'll buy a beer to anyone who'll point me to an application of descriptive set theory to any real life problem.
Almost all of the mathematics existing is purely abstract, and not applicable to real world, and I find it hard to even imagine anyone trying to argue otherwise. One can argue that what now is considered abstract and not applicable can become very useful in real life problems in future, and indeed, it happens quite often, but I think it will not be much, since we're applying to real life problems only a small fraction of 200 years old mathematics today anyway.
I don't think it's fair to say some idea is not applicable because you don't know of an application. Number theory was like that before the advent of computers, and I'm a firm believer that there are thoughts that cannot be thunk until the right framework comes along to allow it.
That being said, a lot of these pure subjects do have applications. For example: algebraic geometry (the crown jewel of pure mathematics, my colleagues would have be believe) has applications to tons of industrial problems in the form of solving systems of polynomial equations (see homotopy continuation). Algebraic geometry has also been applied to robot motion planning, etc.
Descriptive set theory is applied in functional analysis and in ergodic theory, which in turn is applied to statistical physics. Not to mention that descriptive set theory is the pure-logic equivalent of computational complexity theory, and that there is potential to connect the two fields and resolve some big open problems (though it's doubtful that P vs NP will be resolved this way).
And almost all of modern physics is based on more or less modern mathematics: tensor analysis and other flavors of linear algebra, lie theory, etc. Algebraic topology is starting to find some traction in the subfield of persistent homology, which aims to study high-dimensional data sets in the context of homological algebra. I even gave a talk earlier this year on the concrete attempts people have made to apply persistent homology to real-world problems [1]. It's still an extremely young field, but shows some promise.
I'll give you that mathematics is extremely abstract, because I believe it. But to say that it's not applicable and being applied wherever possible is a bit naive. And to say that it's only 200 year old mathematics is to ignore the most applicable fields which did not exist even a hundred years ago: combinatorial optimization, mathematical computer science, and modern statistics and probability theory.
I must have not communicated what I meant clearly, because you missed my point.
Only a very, very small fraction of results in number theory are applicable. Only a very, very small fraction of stuff done in algebraic topology is applicable. While descriptive set theory is indeed sometimes applied in functional analysis, the intersection between these two is not a significant fraction of each one of them, and by the time you apply (a very small fraction of) functional analysis to statistical physics, you're already too far from descriptive set theory to even see it on the horizon.
I'm not saying that none of the mathematics is applicable, because this is obviously not the case. What I'm saying is that the stuff that gets applied to real life problems is surprisingly small, even more so when you're not a mathematician.
When I first started to learn mathematics, I was completely overwhelmed, when I found out just how much knowledge is out there in this field of human activity. The planes of mathematics are so vast, there's almost nothing else in sight when you stand atop of the Mount Bourbaki. I realized that even if I get a PhD in pure mathematics, I will still only be able to learn less 1% of mathematics ever created in my whole lifetime. That's why when I hear people saying that all math can be applied, I think that they must have not realized just how much of the stuff is in there.
I used to specialize in algebraic topology, and when I first learned about persistent homology, I encountered a home page of the professor at some US faculty, who specialized in applying algebraic topology in real life, and the first reaction of me and my classmates whom I have shown his website was not appreciation of his results. We were totally amazed that this stuff can be applied to anything _at all_. Yeah, some of these applications were to the problems that are usually solved in a better way, some problems were very contrived and seemed to actually be tailored so that one can apply algebraic topology to them, but still these were very fine and interesting results, and we were very surprised by them.
The field of algebraic geometry actually makes an interesting example. Indeed, it is considered by many one of the most, if not the most abstract field of the mathematics. Initially, at the beginning of the previous century, people were mostly concerned with studying the sets of solutions of systems of polynomial equations. David Hilbert with his landmark results being his basis theorem and Nullstellensatz laid fundamentals to this field, and because of an essential assumption in the Nullstellensatz theorem that creates a bridge between algebra and geometry, for many decades most of the results concerned only polynomials and sets of solutions in algebraically closed fields, because for anything else, the apparatus was just lacking. Algebraic geometry didn't have a reputation of an extremely abstract field, especially since sets of solutions to systems of polynomial equations are quite natural objects, and it's easy enough to imagine their occurrence in real life problems.
In 1950s and 1960s, though, the field was completely and utterly revolutionized by Alexander Grothendieck and his school, and that's when the field gained its fame of being very esoteric. Grothendieck's methods and approach allowed algebraic geometers to tackle vastly bigger range of problems, and ultimately to efface the distinction between algebra and geometry, at the price of making things much more abstract and distanced from more concrete considerations. That's when algebraic geometry expanded its reach, to encompass a large amount of research in abstract algebra and number theory.
In the meantime, another interesting thing happened in the field: the advent of computational techniques. Things like Groebner bases really pushed things forward and made a lot of theoretical stuff actually doable in practice. This is mainly what made things you mention in your post possible: while many industry problems could be formulated in geometric terms earlier, only rise of computational methods actually allowed us to solve them.
The point here is this: algebraic geometry field consists of two parts, the older and more concrete, which can and is (relatively) frequently applied, and the newer, but more abstract, of which applied is very little -- with notable exceptions though being for instance finite elliptic curves with well known application to cryptography, or Calabi-Yau manifolds which are intimately connected to the string theory. Nevertheless the applied stuff constitutes only a small part of the field, the rest is just pure and abstract mathematics, without any applications whatsoever, though it's still worth noting that algebraic geometry is still relatively very good in the amount of applicable stuff, and fields like algebraic topology, not to even mention descriptive set theory, fare much, much worse with regard to it. For me, a bit naive is to think that big part of mathematics will be applied, when there's so much of it.
I'm aware of the categorical revolution, since I'm young enough to have been (mathematically) raised on that perspective. Some of the more abstract algebraic geometry is actually coming back to computational applications. See, for example: http://www.researchgate.net/publication/226664601_From_Oil_F...
To say that something is not applicable is hard to argue. And besides giving examples of when it is applied (and you claiming it's still an unimaginably small fraction of mathematics) all I can say is that the understanding of some object can provide insights and applications in unexpected ways. Dynamical systems inspire computer graphics, Mobius bands inspire carburetor belt design, category theory inspires Haskell... I just don't think it's fair to ask for the immediate applications of any given theorem because the ultimate application is understanding what's going on.
Unluckily I know little about homological algebra, algebraic topology and descriptive set theory.
But I surely have already seen papers where methods from algebraic topology are applied for pattern detection or finding whether some high-dimensional data are homeomorphic.
Also simplicial complexes (that are used in homological algebra and algebraic topology) can also come from triangulating high-dimensional point sets (for example from big data). I could well imagine that methods from homological algebra or algebraic topology could give us some insight into some properties of this data.
