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Some days ago, a friend - who's studying civil engineering - told me something about the mathematics of the theoretical mathematics degree:

It's strange, their math is so abstract.

What should be the meaning of this? This reminded me of programming languanges - in which the programming languanges "nearest to the machine" are "less abstract" and when languanges are built upon this ones, they're called layers of abstraction, the more layers, the more abstract is the languange. (Notice that I'm not so sure about this information.)

Considering her argument, is the mathematical meaning of abstract similar to the meaning of abstract in programming languanges?

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    "Abstract mathematics" generally describes mathematics which has no immediately obvious practical application. For example, [this](http://en.wikipedia.org/wiki/Higher_category_theory).2012-09-13
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    The civil engineer might consider very abstract the kinds of mathematics routinely used by people in other branches of engineering.2012-09-13
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    I'm in Engineering Physics: to a civil engineer something like a Hilbert space might seem pointlessly abstract, but for me it's essential in the design of anything in my field. There is (currently) useless math, but aside from very specific examples most people (including myself) couldn't read the titles of such papers. I can almost guarantee that any math in an undergrad math degree has application.2012-09-13
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    Perhaps this is another perspective: http://www.uiowa.edu/~030116/prelaw/whymath.htm2012-09-13
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    Relevant: [category theory](http://en.wikipedia.org/wiki/Category_theory).2012-09-13
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    Downvotes for no reason. $$\left(\frac{\odot \odot }{\wedge } \right)$$2012-09-13
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    *Downvotes for no reason*... How do you know? Downvotes for no *stated* reason, which is quite different. (I neither upvoted nor downvoted.)2012-09-13
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    @did Yep, it's obviously the second one. Considering I'm not a soothsayer.2012-09-13
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    I do not understand if you mean abstraction of programing languages in a concrete technical sense or a rather loose figurative sense. But, mathematics is no more abstract to a mathematician than playing a (difficult) piece of music to a pianist. Of course to an outside r it looks strange and esoteric.2012-09-13
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    Then why do you see fit to say so?2012-09-13
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    I've met engineers for whom anything above the multiplication table of number 12 and/or any operation involving more than two bottons in a calculator is "abstract mathematics"...2012-09-13

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The answer to the question at the end of the body is yes.

As Gerry said, the story about counting layers as a measure of abstraction doesn't hold much water for mathematics, and I don't think it works very well as a classification of programming language either. In fact, the point of explicit specification abstract of programming languages is that in principle you don't know at the time you write a program how many layers will be involved in executing it, as long as the topmost of them present the same interface to you. There certainly is an (informal!) hierarchy of languages that are more or less close to the raw hardware, but it cannot be expressed as crudely as counting layers.

It still makes excellent sense to think in terms of layers when understanding a computer system, but the number and identity of layers are not so well-defined as certain textbooks (or the infamous OSI reference model) would have you believe. And real life is ripe with instances where a "low-level" service is implemented on top of "high-level" services in underlying layers (protocol tunneling, simulators, virtual machines, virtualization, etc), which again makes counting the layers an extremely inexact science.


However, there's a quite separate concept of "abstraction" in software development within a single programming language, and that matches the mathematical idea of abstraction quite well. In both cases the idea is to explicitly decide on a small set of attributes of the thing under consideration that you care about, and then make sure that your work depends on those attributes only. The meaning of "abstract" in the mathematical idea of an abstract group is very close to the meaning of "abstract" in the programming concept of an abstract list type.

Certainly, mathematicians and programmers tend to have different motivations for using abstractions. In mathematics, the eventual reason for considering an abstraction is often to reach a better understanding of each concrete instance of the abstract structure, by using the abstraction to transfer intuition and arguments between its different instance -- whereas in programming, the instances of an abstract type are just tools, and the eventual goal is what we do with them. Here the reason for abstraction is simply to manage the complexity of large software systems, and make it easier to predict and contain the consequences of changing one piece of it.


But neither of these senses seem to be be directly relevant to characterizing pure mathematics as "strange" and "so abstract". Here I think the matter is that most of the abstractions that contemporary research considers look very unmotivated to outsiders. Outsiders have relatively easy access to the technical definitions -- you can go to Wikipedia and learn exactly what a left semi-symplectic Smith manifold over a topological division algebra is, but the article probably won't even attempt to tell you what it's good for. It can be hard to figure out even one example of such a thing that makes all of the conditions mean something.

For some reason, the motivation for studying things tends to be communicated among mathematicians in a much less structured and accessible way than technical details, and therefore it can easily look from the outside like the technical details are the motivation, complexity for its own sake. And even when you do dig down to find the real motivations, the study of left semi-symplectic Smith manifolds over topological division algebras often turns out to be just a link in a long chain of conjectures that after five or eight similarly arcane steps would lead to a proof of something intuitively interesting. (Here is something that could be parallel to your "layers of abstraction", though the relation between the various steps is not necessarily a well-founded "X builds on Y" kind of thing).

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    Very good. Concerning abstraction in programming languages, I took OP's word for it, since I didn't know enough about it to argue. Clearly, you do.2012-09-13
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    I suppose abstraction of some items as some patterns and similarities between the objects (and adding some structure to the patterns/similarities will lead to the items themselves). Is my view incorrect (with respect to the mathematical abstraction)? What do you mean (some examples would be good) by the word "attribute" the the italic quotation on abstraction ?2016-04-29
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The answer to the question at the end of the body is, no. There is no direct relation between the degree of abstraction in Mathematics, and the number of layers it rests on. HIghly applicable mathematics can be built upon multiple layers of other applicable mathematics, and highly abstract mathematics can be constructed from scratch.

The first question in the body seems to be, what did your friend mean by what he/she said, and I suggest that your friend is the person to ask.

The title has a question mark in it, but I can't tell whether you are asking what is the point of abstract mathematics, or whether you are just quoting someone.