What are they and what are some examples of abstract structures?

Question

I am new not only in the forum but also in the world of mathematics and science in general. I am a complete newbie, but I consider myself brave and persistent. Also I am not a native of English, so I hope you have patience :) For different reasons I am very intrigued with mathematics, and reading a book I had some doubts: In mathematics, what are abstract structures? Not all? Mathematics is the science of structures. Very well yes, but the author of the book mentions that abstract structures are the essence of thought, communication and life in general ... That made me wonder if the language we use is an abstract structure, but according to wikipedia is not, although I am not sure I understand since the translation of that wikipedia page did not seem good to me.

This is the page: https://en.wikipedia.org/wiki/Abstract_structure

So, on that page they use a term that I'm not really familiar with, it is "formal object", but when I searched wikipedia I just got more confused. And looking for more, I found that it is the way in which some subject of a science is studied ... The book that led me to this question is "Language of Mathematics" by Keith Devlin. I just started reading it.

What's more, this is just what it says in that part of the book:

"In today's age, dominated by information, communication, and computation, mathematics is finding new locks to turn. There is scarcely any aspect of our lives that is not affected, to a greater or lesser extent, by mathematics, for abstract patterns are the very essence of thought, of communication, of computation, of society, and of life itself."

I would like you to help me clarify this little issue. I hope you can help me, thank you very much in advance.

Answer 1

Abstraction is everywhere in Mathematics, it's a powerful tool and it allows for generalization of rules and theorems. There is a lot of debates of whether some things mathematicians study are abstract, man-made, or already in nature (for example numbers) but to mathematicians this doesn't really matter, it's more of a philosophical matter. Here is a quote in one of Terrence Tao's books, he is a mathematicians who is considered to be the best worldwide by many:

We have not told you what the natural numbers are (so we do not address such questions as what the numbers are made of, are they physical objects, what do they measure, etc.) - we have only listed some things you can do with them and some of the properties that they have. This is how mathematics works - it treats its objects abstractly, caring only about what properties the objects have, not what the objects are or what they mean. If one wants to do mathematics, it does not matter whether a natural number means a certain arrangement of beads on an abacus, or a certain organization of bits in a computer’s memory, or some more abstract concept with no physical substance; as long as you can increment them, see if two of them are equal, and later on do other arithmetic operations such as add and multiply, they qualify as numbers for mathematical purposes (provided they obey the requisite axioms, of course).

This applies to a lot of other things in Mathematics, mathematicians just don't bother with the philosophical side of things.

It's true that a lot of basic Mathematical things started based on real-world things but it eventually got more abstract and complex (mainly as it helps generalizing them), everything you study in Math is abstract but some of the things can be concrete depending on the context (this is in the branch of Mathematics called "Applied Mathematics" and in other fields such as Engineering and Physics). For example, you have most probably learned about the laws of addition, how to add to numbers, etc... When you get an exercise asking you to find the result of $3+5$ it's still in the abstract sense (what's really $3$ and $5$ here?) but when you go to the grocery store and take $3$ green apples and $5$ red apples and get $8$ apples in total then you are assigning concrete objects and meaning to these numbers but at the end of the day these are just abstract ideas that help us understand the world.

I gave you the example of numbers as that is what everyone learns in Mathematics first, I wanted to show you that even the basic things in Mathematics are in fact abstract, it just happens that you can give concrete meaning to some and not to others. Mathematics is mainly divided into 2 main branches: Applied Mathematics and Pure Mathematics. As the name suggest Applied Mathematics is where you would find Mathematicians giving concrete meaning to these objects and finding real world applications to all of this abstraction (this blends well into subjects like Physics, Engineering, Economics, etc...) while subjects classified as part of Pure Mathematics are usually things we didn't find applications for.

I hope this clarified things a bit, good luck with your journey.

Answer 2

For the sake of aiming to give an answer sufficiently acceptable, I will start by quoting the definition of abstract structure given in Wikipedia, I will then follow it by a couple of observations philosphical in nature and then I will try my best to re-direct these observations onto the realm of Mathematics to support and exemplify the definition.

Definition: An abstract structure is a formal object that is defined by a set of laws, properties and relationships in a way that is logically if not always historically independent of the structure of contingent experiences, for example, those involving physical objects.

You see, one of the characteristic traits of us humans is our power of abstraction; that is, our ability to comprehend not only that which is surrogated to the experience, but also those conceptual and intelligible constructs arising solely in our minds. The key point here is that not only we are able to comprehend the abstract, but we're also able to create, manipulate and transform these constructs at our will, and through the use of speech and writing we can share all of these to other fellow human beings.

This particularly remarkable faculty has created almost since the beginning of the history of thought a dichotomy of reality, namely that of the concrete reality which we experience and the abstract reality which we comprehend. This ontological issue has been widely discussed by various (if not all) philosophical trends, and the reason I'm mentioning it in this question is beacuse the notion of an abstract structure lies at the very bottom of this distinction of realities.

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Right, so before I start talking about abstract structures in Mathematics it would be interesting to see what the definition of a structure is. According to the Cambridge Dictionary:

Definition: a structure is the way in which the parts of a system or object are arranged or organized, or a system arranged in this way.

By our discussion above, we can then infer that an abstract structure is a structure of that intelligible reality mentioned above and not one belonging to our concrete experience. The issue now is the following: how can we organize a system of objects which we cannot even experience? Although it sounds slightly silly, this is a pivotal question. For example, a building, a book, a lamp and a song are all example of structures, but we rely on some kind of physical machinery to construct and experience them. So how do we do this for abstract strcutures? Here is where the formal object part of the definition comes into play; I will stat by giving a natural example of an abstract structure and then I will explain how these constructions are generally done in Mathematics.

In Egypt and Mesopotamia, people needed some kind of tool to keep track of various accountability issues, like taxes or food stock. To so do, they developed an abstract structure consisting of some objects (now called integers) and some operations between them (now called addition and multiplication) and over time, this structure was standardized to the point that we all can now agree on the same notation to refer to the same number system. So what is it exactly this number system? Well, it consists of:

• $\underline{Integers}$ ($\mathbb{Z}$): these are elements of the set $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.
• $\underline{Addition}$ ($+$): this is a binary operation (i.e. it takes two inputs and gives one output) which satisfies the following properties:

$$(a+b)+c = a+(b+c) \ \ \text{ for all integers } \ \ a,b,c.$$

$$a +b = b+a \ \ \text{ for all integers } \ \ a,b.$$

$$a+0 = a \ \ \text{ for all integers } \ \ a.$$

$$a + (-a) = 0 \ \ \text{ for all integers } \ \ a.$$

• $\underline{Multiplication}$ ($\cdot $): this is a binary operation (i.e. it takes two inputs and gives one output) which satisfies the following properties:

$$(a\cdot b)\cdot c = a\cdot (b\cdot c) \ \ \text{ for all integers } \ \ a,b,c.$$ $$a \cdot b = b\cdot a \ \ \text{ for all integers } \ \ a,b.$$ $$a\cdot 1 = a \ \ \text{ for all integers } \ \ a.$$ $$(a+b)\cdot c = (a\cdot c) + (b \cdot c) \ \ \text{ for all integers } \ \ a,b,c.$$ $$a\cdot (b+ c) = (a\cdot b) + (a \cdot c) \ \ \text{ for all integers } \ \ a,b,c.$$

This is an example of an abstract structure! Note that the notion of this being a formal object comes from the fact that in order to define it we have used some symbols (i.e. $+, -, \cdot, 0,1, -3\dots$, and even $(, )$ and $=$) which by their own they don't mean anything; $+$ for example doesn't mean addition unless I tell you what are the rules of addition and which are the objects that you are actually adding (in our case, integers).

Now you might be thinking: but Rick, what do you mean that $+$ doesn't mean addition? Of course it does, it's always been like that! Well, it's always been like that if you consider the time frame of your life span, and maybe even some centuries ago; as I said, this symbol now is so common and standardized that when we see a $+$ we immediatly know how to use it and what rules it obeys. However, I'm pretty sure Egyptians and Mesopotamians didn't use $+$ as a symbol for addition and they neither used the symbols $-1, -4, 0, 1, 23, \dots $ to denote the integers, and yet they were still using the same rules of addition as we use today for $+$ in the integers and they did pretty amazing Mathematics with that and more. After this, I suggest you to go and read again the definition of abstract structure and see if makes more sense now.

More generally, in Mathematics we start by using some kind of formal language or collection of symbols which by their own do not mean anything, just like $\rightarrow, \sim, \oplus, \cup$ (this is where the formal object bit of the definition comes from) and we then impose certain rules (usually called axioms) to these symbols so that this combination of symbols with rules enable us to define certain mathematical objects of our interest. In the case above, a structure obeying the properties described above is called a (commutative and unital) ring, and if we restrict ourselves to $\mathbb{Z}$ together with addition we call the resulting structure an (additive abelian) group; some other examples of rings and groups are given by replacing $\mathbb{Z}$ with other sets of numbers such as the rationals, the reals or the complexes, and all of these are again abstract structures (in fact, they are called algebraic structures).

Just to conclude and to reiterate my point, all mathematical objects are examples of abstract structures since in order to define them one needs to make use of some kind of formal language or collection of symbols and then one needs to specify the rules that these symbols obey; this construction is precisely what makes mathematical objects "independent of contingent experiences" and what in a very big part provides rigour and exactness to all Mathematics.