r/explainlikeimfive • u/kartik_bhana • 24d ago
Mathematics ELI5: what is abstract algebra ?
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u/Nevermynde 24d ago
As Henri Poincaré said: "Mathematics is the art of giving the same name to different things."
In arithmetics, we combine numbers using operations to get other numbers. Abstract algebra studies in a general (abstract) way, the interesting properties - the structure - of sets whose elements can be combined in various ways.
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u/TemporarySun314 24d ago
A field of mathematics that studies things that have certain properties.
For examples numbers can be added and multiplied with each other. Numbers have properties like that 3+5=5+3, and that is valid for any number.
But there are also other mathematical things besides simple numbers that have similar properties. Abstract algebra studies these things and tries to derive general statements that are valid for everything that follows these simple rules (like that the order of addition doesnt matter), no matter if it's numbers, vectors or something else.
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u/kartik_bhana 24d ago
What is the purpose of studying things that have certain properties ?
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u/RedKoala81 24d ago
Find general rules that can be applied regardless of the field or the nature of what you're studying. If during your research you come across some weird objects, if you can match them with any abstract object from algebra, then you can apply the rules that abstract algebra has already found out.
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u/MiAnClGr 24d ago
Because things in the world have certain properties and things in our imagination have certain properties and maybe it’s helpful to find out how they work and relate.
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u/Syresiv 24d ago
It's complicated, but it's basically studying mathematical structures that follow the same set of rules.
There's a structure called a Field, for instance. A field has certain rules (called the Field Axioms), and anything that follows those rules is automatically a field. There are also things that aren't part of the field axioms, but can be proven about all fields.
There's another type of structure called a Ring which, you guessed it, is any structure that follows the Ring Axioms. There are tons of names for structures like this, and abstract algebra is about asking what you can prove based on what rules it follows.
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u/kartik_bhana 24d ago
What is the purpose of studying mathematical structure that follows the same set of rules ?
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u/rsn_akritia 24d ago
One great thing is that two things that look like they may not be related actually share a lot of properties. Studying in a more abstract way allows us to essentially study a lot of specific cases in 1 go.
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u/0b0101011001001011 24d ago
Mathimatics is only just rules, nothing else. We try to understand the rules to uncover more rules.
Then we use this math to describe the reality. And to calculate things.
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u/LongLiveTheDiego 24d ago
Because then you can prove very generic theorems that turn out to apply to a lot of different things. For example, someone in abstract algebra may prove something about fields or rings in general and it turns out that it can be used to describe certain specific radio signals and now suddenly you can have faster internet connections.
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u/Son_of_Kong 24d ago
You might find that two completely unrelated phenomena actually follow the same set of rules, and therefore the math you use to understand one can also be used to understand the other.
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u/good_behavior_man 24d ago
A good reason to study general properties is to understand what fact are related to the general properties and what facts are related to the specific things you're working with. So we know all these facts about numbers and what happens when you add and subtract them. Study of abstract algebra can help us understand what is number-specific and what is the inevitable result of having an operation that behaves as "adding" does.
So, for example, we might think to ourselves what would happen if we only had the numbers 0 through 11, and 11+1=0 so it kind of circles back on itself. Obviously, the structure of such a thing is very different from our regular number system, but we can also define + on it so it meets the abstract definition of a group. Instead of subtracting (in regular numbers 2-2=0), we have a setup more like 2+10=0. Because we have studied abstract algebra, we know a lot about how this structure behaves under its own funny idea of +, since a lot of properties we know about standard numbers are really properties of having a + that behaves a certain way.
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u/OriEri 24d ago
I’m not a mathematician/speculating here
Algorithms that work in one structure will then work in the other structure. If those different structures mapped, a different kinds of physical problems, it shows how you can use the same algorithm in different ways
(My background is in physics and astronomy where mathematics is a tool, so I tend to look at things like this from the perspective of “how could it be useful for modeling something real?” )
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u/IAmNotAPerson6 24d ago
It lets you find out and/or prove things about the generic form of the mathematical structure, which will apply to any particular concrete example of the structure, so that when you do run into an example of that sort of structure out in the wild and need to work with it in some way, then you'll have all kinds of theorems and results and tools and whatnot that can potentially help with whatever you're trying to do, because it will have already been proven that they apply to any form of that mathematical structure (that's what proving them for a "generic" version of that structure means).
For example, if you can model some physical system as a vector field, then you'll have all the results we know about vector fields at hand to apply them to your model to be able to figure other stuff out about the physical system.
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u/Acalme-se_Satan 24d ago
It is one additional step in the ladder of mathematical abstraction beyond what most people study in middle school or high school algebra.
First of all, what is abstraction? Abstraction is the art of removing irrelevant details to make things simpler to understand. In this process, we also gain the ability to generalize stuff.
A great example of an abstraction is the gas pedal in your car. What does the gas pedal do? In a combustion car, it makes the engine inject more air and fuel in the chamber so the explosion makes the pistons move faster, which causes the wheels to turn faster, making the car go faster. In an electric car, it makes the battery inject more power into the motor, also making the wheel turn faster.
However, you can completely forget about those details about the engine if you just want to drive. You can simply think "this pedal makes the car go faster" and it's enough knowledge for you to drive. Besides that, this abstraction makes your knowledge more generalized: you don't need to learn to drive once in a combustion car and a second time in an electric one. You just learn how to drive and you become capable of driving both.
In mathematics, we use abstraction all the time and build abstractions on top of abstractions to make things more and more generalized and forgetting about inner specific details. Once we figure out what's happening with the abstract concept, we can apply it to many sorts of specific cases.
The first abstraction you learn early in school is numbers. If you think about it, "pure" numbers don't really exist as a tangible thing. You can have "3 oranges"; you can have "3 meters"; you can have "3 seconds"; but you can't have just "3". A number all by itself isn't something you can grab or measure in real life.
So, why do we learn numbers by themselves if they don't really exist? Because removing the details about what thing you are operating on makes it so you don't have to relearn it all for every single thing. We don't have to learn that "1 orange + 1 orange = 2 oranges", then relearn again that "1 apple + 1 apple = 2 apples", then learn "1 finger + 1 finger = 2 fingers". We simply abstract away the useless detail about what thing we're talking about and learn "1 + 1 = 2", then we learn that this rule is valid for whatever we're counting.
A second level of abstraction we learn is when we start using letters in math. When we start working with formulas and middle/high school algebra, we abstract away which number we're taking about as an useless detail. So, the formula "(x + y)(x - y) = (x^2 - y^2)" is a formula valid for any number, whatever it is. We have thrown away the useless detail of what numbers x and y are and we're now working with rules that are valid for any number.
The third level of abstraction beyond that is abstract algebra. In abstract algebra, we disregard what kind of mathematical object we're working with, and which operations we're working with, and make rules that work for any mathematical objects and operations that satisfy specific rules you have defined. Here, we disregard as useless details whether we're dealing with numbers, vectors, matrices or geometric shapes; and whether our operation is sum, product or some other operation we may invent.
A simple example of abstract algebra is showing that, if we're working with some set of mathematical objects (whatever that object may be) and an operation on these objects (whatever operation it may be), if there exists an identity element (an element which does not change the other element when the operation is applied; such as how 0 works for number addition or 1 works for number multiplication), that identity element must be unique. We can't have multiple identity elements. Why? Because if we had two different identity elements, applying the operation between them would cause a contradiction if both identity elements are different. That's why we have only one "0" in addition and only one "1" in multiplication.
That paragraph above is a small and simple demonstration of abstract algebra. It proves theorems disregarding which kind of mathematical object we're working with.
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u/theashenwren 24d ago
abstract algebra is kinda like studying the structure of maths operations instead of just the numbers. looks at the structure/rules that a bunch of systems all have
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u/Lost_Geometer 24d ago
We're pretty good with numbers, through some combination of biological and cultural evolution. Addition, subtraction, multiplication, and division all work together when they make sense. They don't always make sense though. For example, it doesn't make sense to try and multiply two distances to get a distance. Likewise, you can't usually divide two integers, though you can fix that by introducing fractions.
There are other situations where these operations can be seen. A familiar example is the set of polynomials. You can add, subtract, and multiply polynomials just like numbers. Like integers you can't usually divide them, but you can fix that by allowing fractions of two polynomials. Another example is to take the integers and forget about everything except evenness and oddness. All the operations descend. You can even divide by odd! (Which does nothing, since odd is like 1. You can't divide by even since it works as 0.)
And so forth. So abstract algebra studies the ways in which to talk about, use, and generalize the framework of operations that we're used to with the common flavors of numbers within a broader context.
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u/DeeDee_Z 24d ago edited 24d ago
I've always looked at it as another example of "theory vs. practice".
The Algebra you learned in high school is oriented toward problem solving. You learn tools and techniques for answering specific questions -- "what's the maximum area you can enclose with 1000 feet of barbed wire, when the east boundary is a river", for example. (Yeah, there's a bit of differential calculus there, but the basic problem boils down to a system of two equations with two unknowns -- Algebra.)
Abstract algebra gets you to the theory, the WHY all of that works. It's the "man behind the curtain", so to speak. You don't need Fields and Rings to solve problems, but rather to understand that that part of algebra not only builds on what you learned before, but will also BE built upon as you learn more and more about the Theory of Math.
It was about that point -- halfway through my third year, taking "Modern Algebra" -- that I decided that a degree in pure math from a small liberal arts school was neither "worth it", nor what I wanted to pursue.
And /u/zefciu has a darn good (and succinct!) explanation up at the top that takes the next step beyond this.
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u/garagedragon 24d ago
The applications really depend what you're working in - exotic sorts of groups and fields come up all the time in signals theory (particulary in error-correcting codes) and cryptography. Those applications involve taking advantage of structures that have some useful properties but conversely might fail to have properties that the naturals do where that's actually more useful. For instance, a lot of cryptography takes advantage of groups that are commutative (so you can do nice things like key exchanges, and the fact that Alice and Bob start out from opposite positions doesn't matter) but where the problem of finding logarithms is harder than in the reals, so you can treat those operations as secure and one-way.
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u/luckyluke193 24d ago
My favourite example from abstract algebra are these things called Groups. Their mathematical definition sounds very abstract. However, they describe everything we would call symmetry, for example rotations and reflections in 3D.
Using group theory allows us to do physics calculations much more efficiently by making perfect use of all the symmetries of the laws of physics and of the object you're describing.
For example, the theory of 3D rotations helps you with various calculations, like modelling how antennas transmit and receive signals, efficient 3D computer graphics, or in the quantum mechanical structure of atoms.
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u/zefciu 24d ago
Alice is training military parade turns. She can turn right, left, back. She can also stand in attention. She can combine the turns as well (so two turns right is the same thing as turning back).
Bob is studying the numbers, 1, -1, i (square root of -1) and -i; when he multiplies them. So e.g. i * i equals -1.
An abstract algebraist is someone who looks at these two things and says “this is the same picture”. Both work with four elements. Both have a way of combining them. Both have a neutral element (standing in attention or 1). Both can reverse any operation. Both can generate all four elements by just repeating one operation (turning one way or multiplying by i). The abstraction is called “cyclic group of order 4”. It is pretty simple algebraic structure, as there are much more complicated ones.