For questions about the motivation behind mathematical concepts and results. These are often "why" questions.

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-3
votes
1answer
58 views

Simple examples for motivation of topology [on hold]

It is easy to see motivation for groups and fields, as abstractions of operations defined on integers, rationals, reals etc. and how the results from those abstractions apply to integers, reals etc. ...
1
vote
0answers
40 views

Motivation of indices of all subgroups of symmetric group $S_n$

In 1858 a prize question of the Acad´emie des Sciences was - What are the indices of all subgroups of symmetric group $S_n$ acting on $n$ objects ? Three submissions was submitted in 1860, no ...
28
votes
5answers
4k views

Jacobi identity - intuitive explanation

I am really struggling with understanding the Jacobi Identity. I am not struggling with verifying it or calculating commutators.. I just can't see through it. I can't see the motivation behind it (as ...
14
votes
3answers
572 views

Motivation behind the definition of tangent vectors

I've been reading the book Gauge, Fields, Knots and Gravity by Baez. A tangent vector at $p \in M$ is defined as function $V$ from $C^{\infty}(M) $ to $\mathbb R$ satisfying the following properties: ...
5
votes
4answers
347 views

Motivation for construction of cross-product (Quaternions?)

I'm trying to present a narrative that brings the (3D) Cross Product into existence. "Given two vectors $\mathbf u$, $\mathbf v$, how to construct a vector perpendicular to both?" ... looks like ...
4
votes
2answers
66 views

Motivation for the study of algebraic structures

I am currently studying group theory and I realized that most concepts we study are just definitions on which we build theory. I do understand that some theorems are beautiful and don't need any ...
1
vote
0answers
42 views

Why is Grobner basis useful? [duplicate]

I don't get the motivation for calculating Grobner bases. What's good by computing a Grobner basis for an ideal of $k[X_1,...,X_n]$? Moreover, is there any theorem whose proof relies on the use of ...
20
votes
3answers
1k views

Motivation for/history of Jacobi's triple product identity

I'm taking a short number theory course this summer. The first topic we covered was Jacobi's triple product identity. I still have no sense of why this is important, how it arises, how it might have ...
30
votes
3answers
2k views

Why was Sheaf cohomology invented?

Sheaf cohomology was first introduced into algebraic geometry by Serre. He used Čech cohomology to define sheaf cohomology. Grothendieck then later gave a more abstract definition of the right derived ...
3
votes
1answer
34 views

Motivation for introducing $A^{\prime} := \frac{1}{\left|G \right|} \sum_{g \in G} \Delta(g^{-1}) A \Delta(g)$ in representation theory proof

Assume $\left|G \right|= \left| G \right| \cdot 1_F$ is invertible in $F$. Let $\Delta:F \to GL_n(F)$ be a representation and $U \subseteq F^n$ be an $F$-subspace that is $\Delta$-invariant. Then ...
1
vote
0answers
32 views

Hopfian and Co-Hopfian Modules. [closed]

Let $M$ be a $R-$module. We say that: $M$ is a Hopfian module, if every epimorphism of $M$ is a monomorphism. $M$ is a Co-Hopfian module, if every monomorphism of $M$ is an epimorphism. Why do we ...
8
votes
0answers
127 views

Why is Fourier Analysis effective for studying uniform distributions

On his great expository article about the naturality of the Zeta function in number theory, Tim Gowers makes the following claim: When it comes to the primes, we find that we do not have a good ...
-1
votes
1answer
101 views

Why were limits introduced in calculus? [closed]

I have got the intuitive of the limits but why were they introduced in calculus, are they really helpful to us?
2
votes
3answers
87 views

What is the 'meaning' of nowhere dense set?

In some books, nowhere dense set is defined to be $int(\bar A)=\emptyset$ but meanwhile is defined to be $int(A)=\emptyset$ in some books(e.g. Munkres). So what is the 'meaning' (i.e motivation, ...
3
votes
0answers
139 views

What are the Big Theorems in Information Geometry?

I am working on preparing a talk on information geometry to a young finance/applied math audience. Motivating this area is turning out to be a little difficult. What are some big theorems or results ...
0
votes
1answer
42 views

Providing motivation for the importance of the concept of a 'basis'.

In a few situations, I found myself being asked by younger students why the concepts of a basis was important. First, in the concept of linear spaces, it's easy to explain that having a basis allows ...
3
votes
1answer
41 views

Motivation behind Arithmetic Mean

I know that the arithmetic mean $(x_1+x_2+...+x_n)/n$ is the value that minimizes $f(x)=\sum_{k=1}^n (x_k-x)^2$; however, I'm looking for an intuitive relationship between the mean and $g(x)=\sum_{k=1}...
3
votes
1answer
32 views

Why are differential equations with sinusoidal source terms easier to solve than others?

I am a software engineer trying to wrap my tiny human brain around Fourier Transforms for a project I'm currently working on. Although I will ultimately use an open source Math library to do all the ...
3
votes
2answers
46 views

Value Proposition of Fourier Analysis?

I am a software engineer trying to wrap his head around Fast Fourier Transform (FFT). Specifically, I need to implement it as part of some software I am writing. Now I can handle the implementation of ...
19
votes
3answers
1k views

Motivation for Eisenstein Criterion

I have been thinking about this for quite sometime. Eisentein Criterion for Irreducibility: Let $f$ be a primitive polynomial over a unique factorization domain $R$, say $$f(x)=a_0 + a_1x + a_2x^2 + ...
42
votes
6answers
3k views

Motivation for spectral graph theory.

Why do we care about eigenvalues of graphs? Of course, any novel question in mathematics is interesting, but there is an entire discipline of mathematics devoted to studying these eigenvalues, so ...
7
votes
2answers
270 views

What is the physical meaning of fractional calculus?

What is the physical meaning of the fractional integral and fractional derivative? And many researchers deal with the fractional boundary value problems, and what is the physical background? What ...
4
votes
2answers
91 views

Do coalgebras arise outside the study of bi/Hopf-algebras?

Hopefully the title is fairly self explanatory. I'm curious as to whether the coalgebra structure (that is, a vector space with a comultiplication and counit) comes up an any area of mathematics not ...
15
votes
2answers
841 views

Motivation behind the definition of Prime Ideal

Can someone explain what's the motivation behind the definition of a prime ideal? Or why is it exactly called a prime ideal? Has it anything to do it prime numbers?
2
votes
2answers
39 views

The definition of a subspace in linear algebra

I'm trying to learn linear algebra on my own but I am stuck on the definition of a linear subspace. Let's assume I want to find out if $S$ is a subspace of $\mathbb{R}^2$, where $ S = [X_1 , X_2] $ ...
1
vote
1answer
58 views

Examples of physical motivation for integrals over scalar field?

I'm looking for good examples of physical motivation for integrals over scalar field. Here is an example I've found (source): A rescue team follows a path in a danger area where for each position ...
18
votes
1answer
826 views

Motivation for abstract harmonic analysis

I am reading Folland's A Course in Abstract Harmonic Analysis and find this book extremely exciting. However it seems Folland does not give many examples to illustrate the motivation behind much of ...
8
votes
2answers
143 views

What is Representation Theory?

I'm beginning a course that uses representation theory, but I do not really understand what that is about. In the text I am following, I have the following definition: A representation of the Lie ...
75
votes
9answers
7k views

Why do mathematicians sometimes assume famous conjectures in their research?

I will use a specific example, but I mean in general. I went to a number theory conference and I saw one thing that surprised me: Nearly half the talks began with "Assuming the generalized Riemann ...
4
votes
1answer
76 views

What is the significance of “Homomorphism”?

Certainly Homomorphism is a prerequisite to establish an “Isomorphism”(Bijection), but what does a Homomorphism tell independently when it is established between two sets? Homomorphism relates ...
6
votes
1answer
111 views

The significance of the composition of an operator and its adjoint

As I read the literature, I have noticed that the composition $T^*T$ of a linear operator $T:H\to H$ and its adjoint frequently turns up in all kind of places. I am aware that it is Hermitian (at ...
4
votes
0answers
79 views

What was the genesis of Hua's identity?

Many resources I have read prove Hua's identity more-or-less mechanically. I have seen there is more than one raison d'être for Hua's identity: e.g. its connection to the fundamental theorem of ...
8
votes
3answers
189 views

Motivation for definition of Mobius function

Why is the Mobius function defined the way it is? \begin{align*} \mu(n) = \begin{cases} (-1)^r & \text{ if $n$ is square-free and is of the form }n=p_1p_2\ldots p_r\\ 0 & \text{ if $n$ is not ...
3
votes
1answer
67 views

Why Frobenis concerned the groups which today called “Frobenius Group”?

From their work, it seems that the Ancient mathematicians were investigating a mathematical object not as a fun, but to solve some problem occurred in earlier work of someone. Lagrange, Galois, Abel ...
14
votes
8answers
1k views

What is the motivation behind the study of sequences?

I was discussing some ideas with my professor and he always says that before you work on something in mathematics, you need to know the motivation for studying/working on it. A better way to put this ...
15
votes
6answers
3k views

Motivation of the Gaussian Integral

I read on Wikipedia that Laplace was the first to evaluate $$\int\nolimits_{-\infty}^\infty e^{-x^2} \, \mathrm dx$$ Does anybody know what he was doing that lead him to that integral? Even better, ...
1
vote
0answers
62 views

Motives, anabelian geometry? What are they?

I am about to delve into one of these subjects, but they had such huge prerequisite that I am hesitant, not sure which choice to make. Try as a might, I don't really understand the big picture behind ...
0
votes
1answer
32 views

Different discriminant ideal, what are their applications?

Lots of texts online on number theory do not even mention the different ideal. Some do, but then it gets ignored and is never mentioned again. I could not find a single application for it, as if it is ...
4
votes
1answer
64 views

Applications of the completeness of $L^1$

I'm teaching a measure theory class. I think one of the main motivations for the development of the Lebesgue integral is that the space $L^1(\mathbb{R})$ of integrable functions on $\mathbb{R}$ is ...
3
votes
0answers
76 views

Motivation of Vieta's transformation

The depressed cubic equation $y^3 +py + q = 0$ can be solved with Vieta's transformation (or Vieta's substitution) $y = z - \frac{p}{3 \cdot z}.$ This reduces the cubic equation to a quadratic ...
0
votes
1answer
23 views

Particular and homogenous differential equation solution

When solving linear nonhomogeneous equations, we deal with two types of solutions: particular homogeneous Why do we have these two types of solutions for differential equations? What does each of ...
17
votes
7answers
2k views

What is the purpose of showing some numbers exist?

For example in my Analysis class the professor showed $\sqrt{2}$ exists using Archimedean properties of $\mathbb{R}$ and we showed $e$ exists. I want to know why it's important to show their existence?...
12
votes
2answers
1k views

How can I motivate myself to study math every morning? [closed]

I am jobless, so I have relatively plenty of time though life is short. I started studying math and some other things every day because I want to become an artificial intelligence researcher. When I'...
2
votes
2answers
115 views

What (previously and currently unsolved) problems motivate the study/development of analysis?

As I had ever know there are at least two (previously unsolved) problems motivate the study/development of abstract algebra: (1) the ancient Greeks' three problems in compass-and-straightedge ...
3
votes
5answers
157 views

How to make sense out of the $\epsilon-\delta$ definition of a limit?

The informal intuition for the limit of a function is this: What is the value of the function $f$ as $x$ gets infinitely close to $c$? How on earth does this monster $$ \lim_{x \to c} f(x) = L ...
6
votes
1answer
211 views

Do expressions like $(-1)^{2/3}$ show up naturally in pure or applied math?

Background. Let $x$ denote an arbitrary real number. Then $x^n$ can be defined for each $n \in \mathbb{N}$ as follows: $$x^n = \underbrace{x \times \cdots \times x}_n$$ If $x$ is furthermore non-...
0
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0answers
14 views

How to naturally justify sigma additivity in Kolmogorov's PT axiomatics?

The last axiom in the axiomatization of probability theory by Kolmogorov, that states: Any countable sequence of disjoint (synonymous with mutually exclusive) events E$_1, E_2$, ... satisfies $$P\...
-2
votes
3answers
74 views

Purpose of a vector space [closed]

I am currently studying linear algebra and I've seen what vector spaces are, but I can't seem to understand what their purpose is. What do they allow us to do?
17
votes
2answers
844 views

Motivation behind the definition of flat module

Can someone explain what is the motivation behind the definition of a flat module? I saw the definition but I don't really know why it is important to work with these structures.
1
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0answers
30 views

Application of abstract algebra for computer science/software engineering [duplicate]

What are some cool applications of abstract algebra for computer science/software engineering? A friend of mine has been invited to my university (most of the students are soon to be software ...