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

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1answer
37 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
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1answer
34 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 ...
3
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1answer
25 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 ...
2
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2answers
40 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 ...
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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 + ...
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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 ...
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2answers
260 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
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2answers
88 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 ...
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2answers
800 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?
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2answers
33 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] $ ...
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1answer
39 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 ...
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2answers
125 views

Motivation for construction of cross-product (Quaternions?)

I just found a very interesting article here: http://www.johndcook.com/blog/2012/02/15/dot-cross-and-quaternion-products/ The author observes that by defining i,j,k s.t. $i^2=j^2=k^2=ijk=-1$, ...
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1answer
774 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 ...
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2answers
122 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 ...
74
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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
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1answer
70 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
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1answer
103 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
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0answers
71 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 ...
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3answers
186 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
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1answer
65 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 ...
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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
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6answers
2k 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, ...
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0answers
56 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 ...
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1answer
27 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
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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 ...
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0answers
69 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 ...
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0answers
32 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 ...
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1answer
19 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
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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 ...
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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 ...
2
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2answers
109 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
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5answers
139 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
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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 ...
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0answers
12 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 ...
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3answers
72 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?
26
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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 ...
16
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2answers
805 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.
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0answers
29 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 ...
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7answers
3k views

What purpose does the use of functions serve in mathematics?

Ok, so I know the overview a bit. I would like to know why one should use one, what they're used for, and maybe even the history behind their purpose. X = X (Y) + 7 ... 5 = 5 (20) + 7 ... 32 is the ...
6
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1answer
463 views

A layman's motivation for non-standard analysis and generalised limits

Disclaimer: My apologies for making such a long question. The question is possibly also rather specific, but I hope that (some parts of) it might be useful in general. Background: I have recently ...
9
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3answers
444 views

What's a good motivating example for the concept of a slice category?

What nice example can one give a beginner to really motivate the idea of a slice category, before they've met the more general notion of a comma category? There's the toy example of a poset category ...
2
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0answers
49 views

Motivation for Definition of Tensor Product of Modules

Let $A$ be a commutative ring and $M,N,P$ be A-modules.I know that tensor product of $M$ and $N$ is a universal object ($ M \otimes N$,u) (where $M \otimes N$ is a $A$-module and $u: M\times N \to ...
3
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1answer
453 views

Intuition behind the definition of Adjoint functors

I think of adjoint functors as some sort of inverses. So, the first part of the definition looks reasonable that there exists natural transformations $$\epsilon : FG \rightarrow 1_C$$ $$\eta : 1_D ...
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1answer
50 views

The (matrix) definition of a positive-definite function

In the definition of a positive-definite function, why are the elements of the n$\times$n matrix chosen as $f(x_i-x_j)$ for $i, j = 1,...,n$? Also, it says that "for any real numbers $x_i$". Does ...
3
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2answers
172 views

Why are special numbers important? (Such as fermat prime, mersenne prime)

Whenever I studied topics in mathematics, I found those topics are important in purely mathematical sense and I could see some motivations. However, I cannot see neither motivation nor importance of ...
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1answer
60 views

The motivation for considering exponential families of distributions [closed]

I saw problems of the form: "show that the distributions ... form an the exponential family". Why is this property, being an exponential family, important?
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9answers
606 views

Motivation for the Definition of Compact Space

A compact topological space is defined as a space, $C$, such that for any set $\mathcal{A}$ of open sets such that $C \subseteq \bigcup_{U\in \mathcal{A}} U$, there is finite set $\mathcal{A'} ...
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0answers
45 views

What does a functional integral evaluation look like?

I've read the Wikipedia page on functional integration, but it really isn't very easy to understand. There don't seem to be any online videos on the subject either. In addition, when I search online, ...
5
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1answer
179 views

What can we actually do with congruence relations, specifically?

Let $T$ denote a Lawvere theory and $X$ denote a $T$-algebra. Under my preferred definitions: A subalgebra of $X$ consists of a $T$-algebra $Y$ together with an injective homomorphism $Y ...
14
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4answers
2k views

Motivation behind the definition of GCD and LCM

According to me, I can find the GCD of two integers (say $a$ and $b$) by finding all the common factors of them, and then finding the maximum of all these common factors. This also justifies the ...