For questions about gaining and achieving motivation.

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4
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1answer
116 views

What is a motivation for this theorem and what is an example this theorem is applied?

If one doesn't know a motivation, it's hard to memorize such a theorem. So do I. Rudin RCA p.30 Let $(X,\Sigma,\mu)$ be a measure space such that $\mu(X)<\infty$ and $f\in L^1(\mu)$. ...
3
votes
1answer
89 views

Why might one be inclined to think that polynomials of the form $\cos(n\arccos{x})$ would minimize error in Lagrange interpolation?

I was first introduced to Chebyshev polynomials (of the first kind) in the form $T_n(x)=\cos\left(n \operatorname{arccos}(x)\right)$. The usual recurrence relation was then derived from using trig ...
8
votes
1answer
115 views

Representation theory approach VS Module theory approach?

Given an associative algebra $A$, there is a correspondence between representations of $A$ and left $A-$ modules. Thus, one can study the representation theory of an associative algebra via its left ...
2
votes
5answers
428 views

Why - not how - do you solve Differential Equations? [closed]

I know HOW to mechanically solve basic diff. equations. To recap, you start out with the derivative $\frac{dy}{dx}=...$ and you aim to find out y=... To do this, you separate the variables, and ...
2
votes
1answer
177 views

What's with conditionals in mathematical logic?

Having a bit of difficulty understanding the conditional ($\rightarrow$) in mathematical logic. I read up on the already-existing questions and it did help me understand it better (the 'promise' ...
3
votes
1answer
107 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 ...
5
votes
6answers
406 views

What is the point of quadratic residues?

What is the most motivating way to introduce quadratic residues? Are there any real life examples of quadratic residues? Why is the Law of Quadratic Reciprocity considered as one of the most ...
11
votes
3answers
156 views

Applications of functions of the form $f(x)^{g(x)}$

Early on in my calculus education, I learned how to take the derivative of $x^x$ by re-writing it in the form $e^{x\ln x}$. More generally, this technique is helpful in finding the derivative of ...
6
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1answer
209 views

Can we capture all domains of discouse in the predicate logic within categorical logic?

In the construction of the bounded quantifiers via adjoints in the fibered category of subsets over a set (see e.g. here on Wikipedia), is there any restriction on the sets - specifically regarding ...
3
votes
2answers
116 views

Motivation for abstractness

I'm seeking examples of concepts or theorems in school mathematics that are better understood when we generalize (when we deal with a more abstract concept where the former concept is a special case ...
2
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1answer
156 views

What motivates the study of Abelian groups?

Monoids arise naturally as endomorphism monoids, and groups arise naturally as automorphism groups. These are among the primary motivators for their study, in my opinion. What are the (main) ...
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10answers
1k views

What are the applications of continued fractions?

What is the most motivating way to introduce continued fractions? Are there any real life applications of continued fractions?
-3
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6answers
1k 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 ...
1
vote
3answers
89 views

Mean Value Theorem Motivation

I am currently practicing presenting mathematics to various audiences and am considering the example of the mean value theorem. I was wondering how would I be able to motivate this theorem to a ...
2
votes
1answer
165 views

Motivations for Prime Factorizaton

I'm at the beginning of some middle school math sessions on divisors, gcd, lcm, and prime numbers. It's the first place in the curriculum that the students encounter the three latter concepts ...
8
votes
2answers
299 views

Why do we want probabilities to be *countably* additive?

In probability theory, it is (as far as I am aware) universal to equate "probability" with a probabilistic measure in the sense of measure theory (possibly a particularly well behaved measure, but ...
3
votes
1answer
82 views

Motivation and application for stochastic geometry.

I am starting a PhD, and there is a good chance that my project will be oriented in the study of random polytopes or/and random mosaics. I was wondering what are the motivations and applications of ...
3
votes
1answer
115 views

New ways to light the fire again

Recently I've been studying a lot of analytic geometry and this subject made my motivation drop. The thing is, the courses aren't stopping and I'm beginning to lose the passion I had before. I need ...
7
votes
1answer
330 views

What is the significance of limit points?

When I had my first taste of topology a couple of years ago, our lecturer emphasized the following notions. closed set, closure, closure point open set, interior, interior point Of course, these ...
3
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0answers
112 views

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

Let $x$ denote an arbitrary real number. Then $x^n$ makes sense for arbitrary $n \in \mathbb{N},$ via the obvious recursive definition. We can extend this definition by asserting that if $x$ is ...
3
votes
0answers
189 views

What are the advantages of proof-relevant mathematics?

I've read that Theorems in HoTT (homotopy type theory) tend to characterize the space of proofs of a proposition, rather than simply state that the corresponding type is inhabited. So, HoTT ...
4
votes
1answer
106 views

Why doesn't the definition of (model-theoretic) conservative extension need strengthening?

In the wikipedia page dedicated to conservative extensions, we find the following sentence: $T_2$ is a model-theoretic conservative extension of $T_1$ if every model of $T_1$ can be expanded to ...
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0answers
85 views

Big theorems in information geometry?

Working on preparing a talk on information geometry to a young finance/applied math audience. Motivating this area is turning out a little difficult. What are some big theorems or results that I ...
15
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1answer
314 views

Motivation for the study of amoebas.

What was the primary motivation for the study of the amoebas?
6
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4answers
516 views

Why the axioms for a topological space are those axioms?

This question might have even been asked here before, I don't really know, so sorry if it's duplicate. I've started to study topological spaces and I've found the axioms for such spaces kind of hard ...
2
votes
1answer
304 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 ...
4
votes
3answers
582 views

Quaternions vs Axis angle

Whats the use of representing rotation with quaternions compared to normal axis angle representation? I've been trying to learn quaternions and they make enough sense but as far as I can tell ...
18
votes
2answers
821 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 ...
2
votes
2answers
106 views

What is the significance of permutable subgroups? (and $X$-permutable subgroups?)

Let $G$ be a group and $H$, $K$, $X$ be subgroups of $G$. We say $H$, $K$ are permutable if $HK=KH$. or we say $H, K$ are X-permutable if $‎\exists x, x\in X$ such that $H^{x}K=KH^{x}.$ Why are ...
6
votes
1answer
98 views

Recovering the structure of an object from its morphism:Yoneda Lemma

I've heard that Yoneda lemma informally states that one can recover the internal structure of an object by looking at the morphism coming out from that object. But this is not clear to me from the ...
8
votes
2answers
436 views

Motivation of stable homotopy theory

A stable homotopy category can be obtained by modifying the category of pointed CW-complexes: objects are pointed CW-complexes, and for two CW-complexes $X$ and $Y$, we take $$\lbrace X,Y \rbrace = ...
2
votes
1answer
212 views

Why do we need continuous random variables since they can be approximated by discrete ones?

I do not understand the motivation of developing the theory of continuous random variables. Given simple discrete random variables, the continuous ones can be well approximated.
6
votes
1answer
208 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 ...
12
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7answers
4k views

Why is it important to study combinatorics?

I was having a discussion with my friend Sayan Mukherjee about why we need to study combinatorics which admittedly, is not our favourite subject because we see very less motivation for it(I am not ...
3
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6answers
156 views

Motivation for creation of complex exponentiation

I am curious how mathematicians came to develop complex exponentiation. How is the rule for complex exponentiation derived?
4
votes
2answers
352 views

Motivation for the study of the Chern connection

Given a Hermitian metric $H$ over a holomorphic vector bundle $E$ with holomorphic structure $\overline{\partial}$, there exists a unique connection $\nabla$ (named afer Chern) satisying the following ...
1
vote
1answer
203 views

Intuition behind symmetric and antisymmetric tensors

I've been studying multilinear algebra on Kostrikin's "Linear Algebra and Geometry" and he says the following. If $V$ is a linear space, $T^q_0(V)=V^{\otimes q}$ and if $f_\sigma :T^{q}_0(V)\to ...
1
vote
1answer
54 views

It is not possible to introduce multiplication in $v_n$(For $n>2$) so as to satisfy all field properties

In the book Calculus Vol 1- Tom M. Apostol .Before beginning to define the dot product of two vectors he tells It can be shown that except $n=1, 2$, it is not possible to introduce multiplication ...
5
votes
3answers
229 views

What is the point of extremal epimorphisms in category theory? Why not just use strong epis instead?

I've been trying to get my head around the various types of epimorphisms you get in category theory, but I can't see why anyone uses "extremal" epis as opposed to the slightly less general notion of ...
8
votes
5answers
2k views

Why is the tensor product constructed in this way?

I've already asked about the definition of tensor product here and now I understand the steps of the construction. I'm just in doubt about the motivation to construct it in that way. Well, if all that ...
30
votes
6answers
2k 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 ...
1
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3answers
143 views

The relation between been the quotient ring of a prime ideal and its localization

Let $A$ be a ring and $\mathfrak{p} \subset R$ be a prime ideal. Set $A_\mathfrak{p}=R[U^{-1}]$, where $U= A-\mathfrak{p}$. What is the relation between $A/\mathfrak{p}$ and $A_\mathfrak{p}$? My ...
6
votes
2answers
173 views

Motivation for the relations defining $H^1(G,A)$ for non-commutative cohomology

First let me review the definition of first non-commutative cohomology. Let $G$ be a group and $A$ a left $G$-group, i.e. for any $\sigma, \tau\in G$ and $a, b\in A$, one has ...
7
votes
4answers
2k views

Order of nontrivial elements is 2 implies Abelian group

If the order of all nontrivial elements in a group is 2, then the group is Abelian. I know of a proof that is just from calculations (see below). I'm wondering if there is any theory or motivation ...
2
votes
4answers
93 views

Motivating complex structure on $\mathbb{R}^2$

I'm giving a talk to a group of bright but not all that mathematically sophisticated students on the subject of complex numbers. I'd like to introduce complex numbers via geometric considerations ...
3
votes
2answers
118 views

Use of $\mathbb N$ & $\omega$ as index sets

Why all the properties of a sequence or a series or a sequence of functions or a series of functions remain unchanged irrespective of which of $\mathbb N$ & $\omega$ we are using as an index set? ...
55
votes
12answers
31k views

Why study linear algebra?

Simply as the title says. I've done some research, but still haven't arrived at an answer I am satisfied with. I know the answer varies in different fields, but in general, why would someone study ...
12
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2answers
500 views

Motivating (iso)morphism of varieties

I am reading course notes on algebraic geometry, where a morphism of varieties is defined as follows ($k$ is an algebraically closed field): Let $X$ be a quasi-affine or quasi-projective ...
3
votes
3answers
227 views

Intuition behind independence & conditional probability

This is a basic question. I have a good intuition that $A$ is independent of $B$ if $P(A \vert B) = P(A)$, and see how you can easily derive from this that it must hold that $P(A,B) = P(A)P(B)$. ...
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2answers
731 views

Grassmann Variables and Complex Conjugate

While dealing with Grassmann Variables, the complex conjugate is defined as $$ (\phi \psi)^{\dagger} = \psi^{\dagger} \phi^\dagger $$ and why not $ \phi^{\dagger} \psi^\dagger $. I want to know the ...