For questions about gaining and achieving motivation.

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5
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1answer
575 views

Why is unique ergodicity important or interesting?

I have a very simple motivational question: why do we care if a measure-preserving transformation is uniquely ergodic or not? I can appreciate that being ergodic means that a system can't really be ...
4
votes
3answers
307 views

Motivation for the Mapping Class Group

Question: What is the motivation for studying the mapping class group? In particular, what types of questions does it attempt to answer and what kind of invariant is it? Motivation for this ...
4
votes
2answers
69 views

Why are functional analysts interested in not only the point spectrum of $f$, but also, its spectrum?

Suppose $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\},$ that $X$ is a Banach space over $\mathbb{K}$, and that $f : X \leftarrow X$ is a bounded linear transform. Then the spectrum of $f$ is defined as the ...
4
votes
2answers
768 views

Why do we need (the abstract concept of) random variables (in discrete probability models)?

What we defined: Suppose we have a (discrete) probability model $\left(\Omega,P\right)$, where $P$ is the probability function (at least, that was the way it was introduced in a course I took; that ...
4
votes
2answers
141 views

Why a norm and not some other function that defines a metric?

If one defines on a $\mathbb{R},\mathbb{C}$-vector space a norm this gives rise to a metric. Why are particularly mappings that satisfy the norm axioms so important that in every book for beginners on ...
4
votes
1answer
107 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 ...
4
votes
1answer
389 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 ...
4
votes
1answer
122 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 $f\in L^1(\mu)$. ...
4
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2answers
371 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 ...
4
votes
1answer
456 views

Motivation for studying quadratic algebras, Koszul algebras, Koszul duality

I'm trying to gain a practical understanding of Koszul duality in different areas of mathematics. Searching the internet, there's lots of homological characterisations and explanations one finds, but ...
4
votes
3answers
638 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 ...
4
votes
0answers
60 views

Why do we care about non-$T_0$ spaces?

(Reminder: A $T_0$ topological space, also known as a Kolmogorov space, is a space where the topological structure "recognizes" that different points are different: No two points have exactly the same ...
4
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0answers
195 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 ...
3
votes
2answers
117 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 ...
3
votes
1answer
116 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 ...
3
votes
2answers
271 views

What is the importance of the spectral theorem?

I know that the spectral theorem tells us that in the case of a real inner product space, an operator is self adjoint if and only if there is an orthonormal basis with only eigenvectors of that ...
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? ...
3
votes
1answer
109 views

Motivation and examples for ramification

I started learning algebraic number theory, but it seems like all the sources I had are too abstract, giving me difficulty understanding the concept and tripping me up frequently. For today it is ...
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?
3
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3answers
252 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 $P(A,B) = P(A)P(B)$. ...
3
votes
1answer
83 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
74 views

Traveling salesman problem: why visit each city only once?

According to wikipedia, the Traveling Salesman Problem (TSP) is: Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city ...
3
votes
1answer
114 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
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 ...
3
votes
0answers
113 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
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0answers
173 views

Algebraic Geometry question

Why do we study projective normality of a projective variety ? Does it have anything to do with non-singularity ? Any other purpose to study this ?
2
votes
5answers
431 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
157 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) ...
2
votes
1answer
311 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 ...
2
votes
1answer
46 views

Why do we require $X$ to be Hausdorff when defining $C(X)$?

In Efton Park's "Complex Topological K-theory", he begins a section by defining the Banach algebra $C(X)$ (continuous functions $f: X \rightarrow \Bbb C$), presupposing that $X$ is Hausdorff and ...
2
votes
2answers
74 views

Motivational example for complex numbers

Years ago I was introduced to complex numbers. In class we had been talking about the cubic polynomial and its solutions. At one point we saw an example where, when using the formula, one had to stop ...
2
votes
2answers
107 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 ...
2
votes
1answer
214 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.
2
votes
2answers
72 views

Ordinals - motivation and rigor at the same time

Can someone provide a description of ordinals within ZFC in a rigorous way that exhibits motivation? Every description or explanation I see in the literature or on the Internet is either too formal ...
2
votes
1answer
85 views

How to understand cocategories

$\newcommand\CC{\mathsf{C}}$The notion of a category is well-known. There are multiple equivalent definitions; small categories can be seen as an internal category in $\mathsf{Set}$, that is a ...
2
votes
1answer
81 views

What is the motivation/applications for the definition of Lebesgue measure on $\mathbb R^n$?

The definition of the Lebesgue measure on $\mathbb R^n$ is fundamentally tied up with the following assumption: The measure of the cartesian product of $n$ intervals should be the product of the ...
2
votes
2answers
68 views

Practical motivation for analysis at school

I want to show pupils at an age of 14 to 17 and who do not like math what they can use analysis/analytic geometry in a job. I could not remember an impressive example for most topics. Let's assume ...
2
votes
1answer
77 views

For what reason, the surface measure represents the surface area?

Let $||\cdot||$ be a norm on $\mathbb{R}^n$ (It's an arbitrary norm, not 2-norm) Define $S^{n-1}=\{x\in\mathbb{R}^n : ||x||=1\}$ Let $\mu$ be the n-dimensional Lebesgue measure. Define ...
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 ...
2
votes
1answer
38 views

Is there a particularly simple example of geometric descent?

I'm looking for a particularly simple and familiar example of descent in geometry or topology in order to motivate the general definition. I'm not counting the definition of the arrow category ...
2
votes
3answers
344 views

The “Empty Tuple” or “0-Tuple”: Its Definition and Properties

(I would like to link to a previous discussion on the subject: What is A Set Raised to the 0 Power? (In Relation to the Definition of a Nullary Operation)) In axiomatic (ZFC) set theory, we define ...
2
votes
1answer
165 views

Why should we accept the existence of subsets $A$ such that neither $A$ nor $A^c$ are recursively ennumerable? And how can we persuade others?

Encode every pair $(t,x)$ (where $t$ is a Turing machine and $x$ is an input string) as a distinct natural number. Then the halting subset $H$ fails to be recursive. $$H := \{(t,x) \in \mathbb{N} ...
2
votes
1answer
173 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 ...
2
votes
0answers
67 views

A good way to explain $\varepsilon$-$\delta$ for chemistry / biology students?

I feel like I have a pretty good way to talk about $\varepsilon$-$\delta$ to physics and engineering students (and possibly students in comp sci). But I am not very sure what I can do for chemistry ...
2
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0answers
59 views

Intuition for the Positive Real Number $\epsilon$ in Topology

Although this question might sound a little too simple, it is a problem that I must get addressed. In addition, there is no way for me to formally describe it. If you have something you can add, by ...
2
votes
0answers
49 views

Set theory, motivating factors [duplicate]

Set theory seems to pop up in many different fields of mathematics. As someone with a CS degree, I've only encountered very basic set theory; dealing with non-specific sets, and their intersections, ...
2
votes
0answers
57 views

What is the motivation behind the study of pattern-avoiding permutations?

There is a ton of research on pattern-avoiding permutations (permutations that do not contain some designated permutation pattern). We're figuring out how to enumerate them, what random ones are ...
2
votes
1answer
178 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' ...
2
votes
2answers
49 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 ...
1
vote
2answers
135 views

Relation between long exact sequences and Derived functors

I know that if i have a short exact sequence of chain complexes $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ then i can extend it to long exact sequence of homology groups as ...