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

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6
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3answers
289 views

Finitely generated ideal in Boolean ring; how do we motivate the generator?

This problem is Exercise 11.3 in Atiyah/Macdonald Commutative Algebra. They ask to prove every finitely generated ideal in a Boolean ring is in fact a principal ideal. The question has been answered ...
6
votes
1answer
141 views

What is the motivation to continuous functions and measurable functions?

In topology the objects of interest are the space open sets, and a function will be continuous if the inverse image of any open set is an open set. In measure theory the objects of interest are the ...
6
votes
1answer
219 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 ...
6
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3answers
1k 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 ...
6
votes
1answer
106 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 ...
6
votes
1answer
79 views

Motivation for and differences between properties of measure, outer measure

This is related to my measure theory class, but it's not homework. The motivation behind this post is to understand the big picture relationship between properties of measure and outer measure. I ...
6
votes
1answer
210 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 ...
6
votes
1answer
101 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 ...
6
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0answers
69 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 ...
6
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1answer
239 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 ...
5
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6answers
537 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 ...
5
votes
5answers
469 views

About Trigonometry

Is there anything cool about trigonometry? I was just curious. I'm learning trig right now and I often find myself asking myself, "What's the point?" I feel if I knew what I was working on and why, ...
5
votes
2answers
1k 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 ...
5
votes
1answer
447 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 ...
5
votes
2answers
452 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 ...
5
votes
3answers
277 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 ...
5
votes
1answer
169 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 ...
5
votes
1answer
748 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 ...
5
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0answers
235 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
3answers
322 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
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5answers
476 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 ...
4
votes
2answers
731 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 ...
4
votes
2answers
82 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
1answer
128 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
2answers
149 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
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1answer
164 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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1answer
265 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 ...
4
votes
1answer
543 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
1answer
56 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
5answers
129 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 ...
3
votes
2answers
126 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
117 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
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1answer
422 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 ...
3
votes
1answer
141 views

What motivates the definition of a ring in abstract algebra? [closed]

I've read a lot of questions about rings but I still don't know why they are useful/ I'm sure they are, but I don't know why. Are their properties somehow used in proofs or as foundations for ...
3
votes
2answers
122 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
2answers
148 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 ...
3
votes
6answers
182 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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1answer
114 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 ...
3
votes
3answers
406 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
97 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
56 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 ...
3
votes
1answer
64 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 ...
3
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1answer
60 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 ...
3
votes
0answers
49 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 ...
3
votes
0answers
66 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 ...
3
votes
2answers
110 views

Motivation for separation axioms

I have recently been studying different separation and countability axioms in topology. I am looking for a motivation for why such a refined division of different axioms was made and is studied. I am ...
3
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0answers
82 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 ...
3
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0answers
123 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 ...
3
votes
1answer
99 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
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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 ?