Mathematics

Questions tagged [motivation]

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

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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 linear algebra?
Aaron
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Motivation for the rigour of real analysis

I am about to finish my first year of studying mathematics at university and have completed the basic linear algebra/calculus sequence. I have started to look at some real analysis and have really enjoyed it so far. One thing I feel I am lacking in…
1729
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Is Bayes' Theorem really that interesting?

I have trouble understanding the massive importance that is afforded to Bayes' theorem in undergraduate courses in probability and popular science. From the purely mathematical point of view, I think it would be uncontroversial to say that Bayes'…
user78270
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91
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6 answers

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 they must be important. I always assumed that…
Alexander Gruber
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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 Hypothesis..." Almost always, the crux of their…
Joseph DiNatale
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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 an axiom for defining a Lie algebra). Could…
aelguindy
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64
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7 answers

What's the point of studying topological (as opposed to smooth, PL, or PDiff) manifolds?

Part of the reason I think algebraic topology has acquired something of a fearsome reputation is that the terrible properties of the topological category (e.g. the existence of space-filling curves) force us to work very hard to prove the main…
Qiaochu Yuan
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59
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7 answers

Uses of quadratic reciprocity theorem

I want to motivate the quadratic reciprocity theorem, which at first glance does not look too important to justify it being one of Gauss' favorites. So far I can think of two uses that are basic enough to be shown immediately when presenting the…
Gadi A
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Nonobvious examples of metric spaces that do not work like $\mathbb{R}^n$

This week, I come to the end of the first year analysis, and suffer from a "crisis of motivation." With this question, I want to chase away my thought, "Why is it important to study the general properties of metric spaces? Is every good example just…
Samuel Handwich
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Why are modular lattices important?

A lattice $(L,\leq)$ is said to be modular when $$ (\forall x,a,b\in L)\quad x \leq b \implies x \vee (a \wedge b) = (x \vee a) \wedge b, $$ where $\vee$ is the join operation, and $\wedge$ is the meet operation. ( Join and meet .) The ideals of a…
user23211
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Big list: books where we "...start from a mathematically amorphous problem and combine ideas from sources to produce new mathematics..."

What books are there like like Radin's Miles of Tiles , which share these particular features: Theme: "In this book, we try to display the value (and joy!) of starting from a mathematically amorphous problem and combining ideas from diverse sources…
bzm3r
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47
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Motivation behind topology

What is the motivation behind topology? For instance, in real analysis, we are interested in rigorously studying about limits so that we can use them appropriately. Similarly, in number theory, we are interested in patterns and structure possessed…
Adhvaitha
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45
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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 functor of the global section functor. What I…
Mohan
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What are the applications of the Mean Value Theorem?

I'm going through my first year of teaching AP Calculus. One of the things I like to do is to impress upon my students why the topics I introduce are interesting and relevant to the big picture of understanding the nature of change. That being said,…
user694818
43
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4 answers

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 $k$-variety, and let $Y$ be a quasi-affine or quasi-projective…
Daan Michiels
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