What is motivation in mathematics?

[Frion]

I keep seeing this word a lot. In reviews of textbooks (apparently some authors provide more motivation), in the textbooks themselves (one of mine said that no proof of the cauchy-schwartz inequality is well-motivated) but despite lots of google searching I can't really get a clear definition of what motivation means in these contexts. I'm kinda understanding it as something that "makes sense" as opposed to "follows from the premises" but that is a very loose definition that I inferred from context. I'd like to know what it really means and whether it's a good thing.

 

[rubrix]

one of the great motivation for me is if we can apply w/e we learn to real life, other classes we are studying, or those classes we will be studying.

for instance, when i first encountered calculus I (limits & derivatives) i was really motivated (and excited) b/c of the fact that i had worked out non-calc based physics problem on velocity, acceleration and i saw correlation between them which is rate of change.

i was motivated to ODE course coz i knew it was very important for PDE. And i want to do PDE at some point.

 

[boboYO]

'motivated' means the author let's you know WHY we should care about the Theorem X or WHY something is defined as it is, otherwise the book is just a dry collection of definitions and theorems.

not saying the latter is bad though, figuring out why something is important, by yourself, is good for you too.

 

[ice109]

it's synonymous with intuition. when someone says this proof will be motivated or motivated as such ___ and then writes some stuff they're giving you intuition.

here's an example - motivation for the theorem that with respect to the usual topology in R2 open balls are convex:

pick a point in R2 and draw a circle around said point. any 2 other points inside that circle can be connected by a line that itself is in the circle.

that's not a proof of the fact that open balls are convex but it is motivation.

 

[lurflurf]

Motivation (in mathematics as well as other areas) is a reason why one would want to do something, and inducement or incentive. In a textbook we might say theorem 172 is well motivated if it allows us to solve problem 412. This begs the question "What is the motivation for solving problem 412?". Often in a treatise it is not so much the amount of motivation that is important as the organization of motivation. So when a result is introduced the reader should feel the result is important, follows from the prevous results, and leads to the future results.

 

[quasar987]

what lurflurf and boboyo said.