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Definitions in mathematics are used to make the language describing abstract concepts elegant, as explained in the entry.

However, definitions aren't chosen all willy-nilly - there are good arguments why definitions are adopted, as should have been seen in the article.

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My favourite algebra professor always said something to the effect of "we always have to be careful about abusing notation but if we make good choices of notation we can abuse it right." Definitions are sort of the same deal.

Sort of like how we use terse symbols for everything because ink and papyrus is expensive.

Not really. We have some very good reasons for using the definitions we use every day. Those reasons are more related to us than our technology, although our tech is getting so good that we already need to think about it. And by the way, people changed almost all the usual definitions at the XX century.

Now, mathematicians do throw the usual definitions away all the time. It's important to know when to reuse other people's coding, or roll your own.

And because if you're Isaac Newton and you're writing pages and pages of proofs, you naturally want to use a shorthand in order to save yourself time, energy, and from repetitive strain injury.

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