"Philosophy is really very useful for expressing commonplace ideas in fancy ways."(Robert Paul Wolff, here)

Meet young Joe.
Joe is a mathematician, although he doesn’t know it.

Joe is a poor
Arts student, doing sundry stuff for money. He woke up and counted how
much money he had left: $30. Luckily, today he has some hours of casual cleaning
scheduled at the pub. He has an exam, too (at 10:30), so he's in a hurry.

Joe started
at the pub at 6:00. He did his thing, finishing by 10:00.

The
publican paid him, but Joe was in such a hurry he just took the
money, shoved it in his pocket, and rode his bike to uni.

Finished
his exam, Joe decided to eat something. That’s when he counted his money: now, he has $70.

*“What was my hourly pay rate?”*Joe wondered. He reasons, as you would, like this:

- Earlier, I had $30.
- I have $70, now.
- I’ve spent nothing, yet.
- So, the boss paid me $40 (=$70 -
$30).
- I worked 4 hours, from 6:00 to
10:00.
- Dividing $40 by 4 hours yields
$10 per hour.
- That’s my hourly wage.

*“Meh,”*readers may think.

*“That’s garden-variety reasoning, and basic arithmetic, not Real Maths.”*

Hold that
thought for a moment. This is what Joe did:

($70 - $30)/4 = $10

Now, brace
yourselves for the revelation: Joe actually solved the linear equation below,
for HourlyWage.

TotalMoney
= InitialMoney + HourlyWage * Hours

The steps
taken to solve this equation (which one learns as rules in school, say "if it multiplies on the right hand side, it divides on the left"), in the order
adopted, parallel the steps Joe took to reason his solution:

HourlyWage
* Hours = TotalMoney – InitialMoney = $70 - $30 = $40.

HourlyWage
= (TotalMoney – InitialMoney)/Hours = ($70 - $30)/4 = $10.

Joe did
that in his mind, without pen or paper (like you did). But it’s not the
use of pen and paper that makes it "Real Maths": maths is a form of
reasoning. The school "rules" are not arbitrary, they have a reason. Knowingly or not, Joe reasoned mathematically, and it didn’t hurt him
a bit.

In simple
cases like this (and simplicity, like beauty, is in the eye of the beholder),
anyone can reason correctly “in their minds”; in more complex cases, it’ll be
wiser to appeal to pen and paper, because the risk of making mistakes is far greater. This does not alter the fact: maths is a reasoning process, no more, no less.

----------

Is this to
say that maths in economics is a guarantee of “scientificity”, or at least that
it’s always appropriate or innocent? Obviously, not. On top, there’s no absolute
rule to judge.

What one can
conclude is that the usage of maths in economics is not inherently wrong, for
some obscure reason that anti-maths people can’t explain in simple terms,
without a generous side of “metaphysics”, “epistemology”, “ontology”,
“teleology”.

Know this: maths and technical language sometimes are necessary; but the unnecessary use of maths (or technical,
specialist words) may indicate a charlatan trying to
bullshit you.