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[Reuben Hersh]

I agree that Tarski can give a first-order proof of everything
in Euclid's Elements.

Nevertheless, mathematicians use geometric proofs, even
if you claim you  can prove they aren't proofs.

There's "Visual Complex Analysis," I forget the author's name,
it should be in your library.

There's Arnold's books on ordinary differential equations.

Mathematics magazine for years has published "proofs without words."

Recently a collection of proofs without words was published
as a book entitled "Proofs without words."  If it's not in
your library, just look at some issues of Math Magazine.

Why do mathematicians do this?

Because a geometric proof is often more convincing, more perspicuous,
more inteeesting, more memorable.

You can say it isn't a proof, but it is recognized as a proof by 
people whose business is proving theorems.

You can take a tree branch and chop is into bits, and then
say you've proved it was really a tree branch.  

To what end?

Reuben Hersh