FOM: precedents for reverse mathematics

Stephen G Simpson simpson at
Mon Aug 9 19:12:35 EDT 1999

Some FOM subscribers have pointed out some early precedents for
reverse mathematics.

Odifreddi 4 Aug 1999 10:06:24:

 > the first example i can think of is john wallis' proposal of
 > replacing the parallel axiom by the theorem that "given any
 > triangle and a segment, one can construct on that segment a
 > triangle similar to the given one".

Podnieks 6 Aug 1999 10:31:55:

 > The invention of groups, rings, fields, modules etc. also was a
 > kind of reverse mathematics - determining of the minimum sets of
 > "axioms" necessary to prove significant sets of properties of
 > different mathematical structures.

Davis 6 Aug 1999 10:47:20:

 > To quote Jacobi: "Man muss immer umkehren". That is: One must
 > always invert.

And in my book ``Subsystems of Second Order Arithmetic'' I quote an
even earlier precedent!  According to the ancient Greek philosopher
Aristotle, ``reciprocation of premisses and conclusion is more
frequent in mathematics, because mathematics takes definitions, but
never an accident, for its premisses ....''

However, when discussing reverse mathematics here on the FOM list, I
have been referring more specifically to the program spelled out under
this name in my book.  Here one sees that specific, well-known
mathematical theorems are logically equivalent to the set-existence
axioms needed to prove them, the axioms being formulated in the
language of second order arithmetic.  I think this program goes back
only to the 1970's.  Some historical remarks are at the end of section
I.9 of my book.

-- Steve

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