FOM: "New results in Mathematics"

Arnon Avron aa at
Wed Sep 15 08:20:01 EDT 1999

In his very agressive posting from September 9, B. Soare made the following 
unbelievable claim:
 "It is time for Reverse Mathematics to add something *new* to
  mathematics, or to get off the pot." 

A few days later J. Pais repeated this argument (in a more civilized way):

 "RFOM3. Is it true that "Reverse Mathematics does NOT produce new results 
  in mathematics (unlike set theory, model theory, and computability theory).  
  It merely takes an existing mathematical result and proves it equivalent 
  to some logical axiom.  This is like stirring a pot of stew and having the 
  pieces [theorem] adhere to the sides of the pot at some level
  [corresponding to the level of proof theoretic strength].  Nothing NEW 
  is ever added to the pot in terms of new mathematical results."

This type of attack was seriously taken even by Simpson, who in a 
recent message to FOM finds it necessary to show that Reverse Mathematics
did produce "Something new" (after explaining that this is not necessary
for RM). 

 After reading all these messages I realized how wrong I have been my whole 
life in my conception of what a "mathematical result" is. Imagine: one of the 
"results" that impressed me most 25 years or so ago, when I read Takeuti
and Zaring Introduction to Set Theory (which just happened to be the first 
book I read on axiomatic set theory) was Sierpinski's proof that GCH implies
AC. I did not understand then that what this stupid Sierpinski is doing
is to derive an obvious axiom,  which almost every mathematician uses 
regularly, from an hypothesis that is even not known to be true!
(Takeuti and Zaring also explicitely took this as one of the major
results in their book. Can you believe this?).
And those guys, Rubin & Rubin, they devoted a whole book (two books,
in fact) for showing that a lot of mathematical facts are equivalent
to AC, wasting half of their book (and their reader's time) proving
things that add nothing new to our Mathematical Knowledge. Not to mention
that idiot, Hilbert, who devoted a full book to Geometry, in which he just
did not produce any new geometrical result! (he only shows that the already
known results all follow from a certain set of axioms- what for??). That same
Hilbert later invested a lot of his energy (as well as the time and energy
of many other excellent mathematicians) trying to prove the consistency
of theories like PA, which nobody suspect to be inconsistent! Obvioiusly, 
he  just had no idea what a new mathematical result is!
 Speaking about Geometry, I now find it very difficult to understand
why people in the past took pain to distinguish between "absolute"
Geometry and general geometry, and found it important to find out
what can be proved in Absolute Geometry and what not, showing useless
related theorems (for example: that if the sum of angles in a triangle
is 180 then Euclide's fifth postulate obtains).  These people add nothing new
to our mathrematical knowledge. They should have consulted Soare
before engaging themselves in such a fruitless activity!

Yes, after reading Soare's message I understand things much better. Only
one thing remains as a mystery for me: how is it possible that 
although an activity like  Reverse Mathemaics is mathematically empty, almost
every mathematician  in my school of mathematical sciences who hears about 
it usually reacts with interest, while I cant imagine raising any
interest here in the structure of r.e. degrees??? (note that
nobody in my school, including myself, has ever done any research in
either RM or r.e. degrees!).

Arnon Avron

Department of Computer Science
School of Mathematical Sciences
Tel-Aviv University.
Tel-Aviv, Israel

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