Certainty In Mathematics and Physics

Rigorous proof (of the kind that supposedly distinguishes mathematics 
from physics) resides only within a formal system.  Each theorem of a 
formal system can be viewed as just a single data point that either does 
or does not contradict some other theorem of the system.  After we 
have explored a given formal system for a long time we may feel very 
confident that it is consistent but, needless to say, no finite number
of contradiction-free theorems can constitute a PROOF of consistency.  
Our confidence in PA, ZFC, or any other formal system is necessarily 
based on an incomplete induction.  It's always possible that a given 
formal system could exhibit an inconsistency at some point.  In fact, 
it's been suggested that EVERY formal system, if pressed far enough, 
is inconsistent.  Nothing guarantees us the existence of a consistent 
formal system with enough complexity to encompass arithmetic.

On the other hand, the existence of a contradiction in a formal 
system need not completely vitiate the system.  The only operative 
consistency in any system, mathematical or physical, is LOCAL.  We 
might define a metric on the space generated by the axioms of a system, 
and find that there is consistency within a certain region of that 
space, even though there may be global inconsistencies.  Everyone is 
familiar with a similar situation in physics, such as the success 
of Newtonian mechanics in the quasi-static limit of low relative 
velocities.  However, we shouldn't overlook examples in mathematics, 
such as the naive set theory of Cantor, Frege, et al, that was developed 
extensively over a period of many years, with many meticulously 
proven theorems, before it was recognized that the system itself 
was ill-founded and inconsistent (unless restrictions analogous to
the quasi-static stipulation for Newtonian mechanics) are imposed.

The Winter 97 issue of Mathematical Intelligencer contained an article 
by Dan Vellman discussing Wiles's proof of Fermat's Last Theorem, and 
pointed out that what has really been proven (assuming Wiless' proof 
has no flaw) is not "FLT" but rather "If ZF is consistent then FLT".
Of course, even though our belief in ZF is based on incomplete
induction, very few mathematicians think it's worth worring about.  
Admittedly a generation of competent mathematicians were deceived by
naive set theory, but we tend to think that our thought processes 
have subsequently become infallible.

Regarding the contrast between physical theories and mathematical
theories, we note that a physical theory such as Newtonian mechanics 
can also be construed reductively as a purely mathematical theory, 
and as such it may be internally consistent, even if its physical
validity is in doubt.  However, I'd venture to say that Newtonian
mechanics is mathematically complex enough that even its purely
mathamatical consistency can't be rigorously proven except perhaps
within a formal system whose consistency is even more dubious.

Still, a physical theory is by definition a theory of knowledge 
that associates (or identifies) some of its elements with observable 
quantities.  That identification is part of the theory, and is 
what makes it a physical rather than a purely mathematical theory.
Newtonian mechanics, interpreted as a physical theory, DOES assert a 
correspondence between its elements and certain measureable quantities, 
and in this sense it fails (as a physical theory, not necessarily as 
mathematical theory) when pressed to extremes of relative velocities.  
Whether it also fails at some point as a purely mathematical theory is 
a separate (and interesting) question.

Perhaps a better example would have been the theory of general 
relativity, which predicts - and breaks down at - singularities in 
the spacetime field.  In this case the breakdown is both physical 
and mathematical.

In response to my assertion that our confidence in PA, ZFC, or any 
other formal system is necessarily based on an incomplete induction,
Torkel Franzen replied

  "There is nothing necessary about it. ...it's doubtful 
   whether anybody actually accepts those theories on the 
   basis of inductive evidence."

On the other hand, the following remark of Dan Velleman in his 
Mathematical Intelligencer article casts doubt on Franzen's claim:

   It might be thought that the consistency of ZF is sufficiently
   well-established that it is not worth worrying about.  After 
   all, mathematicians have been working with ZF for most of this
   century, and no contradictions have been discovered yet.

Needless to say, this is virtually the definition of incomplete 
induction.  Still, Franzen argues that this all misses the point,
because

 "The consistency of arithmetic is a consequence of the truth 
  of its axioms."

One is tempted to add that the "truth" of arithmetic's axioms is a 
consequence of their consistency.  But seriously, labelling a set of 
axioms as "true" doesn't constitute a rigorous formal proof of their 
consistency.  For example, Frege and others regarded the axioms of 
naive set theory as manifestly true, even more so than the axioms of
arithmetic!  But they were wrong.

In response to my suggesting the possibility that EVERY formal system, 
if pressed far enough, would be fond to be inconsistent, and that 
nothing guarantees us the existence of a consistent formal system 
with enough complexity to encompass arithmetic, Franzen replied

 "This is like a suggestion that 0 and 1 might turn out to be 
  identical if we look hard enough."

However, it's worth noting that when it was realized the axioms of 
naive set theory were flawed and led to contradictions, we did not 
conclude (as we were strictly entitled to do) that 0 and 1 were 
identical.  Rather we concluded that the axioms of set theory were 
flawed and needed to be modified.  Some proofs, and even some 
theorems, were rendered invalid in the new system, but happily we
were able to preserve the distinctness of 0 and 1.

When I mentioned Vellman's discussion of Wiles' proof of Fermat's 
Last Theorem, pointing out that what has really been proven is not 
{FLT} but rather {If ZF is consistent then FLT}, Franzen asked
"What does "really" mean here?"  Let me try to explain.  People who 
believe that ZF is manifestly true and consistent tend to regard the
two statements {FLT} and {If ZF is consistent then FLT} as equivalent.
As such they tend to say that Wiles has proved FLT.  The word "really"
was intended to remind those people that the two statements are not 
equivalent, and that Wiles has proved the latter, not the former.  
This is essentially just a paraphrase of Vellman's article, which 
concludes

  For those who believe in the existence of the universe of all
  sets, and who believe that the ZF axioms are true statements
  about this universe, there is no reason to doubt the reliability
  of Wiles' proof.  But for those who are skeptical about the
  existence of such a universe, the question remains:  Should we
  be convinced by Wiles' proof that no counter-example will ever
  be found, or merely that IF ZF is consistent THEN no counter-
  example will be found?

Of course, one could argue that it's a bit unfair to single out Wiles' 
proof to highlight this clarification, since the same qualifier could be 
applied to every mathematical proof carried out within the ZF framework - 
which is essentially all of mathematics.  On the other hand, the issue 
becomes more relevant as the "size" and complexity of a proof increases.
The more thoroughly a proof exercises all the remotest resources of our 
axiomatic system, the more likely it is (arguably) to be at risk if a 
subtle flaw or inconsistency in that axiomatic system ever came to light.

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