Clear and Certain Notions

Salviati: Our confidence in the various formal systems (PA, ZFC,
etc.) and results in mathematics is necessarily based on an 
incomplete induction, just as is our confidence in any proposition 
of physics.

Simplicio: Nonsense! Our confidence in arithmetic is based on 
seeing a model: the integers N, addition, multiplication.  These 
are clear and certain notions, and the PA axioms hold in N; so we 
believe PA is consistent.  This is not a formal proof, nor can it 
be; but it is vivid and compelling.

Salv: You're describing the intuitionist point of view:  To "SEE 
a model", to apprehend clear and CERTAIN notions, and to simply 
KNOW that those notions must be consistent.  For an intuitionist
like Brouwer, mathematics has as its source an *intuition* that 
makes its concepts and inferences immediately clear to us.  You've 
gone further, implying that our chosen axiomatization is automatic-
ally entirely consistent with those intuitive notions, which the 
intuitionist surely doesn't suppose, but in any case, if you concede 
that this intuitive sense ultimately arises from (or is based on) 
multiple individual perceptions, all seeming to conform to a simple 
pattern, then you are subscribing to incomplete induction.

If, on the other hand, you maintain that our intuition is a single 
indivisible whole, arising not from any interactions with external 
objects but only from a pure metaphysical vision, then you are 
subscribing to Platonic mysticism.  In either case, exactly the 
same type of justifications can be (and have been) given for 
believing in various physical theories (cf Pythagoras, Kepler, 
Eddington, Einstein,...)  

The purpose of this discussion was to ascertain whether the way in 
which we "know" mathematics is qualitatively different from the way 
in which we "know" physics.  Nothing in the "I see a model" approach 
gives us any reason to think the two are fundamentally different.
Both rest on necessarily incomplete induction.  Indeed 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.

Simp:  What!?  You're not serious!  It's true that Arithmetic is quite 
complex, and in a certain Godelian sense we cannot hope to formally
prove its consistency (within any formal framework whose consistency
is more indubitable), but neither this nor anything else, I should 
think, establishes there are no consistent systems at all!  

Salv:  Calm down, Simplicio.  There is no claim to have "established"
this result, but simply to point out that nothing guarantees us the 
existence of a consistent formal system with enough complexity to 
encompass arithmetic.  By the way, be assured that 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 can 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 are global 
inconsistencies.

Simp: In mathematics?  I don't think so.  Logics have been proposed 
that profess to 'tolerate' inconsistency, from what I hear; but logic 
as actually used in mathematics does not.  How would it go?... what 
metric?

Salv:  In any given fixed formal system the mathematics can be 
reduced to computation.  Within computation there are varying 
distances between pieces of information.  For example, given the 
integers x and y, the distance to the sum x+y is short, but given 
the integer N (=xy) the distance to the factors x,y is large 
(assuming P~=NP).  These distances can be quantified, as in 
polynomial time, exponential time, etc.  

Now suppose we have a set of axioms encompassing ordinary arithmetic,
and just for fun I add one more axiom, stating that the 20 billionth
decimal digit of pi is 7.  What happens?  Does the system instantly
dissolve before our eyes?  Can we now easily prove 1=0?  Well, we
might be able prove it, but it won't be easy, because (as far as I
know) there is no short path between any existing theorem and the
value of the 20 billionth decimal digit of pi.  (Note that there are
algorithms to produce the nth hexidecimal digit of pi, and in some
other bases, but none is known so far for decimal digits.)  I can 
assume it is 7, or I can assume it isn't 7, and it really has no 
demonstratable effect on any other theorem.  This is an admittedly 
a clunky example, but it makes the point that it's perfectly possible 
to conceive of an axiom system that may well be inconsistent, but 
whose inconsistency is practically inaccessible.  (If 20 billion 
isn't safe enough for you, make it 10^10^10 or whatever.)  If I 
say that my axioms MUST be consistent because I've been working 
with them for many years and have yet to find an inconsistency, 
would this persuade anyone?

Is it conceivable that a more benign looking set of axioms might
nevertheless imply two contradictory things about some computationally
very remote entity?  Speaking of benign-looking axiom systems, 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.

Simp:  Not Cantor.  Nobody has ever found something ill-founded or
inconsistent in Cantor's work.  

Salv:  Your loyalty is admirable, but it's well known that Cantor's
early ideas were ill-founded.  Even the humble "Encyclopedic 
Dictionary of Mathematics" (2nd Ed., MIT Press) tells us that

    "It was G. Cantor who introduced the concept of the
     set as an object of mathematical study...   Meanwhile 
     it was pointed out that Cantor's naive set concept 
     leads to various logical paradoxes..."

Simp: Infallible, we surely are not.  Errors in published papers 
happen all the time; on occasion a mistake survives for years.   
However, I'm not aware of any established body of mathematics, 
however old, having been invalidated and overturned by later 
work, ever.

Salv:  Now, Simplicio, statements like these undermine your previous 
claim that incomplete induction plays no part in your confidence in 
mathematics.  Moreover, the statement is simply not historically
accurate.  For example, Euclid's "Elements" was surely an established 
body of mathematics if ever there was one, displayed as a model of 
mathematical rigor and perfection for two thousand years, but today 
it's widely acknowledged to be shot full of gaps and errors.  You 
may counter that nevertheless the sum of the squares of the legs of 
a right triangle equal the square of the hypotenuse, but that is 
just the raw data.  We could equally well say that Ptolemy's 
astronomy was never overturned, because after all the sun still 
sets in the west.

If you don't count philosophical misconceptions, gross limitations 
on applicability, and blatant explanatory inconsistencies, then no 
established body of knowledge in ANY field has ever been overturned.
On the other hand, if you DO count those things, then mathematics
has been distinguished by almost continuously overturning its past
forms and establishing new modes of thought and standards of rigor.
For example, people had given proofs of the fundamental theorem of
algebra before Gauss, but he showed that all his predecessors had 
been insufficiently rigorous and their proofs were unreliable.  His
proof was then accepted as the first truly rigorous proof.  Today 
any freshman can tell you that Gauss's methods were insufficiently
rigorous, and yet many professors will tell you that no mathematics 
has ever been overturned.  So it goes.

Surely the correct view of Cantor and his naive set theory is that,
like Newton and his fluxions, his intuition was strong enough to 
compensate for a somewhat informal foundation.  All of which supports 
the point that mathematical knowledge is as likely to come from 
intuition as from formal axiomatic reasoning, and therefore it is 
of a kind with scientific thought.

Next: Trial and Error