Trial and Error

Simp:  I must say, Salviati, that your reference to computational
time heirarchies (polynomial time, exponential, etc) astonishes me.
Such time hierarchies only affect the practical issue of intract-
ability, but this is precisely what mathematics is NOT concerned 
with.  "Easily" doesn't matter in mathematics.

Salv: The irreducible formal/computational distances between elements 
of a formal system are becoming more and more conspicuous, mainly due 
to the computer.  The old style of mathematics paid no attention to 
this structure, largely because it was too difficult to study in 
detail wihout computers.  But the interpretation (or modelling if 
you prefer) of formal mathematics as computation beginning (after 
Leibniz) with Godel, Turing, Post, et al, has shown that contemplation 
of the "computer model" of mathematics can yield very profound 
results.  I think it's natural to expect that the "intractable 
computational structure" of formal systems will likewise exert an 
influence on the way we view foundational questions.

Simp:  Well, in any case, I must object to your characterization
of Euclid's Elements as "shot full of holes".  Surely no respectable
mathematician shares your opinion.  Perhaps Euclid used some tacit 
assumptions which he overlooked to list, but nothing more serious
than that. 

Salv:  I'd refer you to any of several excellent books on the history
and philosophy of mathematics.  For example, Michael Crowe remarks

   "...as early as 1892 C.S.Peirce... summarized a conclusion 
    reached by most late-nineteenth-century mathematicians: 
    'The truth is that Euclidean geometry, instead of being 
    the perfection of human reasoning, is riddled with 
    fallacies...'" 

The numerous deficiencies in Euclid as an axiomatic system are
well-documented, and easy to see.  Of course, this doesn't reflect
badly on Euclid because he wasn't in the business of creating
axiomatic systems in the modern sense.  He understood himself to be
engaged in what we would now call physics, i.e., saying things about
spatial relations in the real world.  Read Heath's translation of The
Elements, then read Aristotle on the structure of various fields of
knowledge (including mathematics), and then have a look at Hilbert's
modern axiomatization of geometry.

Simp:  You say Euclid was engaged in physics?!  Lines without breadth, 
points having no parts...  No perfect straight lines or circles in 
Nature...  In short, I don't think so.

Salv:  Idealization is an aspect of physics just as much as of 
mathematics.  There is no perfectly ideal gas, and yet physicists 
work out the implications of ideal relations such as pV=uRT.  This 
doesn't imply that physicists are uninterested in actual gases.  
Similarly Euclid was idealizing real circles and lines, and working 
out the implications of those ideal forms, but his interest in those
particular ideal forms was due to their correspondence with true
spatial relations.  He wasn't interested in formal relations based 
on artificial or arbitrary axiom systems.  His axioms were specifically 
designed to reflect the "truth" about (ideal) spatial relations, just 
as the physicist establishes premises that reflect the truth about 
(ideal) gases.

Simp:  Perhaps so, but let's return to your claim that Gauss over-
turned an established body of knowledge with his proof of the 
fundamental theorem.  A single faulty proof by D' Alembert is not 
exactly an "established body of knowledge".

Salv:  It wasn't just a single proof by D'Alembert.  Gauss began his 
doctoral dissertation in 1799 with a critique of the inadequate rigor 
in the purported "proofs" of the fundamental theorem given by 
D'Alembert (1746), Euler (1749), De Foncenet (1759) and Lagrange 
(1772) over the preceeding half-century.  Ironically, as mentioned
previously, Gauss's proof is now criticized for lack of rigor on
very similar grounds.

Simp: I would have to be shown Gauss's lack of rigor.

Salv: In Part I of Section 19 of his dissertation Gauss says 
"but since the value at x is negative and at y is positive, it
must equal zero somewhere between x and y."  Today this is
known as using visual evidence, and is not accepted as valid
proof because circumstances are known in which it doesn't 
work, i.e., it is not a reliable method of inference.

Simp:  It's a standard property of continuous functions (and of
derivatives too); not visual evidence but valid argument.  Are 
you claiming that Gauss used it for something that wasn't, say, 
continuous?  

Salv: He used it for something that wasn't necessarily continuous, 
i.e., the continuity of the locus in question had not been 
established.  Ironically, it was precisely on these grounds that 
Gauss had criticized some of the prior proofs, but then he commited 
the same error (in another part of the proof) himself.

Sagredo:  Let me interrupt, Salviati.  Your the argument that
our confidence in ZF rests on an "incomplete induction" is 
completely unconvincing.

Salv:  Perhaps you think the induction I'm referring to is limited 
to conscious induction at the level of the axiom system itself.  
That's not the case.  The first and most important use of induction 
is in the formation of your clear and immediate notions of the 
sequence of natural numbers - your mental model of the naturals.  
You may have begun to form those notions when you first opened your 
eyes.  Or, if you prefer, your common notions (to use Euclid's 
phrase) may be a consequence of the inherent wiring of your brain 
which evolved by a process of trial-and-error interactions with 
the environment - again involving incomplete induction, albiet 
unconscious on your part.  Or you may believe that your clear and 
simple notions arise from an entirely metaphysical source, i.e., 
entirely unconditioned by any external factors, which is a form 
of mysticism.  Or etc. etc.  It isn't my purpose to list all the 
possible theories for the sources of knowledge (efficacious grace, 
merely sufficient grace,...) but simply to observe that there is 
no reason to believe mathematical knowledge at the fundamental 
level is essentially different from physical knowledge.

Sagr: Whatever "induction" that may be involved in forming a picture 
of the natural numbers is not at issue. The "incomplete induction" 
on which you claimed that our confidence in a formal system is
"necessarily based" was, according to your own later explanation and
quotation, that expressed in the reflection that "mathematicians have
been working with ZF for most of this century, and no contradictions
have been discovered yet". 

Salv:  No, my claim has been and remains that induction is used 
throughout the establishment of our thought processes, from the 
formation of rudimentary notions to the development of more refined 
intuition to the construction of formal axiomatic systems.  Further-
more, at each of these levels there are historical examples 
illustrating the fallibility of our inductive judgements.

Everyone seems to agree that we arrive at our rudimentary notions, 
if not inductively, then at least by some means other than formal
deductive reasoning.  (One can hardly disagree with this.)  Your
original disagreement was with the idea that induction is also 
applied to assess the validity of high-level axiomatic systems.  
In particular, you claimed that "no one's confidence in ZF is based 
on induction".  To show that your claim was false I presented a 
quote from a mathematical philosopher explicitly citing an 
inductive argument for the consistency of ZF.  

I could quote many others.  For example, when this was discussed on
sci.math last year one of the participants chided me for suggesting
that mathematical knowledge is ultimately based on induction and that
our confidence in results within ZF must always be less than perfect:

   "[This] suggests yet again that you don't know beans about
   what you are talking about.  A proof from ZF brings with it 
   the supreme confidence that a century of working with ZF and
   beyond has given us."

In view of remarks like this it's hard to see how anyone can
disagree that mathematicians routinely invoke inductive arguments
to support the validity of axiomatic systems.  I also cited Frege 
as an example of an axiomatic system that survived the careful 
scrutiny of eminent mathematicians for several years and that 
failed only when confronted with an actual example of a logical
inconsistency.

You go on to assert that "Whatever "induction" may be involved in 
forming a picture of the natural numbers is not at issue."  But
since this in a discussion about the origins of mathematical 
knowledge, I simply have to disagree with you.  The formation of 
our rudimentary notions and intuitions is not only at issue, it is 
THE issue.

Sagr: What I claim is that the actual behavior of mathematicians - 
as opposed to their occasional philosophical asides - does not 
support the view that they base their confidence in any formal 
system on such "incomplete induction".

Salv: You have this exactly backwards.  The actual behavior of most
mathematicians is to rely heavily on the integrity of the mathematical
community for assurance of the correctness of the vast body of
mathematical work they could never personally hope to check in 
detail, often not even completely mastering results on which their 
own work is based, and certainly not worrying about foundational
questions such as the ultimate consistency of the formal system in
which they happen to be working.  It's only in their "philosophical
asides" that most mathematicians are likely to overlook how they
really do business, and claim that mathematical knowledge is certain
and sure because we can see it in all it's irrefutable inevitability
all the way to the ground.

Next: Seeing It