Seeing It

Simp: The view that we "see" N is held by just about everybody
except strict formalists.  ... it was in reply to the question: 
how do we come to believe Consis(PA) or (ZF) I said, by having 
a model of PA, or ZF.  People who believe  Consis(PA) do so 
because they see PA holds in N.

Salv:  So your claim is that we "see" N (the set of natural numbers)
and then we "see" that PA holds in N.  As far as I can tell, you
don't claim this "seeing" is a process of deductive reasoning. You're 
evidently content to simply name it "seeing" and declare that it 
can't be analyzed any further.  The only properties of "seeing" that 
you acknowledge are (1) it is the ultimate source and justification 
of mathematical knowledge and (2) it differs from other sources of 
knowledge in the indubitability of its results and its complete 
independence from experience.  My claim is that this "seeing" is 
ultimately derived from experience, mainly from our most primitive 
perceptions of consistency.  (I trust it goes without saying that 
the distinction between internal and external experience is itself 
a subjective judgement based on experience.)

Simp: I'll ask again: do you consider N, +, *  clear and precise?  

Salv:  "Clear and precise" are terms that can be (and often are) 
applied to various notions in physics, so your question doesn't get 
at a potential difference between physical and mathematical knowledge.
To be relevant to the discussion at hand you must have meant to ask 
a much sharper question, namely, whether N is *perfectly* clear,
transcendentally clear, more clear than anything based on experience
can be.  The answer is no, because N is, in fact, based on experience
at various levels.  Of course, the complete set N is an extrapolation,
because we don't actually have any direct experience of completed
infinite sets, the adoption of which as a foundation of mathematical
knowledge is a relatively recent event.  Completed infinities 
would have (and did) horrify the ancient Greek philosophers and
mathematicians.  This doesn't prove that N is ill-founded, but 
it certainly casts doubt on your comfortable premise that "N" can
claim the universal assent of all sentient beings.  Furthermore, the
alledgedly perfect correspondence between "N" and PA is far from
clear, not least because N itself is an intuitive notion that may 
not even be perfectly capturable as an axiomatic system.

Simp:  Returning to your claim that "the intractable computational 
structure of formal systems will exert an influence on the way 
we view foundational questions", I agree that this is very possible,
but not in the way you suggested -- tolerating contradictions if 
they are 'hard to reach'!   Note: ONE proof of a contradiction 
suffices; so input-length asymptotics, e.g. P, NP, Exponential 
time... hardly apply anyway.

Salv: You should finish your thought, Simplicio.  One proof of a 
contradiction suffices...to do what?  In the context of a discussion 
about whether ZF (for example) might be inconsistent, someone 
invariably raises the objection you have made, which can be 
summarized by the following argument for the consistency of ZF:

 1. If ZF is inconsistent we would be able to prove 1=0 in ZF.
 2. We are not able to prove 1=0 in ZF.
 3. Therefore ZF is consistent.  QED.

The problem with this argument is that [1] is not necessarily true.
There is a similar statement, let's call it [1'], that is true:

 1'. If ZF is inconsistent it would be possible to prove 1=0 in ZF.

Notice that I've changed  "we would be able"  into  "it would be
possible".  This is a distinction that mathematicians have often 
been reluctant to make (cf Hilbert, "we SHALL know"), but the
influence of computers and the increased focus on the reality of
cognitive intractability is forcing mathematicians to view the nature
of mathematical knowledge in a new light.  The complete argument 
based on [1'] becomes

 1'. If ZF is inconsistent it would be possible to prove 1=0 in ZF.
 2'. It is not possible to prove 1=0 in ZF.
 3.  Therefore ZF is consistent.  QED.

In this form, item [1'] is okay, but we have no justification for
asserting [2'].  All we can really claim to know is [2].  Thus,
neither form of the argument is valid, because there is a difference
between what could be done and what we can do.

Simp:  Let me return to an earlier point.  Cantor didn't misuse set 
theory... if the Encyclopedic Dictionary of Mathematics seems to 
indicate such a thing, then the wording is unfortunate, to say the 
least.

Salv:  Cantor's original axiomatization of set theory led to 
inconsistencies.  It's remarkable that in this instance you conceed 
(in fact, insist) that a flawed formal foundation is irrelevant to 
the quality of the mathematics, whereas previously you asserted that 
a single implicit contradiction utterly invalidates every result 
derived within a given formal system.  Your view of Cantor's (and 
Euclid's) work reveals that you know this reductionist/formalist 
view is not justified.

Cantor himself warned others about the potential for pardoxes 
implicit in his original definitions and premises, once he
recognized the problems himself.  In particular, his original
definition of a "set" turned out to be insufficiently restrictive, 
as he himself said.  Thus, by his own admission, his original 
work was based on premises and definitions that were implicitly
inconsistent.

Moreover, the flaw in the premises was not "seen" by thinking harder
about the premises and whether they really do correspond to some
"model" that we know for certain is consistent; it was revealed only
by actually bumping into inconsistencies, and then working backwards
to arrive at the conclusion that the premises must be flawed.  Then 
they dreamed up a new set of premises that would avoid those
inconsistencies.  Moral: the consistency of our results is not due 
to the clarity and precision of our premises; rather we shape (and
reshape) our premises to make the formal system give the results that
we've already determined to be consistent by external means - just as
we do in physics.

Simp:  Well, I would agree to the extent that usually development 
of a mathematical theory comes first, and axiomatization afterwards.

Salv:  Precisely, and this is the hierarchy conceptually as well 
as chronologically.  Math is developed informally, and then someone
gets around to attempting to formalize it, to make it rigorous,
thereby ostensibly setting it apart from informal sciences like
physics.  But it usually happens that the early attempts at
axiomatization are flawed, and we "see" it quickly.  Then we improve
the axioms and they stand up a while longer, but we come across
another flaw.  We fix the axioms again (see Proofs and Refutations,
Lakotos), and so on, until we go for years without anyone finding a
flaw.  Then we declare the project complete.  Clearly the governing
force behind this project has not been formal axiomatic reasoning from
manifestly clear first principles; rather the formal system has been
shaped and corrected at every stage by informal reasoning.  That's why
I began (several posts ago) by saying that we're not justified in
differentiating between math and physics knowledge on the grounds 
that math knowledge is based on formal deductive reasoning while
physics knowledge is based on informal reasoning.  In truth, they are
both based on informal reasoning from incomplete and (for all we know)
flawed premises.  If and when the consequences of any flaws become
apparent, we will revise our premises.

Simp:  It may be true that informal reasoning is often used in
mathematics as well as physics, but it doesn't follow that informal 
mean non-rigorous.  For example, NOBODY carries out proofs as 
Predicate Calculus formal deductions. 

Salv:  But surely the "fact" that no one practices formal proof 
doesn't imply that informal proof is rigorous.

Simp:  Let me take another approach to distinguishing between math
and physics.  The big difference is, what physical sciences study 
is in the world; what math studies isn't.

Salv:  Here you're making a very conventional distinction between 
internal and external experience, and claiming that the former is 
not "in the world".  It isn't clear that more sophisticated theories 
in both math and physics will necessarily bifurcate our experience
along these same lines.  Thus, on a purely experiential level, there 
is no reason to believe mathematical knowledge is essentially 
qualitatively different from physical knowledge.

Simp:  Well, studying N does seem different from studying things 
in the world, like accelerations, electric charges, temperatures...
I don't see what argument you have made that it isn't.

Salv:  As Aristotle observed, in every field of knowledge there is 
an inductive and a deductive component.  I would say that your study 
of N "seems" (to you) to lack an inductive component only because 
you have never reflected on the ultimate source of your primitive
notions about N.

Simp:  Granted that the foundations of knowledge in both mathematics 
and physics contain inductive components, I would say the crucial 
difference is that in mathematics the GOAL is deduction, whatever 
the path that takes us there; the end result has to be a correct 
proof.  The subject-matter is abstract; there is no 'agreement with 
experiment', no perihelion anomalies or anything.

Salv:  So now you're saying the difference between math and physics 
is not in what we do, but why we do it (i.e., what's the goal).  For 
example, you would say that Kepler's study of conics was physics, 
whereas Apollonius's study of conics was math.  Riemann's study of 
geometry was math but Einstein's was physics, and so on.  I would 
argue that in all of those cases the motivations had more in common 
than you might think.  Of course, we have Gauss's comment that 
geometry is actually a part of mechanics.  You might also have 
trouble classifying things like string theory.  Some parts of modern 
physics have little or no connection with the kind of "experiment" 
you have in mind.  On the other hand, in the broader sense of 
*experience*, both math and physics are inextricably bound up in it.