Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?

From: Keith Ramsay (kramsay_at_aol.com)
Date: 02/06/05 

Date: 6 Feb 2005 10:05:01 -0800

poopdeville@gmail.com wrote:
[...]
| Obviously not. Arithmetic has been done for centuries without
| formalization, and Peano drew from centuries' worth of inspiration
when
| formalizing PA. PA, however, is the de facto language of arithmetic,
| in the sense that if one makes an arithmetical claim, it is assumed
to
| be formalizable in PA unless the speaker says he is referring to a
| different form.

I went through quite a bit of mathematical education without
finding this to be so anywhere I've been. You might as well
have said that it was customary for us to greet one another
by pinching each other's cheeks.

Peano had a set of five axioms where the induction axiom was
not a first-order axiom, but stated more like induction is
usually stated. Those axioms are categorical; the natural
numbers are the only model up to isomorphism. In order to
get a first-order axiom system, PA, the induction axiom is
replaced with a scheme restricted to just predicates
expressed in first-order terms.

I suppose some fraction of professional mathematicians would
have heard of first-order logic, but I'm not confident it's
a majority. Induction as restricted to first-order formulas
plays no role in ordinary mathematical discourse.

| > As Tim Chow has nicely pointed out, you can't apply this
| > formal interpretation to *everything*-- you have to start
| > out with at least some concepts that aren't relativized
| > in the same way. Otherwise one has an indefinite regression
| > of interpretations.
|
| And we *do* have an indefinite regression of explanations. However,
| explanations must come to an end somewhere. That happens when you
| understand what a speaker has uttered.

This doesn't explain why you think there's an ambiguity in
the language of number theory. I think I get to the point
where I understand what the speaker is saying, without having
to worry about nonstandard models of PA or ZFC or anything
like that getting in the way.

| > Assuming axiom-system and/or model relativity helps to
| > cloud the real issues, by leading people to worry about
| > the part of mathematics that they interpret in such a
| > way, when what makes the crucial difference is how much
| > of mathematics they regard as meaningful without resorting
| > to such a dodge.
|
| I find this obscure. What are the real issues? Why is
| "model-relativism" a dodge?

Well, think of it like wanting to know at what point a
person is speaking metaphorically, and when they are speaking
literally. If someone says, "I have a guardian angel.", I
usually assume that this means they believe in the existence
of angels (as many people do, in a literal sense), just as
if someone says, "I have an apple.", they mean an actual
apple, not something else. But some have spoken of guardian
angels without meaning it literally; they may go on to explain
that they talk about angels as a metaphor for something else.

Model-relativism represents another way for someone to be
saying something (in this case a mathematical claim) without
meaning the terms being used literally. Suppose someone in
a class says, "There exists a well-ordering of the reals."
A literal interpretation of that means that they believe in
subsets of RxR and so on. On the other hand, they might not
actually believe in the existence of sets at all. They might
mean that every model of ZFC satisfies the formalization of
"there exists a well-ordering of the reals".

Discussions of the philosophy of mathematics seem usually to
be impeded quite a bit when I'm unable to tell how much of
mathematics another person means literally, and how much
as an indirect way of talking about something else. The claims
that they understand literally make up their real view of the
truth in mathematics. This remains obscure as long as I'm
unable to find what kind of facts they do finally accept as
literally true.

You, for example-- I don't actually know whether you believe
in the existence of models of first-order theories, in
character strings, and so on. It seems like you ought to,
given the way you write. But it also seems like you ought
to believe in integers if you believe in any of those
things. Integers can be represented as character strings.
The proofs that there exist models of consistent theories
rely upon the existence either of integers or of something
equivalent to them. But then you write as if it were
appropriate to assume model-relativity for the integers,
which suggests you don't really believe there are such
things as integers (in the sense: you don't think there's
an unambiguous notion of "integer").
The line between literal and metaphorical or reinterpreted
seems to say a lot about a philosophy of mathematics.

In another thread, for example, someone referred to a paper
by Putnam where he apparently discusses a kind of formalist
philosophy where the "real" underlying mathematics is taken
to be finite combinatorial mathematics. (Hence including
elementary number theory.) And then mathematics lying
outside of that arena is treated as a kind of formal game.
That differs from some other kinds of formalism, because
for one thing it introduces an element of finitism. A person
adhering to such a view would accept a statement such as
the twin prime conjecture as having a definite meaning,
because it's about finite combinatorial objects. But not
all formalists would; some would say (I think) that it's only
whether it's a theorem in some formal system that has a
definite meaning. I would consider these two kinds of
formalism considerably different from each other. One is
verificationist; the other is not. (The meaning of a
statement to the one kind of formalist is the same as the
possibility of verifying it.) And so on.

| > | The language
| > | we speak admits *very* different interpretations, as Godel
| > | showed.
| >
| > We don't speak a formal language, but leaving that aside,
|
| But we speak about "numbers." Our talk of numbers admits wildy
| different interpretations.

Name two.

| > the different interpretations of the language of PA exhibited
| > (implicitly at least) by Goedel are mostly irrelevant. They
| > are interpretations whose existence is demonstrated assuming
| > that one has a meaningful interpretation of the language of
| > arithmetic to begin with.
|
| Please explain. From my understanding of it, Godel's theorem is
purely
| syntactic, though it implicitly shows that non-standard models of PRA
| (is that what it was called? -- I don't mean PA) exist.

What most people mean by "purely syntactic" is that one starts
out with an alphabet Sigma, and defines Sigma* to be the set
of functions from {1,...,n} to Sigma for some natural number
n-- oops, we're talking about natural numbers.

[...]
| The existence of non-standard models does not prevent communication
so
| long as one fixes a context in which an utterance is to occur. In
| mathematics, this is done by (1) finding an axiomatization we'll
agree
| to work with (that may not capture all the features of your private
| model) or (2) finding an axiomatization which picks out your private
| model exactly.

I have a hard time coming up with a useful response to your
claims about how we do things in mathematics when I haven't
observed anything of the kind. This is just not how we ever
explained what the natural numbers are. We were taught how
they correspond to strings of characters (although this was
not presented as a definition)... why is this not an
acceptable explanation? Are character strings not an
acceptable concept? If not, how can first-order logic be
understood?

Quite a lot of mathematics can be understood as discussing
structures that are singled out by *second order*
axiomatizations. The natural numbers, for example, have a
categorical second-order axiomatization. The continuum
hypothesis is equivalent to a statement about a structure
containing the continuum that has a categorical second
order axiomatization.

But you're talking about first-order axiomatizations,
aren't you?

I find it strange to think that a person can have a "private"
model. I don't think there's anything going on inside my head
that would enable me to grasp a piece of mathematics, that is
so unlike what is going on between people discussing
mathematics that we would be unable to communicate that grasp.

[...]
| > "Axiomatization" isn't the only way to describe a structure,
| > unless you take "axiomatization" in such a broad sense that
| > it automatically includes all legitimate means of describing
| > a structure, or we place our means of description onto a
| > kind of arbitrary Procrustean bed.
|
| Please explain how else one might describe a structure if not by
giving
| a method to generate a set of relations the structure is to satisfy.

The natural numbers can be specified as the minimal model
of a set of axioms.

How can you understand what a "model" is without understanding
what a subset of a model is?

| > I think the term "ideology" applies much better to the kinds
| > of thinking that tell us that so much of our usual discourse
| > (like in mathematics) is "meaningless" because it fails to
| > satisfy this or that arbitrary criterion.
|
| Mathematical discourse is clearly not meaningless. Just look at
| sci.math. However, voodoo realist talk of truth of propositions
| without fixing a context in which to evaluate them *is* meaningless.

This "without fixing a context" is a straw man. Nobody says
"I don't have to fix a context". They do say, however, that
the context is not fixed by first-order axiomatization.

| (This has actually been dealt with -- Tim et al actually *meant* to
fix
| a model, V, the universe of sets, in which to evaluate the truth of
AC.
| I was not familiar with this usage of "true.")

Note that V is not fixed by even a second-order axiomatization.

Accepting V as a well-defined structure usually is taken as a
sign that the speaker is mathematical Platonist of the most
extreme kind!

Keith Ramsay

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