Re: Formulating sentences in a possibly consistent ZF

Nam Nguyen <namducnguyen@xxxxxxx> writes:

That's right. "The standard" model is what CM et al. defined and any
non-standard one is well ... not standard! It's as trivial as in
communist countries the standard color is red, so any non-standard
color must be understood as not red.

Very trivial indeed.

(I wonder why CM and others had kept spending time on such "profound"
triviality?)

In my case it is due to bafflement over the idea that there is
something ambiguous, vague, indeterminate, unclear in our
understanding of the naturals; and due to the observation that all
purported explanations of what this ambiguity, vagueness,
indeterminateness, unclarity amounts to are very obscure, and
apparently entirely unconnected to anything in our actual mathematical
reasoning, the use we make of mathematical language and notions, the
way we come to understand mathematical language etc.

The issue at hand here might not be of great importance, but it is of
some interest at least as an example of a general phenomenon in
philosophy, that people often feel free to bandy about abstract
notions and pictures without giving any explicit and clear explanation
as to how they relate to the aspects of our actual practice and
experience they purport to shed light on. Your going on about
subjectivity of this or that is a perfect example of this venerable
tradition, this time-honoured way of waxing philosophical. In so far
as we regard the business of philosophy of mathematics as being
understanding our actual mathematical reasoning and practice it is
however in the end rather obscure just what the point to such
exercises is.

In philosophy of mathematics in particular all sorts of logical and
mathematical results are thought to be philosophically
significant. Often no argument why this should be so is offered, which
is baffling since in general technical results in logic and
mathematics tell us absolutely nothing philosophically significant. I
quote from a paper I'm working on, /A note on the characteristic
constant of a theory and its significance/:

It would also be of interest if some philosophical insight could be
squeezed out of Chaitin's results. On the conception of philosophy of
mathematics adopted by the author, and apparently by Raatikainen,
Franzén and many others -- though by no means everyone, as nicely
illustrated by the excitement some people experience when pondering
and reflecting on such expositions as Hofstadter's of Gödel's
results, or Chaitin's of his -- such insight does not at all consist
in inspiring metaphors, waxing philosophical in the abstract, or
basking in the warmth of the mathematical ingenuity that goes into
the proofs; rather, it must come in the form of a tediously detailed
explanation and analysis, expressed in clear and unambiguous terms,
rendering it completely transparent just how these purely
mathematical results relate to this or that aspect of our actual
mathematical reasoning or experience. Repeating the observation found
in a footnote in the introduction, there is of course no a priori
reason to suppose there is any such philosophical insight to be
gleaned from any particular piece of mathematics. A special argument
is needed in each case. In case of the results considered in this
note in particular such an argument is entirely lacking.

I put it to you that also in case of non-standard models, the possible
undecidability of the Goldbach conjecture in this or that formal
theory, and so on, such an argument is also entirely lacking; that
merely blathering about 'subjectivity' and what not, in some vague way
related to such results, is uninteresting unless you explain, in
tedious detail and in banal terms, just why we should consider them as
telling us anything about our understanding of mathematical notions,
about our actual mathematical reasoning.

Btw, would you - knowing what is and isn't standard - believe the standard
one would contain GC? Or would that be in the non-standard one?

Why should someone knowing what is standard and what is not have any
opinion about the Goldbach conjecture? It is a peculiar idea that if
one understands what a natural number is -- or, in more posh and
pointless terms, what 'the standard model of arithmetic' means -- one
should know the truth or falsity of the Goldbach conjecture or any
other given open problem. We would rather expect someone with this
understanding be able to explain why the induction principle is valid,
be capable of following mathematical reasoning based on basic
arithmetical principles, be capable of offering illustrations and
explanations as to what these principles amount to and why they are
evident on basis of our understanding of the naturals as what one
obtains from 0 by repeatedly applying the add-one operation, be
capable of recognising calculations as correct or erroneous, and of
explaining why this or that calculation is correct or erroneous, and
so on; and to be just as stumped as the next man when it comes to
difficult open problems in number theory.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxx)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

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