[FOM] shipman's challenge: the best defense
Vladimir Sazonov
Vladimir.Sazonov at liverpool.ac.uk
Wed Jan 9 19:48:04 EST 2008
Gabriel Stolzenberg wrote:
> On Thu 03 Jan 2008, in Re: Formalization Thesis (Vol 61 Issue 5),
> Joe Shipman wrote:
>
> > I repeat my earlier challenge: can anyone who disputes Chow's
> > Formalization Thesis respond with a SPECIFIC MATHEMATICAL STATEMENT
> > which they are willing to claim is not, despite its expressiblity in
> > English text on the FOM discussion forum, "faithfully representable"
> > or "adequately expressible" as a sentence in the formal system ZFC?
>
> I would expect that, at least initially, in just about every case,
> a more responsible attitude for a mathematician (e.g., me) to take
> would be to refrain from answering either in the affirmative or the
> negative. I say this for two related reasons. First, the expression,
> "adequately expressible," is radically vague and subjective.
I think that the usage of "adequate formalization" is of such a
specific character for which vagueness does not play essential role to
doubt in "adequate". "Adequate formalization" is not the same as
"adequate translation" from one natural language to another (or from a
natural language to formal) as it might be thought.
We always formalize something highly informal, intuitive, NOT YET
MATHEMATICAL. (Mathematical is already formalised and may need only in
further detalization.) Some mathematics appears only after and due to
formalization. (As I wrote many times, I understand "form" in a wide
sense of this word as the opposite to "content".) Such an act of
formalization arises also each time when we read a good mathematical
text which is, strictly speaking, only a way how to communicate ideas
on such a formalization. With the help of the author of such a text the
readers also do the work of formalization in their minds (working like
co-authors). After and due to formalization the initial (vague) idea is
usually radically CHANGED. As the result, we have a renewed intuition +
formalization (content + form). The old intuition almost disappears.
(Almost nothing to compare with the result.) Only by analysis of what
happened we could somehow recall the old state of our mind, if possible
at all - so strong influence has the process of formalization on our
thought. As the result, we have a new mathematical concept or proof
related with other (formerly invented) concepts and proofs in such a
nice and fruitful way that we get a highest feeling of satisfaction,
even euphoria. Asserting that we have got an "adequate formalization"
is a only very modest and sober way to express such a feeling.
Let me repeat: saying that a formalization is adequate does not mean
that there still exists what to compare the result with. We are
satisfied with the RESULT by a different reason - the result is
mathematically excellent! That is the point.
Second,
> because mathematicians are not trained to assess such matters (there
> is no course of training because, as of now, there is nothing to be
> taught),
I think all mathematicians, even those who can only read and understand
mathematical text, not being able to create something serious
mathematical themselves, can have feeling like above. They were trained
by reading good mathematical texts, by listening good teachers and by
doing exercises under their supervision.
they should not assume that just because they do/don't have
> any doubts about adequacy/inadequacy today, they will/won't have any
> tomorrow.
Sorry, I do not understand this fragment at all.
> Having said this, I invite Joe to think about the "adequacy" of
> a formalization of the informal statement, "The set of real numbers
> is uncountable." The claim that it is a "faithful representation"
> of the informal statement might well make a mathematician (e.g., me)
> uncomfortable. I believe this is a familiar point.
Good example. (Joe can say himself, but this is my view.) Before
studying mathematics, in our childhood, we have had some intuition on
discrete objects and continuous physical lines - looking as VERY
different. It was hardly possible to say something more rational
without any mathematics (even if we would be not children at that time).
This pre-intuition is formalized in mathematics at least in two ways,
one of which is the concept of cardinality (bijection). I think, all of
us experienced a very strong feeling of euphoria of the proof by Cantor
that continuum is not countable. Not only we can feel that in some
naive way, we can PROVE that!! And the manner of the proof was quite
unexpected from the point of view of our original intuition. Then it
unexpectedly appeared, against our intuition, that continuous lines and
surfaces have the same cardinality. At this moment (after some internal
struggle with ourselves) our intuition has changed even more strongly
than by Cantor's diagonal method. Later we understood mathematical
reasons why lines and surfaces are still strongly different (by the
formal concept of dimensions). So, the intuition was even ramified.
I think, there is nothing wrong to call the two different
formalizations arisen as highly adequate, although our original naive
intuition was at least split.
If somebody would say that our original intuition was rather on
dimensions than on cardinalities, I would hardly agree. Original
intuition is something too vague (and even hardly possible to recall
after the work of thought done) that it is difficult to assert what it
was exactly about. Just some dim feeling. Mathematical crystallizing of
our intuition is a hard, irreversible work (even if we are only
"passive" readers). We are happy on the result of this work calling it
"adequate".
Other examples:
Church Thesis asserts that known (quivalent) formalization(s) of
computability are adequate.
Epsilon-delta (or topological) definition of continuity is adequate.
Scott-Ershov domains are adequate class of specific partial ordered
sets (which are also specific topological T-0 spaces) to define
semantics of (lambda calculus and) programming languages.
Gurevich's Abstract State Machines are also (quite different) adequate
tool to define semantics of programming languages.
What is "more adequate" and IN WHICH SENSE is an important, but
different question. Each is adequate in its own sense!
Polynomial-time computability (and NP-completenes) is very adequate
approach to feasible computability (however, there can be some doubts).
Everything this is so nice! I see nothing subjective in these
assertions of adequateness. THIS IS MATHEMATICS! ANY mathematician is
able to understand these examples and to get something reasonable from
these assertions of adequateness. (Of course, somebody will suggest
other examples.)
Etc.
Vladimir Sazonov
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